The birth of string theory
Paolo Di Vecchia1
Nordita, Blegdamsvej 17, 2100 Copenhagen Ø, Denmark divecchi@alf.nbi.dk
Summary. In this contribution we go through the developments that in the years
from 1968 to about 1974 led from the Veneziano model to the bosonic string theory.
They include the construction of the N-point amplitude for scalar particles, its
factorization through the introduction of an infinite number of oscillators and the
proof that the physical subspace was a positive definite Hilbert space. We also discuss
the zero slope limit and the calculation of loop diagrams. Lastly, we describe how
it finally was recognized that a quantum relativistic string theory was the theory
underlying the Veneziano model.
1 Introduction
The sixties was a period in which strong interacting processes were studied
in detail using the newly constructed accelerators at Cern and other places.
Many new hadronic states were found that appeared as resonant peaks in var-
ious cross sections and hadronic cross sections were measured with increasing
accuracy. In general, the experimental data for strongly interacting processes
were rather well understood in terms of resonance exchanges in the direct
channel at low energy and by the exchange of Regge poles in the transverse
channel at higher energy. Field theory that had been very successful in de-
scribing QED seemed useless for strong interactions given the big number of
hadrons to accomodate in a Lagrangian and the strength of the pion-nucleon
coupling constant that did not allow perturbative calculations. The only do-
main in which field theoretical techniques were successfully used was current
algebra. Here, assuming that strong interactions were described by an almost
chiral invariant Lagrangian, that chiral symmetry was spontaneously broken
and that the pion was the corresponding Goldstone boson, field theoretical
methods gave rather good predictions for scattering amplitudes involving pi-
ons at very low energy. Going to higher energy was, however, not possible
with these methods.
Because of this, many people started to think that field theory was use-
less to describe strong interactions and tried to describe strong interacting
http://arxiv.org/abs/0704.0101v1
2 Paolo Di Vecchia
processes with alternative and more phenomenological methods. The basic
ingredients for describing the experimental data were at low energy the ex-
change of resonances in the direct channel and at higher energy the exchange
of Regge poles in the transverse channel. Sum rules for strongly interacting
processes were saturated in this way and one found good agreement with the
experimental data that came from the newly constructed accelerators. Be-
cause of these successes and of the problems that field theory encountered to
describe the data, it was proposed to construct directly the S matrix without
passing through a Lagrangian. The S matrix was supposed to be constructed
from the properties that it should satisfy, but there was no clear procedure on
how to implement this construction1. The word “bootstrap” was often used
as the way to construct the S matrix, but it did not help very much to get an
S matrix for the strongly interacting processes.
One of the basic ideas that led to the construction of an S matrix was
that it should include resonances at low energy and at the same time give
Regge behaviour at high energy. But the two contributions of the resonances
and of the Regge poles should not be added because this would imply double
counting. This was called Dolen, Horn and Schmidt duality [2]. Another idea
that helped in the construction of an S matrix was planar duality [3] that
was visualized by associating to a certain process a duality diagram, shown in
Fig. (1), where each meson was described by two lines representing the quark
and the antiquark. Finally, also the requirement of crossing symmetry played
a very important role.
Fig. 1. Duality diagram for the scattering of four mesons
Starting from these ideas Veneziano [4] was able to construct an S matrix
for the scattering of four mesons that, at the same time, had an infinite number
of zero width resonances lying on linearly rising Regge trajectories and Regge
behaviour at high energy. Veneziano originally constructed the model for the
1 For a discussion of S matrix theory see Ref.s [1]
The birth of string theory 3
process ππ → πω, but it was immediately extended to the scattering of four
scalar particles.
In the case of four identical scalar particles, the crossing symmetric scat-
tering amplitude found by Veneziano consists of a sum of three terms:
A(s, t, u) = A(s, t) +A(s, u) +A(t, u) (1)
where
A(s, t) =
Γ (−α(s))Γ (−α(t))
Γ (−α(s)− α(t))
dxx−α(s)−1(1− x)−α(t)−1 (2)
with linearly rising Regge trajectories
α(s) = α0 + α
′s (3)
This was a very important property to implement in a model because it was
in agreement with the experimental data in a wide range of energies. s, t and
u are the Mandelstam variables:
s = −(p1 + p2)2 , t = −(p3 + p2)2 , u = −(p1 + p3)2 (4)
The three terms in Eq. (1) correspond to the three orderings of the four parti-
cles that are not related by a cyclic or anticyclic 2 permutation of the external
legs. They correspond, respectively, to the three permutations: (1234), (1243)
and (1324) of the four external legs. They have only simple pole singularities.
The first one has only poles in the s and t channels, the second only in the s
and u channels and the third only in the t and u channels. This property fol-
lows directly from the duality diagram that is associated to each inequivalent
permutation of the external legs. In fact, at that time one used to associate
to each of the three inequivalent permutations a duality diagram where each
particle was drawn as consisting of two lines that rappresented the quark and
antiquark making up a meson. Furthermore, the diagram was supposed to
have only poles singularities in the planar channels which are those involving
adjacent external lines. This means that, for instance, the duality diagram
corresponding to the permutation (1234) has only poles in the s and t chan-
nels as one can see by deforming the diagram in the plane in the two possible
ways shown in figure (2).
This was a very important property of the duality diagram that makes
it qualitatively different from a Feynman diagram in field theory where each
diagram has only a pole in one of the three s, t and u channels and not
simultaneously in two of them. If we accept the idea that each term of the
sum in Eq. (1) is described by a duality diagram, then it is clear that we
2 An anticyclic permutation corresponding, for instance, to the ordering (1234) is
obtained by taking the reverse of the original ordering (4321) and then performing
a cyclic permutation.
4 Paolo Di Vecchia
Fig. 2. The duality diagram contains both s and t channel poles
do not need to add terms corresponding to equivalent diagrams because the
corresponding duality diagram is the same and has the same singularities. It
is now clear that it was in some way implicit in this picture the fact that the
Veneziano model corresponds to the scattering of relativistic strings. But at
that time the connection was not obvious at all. The only S matrix property
that the Veneziano model failed to satisfy was the unitarity of the S matrix.
because it contained only zero width resonances and did not have the various
cuts required by unitarity. We will see how this property will be implemented.
Immediately after the formulation of the Veneziano model, Virasoro [5]
proposed another crossing symmetric four-point amplitude for scalar particles
that consisted of a unique piece given by:
A(s, t, u) ∼
Γ (−α(u)
)Γ (−α(s)
)Γ (−α(t)
Γ (1 +
)Γ (1 +
)Γ (1 +
where
α(s) = α0 + α
′s (6)
The model had poles in all three s, t and u channels and could not be written
as sum of three terms having poles only in planar diagrams. In conclusion,
the Veneziano model satisfies the principle of planar duality being a crossing
symmetric combination of three contributions each having poles only in the
planar channels. On the other hand, the Virasoro model consists of a unique
crossing symmetric term having poles in both planar and non-planar channels.
The attempts to construct consistent models that were in good agreement
with the strong interaction phenomenology of the sixties boosted enormously
the activity in this research field. The generalization of the Veneziano model to
the scattering ofN scalar particles was built, an operator formalism consisting
of an infinite number of harmonic oscillators was constructed and the complete
spectrum of mesons was determined. It turned out that the degeneracy of
states grew up exponentially with the mass. It was also found that the N
point amplitude had states with negative norm (ghosts) unless the intercept
of the Regge trajectory was α0 = 1 [6]. In this case it turned out that the
model was free of ghosts but the lowest state was a tachyon. The model was
called in the literature the “dual resonance model”.
The birth of string theory 5
The model was not unitary because all the states were zero width reso-
nances and the various cuts required by unitarity were absent. The unitarity
was implemented in a perturbative way by adding loop diagrams obtained by
sewing some of the external legs together after the insertion of a propagator.
The multiloop amplitudes showed a structure of Riemann surfaces. This be-
came obvious only later when the dual resonance model was recognized to
correspond to scattering of strings.
But the main problem was that the model had a tachyon if α0 = 1 or had
ghosts for other values of α0 and was not in agreement with the experimental
data: α0 was not equal to about
as required by experiments for the ρ
Regge trajectory and the external scalar particles did not behave as pions
satisfying the current algebra requirements. Many attempts were made to
construct more realistic dual resonance models, but the main result of these
attempts was the construction of the Neveu-Schwarz [7] and the Ramond [8]
models, respectively, for mesons and fermions. They were constructed as two
independent models and only later were recognized to be two sectors of the
same model. The Neveu-Schwarz model still contained a tachyon that only in
1976 through the GSO projection was eliminated from the physical spectrum.
Furthermore, it was not properly describing the properties of the physical
pions.
Actually a model describing ππ scattering in a rather satisfactory way
was proposed by Lovelace and Shapiro [9] 3. According to this model the
three isospin amplitudes for pion-pion scattering are given by:
[A(s, t) +A(s, u)]− 1
A(t, u)
A1 = A(s, t)−A(s, u) A2 = A(t, u) (7)
where
A(s, t) = β
Γ (1− α(s))Γ (1 − α(t))
Γ (1− α(t)− α(s))
; α(s) = α0 + α
′s (8)
The amplitudes in eq.(7) provide a model for ππ scattering with linearly rising
Regge trajectories containing three parameters: the intercept of the ρ Regge
trajectory α0, the Regge slope α
′ and β. The first two can be determined by
imposing the Adler’s self-consistency condition, that requires the vanishing of
the amplitude when s = t = u = m2π and one of the pions is massless, and the
fact that the Regge trajectory must give the spin of the ρ meson that is equal
to 1 when
s is equal to the mass of the ρ meson mρ. These two conditions
determine the Regge trajectory to be:
α(s) =
s−m2π
m2ρ −mπ2
= 0.48 + 0.885s (9)
3 See also Ref. [10].
6 Paolo Di Vecchia
Having fixed the parameters of the Regge trajectory the model predicts the
masses and the couplings of the resonances that decay in ππ in terms of a
unique parameter β. The values obtained are in reasonable agreement with
the experiments. Moreover, one can compute the ππ scattering lenghts:
a0 = 0.395β a2 = −0.103β (10)
and one finds that their ratio is within 10% of the current algebra ratio given
by a0/a2 = −7/2. The amplitude in eq.(8) has exactly the same form as that
for four tachyons of the Neveu-Schwarz model with the only apparently minor
difference that α0 = 1/2 (for mπ = 0) instead of 1 as in the Neveu-Schwarz
model. This difference, however, implies that the critical space-time dimension
of this model is d = 4 4 and not d = 10 as in the Neveu-Schwarz model. In
conclusion this model seems to be a perfectly reasonable model for describing
low-energy ππ scattering. The problem is, however, that nobody has been able
to generalize it to the multipion scattering and therefore to get the complete
meson spectrum.
As we have seen the S matrix of the dual resonance model was constructed
using ideas and tools of hadron phenomenology of the end of the sixties.
Although it did not seem possible to write a realistic dual resonance model
describing the pions , it was nevertheless such a source of fascination for those
who actively worked in this field at that time for its beautiful internal structure
and consistency that a lot of energy was used to investigate its properties and
for understanding its basic structure. It turned out with great surprise that
the underlying structure was that of a quantum relativistic string.
The aim of this contribution is to explain the logic of the work that was
done in the years from 1968 to 1974 5 in order to uncover the deep properties of
this model that appeared from the beginning to be so beautiful and consistent
to deserve an intensive study.
This seems to me a very good way of celebrating the 65th anniversary of
Gabriele who is the person who started and also contributed to develop the
whole thing with his deep physical intuition.
2 Construction of the N -point amplitude
We have seen that the construction of the four-point amplitude is not sufficient
to get information on the full hadronic spectrum because it contains only
those hadrons that couple to two ground state mesons and does not see those
intermediate states which only couple to three or to an higher number of
ground state mesons [12]. Therefore, it was very important to construct the
N -point amplitude involving identical scalar particles. The construction of
4 This can be checked by computing the coupling of the spinless particle at the
level α(s) = 2 and seeing that it vanishes for d = 4.
5 Reviews from this period can be found in Ref. [11]
The birth of string theory 7
the N -point amplitude was done in Ref. [13] (extending the work of Ref. [14])
by requiring the same principles that have led to the construction of the
Veneziano model, namely the fact that the axioms of S-matrix theory be
satisfied by an infinite number of zero width resonances lying on linearly
rising Regge trajectories and planar duality.
The fully crossing symmetric scattering amplitude of N identical scalar
particles is given by a sum of terms corresponding to the inequivalent permu-
tations of the external legs:
An (11)
Also in this case two permutations of the external legs are inequivalent if they
are not related by a cyclic or anticyclic permutation. Np is the number of
inequivalent permutations of the external legs and is equal to Np =
(N−1)!
and each term has only simple pole singularities in the planar channels. Each
planar channel is described by two indices (i, j), to mean that it includes the
legs i, i+ 1, i+ 2 . . . j − 1, j, by the Mandelstam variable
sij = −(pi + pi+1 + . . .+ pj)2 (12)
and by an additional variable uij whose role will become clear soon. It is
clear that the channels (ij) and (j + 1, i− 1) 6 are identical and they should
be counted only once. In the case of N identical scalar particles the number
of planar channels is equal to
N(N−3)
. This can be obtained as follows. The
independent planar diagrams involving the particle 1 are of the type (1, i)
where i = 2 . . .N − 2. Their number is N − 3. This is also the number of
planar diagrams involving the particle 2 and not the 1. The number of planar
diagrams involving the particle 3 and not the particles 1 and 2 is equal to
N − 4. In general the number of planar diagrams involving the particle i and
not the previous ones from 1 to i-1 is equal to N − 1− i. This means that the
total number of planar diagram is equal to:
2(N − 3) +
(N − 1− i) = 2(N − 3) +
= 2(N − 3) +
(N − 4)(N − 3)
N(N − 3)
If one writes down the duality diagram corresponding to a certain planar
ordering of the external particles, it is easy to see that the diagram can have
simultaneous pole singularities only in N − 3 channels. The channels that
allow simultaneous pole singularities are called compatible channels, the other
6 This channel includes the particles (j + 1, . . . , N, 1, . . . i− 1).
8 Paolo Di Vecchia
are called incompatible. Two channels (i,j) and (h,k) are incompatible if the
following inequalities are satisfied:
i ≤ h ≤ j ; j + 1 ≤ k ≤ i− 1 (14)
The aim is to construct the scattering amplitude for each inequivalent per-
mutation of the external legs that has only pole singularities in the
N(N−3)
planar channels. We have also to impose that the amplitude has simultaneous
poles only in N − 3 compatible channels. In order to gain intuition on how to
proceed we rewrite the four-point amplitude in Eq. (2) as follows:
A(s, t) =
du23 u
−α(s12)−1
−α(s23)−1
23 δ(u12 + u23 − 1) (15)
where u12 and u23 are the variables corresponding to the two planar chan-
nels (12) and (23) and the cancellation of simultaneous poles in incompatible
channels is provided by the δ-function which forbids u12 and u23 to vanish
simultaneously.
We will now extend this procedure to the N -point amplitude. But for the
sake of clarity let us start with the case of N = 5 [14]. In this case we have 5
planar channels described by u12, u13, u23, u24 and u34. Since we have only two
compatible channels only two of the previous five variables are independent.
We can choose them to be u12 and u13. In order to determine the depen-
dence of the other three variables on the two independent ones, we exclude
simultaneous poles in incompatible channels. This can be done by imposing
relations that prevent variables corresponding to incompatible channels to
vanish simultaneously. A sufficient condition for excluding simultaneous poles
in incompatible channels is to impose the conditions:
uP = 1−
uP̄ (16)
where the product is over the variables P̄ corresponding to channels that
are incompatible with P . In the case of the five-point amplitude we get the
following relations:
u23 = 1− u34u12 ; u24 = 1− u13u12
u13 = 1− u34u24 ; u34 = 1− u23u13 ; u12 = 1− u24u23 (17)
Solving them in terms of the two independent ones we get:
u23 =
1− u12
1− u12u13
; u34 =
1− u13
1− u12u13
; u24 = 1− u12u13 (18)
In analogy with what we have done for the four-point amplitude in Eq. (15)
we write the five-point amplitude as follows:
The birth of string theory 9
du34u
−α(s12)−1
−α(s13)−1
×u−α(s24)−124 u
−α(s23)−1
−α(s34)−1
δ(u23 + u12u34 − 1)δ(u24 + u12u13 − 1)δ(u34 + u13u23 − 1) (19)
Performing the integral over the variables u23, u24 and u34 we get:
du13u
−α(s12)−1
−α(s13)−1
× (1− u12)−α(s23)−1(1 − u13)−α(s13)−1(1− u12u13)−α(s24)+α(s23)+α(s34)(20)
We have implicitly assumed that the Regge trajectory is the same in all chan-
nels and that the external scalar particles have the same common mass m
and are the lowest lying states on the Regge trajectory. This means that their
mass is given by:
α0 − α′p2i = 0 ; p2i ≡ −m2 (21)
Using then the relation:
α(s23) + α(s34)− α(s24) = 2α′p2 · p4 (22)
we can rewrite Eq. (20) as follows:
−α(s2)−1
−α(s3)−1
3 (1− u2)−α(s23)−1×
× (1 − u3)−α(s34)−1
(1− xij)2α
′pi·pj (23)
where
si ≡ s1i , ui ≡ u1i ; i = 2, 3 ; xij = uiui+1 . . . uj−1. (24)
We are now ready to construct the N -point function [13]. In analogy with
what has been done for the four and five-point amplitudes we can write the
N -point amplitude as follows:
. . .
−α(sP )−1
δ(uQ − 1 +
uQ̄) (25)
10 Paolo Di Vecchia
where the first product is over the
N(N−3)
variables corresponding to all
planar channels, while the second one is over the
(N−3)(N−2)
independent
δ-functions. The product in the δ-function is defined in Eq. (16).
The solution of all the non-independent linear relations imposed by the
δ-functions is given by
uij =
(1 − xij)(1− xi−1,j+1)
(1− xi−1,j)(1 − xi,j+1)
where the variables xij are given in Eq. (24). Eliminating the δ-function from
Eq. (25) one gets:
−α(si)−1
i (1 − ui)
−α(si,i+1)−1
j=i+2
(1− xij)−γij(27)
where
γij = α(sij) + α(si+1;j−1)− α(si;j−1)− α(si+1;j) ; j ≥ i+ 2 (28)
It is easy to see that
α(si,i+1) = −α0 − 2α′pi · pi+1 ; γij = −2α′pi · pj ; j ≥ i+ 2 (29)
Inserting them in Eq. (27) we get:
−α(si)−1
i (1− ui)
j=i+1
(1− xij)2α
′pi·pj (30)
This is the form of the N -point amplitude that was originally constructed.
Then Koba and Nielsen [15] put it in the form that is more known nowadays.
They constructed it using the following rules. They associated a real variable
zi to each leg i. Then they associated to each channel (i, j) an anharmonic
ratio constructed from the variables zi, zi−1, zj, zj+1 in the following way
(zi, zi+1, zj, zj+1)
−α(sij)−1 =
(zi − zj)(zi−1 − zj+1)
(zi−1 − zj)(zi − zj+1)
]−α(sij)−1
and finally they gave the following expression for the N -point amplitude:
dV (z)
(i,j)
(zi, zi+1, zj, zj+1)
−α(sij)−1 (32)
where
dV (z) =
1 [θ(zi − zi+1)dzi]
i=1(zi − zi+2)dVabc
; dVabc =
dzadzbdzc
(zb − za)(zc − zb)(za − zc)
The birth of string theory 11
and the variables zi are integrated along the real axis in a cyclically ordered
way: z1 ≥ z2 . . . ≥ zN with a, b, c arbitrarily chosen.
The integrand of the N -point amplitude is invariant under projective
transformations acting on the leg variables zi:
αzi + β
γzi + δ
; i = 1 . . .N ; αδ − βγ = 1 (34)
This is because both the anharmonic ratio in Eq. (31) and the measure dVabc
are invariant under a projective transformation. Since a projective transfor-
mation depends on three real parameters, then the integrand of the N -point
amplitude depends only on N − 3 variables zi. In order to avoid infinities, one
has then to divide the integration volume with the factor dVabc that is also
invariant under the projective transformations. The fact that the integrand
depends only on N − 3 variables is in agreement with the fact that N − 3 is
also the maximal number of simultaneous poles allowed in the amplitude.
It is convenient to write the N -point amplitude in a form that involves the
scalar product of the external momenta rather than the Regge trajectories.
We distinguish three kinds of channels. The first one is when the particles
i and j of the channel (i, j) are separated by at least two particles. In this
case the channels that contribute to the exponent of the factor (zi − zj) are
the channels (i, j) with exponent equal to −α(sij) − 1, (i + 1, j − 1) with
exponent −α(si+1,j−1)− 1, (i+1, j) with exponent α(si+1,j)+1 and (i, j− 1)
with exponent α(si,j−1) + 1. Adding these four contributions one gets for the
channels where i and j are separated by at least two particles
− α(sij)− α(si+1,j−1) + α(si+1,j) + α(si,j−1) = 2α′pi · pj (35)
The second one comes from the channels that are separated by only one
particle. In this case only three of the previous four channels contribute. For
instance if j = i+2 the channel (i+1, j− 1) consists of only one particle and
therefore should not be included. This means that we would get:
− α(si;i+2)− 1 + α(s1+1;i+2) + 1 + α(si;i+1 + 1) = 1 + 2α′pi · pi+2 (36)
Finally the third one that comes from the channels whose particles are adja-
cent, gets only contribution from:
− α(si;i+1)− 1 = α0 − 1 + 2α′pi · pi+1 (37)
Putting all these three terms together in Eq. (32) and remembering the factor
in the denominator in the first equation of (33) we get:
1 dziθ(zi − zi+1)
dVabc
(zi − zi+1)α0−1
(zi − zj)2α
′pi·pj(38)
A convenient choice for the three variables to keep fixed is:
12 Paolo Di Vecchia
za = z1 = ∞ ; zb = z2 = 1 ; zc = zN = 0 (39)
With this choice the previous equation becomes:
dziθ(zi − zi+1)
(zi − zi+1)α0−1×
j=i+1
(zi − zj)2α
′pi·pj (40)
We now want to show that this amplitude is identical to the one given in Eq.
(30). This can be done by performing the following change of variables:
; i = 2, 3 . . .N − 2 (41)
that implies
zi = u2u3 . . . ui−1 ; i = 3, 4 . . .N − 1 (42)
Taking into account that the Jacobian is equal to:
uN−2−ii (43)
using the following two relations:
(zi − zi+1)α0−1 =
(N−1−i)α0−1
(1− ui)α0−1 (44)
j=i+1
(zj − zi)2α
′pi·pj =
j=i+1
(1− xij)2α
′pi·pj
−α(si)−(N−i−1)α0
i (45)
and the conservation of momentum
pi = 0 (46)
together with Eq. (21), one can easily see that Eq.s (30) and (40) are equal.
The birth of string theory 13
The N -point amplitude that we have constructed in this section corre-
sponds to the scattering of N spinless particles with no internal degrees of
freedom. On the other hand it was known that the mesons were classified
according to multiplets of an SU(3) flavour symmetry. This was implemented
by Chan and Paton [16] by multiplying the N -point amplitude with a factor,
called Chan-Paton factor, given by
Tr(λa1λa2 . . . λaN ) (47)
where the λ’s are matrices of a unitary group in the fundamental representa-
tion. Including the Chan-Paton factors the total scattering amplitude is given
Tr(λa1λa2 . . . λaN )BN (p1, p2, . . . pN ) (48)
where the sum is extended to the (N − 1)! permutations of the external legs,
that are not related by a cyclic permutations. Originally when the dual reso-
nance model was supposed to describe strongly interacting mesons, this factor
was introduced to represent their flavour degrees of freedom. Nowadays the
interpretation is different and the Chan-Paton factor represents the colour
degrees of freedom of the gauge bosons and the other massive particles of the
spectrum.
The N -point amplitude BN that we have constructed in this section con-
tains only simple pole singularities in all possible planar channels. They cor-
respond to zero width resonances located at non-negative integer values n of
the Regge trajectory α(M2) = n. The lowest state located at α(m2) = 0 cor-
responds to the particles on the external legs of BN . The spectrum of excited
particles can be obtained by factorizing the N -point amplitude in the most
general channel with any number of particles. This was done in Ref.s [17] and
[18] finding a spectrum of states rising exponentially with the mass M . Being
the model relativistic invariant it was found that many states obtained by
factorizing the N -point amplitude were ”ghosts”, namely states with negative
norm as one finds in QED when one quantizes the electromagnetic field in a
covariant gauge. The consistency of the model requires the existence of rela-
tions satisfied by the scattering amplitudes that are similar to those obtained
through gauge invariance in QED. If the model is consistent they must decou-
ple the negative norm states leaving us with a physical spectrum of positive
norm states. In order to study in a simple way these issues, we discuss in the
next section the operator formalism introduced already in 1969 [19, 20, 21].
Before concluding this section let us go back to the non-planar four-point
amplitude in Eq. (5) and discuss its generalization to an N -point amplitude.
Using the technique of the electrostatic analogue on the sphere instead of on
the disk Shapiro [22] was able to obtain a N -point amplitude that reduces
to the four-point amplitude in Eq. (5) with intercept α0 = 2. The N -point
amplitude found in Ref. [22] is:
14 Paolo Di Vecchia
i=1 d
dVabc
|zi − zj |α
′pi·pj (49)
where
dVabc =
d2zad
|za − zb|2|za − zc|2|zb − zc|2
The integral in Eq. (49) is performed in the entire complex plane.
3 Operator formalism and factorization
The factorization properties of the dual resonance model were first studied by
factorizing by brute force the N-point amplitude at the various poles [17, 18].
The number of terms that factorize the residue of the pole at α(s) = n,
increases rapidly with the value of n. In order to find their degeneracy it turned
out to be convenient to first rewrite the N-point amplitude in an operator
formalism. In this section we introduce the operator formalism and we rewrite
the N -point amplitude derived in the previous section in this formalism.
The key idea [19, 20, 21] is to introduce an infinite set of harmonic oscil-
lators and a position and momentum operators 7 which satisfy the following
commutation relations:
[anµ, a
mν ] = ηµνδnm ; [q̂µ, p̂ν ] = iηµν (51)
where ηµν is the flat Minkowski metric that we take to be ηµν = (−1, 1, . . . 1).
A state with momentum p is constructed in terms of a state with zero mo-
mentum as follows:
p̂|p〉 ≡ p̂eip·q̂|0〉 = p|p〉 ; p̂ |0〉 = 0 (52)
normalized as 8
〈p|p′〉 = (2π)dδ(d)(p+ p′) (53)
In order to avoid minus signs we use the convention that
〈p| = 〈0|eip·q̂ (54)
A complete and orthonormal basis of vectors in the harmonic oscillator space
is given by
|λ1, λ2, . . . λi; p〉 =
(a†µn;n)
λn;µn
λn,µn !
eipq̂|0, 0〉 (55)
7 Actually the position and momentum operators were introduced in Ref. [23].
8 Although we now use an arbitrary d we want to remind you that all original
calculations were done for d = 4.
The birth of string theory 15
where the first |0〉 corresponds to the one annihilated by all annihilation op-
erators and the second one to the state of zero momentum:
aµn;n|0, 0〉 = p̂|0, 0〉 = 0 (56)
Notice that Lorentz invariance forces to introduce also oscillators that create
states with negative norm due to the minus sign in the flat Minkowski metric.
This implies that the space spanned by the states in Eq. (55) is not positive
definite. This is, however, not allowed in a quantum theory and therefore if
the dual resonance model is a consistent quantum-relavistic theory we expect
the presence of relations of the kind of those provided by gauge invariance in
Let us introduce the Fubini-Veneziano [23] operator:
Qµ(z) = Q
µ (z) +Q
µ (z) +Q
µ (z) (57)
where
Q(+) = i
z−n ; Q(−) = −i
Q(0) = q̂ − 2iα′p̂ log z (58)
In terms of Q we introduce the vertex operator corresponding to the external
leg with momentum p:
V (z; p) =: eip·Q(z) :≡ eip·Q
(−)(z)eipq̂e+2α
′p̂·p log zeip·Q
(+)(z) (59)
and compute the following vacuum expectation value:
〈0, 0|
V (zi, pi)|0, 0〉 (60)
It can be easily computed using the Baker-Haussdorf relation
eAeB = eBeAe[A,B] (61)
that is valid if the commutator, as in our case, [A,B] is a c-number. In our
case the commutation relations to be used are:
[Q(+)(z), Q(−)(w)] = −2α′ log
and the second one in Eq. (51). Using them one gets:
V (z; p)V (w; k) =: V (z; p)V (w; k) : (z − w)2α
′p·k (63)
16 Paolo Di Vecchia
〈0, 0|
V (zi, pi)|0, 0〉 =
(zi − zj)2α
′pi·pj (2π)dδ(d)(
pi) (64)
where the normal ordering requires that all creation operators be put on the
left of the annihilation one and the momentum operator p̂ be put on the right
of the position operator q̂. This means that
(2π)dδ(d)(
pi)BN =
1 dziθ(zi − zi+1)
dVabc
(zi − zi+1)α0−1
× 〈0, 0|
V (zi, pi)|0, 0〉 (65)
By choosing the three variables za, zb and zc as in Eq. (39) we can rewrite the
previous equation as follows:
(2π)dδ(d)(
pi)BN =
θ(zi − zi+1)×
(zi − zi+1)α0−1
〈0, p1|
V (zi; pi)|0, pN 〉 (66)
where we have taken z2 = 1 and we have defined (α0 ≡ α′p2i ; i = 1 . . .N) :
V (zN ; pN)|0, 0〉 ≡ |0; pN〉 ; 〈0; 0| lim
z2α01 V (z1; p1) = 〈0, p1| (67)
Before proceeding to factorize the N -point amplitude let us study the prop-
erties under the projective group of the operators that we have introduced.
We have already seen that the projective group leaves the integrand of the
Koba-Nielsen representation of the N -point amplitude invariant. The projec-
tive group has three generators L0, L1 and L−1 corresponding respectively to
dilatations, inversions and translations. Assuming that the Fubini-Veneziano
fields Q(z) transforms as a field with weight 0 (as a scalar) we can immedi-
ately write the commutation relations that Q(z) must satisfy. This means in
fact that, under a projective transformation, Q(z) transforms as follows:
Q(z) → QT (z) = Q
αz + β
γz + δ
; αδ − βγ = 1 (68)
Expanding for small values of the parameters we get:
QT (z) = Q(z) + (ǫ1 + ǫ2z + ǫ3z
dQ(z)
+ o(ǫ2) (69)
The birth of string theory 17
This means that the three generators of the projective group must satisfy the
following commutation relations with Q(z):
[L0, Q(z)] = z
; [L−1, Q(z)] =
; [L1, Q(z)] = z
They are given by the following expressions in terms of the harmonic oscilla-
tors:
L0 = α
′p̂2 +
na†n · an ; L1 =
2α′p̂ · a1 +
n(n+ 1)an+1 · a†n (71)
L−1 = L
2α′p̂ · a†1 +
n(n+ 1)a
n+1 · an (72)
They annihilate the vacuum
L0|0, 0〉 = L1|0, 0〉 = L−1|0, 0〉 = 0 (73)
that is therefore called the projective invariant vacuum, and satisfy the algebra
that is called Gliozzi algebra [24]9:
[L0, L1] = −L1 ; [L0, L−1] = L−1 ; [L1, L−1] = 2L0 (74)
The vertex operator with momentum p is a projective field with weight equal
to α0 = α
′p2. It transforms in fact as follows under the projective group:
[Ln, V (z, p)] = z
n+1 dV (z, p)
+ α0(n+ 1)z
nV (z, p) ; n = 0,±1 (75)
or in finite form as follows:
UV (z, p)U−1 =
(γz + δ)2α0
αz + β
γz + δ
where U is the generator of an arbitrary finite projective transformation.
Since U leaves the vacuum invariant, by using Eq. (76) it is easy to show
that:
〈0, 0|
V (z′i, p)|0, 0〉 =
(γzi + δ)
2α0〈0, 0|
V (zi, p)|0, 0〉 (77)
that together with the following equation:
(z′i − z′i+1)α0−1 =
(zi − zi+1)α0−1
(γzi + δ)
−2α0(78)
9 See also Ref. [25].
18 Paolo Di Vecchia
implies that the integrand of the N -point amplitude in Eq. (65) is invariant
under projective transformations.
We are now ready to factorize the N -point amplitude and find the spec-
trum of mesons.
From Eq.s (75) and (76) it is easy to derive the transformation of the
vertex operator under a finite dilatation:
zL0V (1, p)z−L0 = V (z, p)zα0 (79)
Changing the integration variables as follows:
; i = 2, 3 . . .N − 2 ; det
= z3z4 . . . zN−2 (80)
where the last term is the jacobian of the trasformation from zi to xi, we get
from Eq.(66) the following expression:
AN ≡ 〈0, p1|V (1, p2)DV (1, p3) . . .DV (1, pN−1)|0, pN〉 (81)
where the propagator D is equal to:
dxxL0−1−α0(1− x)α0−1 = Γ (L0 − α0)Γ (α0)
Γ (L0)
AN = (2π)
dδ(d)
BN (83)
The factorization properties of the amplitude can be studied by inserting in
the channel (1,M) or equivalently in the channel (M +1, N) described by the
Mandelstam variable
s = −(p1 + p2 + . . . pM )2 = −(pM+1 + pM+2 . . .+ pN )2 ≡ −P 2 (84)
the complete set of states given in Eq. (55):
〈p(1,M)|λ, P 〉〈λ, P |D|µ, P 〉〈µ, P |p(M+1,N)〉 (85)
where
〈p(1,M)| = 〈0, p1|V (1, p2)DV (1, p3) . . . V (1, pM ) (86)
|p(M+1,N)〉 = V (1, pM+1)D . . . V (1, pN−1)|pN , 0〉 (87)
Introducing the quantity:
The birth of string theory 19
na†n · an (88)
it is possible to rewrite
〈λ, P |D|µ, P 〉 =
〈λ, P |
(−1)m
α0 − 1
R+m− α(s)
|µ, P 〉 (89)
where s is the variable defined in Eq. (84). Using this equation we can rewrite
Eq. (85) as follows
〈p(1,M)|λ, P 〉
〈λ, P |
(−1)m
α0 − 1
R+m− α(s)
|µ, P 〉〈µ, P |p(M+1,N)〉(90)
This expression shows that amplitude AN has a pole in the channel (1,M)
when α(s) is equal to an integer n ≥ 0 and the states |λ〉 that contribute to
its residue are those satisfying the relation:
R|λ〉 = (n−m)|λ〉 ; m = 0, 1 . . . n (91)
The number of independent states |λ〉 contributing to the residue gives the
degeneracy of states for each level n.
Because of manifest relativistic invariance the space spanned by the com-
plete system of states in Eq. (55) contains states with negative norm corre-
sponding to those states having an odd number of oscillators with timelike
directions (see Eq. (51)). This is not consistent in a quantum theory where
the states of a system must span a positive definite Hilbert space. This means
that there must exist a number of relations satisfied by the external states that
decouple a number of states leaving with a positive definite Hilbert space. In
order to find these relations we rewrite the state in Eq. (87) going back to the
Koba-Nielsen variables:
|p(1,M)〉 =
dziθ(zi − zi+1)]
(zi − zi+1)α0−1×
× V (1, p1)V (z2, p2) . . . V (zM−1, pM−1)|0, pM 〉 (92)
Let us consider the operator U(α) that generate the projective transformation
that leaves the points z = 0, 1 invariant:
1− α(z − 1) = z + α(z
2 − z) + o(α2) (93)
From the transformation properties of the vertex operators in Eq. (76) it
is easy to see that the previous transformation leaves the state in Eq. (92)
invariant:
20 Paolo Di Vecchia
U(α)|p(1,M)〉 = |p(1,M)〉 (94)
This means that the generator of the previous transformation annihilates the
state in Eq. (92):
W1|p(1,M)〉 = 0 ; W1 = L1 − L0 (95)
The explicit form of W1 follows from the infinitesimal form of the transforma-
tion in Eq. (93). This condition that is of the same kind of the relations that
on shell amplitudes with the emission of photons satisfy as a consequence of
gauge invariance, implies that the residue at the pole in Eq. (90) can be fac-
torized with a smaller number of states. It turns out, however, that a detailed
analysis of the spectrum shows that negative norm states are still present.
This can be qualitatively understood as follows. Due to the Lorentz metric
we have a negative norm component for each oscillator. In order to be able
to decouple all negative norm states we need to have a gauge condition of
the type as in Eq. (95) for each oscillator. But the number of oscillators is
infinite and, therefore, we need an infinite number of conditions of the type
as in Eq. (95). It was found in Ref. [6] that, if we take α0 = 1, then one can
easily construct an infinite number of operators that leave the state in Eq.
(92) invariant. In the next section we will concentrate on this case.
4 The case α0 = 1
If we take α0 = 1 many of the formulae given in the previous section simplify.
The N -point amplitude in Eq. (38) becomes:
1 dziθ(zi − zi+1)
dVabc
(zi − zj)2α
′pi·pj (96)
that can be rewritten in the operator formalism as follows:
(2π)4δ(
pi)BN =
1 dziθ(zi − zi+1)
dVabc
〈0, 0|
V (zi, pi)|0, 0〉 (97)
By choosing z1 = ∞, z2 = 1 and zN = 0 it becomes
(2π)4δ(
pi)BN =
θ(zi − zi+1)〈0, p1|
V (zi; pi)|0, pN 〉 (98)
The birth of string theory 21
where
V (zN ; pN )|0, 0〉 ≡ |0; pN 〉 ; 〈0; 0| lim
z21V (z1; p1) = 〈0, p1| (99)
Eq. (81) is as before, but now the propagator becomes:
dxxL0−2 =
L0 − 1
(100)
This means that Eq. (89) becomes:
〈λ, P |D|µ, P 〉 = 〈λ, P | 1
L0 − 1
|µ, P 〉 (101)
and Eq. (90) has the simpler form:
〈p(1,M)|λ, P 〉〈λ, P |
R − α(s)
|λ, P 〉〈λ, P |p(M+1,N)〉 (102)
BN has a pole in the channel (1,M) when α(s) is equal to an integer n ≥ 0 and
the states |λ〉 that contribute to its residue are those satisfying the relation:
R|λ〉 = n|λ〉 (103)
Their number gives the degeneracy of the states contributing to the pole at
α(s) = n. The N -point amplitude can be written as:
BN = 〈p(1,M)|D|p(M+1,N)〉 (104)
where
|p(1,M)〉 =
∫ M−1
[dziθ(zi − zi+1)]×
× V (1, p1)V (z2, p2) . . . V (zM−1, pM−1|0, pM 〉 (105)
Using Eq. (79) and changing variables from zi, i = 2 . . .M−1 to xi = zi+1zi , i =
1 . . .M − 2 with z1 = 1 we can rewrite the previous equation as follows:
|p(1,M)〉 = V (1, p1)DV (1, p2) . . .DV (1, pM−1)|0, pM 〉 (106)
where the propagator D is defined in Eq. (100).
We want now to show that the state in Eq.s (105) and (106) is not only
annihilated by the operator in Eq. (95), but, if α0 = 1 [6], by an infinite set
of operators whose lowest one is the one in Eq. (95). We will derive this by
using the formalism developed in Ref. [26] and we will follow closely their
derivation.
Starting from Eq.s (70) Fubini and Veneziano realized that the generators
of the projective group acting on a function of z are given by:
22 Paolo Di Vecchia
L0 = −z
; L−1 = −
; L1 = −z2
(107)
They generalized the previous generators to an arbitrary conformal transfor-
mation by introducing the following operators, called Virasoro operators:
Ln = −zn+1
(108)
that satisfy the algebra:
[Ln, Lm] = (n−m)Ln+m (109)
that does not contain the term with the central charge! They also showed that
the Virasoro operators satisfy the following commutation relations with the
vertex operator:
[Ln, V (z, p)] =
zn+1V (z, p)
(110)
More in general actually they define an operator Lf corresponding to an
arbitrary function f(ξ) and Lf = Ln if we choose f(ξ) = ξ
n. In this case the
commutation relation in Eq. (110) becomes:
[Lf , V (z, p)] =
(zf(z)V (z, p)) (111)
By introducing the variable:
ξf(ξ)
(112)
where A is an arbitrary constant, one can rewrite Eq. (111) in the following
form:
[Lf , zf(z)V (z, p)] =
(zf(z)V (z, p)) (113)
This implies that, under an arbitrary conformal transformation z → f(z),
generated by U = eαLf , the vertex operator transforms as:
eαLfV (z, p) zf(z) e−αLf = V (z′, p)z′f(z′) (114)
where the parameter α is given by:
ξf(ξ)
(115)
On the other hand, this equation implies:
zf(z)
z′f(z′)
(116)
The birth of string theory 23
that, inserted in Eq. (114), implies that the quantity V (z, p) dz is left invariant
by the transformation z → f(z):
eαLfV (z, p)dze−αLf = V (z′, p)dz′ (117)
Let us now act with the previous conformal transformation on the state in
Eq. (105). We get:
eαLf |p(1,M)〉 =
[dziθ(zi − zi+1)] eαLfV (1, p1)e−αLf×
×eαLfV (z2, p2)e−αLf . . . . . . eαLfV (zM−1, pM−1)e−αLf eαLf |0, pM 〉 =
θ(zi − zi+1)× eαLfV (1, p1)e−αLf×
× V (z′2, p2)dz′2 . . . V (z′M−1, pM−1)dz′M−1eαLf |0, pM 〉 (118)
where we have used Eq. (117). The previous transformation leaves the state
invariant if both z = 0 and z = 1 are fixed points of the conformal transfor-
mation. This happens if the denominator in Eq. (115) vanishes when ξ = 0, 1.
This requires the following conditions:
f(1) = 0 ; lim
ξf(ξ) = 0 (119)
Expanding ξ near the poinr ξ = 1 we can determine the relation between z
and z′ near z = z′ = 1. We get:
ze−αf
1− z + ze−αf ′(1)
(120)
and from it we can determine the conformal factor:
(1 − z + ze−αf ′(1))2
→ eαf
′(1) (121)
in the limit z → 1. Proceeding in the same near the point z = z′ = 0 we get:
zf(0)eαf(0)
f(0) + zf ′(0)(1− eαf(0)
→ zeαf(0) (122)
in the limit z → 0. This means that Eq. (118) becomes
eα(Lf−f
′(1)−f(0))|p(1,M)〉 = |p(1,M)〉 (123)
A choice of f that satisfies Eq.s (119) is the following:
24 Paolo Di Vecchia
f(ξ) = ξn − 1 (124)
that gives the following gauge operator:
Wn = Ln − L0 − (n− 1) (125)
that annihilates the state in Eq. (105):
Wn|p1...M 〉 = 0 ; n = 1 . . .∞ (126)
These are the Virasoro conditions found in Ref. [6]. There is one condition for
each negative norm oscillator and, therefore, in this case there is the possibility
that the physical subspace is positive definite. An alternative more direct
derivation of Eq. (126) can be obtained by acting with Wn on the state in Eq.
(106) and using the following identities:
WnV (1, p) = V (1, p)(Wn + n) ; (Wn + n)D = [L0 + n− 1]−1Wn (127)
The second equation is a consequence of the following equation:
L0 = xL0+nLn (128)
Eq.s (127) imply
WnV (1, p)D = V (1, p)[L0 + n− 1]−1Wn (129)
This shows that the operator Wn goes unchanged through all the product
of terms V D until it arrives in front of the term V (1, pM−1)|0, pM 〉. Going
through the vertex operator it becomes Ln − L0 + 1 that then annihilate the
state
(Ln − L0 + 1)|pM , 0〉 = 0 (130)
This proves Eq. (126).
Using the representation of the Virasoro operators given in Eq. (108) Fu-
bini and Veneziano showed that they satisfy the algebra given in eq. (109)
without the central charge. The presence of the central charge was recognized
by Joe Weis10 in 1970 and never published. Unlike Fubini and Veneziano [26]
he used the expression of the Ln operators in terms of the harmonic oscillators:
2α′np̂ · an +
m(n+m)an+m · am+
m(n−m)am−n · am ;n ≥ 0 Ln = L†n (131)
10 See noted added in proof in Ref. [26].
The birth of string theory 25
He got the following algebra:
[Ln, Lm] = (n−m)Ln+m +
n(n2 − 1)δn+m;0 (132)
where d is the dimension of the Minkowski space-time. We write here d for the
dimension of the Minkowski space, but we want to remind you that almost
everybody working in a model for mesons at that time took for granted that
the dimension of the space-time was d = 4. As far as I remember the first
paper where a dimension d 6= 4 was introduced was Ref. [27] where it was
shown that the unitarity violating cuts in the non-planar loop become poles
that were consistent with unitarity if d = 26.
In the last part of this section we will generalize the factorization procedure
to the Shapiro-Virasoro model whose N -point amplitude is given in Eq. (49).
In this case we must introduce two sets of harmonic oscillators commuting
with each other and only one set of zero modes satisfying the algebra [28] :
[anµ, a
mν ] = [ãnµ, ã
mν ] = ηµνδnm ; [q̂µ, p̂ν ] = iηµν (133)
In terms of them we can introduce the Fubini-Veneziano operator
Q(z, z̄) = q̂ − 2α′p̂ log(zz̄) + i
−n − a†nzn
ãnz̄
−n − ã†nz̄n
(134)
We can then introduce the vertex operator:
V (z, z̄; p) =: eip·Q(z,z̄) : (135)
and write the N -point amplitude in Eq. (95) in the following factorized form:
i=1 d
dVabc
V (zi, z̄i, pi))
|0〉 =
= (2π)4δ(4)(
i=1 d
dVabc
|zi − zj|α
′pi·pj (136)
where the radial ordered product is given by
V (zi, z̄i, pi))
V (zi, z̄i, pi))
θ(|zi| − |zi+1|) + . . . (137)
26 Paolo Di Vecchia
and the dots indicate a sum over all permutations of the vertex operators.
By fixing z1 = ∞, z2 = 1, zN = 0 we can rewrite the previous expression
as follows:
∫ N−1
d2zi〈0, p1|R
V (zi, z̄i, pi))
|0, pN〉 (138)
For the sake of simplicity let us consider the term corresponding to the per-
mutation 1, 2, . . .N . In this case the Koba-Nielsen variables are ordered in
such a way that |zi| ≥ |zi+1| for i = 1, . . .N −1. We can then use the formula:
V (zi, z̄i, pi)) = z
L̃0−1
i V (1, 1, pi)z
i (139)
and change variables:
; |wi| ≤ 1 (140)
to rewrite Eq. (138) as follows:
〈0, p1|V (1, 1, pi1)DV (1, 1, p2)D . . . V (1, 1, pN−1)|0, pN 〉 (141)
where
wL0−1w̄L̃0−1 =
L0 + L̃0 − 2
· sinπ(L0 − L̃0)
L0 − L̃0
(142)
We can now follow the same procedure for all permutations arriving at the
following expression:
〈0, p1|P [V (1, 1, p2)DV (1, 1, p3)D . . . V (1, 1, pN−1)]|0, pN〉 (143)
where P means a sum of all permutations of the particles.
If we want to consider the factorization of the amplitude on the pole at
s = −(p1 + . . . pM )2 we get only the following contribution:
〈p(1...M)|D|p(M+1...N)〉 (144)
where
|p(M+1...N)〉 = P [V (1, 1, pM+1)D . . . V (1, 1, pN−1]|0, pN 〉 (145)
〈p(1...M)| = 〈0, p1|P [V (1, 1, p2)D . . . V (1, 1, pM)] (146)
The amplitude is factorized by introducing a complete set of states and rewrit-
ing Eq. (141) as follows:
The birth of string theory 27
〈p1...M |λ, λ̃〉
2π〈λ, λ̃|δL0,L̃0|λ, λ̃〉
L0 + L̃0 − 2
〈λ, λ̃|p(M+1,...N)〉 (147)
By writing
p̂2 +R ; L̃0 =
p̂2 + R̃ (148)
na†n · an ; R̃ =
nã†n · ãn (149)
we can rewrite Eq. (147) as follows
〈p1...M |λ, λ̃〉
2π〈λ, λ̃|δR,R̃|λ, λ̃〉
R + R̃− α(s)
〈λ, λ̃|p(M+1,...N)〉 (150)
We see that the amplitude for the Shapiro-Virasoro model has simple poles
only for even integer values of αSV (s) = 2 +
s = 2n ≥ 0 and the residue at
the poles factorizes in a sum with a finite number of terms. Notice that the
Regge trajectory of the Shapiro-Virasoro model has double intercept and half
slope of that of the generalized Veneziano model.
5 Physical states and their vertex operators
In the previous section, we have seen that the residue at the poles of the N -
point amplitudes factorizes in a sum of a finite number of terms. We have also
seen that some of these terms, due to the Lorentz metric, correspond to states
with negative norm. We have also derived a number of ”Ward identities” given
in Eq. (126) that imply that some of the terms of the residue decouple. The
question to be answered now is: Is the space spanned by the physical states
a positive norm Hilbert space? In order to answer this question we need first
to find the conditions that characterize the on shell physical states |λ, P 〉
and then to determine which are the states that contribute to the residue
of the pole at α(s = −P 2) = n. In other words, we have to find a way of
characterizing the physical states and of eliminating the spurious states that
decouple in Eq. (102) as a consequence of Eq.s (126). A state |λ.P 〉 contributes
at the residue of the pole in Eq.(102) for α(s = −P 2) = n if it is on shell,
namely if it satisfies the following equations:
R|λ, P 〉 = n|λ, P 〉 ; α(−P 2) = 1− α′P 2 = n (151)
that can be written in a unique equation:
28 Paolo Di Vecchia
(L0 − 1)|λ, P 〉 = 0 (152)
Because of Eq. (126) we also know that a state of the type:
|s, P 〉 = W †m|µ, P 〉 (153)
is not going to contribute to the residue of the pole. We call it a spurious or
unphysical state. We start constructing the subspace of spurious states that
are on shell at the level n. Let us consider the set of orthogonal states |µ, P 〉
such that
R|µ, P 〉 = nµ|µ, P 〉 ; L0|µ, P 〉 = (1−m)|µ, P 〉 ; 1− α′P 2 = n (154)
where
m = n+ nµ (155)
In terms of these states we can construct the most general spurious state that
is on shell at the level n. It is given by
|s, P 〉 = W †m|µ, P 〉 ; (L0 − 1)|s, P 〉 = 0 (156)
per any positive integer m. Using Eq. (154), eq. (156) becomes:
|s, P 〉 = L†m|µ, P 〉 (157)
where |µ, P 〉 is an arbitrary state satisfying Eq.s (154).
A physical state |λ, P 〉 is defined as the one that is orthogonal to all spuri-
ous states appearing at a certain level n. This means that it must satisfy the
following equation:
〈λ.P |L†ℓ |µ, P 〉 = 0 (158)
for any state |µ, P 〉 satisfying Eq.s (154). In conclusion, the on shell physical
states at the level n are characterized by the fact that they satisfy the following
conditions:
Lm|λ, P 〉 = (L0 − 1)|λ, P 〉 = 0 ; 1− α′P 2 = n (159)
These conditions characterizing the physical subspace were first found by Del
Giudice and Di Vecchia [28] where the analysis described here was done.
In order to find the physical subspace one starts writing the most general
on shell state contributing to the residue of the pole at level n in Eq. (154).
Then one imposes Eq.s (159) and determines the states that span the physical
subspace. Actually, among these states one finds also a set of zero norm states
that are physical and spurious at the same time. Those states are of the form
given in Eq. (157), but also satisfy Eq.s (159). It is easy to see that they are
not really physical because they are not contributing to the residue of the pole
The birth of string theory 29
at the level n. This follows from the form of the unit operator given in the
space of the physical states by:
norm 6=0
|λ, P 〉〈λ, P |+
[|λ0, P 〉〈µ0, P |+ |µ0, P 〉〈λ0, P |] (160)
where |λ0, P 〉 is a zero norm physical and spurious state and |µ0, P 〉 its con-
jugate state. A conjugate state of a zero norm state is obtained by changing
the sign of the oscillators with timelike direction. Since |λ0, P 〉 is a spurious
state when we insert the unit operator, given in Eq. (160), in Eq. (102) we
see that the zero norm states never contribute to the residue because their
contribution is annihilated either from the state 〈p(1,M)| or from the state
|p(M+1,N)〉. In conclusion, the physical subspace contains only the states in
the first term in the r.h.s. of Eq. (160).
Let us analyze the first two excited levels. The first excited level corre-
sponds to a massless gauge field. It is spanned by the states ǫµa
1µ|0, P 〉. In
this case the only condition that we must impose is:
1µ|0, P 〉 = 0 =⇒ P · ǫ = 0 (161)
Choosing a frame of reference where the momentum of the photon is given by
Pµ ≡ (P, 0....0, P ) , Eq. (161) implies that the only physical states are:
1i |0, P 〉+ ǫ(a
1;0 − a
1;d−1)|0, P 〉 ; i = 1 . . . d− 2 (162)
where ǫi and ǫ are arbitrary parameters. The state in Eq. (162) is the most
general state of the level N = 1 satisfying the conditions in Eq. (159). The
first state in eq. (162) has positive norm, while the second one has zero norm
that is orthogonal to all other physical states since it can be written as follows:
1;0 − a
1;D−1)|0, P 〉 = L
1|0, P 〉 (163)
in the frame of reference where Pµ ≡ (P, ...0, P ). Because of the previous
property it is decoupled from the physical states together with its conjugate:
1,0 + a
1,d−1)|0, P 〉 (164)
In conclusion, we are left only with the transverse d− 2 states corresponding
to the physical degrees of freedom of a massless spin 1 state. At the next level
n = 2 the most general state is given by:
[αµνa
1,ν + β
2,µ]|0, P 〉 (165)
If we work in the center of mass frame where Pµ = (M,0) we get the following
most general physical state:
|Phys >= αij [a†1,ia
1,j −
(d− 1)
1,k]|0, P 〉+
30 Paolo Di Vecchia
+βi[a
2,i + a
1,i]|0, P >〉+
1,i +
1,0 − 2a
|0, P 〉 (166)
where the indices i, j run over the d− 1 space components. The first term in
(166) corresponds to a spin 2 in (d− 1) dimensional space and has a positive
norm being made with space indices. The second term has zero norm and is
orthogonal to the other physical states since it can be written as L+1 a
1,i|0, P 〉.
Therefore it must be eliminated from the physical spectrum together with its
conjugate, as explained above. Finally, the last state in (166) is spinless and
has a norm given by:
2(d− 1)(26− d) (167)
If d < 26 it corresponds to a physical spin zero particle with positive norm. If
d > 26 it is a ghost. Finally, if d = 26 it has a zero norm and is also orthogonal
to the other physical states since it can be written in the form:
2 + 3L
1 )|0 > (168)
It does not belong, therefore, to the physical spectrum. The analysis of this
level was done in Ref. [29] with d = 4. This did not allow the authors of
Ref. [29] to see that there was a critical dimension.
The analysis of the physical states can be easily extended [28] to the
Shapiro-Virasoro model. In this case the physical conditions given in Eq. (159)
for the open string, become [28]:
Lm|λ, λ̃〉 = L̃m|λ, λ̃〉 = (L0 − 1)|λ, λ̃〉 = (L̃0 − 1)|λ, λ̃〉 = 0 (169)
for any positive integer m. It can be easily seen from the previous equations
that the lowest state of the Shapiro-Virasoro model is the vacuum |0a, 0ã, p〉
corresponding to a tachyon with mass α′p2 = 4, while the next level described
by the state a
1ν |0a, 0ã, p〉 contains massless states corresponding to the
graviton, a dilaton and a two-index antisymmetric tensor Bµν .
Having characterized the physical subspace one can go on and construct
a N -point scattering amplitude involving arbitrary physical states. This was
done by Campagna, Fubini, Napolitano and Sciuto [30] where the vertex oper-
ator for an arbitrary physical state was constructed in analogy with what has
been done for the ground tachyonic state. They associated to each physical
state |α, P 〉 a vertex operator Vα(z, P ) that is a conformal field with conformal
dimension equal to 1:
[Ln, Vα(z, p)] =
zn+1Vα(z, p)
(170)
and reproduces the corresponding state acting on the vacuum as follows:
Vα(z; p)|0, 0〉 ≡ |α; p〉 ; 〈0; 0| lim
z2Vα(z; p) = 〈α, p| (171)
The birth of string theory 31
It satisfies, in addition, the hermiticity relation:
V †α (z, P ) = Vα(
,−P )(−1)α(−P
2) (172)
An excited vertex that will play an important role in the next section is the
one associated to the massless gauge field. It is given by:
Vǫ(z, k) ≡ ǫ ·
dQ(z)
eik·Q(z) ; k · ǫ = k2 = 0 (173)
Because of the last two conditions in Eq. (173) the normal order is not neces-
sary. It is convenient to give the expression of
dQ(z)
in terms of the harmonic
oscillators:
P (z) ≡ dQ(z)
−n−1 (174)
It is a conformal field with conformal dimension equal to 1. The rescaled
oscillators αn are given by:
nan ; α−n =
na†n ; n > 0 ; α0 =
2α′p̂ (175)
In terms of the vertex operators previously introduced the most general
amplitude involving arbitrary physical states is given by [30]:
(2π)4δ(
1 dziθ(zi − zi+1)
dVabc
〈0, 0|
Vαi(zi, pi)|0, 0〉(176)
In the case of the Shapiro-Virasoro model the tachyon vertex operator is
given in Eq. (135). By rewriting Eq. (134) as follows:
Q(z, z̄) = Q(z) + Q̃(z̄) (177)
where
Q(z) =
q̂ − 2α′p̂ log(z) + i
−n − a†nzn
(178)
Q̃(z̄) =
q̂ − 2α′p̂ log(z̄) + i
ãnz̄
−n − ã†nz̄n
(179)
we can write the tachyon vertex operator in the following way:
V (z, z̄, p) =: eip·Q(z)eip·Q̃(z̄) : (180)
32 Paolo Di Vecchia
This shows that the vertex operator corresponding to the tachyon of the
Shapiro-Virasoro model can be written as the product of two vertex oper-
ators corresponding each to the tachyon of the generalized Veneziano model.
Analogously the vertex operator corresponding to an arbitrary physical
state of the Shapiro-Virasoro model can always be written as a product of
two vertex operators of the generalized Veneziano model:
Vα,β(z, z̄, p) = Vα(z,
)Vβ(z̄,
) (181)
The first one contains only the oscillators αn, while the second one only the
oscillators α̃n. They both contain only half of the total momentum p and
the same zero modes p̂ and q̂. The two vertex operators of the generalized
Veneziano model are both conformal fields with conformal dimension equal
to 1. If they correspond to physical states at the level 2n, they satisfy the
following relation (n = ñ):
+ n = 1 (182)
They lie on the following Regge trajectory:
p2 ≡ αSV (−p2) = 2n (183)
as we have already seen by factorizing the amplitude in Eq. (150).
6 The DDF states and absence of ghosts
In the previous section we have derived the equations that characterize the
physical states and their corresponding vertex operators. In this section we
will explicitly construct an infinite number of orthonormal physical states with
positive norm.
The starting point is the DDF operator introduced by Del Giudice, Di Vec-
chia and Fubini [31] and defined in terms of the vertex operator corresponding
to the massless gauge field introduced in eq. (173):
Ai,n =
i Pµ(z)e
ik·Q(z) (184)
where the index i runs over the d−2 transverse directions, that are orthogonal
to the momentum k. We have also taken
= 1. Because of the log z term
appearing in the zero mode part of the exponential, the integral in Eq. (184),
that is performed around the origin z = 0, is well defined only if we constrain
the momentum of the state, on which Ai,n acts, to satisfy the relation:
2α′p · k = n (185)
The birth of string theory 33
where n is a non-vanishing integer.
The operator in Eq. (184) will generate physical states because it com-
mutes with the gauge operators Lm:
[Lm, An;i] = 0 (186)
since the vertex operator transforms as a primary field with conformal dimen-
sion equal to 1 as it follows from Eq. (170).
On the other hand it also satisfies the algebra of the harmonic oscillator
as we are now going to show. From Eq. (184) we get:
[An,i, Am,j] = −
dzǫi · P (z)eik·Q(ζ)ǫj · P (ζ)eik
′ ·Q(ζ) (187)
where
2α′p · k = n ; 2α′p · k′ = m (188)
and k and k′ are supposed to be in the same direction, namely
kµ = nk̂µ ; k
µ = mk̂µ (189)
2α′p · k̂ = 1 (190)
Finally the polarizations are normalized as:
ǫi · ǫj = δij (191)
Since k̂ · ǫi = k̂ · ǫj = k̂2 = 0 a singularity for z = ζ can appear only from the
contraction of the two terms P (ζ) and P ((z) that is given by:
〈0, 0|ǫi · P (z)ǫj · P (ζ)|0, 0〉 = −
2α′δij
(z − ζ)2
(192)
Inserting it in Eq. (187) we get:
[An,i, Am,j ] = δij in
dζk̂ · P (ζ)e−i(n+m))k̂·Q(ζ) =
= inδijδn+m;0
dζk̂ · P (ζ) (193)
where we have used the fact that the integrand is a total derivative and
therefore one gets a vanishing contribution unless n + m = 0. If n + m = 0
from Eq.s (174) and (190) we get:
[An,i, Am,j ] = nδijδn+m;0 ; i, j = 1 . . . d− 2 (194)
34 Paolo Di Vecchia
Eq. (194) shows that the DDF operators satisfy the harmonic oscillator alge-
In terms of this infinite set of transverse oscillators we can construct an
orthonormal set of states:
|i1, N1; i2, N2; . . . im, Nm〉 =
Aik,−Nk√
|0, p〉 (195)
where λh is the multiplicity of the operator Aih,−Nh in the product in Eq.
(195) and the momentum of the state in Eq. (195) is given by
P = p+
k̂Ni (196)
They were constructed in four dimensions where they were not a complete
system of states 11 and it took some time to realize that in fact they were a
complete system of states if d = 26 [32, 33] 12. Brower [32] and Goddard and
Thorn [33] showed also that the dual resonance model was ghost free for any
dimension d ≤ 26. In d = 26 this follows from the fact that the DDF operators
obviously span a positive definite Hilbert space (See Eq. (194)). For d < 26
there are extra states called Brower states [32]. The first of these states is
the last state in Eq. (166) that becomes a zero norm state for d = 26. But
also for d < 26 there is no negative norm state among the physical states.
The proof of the no-ghost theorem in the case α0 = 1 is a very important
step because it shows that the dual resonance model constructed generalizing
the four-point Veneziano formula, is a fully consistent quantum-relativistic
theory! This is not quite true because, when the intercept α0 = 1, the lowest
state of the spectrum corresponding to the pole in the N -point amplitude for
α(s) = 0, is a tachyon with mass m2 = − 1
. A lot of effort was then made
to construct a model without tachyon and with a meson spectrum consistent
with the experimental data. The only reasonably consistent models that came
out from these attempts, were the Neveu-Schwarz [7] for mesons and the
Ramond model [8] for fermions that only later were recognized to be part of a
unique model that nowadays is called the Neveu-Schwarz-Ramond model. But
this model was not really more consistent than the original dual resonance
11 Because of this Fubini did not want to publish our result, but then he went to a
meeting in Israel in spring 1971 giving a talk on our work where he found that
the audience was very interested in our result and when he came back to MIT we
decided to publish our result.
12 I still remember Charles Thorn coming into my office at Cern and telling me:
Paolo, do you know that your DDF states are complete if d = 26? I quickly redid
the analysis done in Ref. [29] with an arbitrary value of the space-time dimension
obtaining Eq.s (166) and (167) that show that the spinless state at the level
α(s) = 2 is decoupled if d = 26. I strongly regretted not to have used an arbitrary
space-time dimension d in the analysis of Ref. [29] .
The birth of string theory 35
model because it still had a tachyon with mass m2 = − 1
. The tachyon
was eliminated from the spectrum only in 1976 through the GSO projection
proposed by Gliozzi, Scherk and Olive [34].
Having realized that, at least for the critical value of the space-time dimen-
sion d = 26, the physical states are described by the DDF states having only
d− 2 = 24 independent components, open the way to Brink and Nielsen [35]
to compute the value α0 = 1 of the Regge trajectory with a very physical ar-
gument. They related the intercept of the Regge trajectory to the zero point
energy of a system with an infinite number of oscillators having only d − 2
independent components:
α0 = −
n (197)
This quantity is obviously infinite and, in order to make sense of it, they in-
troduced a cutoff on the frequencies of the harmonic oscillators obtaining an
infinite term that they eliminated by renormalizing the speed of light and a
finite universal constant term that gave the intercept of the Regge trajectory.
Instead of following their original approach we discuss here an alternative ap-
proach due to Gliozzi [36] that uses the ζ-function regularization. He rewrites
Eq. (197) as follows:
α0 = −
n = −
n−s = −
ζR(−1) = 1 (198)
where in the last equation we have used the identity ζR(−1) = − 112 and we
have put d = 26. Since the Shapiro-Virasoro model has two sets of trans-
verse harmonic oscillators it is obvious that its intercept is twice that of the
generalized Veneziano model.
Using the rules discussed in the previous section we can construct the
vertex operator corresponding to the state in Eq. (195). It is given by:
V(i;Ni)(z, P ) =
dziǫi · P (zi)eiNik̂·Q(zi) : eip·Q(z) : (199)
where the integral on the variable zi is evaluated along a curve of the complex
plane zi containing the point z. The singularity of the integrand for zi = z is
a pole provided that the following condition is satisfied.
2α′p · k̂ = 1 (200)
The last vertex in Eq. (199) is the vertex operator corresponding to the ground
tachyonic state given in Eq. (59) with α′p2 = 1.
Using the general form of the vertex one can compute the three-point
amplitude involving three arbitrary DDF vertex operators. This calculation
36 Paolo Di Vecchia
has been performed in Ref. [37] and since the vertex operators are conformal
fields with dimension equal to 1 one gets:
〈0, 0|V
(z1, P1)V(i(2)
(z2, P2)V(i(3)
(z3, P3)|0, 0〉 =
(z1 − z2)(z1 − z3)(z2 − z3)
(201)
where the explicit form of the coefficient C123 is given by:
C123 = 1〈0, 0|2〈0, 0|3〈0, 0|e
r.s=1
n,m=1
−n;iN
−m;i+
−n;i×
× eτ0
(α′Π2r−1)|N (1)k1 , i
〉1|N (2)k2 , i
〉2|N (3)k3 , i
〉3 (202)
where
N rsnm = −N rnNsm
nmα1α2α3
nαs +mαr
; N rn =
Γ (−nαr+1
αrn!Γ (1− nαr+1αr − n)
(203)
Π = Pr+1αr − Prαr+1 ; r = 1, 2, 3 (204)
Π is independent on the value of r chosen as a consequence of the equations:
Pr = 0 (205)
7 The zero slope limit
In the introduction we have seen that the dual resonance model has been
constructed using rules that are different from those used in field theory.
For instance, we have seen that planar duality implies that the amplitude
corresponding to a certain duality diagram, contains poles in both s and t
channels, while the amplitude corresponding to a Feynman diagram in field
theory contains only a pole in one of the two channels. Furthermore, the
scattering amplitude in the dual resonance model contains an infinite number
of resonant states that, at high energy, average out to give Regge behaviour.
Also this property is not observed in field theory. The question that was
natural to ask, was then: is there any relation between the dual resonance
model and field theory? It turned out, to the surprise of many, that the dual
resonance model was not in contradiction with field theory, but was instead
an extension of a certain number of field theories. We will see that the limit in
The birth of string theory 37
which a field theory is obtained from the dual resonance model corresponds
to taking the slope of the Regge trajectory α′ to zero.
Let us consider the scattering amplitude of four ground state particles in
Eq. (1) that we rewrite here with the correct normalization factor:
A(s, t, u) = C0N
0 (A(s, t) +A(s, u) +A(t, u)) (206)
where
2g(2α′)
4 (207)
is the correct normalization factor for each external leg, g is the dimensionless
open string coupling constant that we have constantly ignored in the previous
sections and C0 is determined by the following relation:
′ = 1 (208)
that is obtained by requiring the factorization of the amplitude at the pole
corresponding to the ground state particle whose mass is given in Eq. (21).
Using Eq. (21) in order to rewrite the intercept of the Regge trajectory in
terms of the mass of the ground state particle m2 and the following relation
satisfied by the Γ - function:
Γ (1 + z) = zΓ (z) (209)
we can easily perform the limit for α′ → 0 of A(s, t) obtaining:
A(s, t) =
m2 − s
m2 − s
(210)
Performing the same limit on the other two planar amplitudes we get the
following expression for the total amplitude in Eq. (206):
A(s, t, u) =
2g(2α′)
(α′)2
m2 − s
m2 − s
m2 − u
(211)
By introducing the coupling constant:
g3 = 4g(2α
4 (212)
Eq. (211) becomes
A(s, t, u) = g23
m2 − s
m2 − s
m2 − u
(213)
that is equal to the sum of the tree diagrams for the scattering of four particles
with mass m of Φ3 theory with coupling constant equal to g3. We have shown
that, by keeping g3 fixed in the limit α
′ → 0, the scattering amplitude of four
38 Paolo Di Vecchia
ground state particles of the dual resonance model is equal to the tree diagrams
of Φ3 theory. This proof can be extended to the scattering of N ground state
particles recovering also in this case the tree diagrams of Φ3 theory. It is also
valid for loop diagrams that we will discuss in the next section. In conclusion,
the dual resonance model reduces in the zero slope limit to Φ3 theory. The
proof that we have presented here is due to J. Scherk [38] 13
A more interesting case to study is the one with intercept α0 = 1. We will
see that, in this case, one will obtain the tree diagrams of Yang-Mills theory,
as shown by Neveu and Scherk [40] 14.
Let us consider the three-point amplitude involving three massless gauge
particles described by the vertex operator in Eq. (173). It is given by the sum
of two planar diagrams. The first one corresponding to the ordering (123) is
given by:
3Tr (λa1λa2λa3)
〈0, 0|Vǫ1(z1, p1)Vǫ2(z2, p2)Vǫ3(z3, p3)|0, 0〉
[(z1 − z2)(z2 − z3)(z1 − z3)]−1
(214)
Using momentum conservation p1+ p2+ p3 = 0 and the mass shell conditions
p2i = pi · ǫi = 0 one can rewrite the previous equation as follows:
0Tr(λ
a1λa2λa3)
× [(ǫ1 · ǫ2)(p1 · ǫ3) + (ǫ1 · ǫ3)(p3 · ǫ2) + (ǫ2 · ǫ3)(p2 · ǫ1)] (215)
The second contribution comes from the ordering 132 that can be obtained
from the previous one by the substitution
Tr(λa1λa2λa3) → −Tr(λa1λa3λa2) (216)
Summing the two contributions one gets
oTr(λ
a1 [λa2 , λa3 ])
× [(ǫ1 · ǫ2)(p1 · ǫ3) + (ǫ1 · ǫ3)(p3 · ǫ2) + (ǫ2 · ǫ3)(p2 · ǫ1)] (217)
The factor
N0 = 2g(2α
′)(d−2)/4 (218)
is the correct normalization factor for each vertex operator if we normalize
the generators of the Chan-Paton group as follows:
δij (219)
13 See also Ref. [39].
14 See also Ref. [41].
The birth of string theory 39
It is related to C0 through the relation
′ = 2 (220)
g is the dimensionless open string coupling constant. Notice that Eq.s (218)
and (220) differ from Eq.s (207) and (208) because of the presence of the
Chan-Paton factors that we did not include in the case of Φ3 theory.
By using the commutation relations:
[λa, λb] = ifabcλc (221)
and the previous normalization factors we get for the three-gluon amplitude:
igYMf
a1a2a3 [(ǫ1 · ǫ2)((p1 − p2) · ǫ3 +
+(ǫ1 · ǫ3)((p3 − p1) · ǫ2) + (ǫ2 · ǫ3)((p2 − p3) · ǫ1)] (222)
that is equal to the 3-gluon vertex that one obtains from the Yang-Mills action
LYM = −
F aαβF
a , F
αβ = ∂αA
β − ∂βAaα + gYMfabcAbαAcβ (223)
where
gYM = 2g(2α
4 (224)
The previous procedure can be extended to the scattering of N gluons finding
the same result that one gets from the tree diagrams of Yang-Mills theory.
In the next section, we will discuss the loop diagrams. Also, in this case one
finds that the h-loop diagrams involving N external gluons reproduces in the
zero slope limit the sum of the h-loop diagrams with N external gluons of
Yang-Mills theory.
We conclude this section mentioning that one can also take the zero slope
limit of a scattering amplitude involving three and four gravitons obtaining
agreement with what one gets from the Einstein Lagrangian of general rela-
tivity. This has been shown by Yoneya [43].
8 Loop diagrams
The N -point amplitude previously constructed satisfies all the axioms of S-
matrix theory except unitarity because its only singularities are simple poles
corresponding to zero width resonances lying on the real axis of the Mandel-
stam variables and does not contain the various cuts required by unitarity [1].
15 The determination of the previous normalization factors can be found in the
Appendix of Ref. [42].
40 Paolo Di Vecchia
In order to eliminate this problem it was proposed already in the early days of
dual theories to assume, in analogy with what happens for instance in pertur-
bative field theory, that the N -point amplitude was only the lowest order (the
tree diagram) of a perturbative expansion and, in order to implement unitar-
ity, it was necessary to include loop diagrams. Then, the one-loop diagrams
were constructed from the propagator and vertices that we have introduced
in the previous sections [44]. The planar one-loop amplitude with M external
particles was computed by starting from a (M + 2)-point tree amplitude and
then by sewing two external legs together after the insertion of a propagator
D given in Eq. (100). In this way one gets:
(2α′)d/2(2π)d
〈P, λ|V (1, p1)DV (1, p2) . . . V (1, pN)D|P, λ〉 (225)
where the sum over λ corresponds to the trace in the space of the harmonic os-
cillators and the integral in ddP corresponds to integrate over the momentum
circulating in the loop. The previous expression for the one-loop amplitude
cannot be quite correct because all states of the space generated by the oscil-
lators in Eq. (51) are circulating in the loop, while we know that we should
include only the physical ones. This was achieved first by cancelling by hand
the time and one of the space components of the harmonic oscillators reducing
the degrees of freedom of each oscillator from d to d − 2 as suggested by the
DDF operators at least for d = 26. This procedure was then shown to be cor-
rect by Brink and Olive [45]. They constructed the operator that projects over
the physical states and, by inserting it in the loop, showed that the reduction
of the degrees of freedom of the oscillators from d to d− 2 was indeed correct.
This was, at that time, the only procedure available to let only the physical
states circulate in the loop because the BRST procedure was discovered a bit
later also in the framework of the gauge field theories!
To be more explicit let us compute the trace in Eq. (225) adding also the
Chan-Paton factor. We get:
(2π)dδ(d)
NTr(λa1 . . . λaM )
(8π2α′)d/2
τd/2+1
[f1(k)]
12 (2π)M×
dνM−1 . . .
dν2 τ
eG(νji)
]2α′pi·pj
; k ≡ e−πτ(226)
where νji ≡ νj − νi,
G(ν) = log
ie−πν
2τ Θ1(iντ |iτ)
f31 (k)
; f1(k) = k
(1− k2n) (227)
The birth of string theory 41
Θ1(ν|iτ) = −2k1/4 sinπν
1− e2iπνk2n
1− e−2iπνk2n
(1− k2n)(228)
Finally the normalization factor N0 is given in Eq. (218). We have performed
the calculation for an arbitrary value of the space-time dimension d. However,
in this way one gets also the extra factor of k
12 appearing in the first line
of Eq. (226) that implies that our calculation is actually only consistent if
d = 26. In fact, the presence of this factor does not allow one to rewrite the
amplitude, originally obtained in the Reggeon sector, in the Pomeron sector
as explained below. In the following we neglect this extra factor, implicitly
assuming that d = 26, but, on the other hand, still keeping an arbitrary d.
Using the relations:
f1(k) =
tf1(q) ; Θ1(iντ |iτ) = iΘ1(ν|it)t1/2eπν
2/t (229)
where t = 1
and q ≡ e−πt, we can rewrite the one-loop planar diagram in the
Pomeron channel. We get:
(2π)dδ(d)
NTr(λa1 . . . λaM )
(8π2α′)d/2
dt[f1(q)]
2−d(2π)M×
dνM−1 . . .
−Θ1(νji|it)
f31 (q)
]2α′pi·pj
(230)
Notice that, by factorizing the planar loop in the Pomeron channel, one con-
structed for the first time what we now call the boundary state [46] 16. This
can be easily seen in the way that we are now going to describe. First of all,
notice that the last quantity in Eq. (230) can be written as follows:
Θ1(νji|it)
f31 (q)
]2α′pi·pj
−2 sin(πνji)
1− q2ne2πiνji
1− q2ne−2πiνji
(1− q2n)2
]2α′pi·pj
(231)
This equation can be rewritten as follows:
〈p = 0|q2R
i=1 : e
ipi·Q(e2iπνi ) : |p = 0〉
Tr (〈p = 0|q2N |p = 0〉)
; R =
na†n · an (232)
16 See also the first paper in Ref. [47].
42 Paolo Di Vecchia
where the trace is taken only over the non-zero modes and momentum con-
servation has been used. It must also be stressed that the normal ordering of
the vertex operators in the previous equation is such that the zero modes are
taken to be both in the same exponential instead of being ordered as in Eq.
(59). By bringing all annihilation operators on the left of the creation ones,
from the expression in Eq. (232) one gets (zi ≡ e2πiνi):
(2π)dδ(d)
(−2 sinπνji)2α
′pi·pj×
n=1 Tr
n·ane
2α′pj ·
znj e
2α′pi· an√
Tr (〈p = 0|q2N |p = 0〉)
(233)
The trace can be computed by using the completeness relation involving co-
herent states |f〉 = efa† |0〉:
e−|f |
|f〉〈f | = 1 (234)
Inserting the previous identity operator in Eq. (233) one gets after some cal-
culation:
(2π)dδ(d)
(−2 sinπνji)2α
′pi·pj×
i.j=1
−2α′pi·pje2πinνji q
n(1−q2n) (235)
Expanding the denominator in the last exponent and performing the sum over
n one gets:
(2π)dδ(d)
(−2 sinπνji)2α
′pi·pj×
2α′pi·pj
log(1−e2πiνji q2(m+1)) (236)
that is equal to the last line of Eq. (231) apart from the δ-function for mo-
mentum conservation. In conclusion, we have shown that Eq.s (231) and (232)
are equal.
Using Eq. (231) we can rewrite Eq. (230) as follows:
NNM0 Tr(λ
a1 . . . λaM )
(8π2α′)d/2
dt[f1(q)]
2−d(2πi)M
dνM−1 . . .
The birth of string theory 43
. . .
λ〈p = 0, λ|q2R
i=1 : e
ipi·Q(e2iπνi ) : |p = 0, λ〉
λ〈p = 0, λ|q2N |p = 0, λ〉
(237)
where the sum over any state |λ〉 corresponds to taking the trace over the
non-zero modes. If d = 26 we can rewrite Eq. (237) in a simpler form:
NNM0 Tr(λ
a1 . . . λaM )
(8π2α′)d/2
dt (2πi)M
dνM−1 . . .
〈p = 0, λ|q2R−2
: eipi·Q(e
2iπνi ) : |p = 0, λ〉 (238)
The previous equation contains the factor
dtq2R−2 that is like the propa-
gator of the Shapiro-Virasoro model, but with only one set of oscillators as
in the generalized Veneziano model. In the following we will rewrite it com-
pletely with the formalism of the Shapiro-Virasoro model. This can be done
by introducing the Pomeron propagator:
dt q2N−2 =
D̂ ; D̂ ≡ α
zL0−1z̄L̃0−1; |z| ≡ q = e−πt(239)
and rewriting the planar loop in the following compact form:
〈B0|D̂|BM 〉 ; |B0〉 ≡
n |p = 0, 0a, 0ã〉 (240)
where |B0〉 is the boundary state without any Reggeon on it,
Td−1 =
2(d−10)/4
α′)−d/2−1 (241)
and |BM 〉 is instead the one with M Reggeons given by:
|BM 〉 = NM0 Tr(λa1 . . . λaM )(2πi)M
dνM−1 . . .
: eipi·Q(e
2iπνi ) : |B0〉 (242)
We want to stress once more that the normal ordering in the previous equa-
tion is defined by taking the zero modes in the same exponential. Both the
boundary states and the propagator are now states of the Shapiro-Virasoro
model. This means that we have rewritten the one-loop planar diagram, where
the states of the generalized Veneziano model circulate in the loop, as a tree
44 Paolo Di Vecchia
diagram of the Shapiro-Virasoro model involving two boundary states and a
propagator. This is what nowadays is called open/closed string duality.
Besides the one-loop planar diagram in Eq. (225), that is nowadays called
the annulus diagram, also the non-planar and the non-orientable diagrams
were constructed and studied. In particular the non-planar one, that is ob-
tained as the planar one in Eq. (225) but with two propagators multiplied
with the twist operator
Ω = eL−1(−1)R , (243)
had unitarity violating cuts that disappeared [27] if the dimension of the
space-time d = 26, leaving behind additional pole singularities. The explicit
form of the non-planar loop can be obtained following the same steps done
for the planar loop. One gets for the non-planar loop the following amplitude:
〈BR|D̂|BM 〉 (244)
where now both boundary states contain, respectively, R and M Reggeon
states. The additional poles found in the non-planar loop were called Pomerons
because they occur in the Pomeron sector, that today is called the closed string
channel, to distinguish them from the Reggeons that instead occur in the
Reggeon sector, that today is called the open string sector of the planar and
non-planar loop diagrams. At that time in fact, the states of the generalized
Veneziano models were called Reggeons, while the additional ones appearing
in the non-planar loop were called Pomerons. The Reggeons correspond nowa-
days to open string states, while the Pomerons to closed string states. These
things are obvious now, but at that time it took a while to show that the
additional states appearing in the Pomeron sector have to be identified with
those of the Shapiro-Virasoro model. The proof that the spectrum was the
same came rather early. This was obtained by factorizing the non-planar dia-
gram in the Pomeron channel [46] as we have done in Eq. (244). It was found
that the states of the Pomeron channel lie on a linear Regge trajectory that
has double intercept and half slope of the one of the Reggeons. This follows
immediately from the propagator D̂ in Eq. (239) that has poles for values of
the momentum of the Pomeron exchanged given by:
p2 = 2n (245)
that are exactly the values of the masses of the states of the Shapiro-Virasoro
model [48], while the Reggeon propagator in Eq. (100) has poles for values of
momentum equal to:
1− α′p2 = n (246)
However, it was still not clear that the Pomeron states interact among them-
selves as the states of the Shapiro-Virasoro model. To show this it was first
The birth of string theory 45
necessary to construct tree amplitudes containing both states of the general-
ized Veneziano model and of the Shapiro-Virasoro model [49]. They reduced
to the amplitudes of the generalized Veneziano (Shapiro-Virasoro) model if
we have only external states of the generalized Veneziano (Shapiro-Virasoro)
model. Those amplitudes are called today disk amplitudes containing both
open and closed string states. They were constructed [49] by using for the
Reggeon states the vertex operators that we have discussed in Sect. (5) in-
volving one set of harmonic oscillators and for the Pomeron states the vertex
operators given in Eq. (181) that we rewrite here:
Vα,β(z, z̄, p) = Vα(z,
)Vβ(z̄,
) (247)
because now both component vertices contain the same set of harmonic os-
cillators as in the generalized Veneziano model. Furthermore, each of the two
vertices is separately normal ordered, but their product is nor normal ordered.
The amplitude involving both kinds of states is then constructed by taking
the product of all vertices between the projective invariant vacuum and inte-
grating the Reggeons on the real axis in an ordered way and the Pomerons in
the upper half plane, as one does for a disk amplitude.
We have mentioned above that the two vertices are separately normal
ordered, but their product is not normal ordered. When we normal order
them we get, for instance for the tachyon of the Pomeron sector, a factor
(z − z̄)α′p2/2 that describes the Reggeon-Pomeron transition. This implies a
direct coupling [51] between the U(1) part of gauge field and the two-index
antisymmetric field Bµν , called Kalb-Ramond field [50], of the Pomeron sector,
that makes the gauge field massive [51].
It was then shown that, by factorizing the non-planal loop in the Pomeron
channel, one reproduced the scattering amplitude containing one state of
the Shapiro-Virasoro and a number of states of the generalized Veneziano
model [52]. If we have also external states belonging to the generalized
Shapiro-Virasoro model, then by factorizing the non-planar one loop ampli-
tude in the pure Pomeron channel, one would obtain the tree amplitudes of
the Shapiro-Virasoro model [52].
All this implies that the generalized Veneziano model and the Shapiro-
Virasoro model are not two independent models, but they are part of the
same and unique model. In fact, if one started with the generalized Veneziano
model and added loop diagrams to implement unitarity, one found the ap-
pearence in the non-planar loop of additional states that had the same mass
and interaction of those of the Shapiro-Virasoro model.
The planar diagram, written in Eq. (230) in the closed string channel, is
divergent for large values of t. This divergence was recognized to be due to
exchange, in the Pomeron channel, of the tachyon of the Shapiro-Virasoro
model and of the dilaton [47]. They correspond, respectively, to the first two
terms of the expansion:
[f1(q)]
−24 = e2πt + 24 +O
e−2πt
(248)
46 Paolo Di Vecchia
The first one could be cancelled by an analytic continuation, while the second
one could be eliminated through a renormalization of the slope of the Regge
trajectory α′ [47].
We conclude the discussion of the one-loop diagrams by mentioning
that the one-loop diagram for the Shapiro-Virasoro model was computed by
Shapiro [53] who also found that the integrand was modular invariant.
The computation of multiloop diagrams requires a more advanced tech-
nology that was also developed in the early days of the dual resonance model
few years before the discovery of its connection to string theory. In order to
compute multiloop diagrams one needs first to construct an object that was
called the N -Reggeon vertex and that has the properties of containing N sets
of harmonic oscillators, one for each external leg, and is such that, when we
saturate it with N physical states, we get the corresponding N -point ampli-
tude. In the following we will discuss how to determine the N -Reggeon vertex.
The first step toward the N -Reggeon vertex is the Sciuto-Della Selva-
Saito [54] vertex that includes two sets of harmonic oscillators that we denote
with the indices 1 and 2. It is equal to:
VSDS = 2〈x = 0, 0| : exp
dzX ′2(z) ·X1(1− z)
: (249)
where X is the quantity that we have called Q in Eq. (57) and the prime
denotes a derivative with respect to z. It satisfies the important property
of giving the vertex operator Vα(z = 1) of an arbitrary state |α〉 when we
saturate it with the corresponding state:
VSDS |α〉2 = Vα(z = 1) (250)
A shortcoming of this vertex is that it is not invariant under a cyclic permu-
tation of the three legs. A cyclic symmetric vertex has been constructed by
Caneschi, Schwimmer and Veneziano [55] by inserting the twist operator in
Eq. (243). But the 3-Reggeon vertex is not enough if we want to compute an
arbitrary multiloop amplitude. We must generalize it to an arbitrary number
of external legs. Such a vertex, that can be obtained from the one in Eq. (249)
with a very direct procedure, or that can also be obtained by sewing together
three-Reggeon vertices, has been written in its final form by Lovelace [56] 17.
Here we do not derive it, but we give directly its expression written in Ref. [56]:
VN,0 =
i=1 dzi
dVabc
i=1[V
i (0)]
[i<x = 0, Oa|] δ(
i,j=1
n,m=0
a(i)n Dnm(ΓV
i Vj) a
(251)
17 See also Ref. [57]. Earlier papers on the N-Reggeon can be found in Ref.s [58].
The birth of string theory 47
where a
0 ≡ αi0 =
2α′p̂i is the momentum of particle i and the infinite
matrix:
Dnm(γ) =
∂mz [γ(z)]
n|z=0 ; n,m = 1.. : D00(γ) = − log |
AD −BC
Dn0 =
)n ; D0n =
)n ; γ(z) =
Az +B
Cz +D
(252)
is a ”representation” of the projective group corresponding to the conformal
weight ∆ = 0, that satisfies the eqs.:
Dnm(γ1γ2) =
Dnl(γ1)Dlm(γ2) +Dn0(γ1)δ0m +D0m(γ2)δn0 (253)
Dnm(γ) = Dmn(Γγ
−1Γ ) Γ (z) =
(254)
Finally Vi is a projective transformation that maps 0, 1 and ∞ into zi−1, zi
and zi+1.
The previous vertex can be written in a more elegant form as follows:
VN,0 =
i=1 dzi
dVabc
i=1[V
i (0)]
[i<x = 0, Oa|] δ(
dz∂X(i)(z)p̂i logV
i (z)
i,j=1
dy∂X(i)(z) log[Vi(z)− Vj(y)]∂X(j)(y)
(255)
where the quantities X(i) are what we called Q, namely the Fubini-Veneziano
field, in the previous sections. The N -Reggeon vertex that satisfies the impor-
tant property of giving the scattering amplitude of N physical particle when
we saturate it with their corresponding states, is the fundamental object for
computing the multiloop amplitudes. In fact, if we want to compute a M -loop
amplitude withN external states, we need to start from the (N+2M)-Reggeon
vertex and then we have to sew the M pairs together after having inserted a
propagator D. In this way we obtain an amplitude that is not only integrated
over the punctures zi (i = 1 . . . N) of the N external states, but also over
the additional 3h− 3 moduli corresponding to the punctures variables of the
48 Paolo Di Vecchia
states that we sew together and the integration variable of the M propaga-
tors. h is the number of loops. The multiloop amplitudes have been obtained
in this way already in 1970 [59, 60, 61] and, through the sewing procedure,
one obtained functions, as the period matrix, the abelian differentials, the
prime form, etc., that are well defined on Riemann surface! The only thing
that was missing, was the correct measure of integrations over the 3h−3 vari-
ables because it was technically not possible to let only the physical states to
circulate in the loops. This problem was solved only much later [62, 63] when
a BRST invariant formulation of string theory and the light-cone functional
integral could be used for computing multiloops. They are two very different
approaches that, however, gave the same result. For the sake of completeness
we write here the planar h-loop amplitude involving M tachyons:
M (p1, . . . , pM ) = N
h Tr(λa1 · · ·λaM ) Ch
2gs (2α
(d−2)/4
[dm]Mh
G(h)(zi, zj)
V ′i (0)V
j (0)
2α′pi·pj
, (256)
where Nh Tr(λa1 · · ·λaM ) is the appropriate U(N) Chan-Paton factor, g is
the dimensionless open string coupling constant, Ch is a normalization factor
given by
(2π)dh
g2h−2s
(2α′)d/2
, (257)
and G(h) is the h-loop bosonic Green function
G(h)(zi, zj) = logE(h)(zi, zj)−
ωµ (2πImτµν)
ων , (258)
with E(h)(zi, zj) being the prime form, ω
µ (µ = 1, . . . , h) the abelian differen-
tials and τµν the period matrix. All these objects, as well as the measure on
moduli space [dm]Mh , can be explicitly written in the Schottky parametrization
of the Riemann surface, and their expressions for arbitrary h can be found for
example in Ref. [64]. It is given by
[dm]Mh =
dVabc
V ′i (0)
dkµ dξµ dηµ
k2µ (ξµ − ηµ)2
(1− kµ)2
(259)
× [det (−iτµν)]−d/2
(1− knα)−d
(1− knα)2
where kµ are the multipliers, ξµ and ηµ are the fixed points of the generators
of the Schottky group,
The birth of string theory 49
9 From dual models to string theory
The approach presented in the previous sections is a real bottom-up ap-
proach. The experimental data were the driving force in the construction of
the Veneziano model and of its generalization to N external legs. The rest of
the work that we have described above consisted in deriving its properties. The
result is, except for a tachyon, a fully consistent quantum-relativistic model
that was a source of fascination for those who worked in the field. Although
the model grew out of S-matrix theory where the scattering amplitude is the
only observable object, while the action or the Lagrangian have not a central
role, some people nevertheless started to investigate what was the underly-
ing microscopic structure that gave rise to such a consistent and beautiful
model. It turned out, as we know today, that this underlying structure is that
of a quantum-relativistic string. However, the process of connecting the dual
resonance model (actually two of them the generalized Veneziano and the
Shapiro-Virasoro model) to string theory took several years from the origi-
nal idea to a complete and convincing proof of the conjecture. The original
conjecture was independently formulated by Nambu [20, 65], Nielsen [66] and
Susskind [21] 18. If we look at it in retrospective, it was at that time a fantastic
idea that shows the enormous physical intuition of those who formulated it.
On the other hand, it took several years to digest it before one was able to
derive from it all the deep features of the dual resonance model. Because of
this, the idea that the underlying structure was that of a relativistic string,
did not really influence most of the research in the field up to 1973. Let me
try to explain why.
A common feature of the work of Ref.s [20, 66, 21] is the suggestion that
the infinite number of oscillators, that one got through the factorization of
the dual resonance model, naturally comes out from a two-dimensional free
Lagrangian for the coordinate Xµ(τ, σ) of a one-dimensional string, that is an
obvious generalization of the Lagrangian that one writes for the coordinate
Xµ(τ) of a pointlike object in the proper-time gauge:
=⇒ L ∼
(260)
Being this theory conformal invariant the Virasoro operators were also con-
structed together with their algebra. In this very first formulation, however,
the Virasoro generators Ln were just the generators associated to the confor-
mal symmetry of the string world-sheet Lagrangian given in Eq. (260) as in
any conformal field theory. It was not clear at all why they should imply the
gauge conditions found by Virasoro or, in modern terms, why they should be
zero classically. The basic ingredient to solve this problem was provided by
Nambu [65] and Goto [68] who wrote the non-linear Lagrangian proportional
18 See also Ref. [67].
50 Paolo Di Vecchia
to the area spanned by the string in the external target space. They proceeded
in analogy with the point particle and wrote the following action:
−dσµνdσµν (261)
where
dσµν =
dζα ∧ dζβ = ∂Xµ
ǫαβdσdτ (262)
Xµ(σ, τ) is the string coordinate and ζ
0 = τ and ζ1 = σ are the coordinates of
the string worldsheet. ǫαβ is an antisymmetric tensor with ǫ01 = 1. Inserting
eq. (262) in (261) and fixing the proportionality constant one gets the Nambu-
Goto action [65, 68]:
S = −cT
(Ẋ ·X ′)2 − Ẋ2X ′2 (263)
where
Ẋµ ≡ ∂X
µ ≡ ∂X
(264)
and T ≡ 1
is the string tension, that replaces the mass appearing in the
case of a point particle. In this formulation, the string Lagrangian is invariant
under any reparametrization of the world-sheet coordinates σ and τ and not
only under the conformal transformations. This, in fact, implies that the two-
dimensional world-sheet energy-momentum tensor of the string is actually zero
as we will show later on. But it took still a few years to connect the Nambu-
Goto action to the properties of the dual resonance model. In the meantime
an analogue model was formulated [69] that reproduced the tree and loop
amplitudes of the generalized Veneziano model. This approach anticipated
by several years the path integral derivation of dual amplitudes. It was very
closely related to the functional integral formulation of Ref.s [70].
However, one needed to wait until 1973 with the paper of Goddard,
Goldstone, Rebbi and Thorn [71], where the Nambu-Goto action was cor-
rectly treated, all its consequences were derived and it became completely
clear that the structure underlying the dual resonance model was that of a
quantum-relativistic string. The equation of motion for the string were de-
rived from the action in Eq. (263) by imposing δS = 0 for variations such
that δXµ(τi) = δX
µ(τf ) = 0. One gets:
∂X ′µ
δXµ +
∂X ′µ
δXµ|σ=πσ=0
(265)
where L is the Lagrangian in Eq. (263). Since δXµ is arbitrary, from eq. (265)
one gets the Euler-Lagrange equation of motion
The birth of string theory 51
∂X ′µ
= 0 (266)
and the boundary conditions
∂X ′µ
= 0 or δXµ = 0 at σ = 0, π (267)
for an open string and
Xµ(τ, 0) = Xµ(τ, π) (268)
for a closed string. In the case of an open string, the first kind of boundary
condition in Eq.(267) corresponds to Neumann boundary conditions, while
the second one to Dirichlet boundary conditions. Only the Neumann bound-
ary conditions preserve the translation invariance of the theory and, there-
fore, they were mostly used in the early days of string theory. It must be
stressed, however, that Dirichlet boundary conditions were already discussed
and used in the early days of string theory for constructing models with off-
shell states [72].
From Eq. (263) one can compute the momentum density along the string:
≡ Pµ = cT
′2 −X ′µ(Ẋ ·X ′)
(Ẋ ·X ′)2 − Ẋ2X ′2
(269)
and obtain the following constraints between the dynamical variables Xµ and
c2T 2x′
+ P 2 = x′ · P = 0 (270)
They are a consequence of the reparametrization invariance of the string La-
grangian. Because of this one can choose the orthonormal gauge specified by
the conditions:
Ẋ2 +X ′
= Ẋ ·X ′ = 0 (271)
that nowadays is called conformal gauge. In this gauge eq. (269) becomes:
Pµ = cT Ẋµ
∂X ′µ
= −cTX ′µ (272)
and therefore the eq. of motion in eq.(266) becomes:
Ẍµ −X ′′µ = 0 (273)
while the boundary condition in eq.(267) becomes:
X ′µ(σ = 0, π) = 0 (274)
52 Paolo Di Vecchia
The most general solution of the eq. of motion and of the boundary conditions
can be written as follows:
Xµ(τ, σ) = qµ + 2α′pµτ + i
[aµne
−inτ − a+µn einτ ]
cosnσ√
(275)
for an open string and
Xµ(τ, σ) = qµ + 2α′pµτ +
[ãµne
−2in(τ+σ) − ã+µn e2in(τ+σ)]
[aµne
−2in(τ−σ) − a+µn e2in(τ−σ)]
(276)
for a closed string. This procedure really shows that, starting from the
Nambu-Goto action, one can choose a gauge (the orthonormal or confor-
mal gauge) where the equation of motion of the string becomes the two-
dimensional D’Alembert equation in Eq. (273). Furthermore, the invariance
under reparametrization of the Nambu-Goto action implies that the two-
dimensional energy-momentum tensor is identically zero at the classical level
(See Eq. (271)).
As the Lorentz gauge in QED the orthonormal gauge does not fix com-
pletely the gauge. We can still perform reparametrizations that leave in the
conformal gauge: they are conformal transformatiuons. Introducing the vari-
able z = eiτ the generators of the conformal transformations for the open
string can be written as follows:
dzzn+1
αn−m · αm = 0 (277)
where
αµn =
naµn if n > 0√
2α′pµ if n = 0√
na†µn if n < 0
(278)
They are zero as a consequence of Eq.s (270) that in the conformal gauge
become Eq.s (271). In the case of a closed string we get instead:
L̃n =
dzzn+1
= 0 (279)
dz̄z̄n+1
= 0 (280)
The birth of string theory 53
In terms of the harmonic oscillators introduced in eq. (276) we get
αm · αn−m = 0 ; L̃n =
α̃m · α̃n−m = 0 (281)
where for the non-zero modes we have used the convention in (278), while the
zero mode is given by:
0 = α̃
(282)
In conclusion, the fact that we have reparametrization invariance implies that
the Virasoro generators are classically identically zero. When we quantize the
theory one cannot and also does not need to impose that they are vanishing at
the operator level. They are imposed as conditions characterizing the physical
states.
〈Phys′|Ln|Phys〉 = 〈Phys′|(L0 − 1)|Phys〉 = 0 ; n 6= 0 (283)
These equations are satisfied if we require:
Ln|Phys >= (L0 − 1)|Phys >= 0 (284)
The extra factor −1 in the previous equations comes from the normal ordering
as explained in Eq. (198).
The authors of Ref. [71] further specified the gauge by fixing it completely.
They introduced the light-cone gauge specified by imposing the condition:
X+ = 2α′p+τ (285)
where
X0 ±Xd−1√
X0 ±Xd−1√
(286)
In this gauge the only physical degrees of freedom are the transverse ones.
In fact the components along the directions 0 and d − 1 can be expressed in
terms of the transverse ones by inserting Eq. (285) in the constraints in Eq.
(271) and getting:
Ẋ− =
4α′p+
(Ẋ2i +X
i ) X
2α′p+
Ẋi ·X ′i (287)
that up to a constant of integration determine completely X− as a function
of X i. In terms of oscillators we get
α+n = 0 ;
2α′α−n =
αin−mα
m n 6= 0 (288)
54 Paolo Di Vecchia
for an open string and
α+n = α̃
n = 0 n 6= 0 (289)
together with
2α′α−n =
αin−mα
2α′α̃−n =
α̃in−mα̃
m (290)
in the case of a closed string.
This shows that the physical states are described only by the transverse
oscillators having only d − 2 components. Those transverse oscillators corre-
spond to the transverse DDF operators that we have discussed in Section 6.
The authors of Ref. [71] also constructed the Lorentz generators only in terms
of the transverse oscillators and they showed that they satisfy the correct
Lorentz algebra only if the space-time dimension is d = 26. In this way the
spectrum of the dual resonance model was completely reproduced starting
from the Nambu-Goto action if d = 26! On the other hand, the choice of
d = 26 is a necessity if we want to keep Lorentz invariance!
Immediately after this, the interaction was also included either by adding
a term describing the interaction of the string with an external gauge field [73]
or by using a functional formalism [74, 75].
In the following we will give some detail only of the first approach for the
case of an open string. A way to describe the string interaction is by adding
to the free string action an additional term that describes the interaction of
the string with an external field.
SINT =
dDyΦL(y)JL(y) (291)
where ΦL(y) is the external field and JL is the current generated by the string.
The index L stands for possible Lorentz indices that are saturated in order to
have a Lorentz invariant action.
In the case of a point particle, such an interaction term will not give any
information on the self-interaction of a particle.
In the case of a string, instead, we will see that SINT will describe the
interaction among strings because the external fields that can consistently
interact with a string are only those that correspond to the various states of
the string, as it will become clear in the discussion below.
This is a consequence of the fact that, for the sake of consistency, we must
put the following restrictions on SINT :
• It must be a well defined operator in the space spanned by the string
oscillators.
The birth of string theory 55
• It must preserve the invariances of the free string theory. In particular, in
the ”conformal gauge” it must be conformal invariant.
• In the case of an open string, the interaction occurs at the end point of
a string (say at σ = 0). This follows from the fact that two open strings
interact attaching to each other at the end points.
The simplest scalar current generated by the motion of a string can be written
as follows
J(y) =
dσδ(σ)δ(d)[yµ − xµ(τ, σ)] (292)
where δ(σ) has been introduced because the interaction occurs at the end of
the string. For the sake of simplicity we omit to write a coupling constant g
in (292).
Inserting (292) in (291) and using for the scalar external field Φ(y) = eik·y
a plane wave, we get the following interaction:
SINT =
dτ : eik·X(τ,0) : (293)
where the normal ordering has been introduced in order to have a well defined
operator. The invariance of (293) under a conformal transformation τ → w(τ)
requires the following identity:
SINT =
dτ : eik·X(τ,0) : =
dw : eik·X(w,0) : (294)
or, in other words, that
: eik·X(τ,0) :=⇒ w′(τ) : eik·X(w,0) : (295)
This means that the integrand in Eq. (294) must be a conformal field with
conformal dimension equal to one and this happens only if α′k2 = 1. The
external field corresponds then to the tachyonic lowest state of the open string.
Another simple current generated by the string is given by:
Jµ(y) =
dσδ(σ)Ẋµ(τ, σ)δ
(d)(y −X(τ, σ)) (296)
Inserting (296) in (291) we get
SINT =
dτẊµ(τ, 0)ǫ
µeik·X(τ,0) (297)
if we use a plane wave for Φµ(y) = ǫµe
ik·y. The vertex operator in eq. (297)
is conformal invariant only if
k2 = ǫ · k = 0 (298)
56 Paolo Di Vecchia
and, therefore, the external vector must be the massless photon state of the
string. We can generalize this procedure to an arbitrary external field and
the result is that we can only use external fields that correspond to on shell
physical states of the string.
This procedure has been extended in Ref. [73] to the case of external
gravitons by introducing in the Nambu-Goto action a target space metric and
obtaining the vertex operator for the graviton that is a massless state in the
closed string theory. Remember that, at that time, this could have been done
only with the Nambu-Goto action because the σ-model action was introduced
only in 1976 first for the point particle [76] and then for the string [77]. As
in the case of the photon it turned out that the external field corresponding
to the graviton was required to be on shell. This condition is the precursor of
the equations of motion that one obtains from the σ-model action requiring
the vanishing of the β-function [78].
One can then compute the probability amplitude for the emission of a
number of string states corresponding to the various external fields, from an
initial string state to a final one. This amplitude gives precisely the N -point
amplitude that we discussed in the previous sections [73]. In particular, one
learns that, in the case of the open string, the Fubini-Veneziano field is just
the string coordinate computed at σ = 0:
Qµ(z) ≡ Xµ(z, σ = 0) ; z = eiτ (299)
In the case of a closed string we get instead:
Qµ(z, z̄) ≡ Xµ(z, z̄) ; z = e2i(τ−σ) , z̄ = e2i(τ+σ) (300)
Finally, let me mention that with the functional approach Mandelstam [74]
and Cremmer and Gervais [79] computed the interaction between three arbi-
trary physical string states and reproduced in this way the coupling of three
DDF states given in Eq. (202) and obtained in Ref. [37] by using the operator
formalism. At this point it was completely clear that the structure underlying
the generalized Veneziano model was that of an open relativistic string, while
that underlying the Shapiro-Virasoro model was that of a closed relativistic
string. Furthermore, these two theories are not independent because, if one
starts from an open string theory, one gets automatically closed strings by
loop corrections.
10 Conclusions
In this contribution, we have gone through the developments that led from
the construction of the dual resonance model to the bosonic string theory
trying as much as possible to include all the necessary technical details. This
is because we believe that they are not only important from an historical point
of view, but are also still part of the formalism that one uses today in many
The birth of string theory 57
string calculations. We have tried to be as complete and objective as possible,
but it could very well be that some of those who participated in the research
of these years, will not agree with some or even many of the statements we
made. We apologize to those we have forgotten to mention or we have not
mentioned as they would have liked.
Finally, after having gone through the developments of these years, my
thoughts go to Sergio Fubini who shared with me and Gabriele many of the
ideas described here and who is deeply missed, and to my friends from Flo-
rence, Naples and Turin for a pleasant collaboration in many papers discussed
here.
Acknowledgments
I thank R. Marotta and I. Pesando for a critical reading of the manuscript.
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	The birth of string theory
	Paolo Di Vecchia

  In this contribution we go through the developments that in the years 1968 to
1974 led from the Veneziano model to the bosonic string.

Introduction
The sixties was a period in which strong interacting processes were studied
in detail using the newly constructed accelerators at Cern and other places.
Many new hadronic states were found that appeared as resonant peaks in var-
ious cross sections and hadronic cross sections were measured with increasing
accuracy. In general, the experimental data for strongly interacting processes
were rather well understood in terms of resonance exchanges in the direct
channel at low energy and by the exchange of Regge poles in the transverse
channel at higher energy. Field theory that had been very successful in de-
scribing QED seemed useless for strong interactions given the big number of
hadrons to accomodate in a Lagrangian and the strength of the pion-nucleon
coupling constant that did not allow perturbative calculations. The only do-
main in which field theoretical techniques were successfully used was current
algebra. Here, assuming that strong interactions were described by an almost
chiral invariant Lagrangian, that chiral symmetry was spontaneously broken
and that the pion was the corresponding Goldstone boson, field theoretical
methods gave rather good predictions for scattering amplitudes involving pi-
ons at very low energy. Going to higher energy was, however, not possible
with these methods.
Because of this, many people started to think that field theory was use-
less to describe strong interactions and tried to describe strong interacting
http://arxiv.org/abs/0704.0101v1
2 Paolo Di Vecchia
processes with alternative and more phenomenological methods. The basic
ingredients for describing the experimental data were at low energy the ex-
change of resonances in the direct channel and at higher energy the exchange
of Regge poles in the transverse channel. Sum rules for strongly interacting
processes were saturated in this way and one found good agreement with the
experimental data that came from the newly constructed accelerators. Be-
cause of these successes and of the problems that field theory encountered to
describe the data, it was proposed to construct directly the S matrix without
passing through a Lagrangian. The S matrix was supposed to be constructed
from the properties that it should satisfy, but there was no clear procedure on
how to implement this construction1. The word “bootstrap” was often used
as the way to construct the S matrix, but it did not help very much to get an
S matrix for the strongly interacting processes.
One of the basic ideas that led to the construction of an S matrix was
that it should include resonances at low energy and at the same time give
Regge behaviour at high energy. But the two contributions of the resonances
and of the Regge poles should not be added because this would imply double
counting. This was called Dolen, Horn and Schmidt duality [2]. Another idea
that helped in the construction of an S matrix was planar duality [3] that
was visualized by associating to a certain process a duality diagram, shown in
Fig. (1), where each meson was described by two lines representing the quark
and the antiquark. Finally, also the requirement of crossing symmetry played
a very important role.
Fig. 1. Duality diagram for the scattering of four mesons
Starting from these ideas Veneziano [4] was able to construct an S matrix
for the scattering of four mesons that, at the same time, had an infinite number
of zero width resonances lying on linearly rising Regge trajectories and Regge
behaviour at high energy. Veneziano originally constructed the model for the
1 For a discussion of S matrix theory see Ref.s [1]
The birth of string theory 3
process ππ → πω, but it was immediately extended to the scattering of four
scalar particles.
In the case of four identical scalar particles, the crossing symmetric scat-
tering amplitude found by Veneziano consists of a sum of three terms:
A(s, t, u) = A(s, t) +A(s, u) +A(t, u) (1)
where
A(s, t) =
Γ (−α(s))Γ (−α(t))
Γ (−α(s)− α(t))
dxx−α(s)−1(1− x)−α(t)−1 (2)
with linearly rising Regge trajectories
α(s) = α0 + α
′s (3)
This was a very important property to implement in a model because it was
in agreement with the experimental data in a wide range of energies. s, t and
u are the Mandelstam variables:
s = −(p1 + p2)2 , t = −(p3 + p2)2 , u = −(p1 + p3)2 (4)
The three terms in Eq. (1) correspond to the three orderings of the four parti-
cles that are not related by a cyclic or anticyclic 2 permutation of the external
legs. They correspond, respectively, to the three permutations: (1234), (1243)
and (1324) of the four external legs. They have only simple pole singularities.
The first one has only poles in the s and t channels, the second only in the s
and u channels and the third only in the t and u channels. This property fol-
lows directly from the duality diagram that is associated to each inequivalent
permutation of the external legs. In fact, at that time one used to associate
to each of the three inequivalent permutations a duality diagram where each
particle was drawn as consisting of two lines that rappresented the quark and
antiquark making up a meson. Furthermore, the diagram was supposed to
have only poles singularities in the planar channels which are those involving
adjacent external lines. This means that, for instance, the duality diagram
corresponding to the permutation (1234) has only poles in the s and t chan-
nels as one can see by deforming the diagram in the plane in the two possible
ways shown in figure (2).
This was a very important property of the duality diagram that makes
it qualitatively different from a Feynman diagram in field theory where each
diagram has only a pole in one of the three s, t and u channels and not
simultaneously in two of them. If we accept the idea that each term of the
sum in Eq. (1) is described by a duality diagram, then it is clear that we
2 An anticyclic permutation corresponding, for instance, to the ordering (1234) is
obtained by taking the reverse of the original ordering (4321) and then performing
a cyclic permutation.
4 Paolo Di Vecchia
Fig. 2. The duality diagram contains both s and t channel poles
do not need to add terms corresponding to equivalent diagrams because the
corresponding duality diagram is the same and has the same singularities. It
is now clear that it was in some way implicit in this picture the fact that the
Veneziano model corresponds to the scattering of relativistic strings. But at
that time the connection was not obvious at all. The only S matrix property
that the Veneziano model failed to satisfy was the unitarity of the S matrix.
because it contained only zero width resonances and did not have the various
cuts required by unitarity. We will see how this property will be implemented.
Immediately after the formulation of the Veneziano model, Virasoro [5]
proposed another crossing symmetric four-point amplitude for scalar particles
that consisted of a unique piece given by:
A(s, t, u) ∼
Γ (−α(u)
)Γ (−α(s)
)Γ (−α(t)
Γ (1 +
)Γ (1 +
)Γ (1 +
where
α(s) = α0 + α
′s (6)
The model had poles in all three s, t and u channels and could not be written
as sum of three terms having poles only in planar diagrams. In conclusion,
the Veneziano model satisfies the principle of planar duality being a crossing
symmetric combination of three contributions each having poles only in the
planar channels. On the other hand, the Virasoro model consists of a unique
crossing symmetric term having poles in both planar and non-planar channels.
The attempts to construct consistent models that were in good agreement
with the strong interaction phenomenology of the sixties boosted enormously
the activity in this research field. The generalization of the Veneziano model to
the scattering ofN scalar particles was built, an operator formalism consisting
of an infinite number of harmonic oscillators was constructed and the complete
spectrum of mesons was determined. It turned out that the degeneracy of
states grew up exponentially with the mass. It was also found that the N
point amplitude had states with negative norm (ghosts) unless the intercept
of the Regge trajectory was α0 = 1 [6]. In this case it turned out that the
model was free of ghosts but the lowest state was a tachyon. The model was
called in the literature the “dual resonance model”.
The birth of string theory 5
The model was not unitary because all the states were zero width reso-
nances and the various cuts required by unitarity were absent. The unitarity
was implemented in a perturbative way by adding loop diagrams obtained by
sewing some of the external legs together after the insertion of a propagator.
The multiloop amplitudes showed a structure of Riemann surfaces. This be-
came obvious only later when the dual resonance model was recognized to
correspond to scattering of strings.
But the main problem was that the model had a tachyon if α0 = 1 or had
ghosts for other values of α0 and was not in agreement with the experimental
data: α0 was not equal to about
as required by experiments for the ρ
Regge trajectory and the external scalar particles did not behave as pions
satisfying the current algebra requirements. Many attempts were made to
construct more realistic dual resonance models, but the main result of these
attempts was the construction of the Neveu-Schwarz [7] and the Ramond [8]
models, respectively, for mesons and fermions. They were constructed as two
independent models and only later were recognized to be two sectors of the
same model. The Neveu-Schwarz model still contained a tachyon that only in
1976 through the GSO projection was eliminated from the physical spectrum.
Furthermore, it was not properly describing the properties of the physical
pions.
Actually a model describing ππ scattering in a rather satisfactory way
was proposed by Lovelace and Shapiro [9] 3. According to this model the
three isospin amplitudes for pion-pion scattering are given by:
[A(s, t) +A(s, u)]− 1
A(t, u)
A1 = A(s, t)−A(s, u) A2 = A(t, u) (7)
where
A(s, t) = β
Γ (1− α(s))Γ (1 − α(t))
Γ (1− α(t)− α(s))
; α(s) = α0 + α
′s (8)
The amplitudes in eq.(7) provide a model for ππ scattering with linearly rising
Regge trajectories containing three parameters: the intercept of the ρ Regge
trajectory α0, the Regge slope α
′ and β. The first two can be determined by
imposing the Adler’s self-consistency condition, that requires the vanishing of
the amplitude when s = t = u = m2π and one of the pions is massless, and the
fact that the Regge trajectory must give the spin of the ρ meson that is equal
to 1 when
s is equal to the mass of the ρ meson mρ. These two conditions
determine the Regge trajectory to be:
α(s) =
s−m2π
m2ρ −mπ2
= 0.48 + 0.885s (9)
3 See also Ref. [10].
6 Paolo Di Vecchia
Having fixed the parameters of the Regge trajectory the model predicts the
masses and the couplings of the resonances that decay in ππ in terms of a
unique parameter β. The values obtained are in reasonable agreement with
the experiments. Moreover, one can compute the ππ scattering lenghts:
a0 = 0.395β a2 = −0.103β (10)
and one finds that their ratio is within 10% of the current algebra ratio given
by a0/a2 = −7/2. The amplitude in eq.(8) has exactly the same form as that
for four tachyons of the Neveu-Schwarz model with the only apparently minor
difference that α0 = 1/2 (for mπ = 0) instead of 1 as in the Neveu-Schwarz
model. This difference, however, implies that the critical space-time dimension
of this model is d = 4 4 and not d = 10 as in the Neveu-Schwarz model. In
conclusion this model seems to be a perfectly reasonable model for describing
low-energy ππ scattering. The problem is, however, that nobody has been able
to generalize it to the multipion scattering and therefore to get the complete
meson spectrum.
As we have seen the S matrix of the dual resonance model was constructed
using ideas and tools of hadron phenomenology of the end of the sixties.
Although it did not seem possible to write a realistic dual resonance model
describing the pions , it was nevertheless such a source of fascination for those
who actively worked in this field at that time for its beautiful internal structure
and consistency that a lot of energy was used to investigate its properties and
for understanding its basic structure. It turned out with great surprise that
the underlying structure was that of a quantum relativistic string.
The aim of this contribution is to explain the logic of the work that was
done in the years from 1968 to 1974 5 in order to uncover the deep properties of
this model that appeared from the beginning to be so beautiful and consistent
to deserve an intensive study.
This seems to me a very good way of celebrating the 65th anniversary of
Gabriele who is the person who started and also contributed to develop the
whole thing with his deep physical intuition.
2 Construction of the N -point amplitude
We have seen that the construction of the four-point amplitude is not sufficient
to get information on the full hadronic spectrum because it contains only
those hadrons that couple to two ground state mesons and does not see those
intermediate states which only couple to three or to an higher number of
ground state mesons [12]. Therefore, it was very important to construct the
N -point amplitude involving identical scalar particles. The construction of
4 This can be checked by computing the coupling of the spinless particle at the
level α(s) = 2 and seeing that it vanishes for d = 4.
5 Reviews from this period can be found in Ref. [11]
The birth of string theory 7
the N -point amplitude was done in Ref. [13] (extending the work of Ref. [14])
by requiring the same principles that have led to the construction of the
Veneziano model, namely the fact that the axioms of S-matrix theory be
satisfied by an infinite number of zero width resonances lying on linearly
rising Regge trajectories and planar duality.
The fully crossing symmetric scattering amplitude of N identical scalar
particles is given by a sum of terms corresponding to the inequivalent permu-
tations of the external legs:
An (11)
Also in this case two permutations of the external legs are inequivalent if they
are not related by a cyclic or anticyclic permutation. Np is the number of
inequivalent permutations of the external legs and is equal to Np =
(N−1)!
and each term has only simple pole singularities in the planar channels. Each
planar channel is described by two indices (i, j), to mean that it includes the
legs i, i+ 1, i+ 2 . . . j − 1, j, by the Mandelstam variable
sij = −(pi + pi+1 + . . .+ pj)2 (12)
and by an additional variable uij whose role will become clear soon. It is
clear that the channels (ij) and (j + 1, i− 1) 6 are identical and they should
be counted only once. In the case of N identical scalar particles the number
of planar channels is equal to
N(N−3)
. This can be obtained as follows. The
independent planar diagrams involving the particle 1 are of the type (1, i)
where i = 2 . . .N − 2. Their number is N − 3. This is also the number of
planar diagrams involving the particle 2 and not the 1. The number of planar
diagrams involving the particle 3 and not the particles 1 and 2 is equal to
N − 4. In general the number of planar diagrams involving the particle i and
not the previous ones from 1 to i-1 is equal to N − 1− i. This means that the
total number of planar diagram is equal to:
2(N − 3) +
(N − 1− i) = 2(N − 3) +
= 2(N − 3) +
(N − 4)(N − 3)
N(N − 3)
If one writes down the duality diagram corresponding to a certain planar
ordering of the external particles, it is easy to see that the diagram can have
simultaneous pole singularities only in N − 3 channels. The channels that
allow simultaneous pole singularities are called compatible channels, the other
6 This channel includes the particles (j + 1, . . . , N, 1, . . . i− 1).
8 Paolo Di Vecchia
are called incompatible. Two channels (i,j) and (h,k) are incompatible if the
following inequalities are satisfied:
i ≤ h ≤ j ; j + 1 ≤ k ≤ i− 1 (14)
The aim is to construct the scattering amplitude for each inequivalent per-
mutation of the external legs that has only pole singularities in the
N(N−3)
planar channels. We have also to impose that the amplitude has simultaneous
poles only in N − 3 compatible channels. In order to gain intuition on how to
proceed we rewrite the four-point amplitude in Eq. (2) as follows:
A(s, t) =
du23 u
−α(s12)−1
−α(s23)−1
23 δ(u12 + u23 − 1) (15)
where u12 and u23 are the variables corresponding to the two planar chan-
nels (12) and (23) and the cancellation of simultaneous poles in incompatible
channels is provided by the δ-function which forbids u12 and u23 to vanish
simultaneously.
We will now extend this procedure to the N -point amplitude. But for the
sake of clarity let us start with the case of N = 5 [14]. In this case we have 5
planar channels described by u12, u13, u23, u24 and u34. Since we have only two
compatible channels only two of the previous five variables are independent.
We can choose them to be u12 and u13. In order to determine the depen-
dence of the other three variables on the two independent ones, we exclude
simultaneous poles in incompatible channels. This can be done by imposing
relations that prevent variables corresponding to incompatible channels to
vanish simultaneously. A sufficient condition for excluding simultaneous poles
in incompatible channels is to impose the conditions:
uP = 1−
uP̄ (16)
where the product is over the variables P̄ corresponding to channels that
are incompatible with P . In the case of the five-point amplitude we get the
following relations:
u23 = 1− u34u12 ; u24 = 1− u13u12
u13 = 1− u34u24 ; u34 = 1− u23u13 ; u12 = 1− u24u23 (17)
Solving them in terms of the two independent ones we get:
u23 =
1− u12
1− u12u13
; u34 =
1− u13
1− u12u13
; u24 = 1− u12u13 (18)
In analogy with what we have done for the four-point amplitude in Eq. (15)
we write the five-point amplitude as follows:
The birth of string theory 9
du34u
−α(s12)−1
−α(s13)−1
×u−α(s24)−124 u
−α(s23)−1
−α(s34)−1
δ(u23 + u12u34 − 1)δ(u24 + u12u13 − 1)δ(u34 + u13u23 − 1) (19)
Performing the integral over the variables u23, u24 and u34 we get:
du13u
−α(s12)−1
−α(s13)−1
× (1− u12)−α(s23)−1(1 − u13)−α(s13)−1(1− u12u13)−α(s24)+α(s23)+α(s34)(20)
We have implicitly assumed that the Regge trajectory is the same in all chan-
nels and that the external scalar particles have the same common mass m
and are the lowest lying states on the Regge trajectory. This means that their
mass is given by:
α0 − α′p2i = 0 ; p2i ≡ −m2 (21)
Using then the relation:
α(s23) + α(s34)− α(s24) = 2α′p2 · p4 (22)
we can rewrite Eq. (20) as follows:
−α(s2)−1
−α(s3)−1
3 (1− u2)−α(s23)−1×
× (1 − u3)−α(s34)−1
(1− xij)2α
′pi·pj (23)
where
si ≡ s1i , ui ≡ u1i ; i = 2, 3 ; xij = uiui+1 . . . uj−1. (24)
We are now ready to construct the N -point function [13]. In analogy with
what has been done for the four and five-point amplitudes we can write the
N -point amplitude as follows:
. . .
−α(sP )−1
δ(uQ − 1 +
uQ̄) (25)
10 Paolo Di Vecchia
where the first product is over the
N(N−3)
variables corresponding to all
planar channels, while the second one is over the
(N−3)(N−2)
independent
δ-functions. The product in the δ-function is defined in Eq. (16).
The solution of all the non-independent linear relations imposed by the
δ-functions is given by
uij =
(1 − xij)(1− xi−1,j+1)
(1− xi−1,j)(1 − xi,j+1)
where the variables xij are given in Eq. (24). Eliminating the δ-function from
Eq. (25) one gets:
−α(si)−1
i (1 − ui)
−α(si,i+1)−1
j=i+2
(1− xij)−γij(27)
where
γij = α(sij) + α(si+1;j−1)− α(si;j−1)− α(si+1;j) ; j ≥ i+ 2 (28)
It is easy to see that
α(si,i+1) = −α0 − 2α′pi · pi+1 ; γij = −2α′pi · pj ; j ≥ i+ 2 (29)
Inserting them in Eq. (27) we get:
−α(si)−1
i (1− ui)
j=i+1
(1− xij)2α
′pi·pj (30)
This is the form of the N -point amplitude that was originally constructed.
Then Koba and Nielsen [15] put it in the form that is more known nowadays.
They constructed it using the following rules. They associated a real variable
zi to each leg i. Then they associated to each channel (i, j) an anharmonic
ratio constructed from the variables zi, zi−1, zj, zj+1 in the following way
(zi, zi+1, zj, zj+1)
−α(sij)−1 =
(zi − zj)(zi−1 − zj+1)
(zi−1 − zj)(zi − zj+1)
]−α(sij)−1
and finally they gave the following expression for the N -point amplitude:
dV (z)
(i,j)
(zi, zi+1, zj, zj+1)
−α(sij)−1 (32)
where
dV (z) =
1 [θ(zi − zi+1)dzi]
i=1(zi − zi+2)dVabc
; dVabc =
dzadzbdzc
(zb − za)(zc − zb)(za − zc)
The birth of string theory 11
and the variables zi are integrated along the real axis in a cyclically ordered
way: z1 ≥ z2 . . . ≥ zN with a, b, c arbitrarily chosen.
The integrand of the N -point amplitude is invariant under projective
transformations acting on the leg variables zi:
αzi + β
γzi + δ
; i = 1 . . .N ; αδ − βγ = 1 (34)
This is because both the anharmonic ratio in Eq. (31) and the measure dVabc
are invariant under a projective transformation. Since a projective transfor-
mation depends on three real parameters, then the integrand of the N -point
amplitude depends only on N − 3 variables zi. In order to avoid infinities, one
has then to divide the integration volume with the factor dVabc that is also
invariant under the projective transformations. The fact that the integrand
depends only on N − 3 variables is in agreement with the fact that N − 3 is
also the maximal number of simultaneous poles allowed in the amplitude.
It is convenient to write the N -point amplitude in a form that involves the
scalar product of the external momenta rather than the Regge trajectories.
We distinguish three kinds of channels. The first one is when the particles
i and j of the channel (i, j) are separated by at least two particles. In this
case the channels that contribute to the exponent of the factor (zi − zj) are
the channels (i, j) with exponent equal to −α(sij) − 1, (i + 1, j − 1) with
exponent −α(si+1,j−1)− 1, (i+1, j) with exponent α(si+1,j)+1 and (i, j− 1)
with exponent α(si,j−1) + 1. Adding these four contributions one gets for the
channels where i and j are separated by at least two particles
− α(sij)− α(si+1,j−1) + α(si+1,j) + α(si,j−1) = 2α′pi · pj (35)
The second one comes from the channels that are separated by only one
particle. In this case only three of the previous four channels contribute. For
instance if j = i+2 the channel (i+1, j− 1) consists of only one particle and
therefore should not be included. This means that we would get:
− α(si;i+2)− 1 + α(s1+1;i+2) + 1 + α(si;i+1 + 1) = 1 + 2α′pi · pi+2 (36)
Finally the third one that comes from the channels whose particles are adja-
cent, gets only contribution from:
− α(si;i+1)− 1 = α0 − 1 + 2α′pi · pi+1 (37)
Putting all these three terms together in Eq. (32) and remembering the factor
in the denominator in the first equation of (33) we get:
1 dziθ(zi − zi+1)
dVabc
(zi − zi+1)α0−1
(zi − zj)2α
′pi·pj(38)
A convenient choice for the three variables to keep fixed is:
12 Paolo Di Vecchia
za = z1 = ∞ ; zb = z2 = 1 ; zc = zN = 0 (39)
With this choice the previous equation becomes:
dziθ(zi − zi+1)
(zi − zi+1)α0−1×
j=i+1
(zi − zj)2α
′pi·pj (40)
We now want to show that this amplitude is identical to the one given in Eq.
(30). This can be done by performing the following change of variables:
; i = 2, 3 . . .N − 2 (41)
that implies
zi = u2u3 . . . ui−1 ; i = 3, 4 . . .N − 1 (42)
Taking into account that the Jacobian is equal to:
uN−2−ii (43)
using the following two relations:
(zi − zi+1)α0−1 =
(N−1−i)α0−1
(1− ui)α0−1 (44)
j=i+1
(zj − zi)2α
′pi·pj =
j=i+1
(1− xij)2α
′pi·pj
−α(si)−(N−i−1)α0
i (45)
and the conservation of momentum
pi = 0 (46)
together with Eq. (21), one can easily see that Eq.s (30) and (40) are equal.
The birth of string theory 13
The N -point amplitude that we have constructed in this section corre-
sponds to the scattering of N spinless particles with no internal degrees of
freedom. On the other hand it was known that the mesons were classified
according to multiplets of an SU(3) flavour symmetry. This was implemented
by Chan and Paton [16] by multiplying the N -point amplitude with a factor,
called Chan-Paton factor, given by
Tr(λa1λa2 . . . λaN ) (47)
where the λ’s are matrices of a unitary group in the fundamental representa-
tion. Including the Chan-Paton factors the total scattering amplitude is given
Tr(λa1λa2 . . . λaN )BN (p1, p2, . . . pN ) (48)
where the sum is extended to the (N − 1)! permutations of the external legs,
that are not related by a cyclic permutations. Originally when the dual reso-
nance model was supposed to describe strongly interacting mesons, this factor
was introduced to represent their flavour degrees of freedom. Nowadays the
interpretation is different and the Chan-Paton factor represents the colour
degrees of freedom of the gauge bosons and the other massive particles of the
spectrum.
The N -point amplitude BN that we have constructed in this section con-
tains only simple pole singularities in all possible planar channels. They cor-
respond to zero width resonances located at non-negative integer values n of
the Regge trajectory α(M2) = n. The lowest state located at α(m2) = 0 cor-
responds to the particles on the external legs of BN . The spectrum of excited
particles can be obtained by factorizing the N -point amplitude in the most
general channel with any number of particles. This was done in Ref.s [17] and
[18] finding a spectrum of states rising exponentially with the mass M . Being
the model relativistic invariant it was found that many states obtained by
factorizing the N -point amplitude were ”ghosts”, namely states with negative
norm as one finds in QED when one quantizes the electromagnetic field in a
covariant gauge. The consistency of the model requires the existence of rela-
tions satisfied by the scattering amplitudes that are similar to those obtained
through gauge invariance in QED. If the model is consistent they must decou-
ple the negative norm states leaving us with a physical spectrum of positive
norm states. In order to study in a simple way these issues, we discuss in the
next section the operator formalism introduced already in 1969 [19, 20, 21].
Before concluding this section let us go back to the non-planar four-point
amplitude in Eq. (5) and discuss its generalization to an N -point amplitude.
Using the technique of the electrostatic analogue on the sphere instead of on
the disk Shapiro [22] was able to obtain a N -point amplitude that reduces
to the four-point amplitude in Eq. (5) with intercept α0 = 2. The N -point
amplitude found in Ref. [22] is:
14 Paolo Di Vecchia
i=1 d
dVabc
|zi − zj |α
′pi·pj (49)
where
dVabc =
d2zad
|za − zb|2|za − zc|2|zb − zc|2
The integral in Eq. (49) is performed in the entire complex plane.
3 Operator formalism and factorization
The factorization properties of the dual resonance model were first studied by
factorizing by brute force the N-point amplitude at the various poles [17, 18].
The number of terms that factorize the residue of the pole at α(s) = n,
increases rapidly with the value of n. In order to find their degeneracy it turned
out to be convenient to first rewrite the N-point amplitude in an operator
formalism. In this section we introduce the operator formalism and we rewrite
the N -point amplitude derived in the previous section in this formalism.
The key idea [19, 20, 21] is to introduce an infinite set of harmonic oscil-
lators and a position and momentum operators 7 which satisfy the following
commutation relations:
[anµ, a
mν ] = ηµνδnm ; [q̂µ, p̂ν ] = iηµν (51)
where ηµν is the flat Minkowski metric that we take to be ηµν = (−1, 1, . . . 1).
A state with momentum p is constructed in terms of a state with zero mo-
mentum as follows:
p̂|p〉 ≡ p̂eip·q̂|0〉 = p|p〉 ; p̂ |0〉 = 0 (52)
normalized as 8
〈p|p′〉 = (2π)dδ(d)(p+ p′) (53)
In order to avoid minus signs we use the convention that
〈p| = 〈0|eip·q̂ (54)
A complete and orthonormal basis of vectors in the harmonic oscillator space
is given by
|λ1, λ2, . . . λi; p〉 =
(a†µn;n)
λn;µn
λn,µn !
eipq̂|0, 0〉 (55)
7 Actually the position and momentum operators were introduced in Ref. [23].
8 Although we now use an arbitrary d we want to remind you that all original
calculations were done for d = 4.
The birth of string theory 15
where the first |0〉 corresponds to the one annihilated by all annihilation op-
erators and the second one to the state of zero momentum:
aµn;n|0, 0〉 = p̂|0, 0〉 = 0 (56)
Notice that Lorentz invariance forces to introduce also oscillators that create
states with negative norm due to the minus sign in the flat Minkowski metric.
This implies that the space spanned by the states in Eq. (55) is not positive
definite. This is, however, not allowed in a quantum theory and therefore if
the dual resonance model is a consistent quantum-relavistic theory we expect
the presence of relations of the kind of those provided by gauge invariance in
Let us introduce the Fubini-Veneziano [23] operator:
Qµ(z) = Q
µ (z) +Q
µ (z) +Q
µ (z) (57)
where
Q(+) = i
z−n ; Q(−) = −i
Q(0) = q̂ − 2iα′p̂ log z (58)
In terms of Q we introduce the vertex operator corresponding to the external
leg with momentum p:
V (z; p) =: eip·Q(z) :≡ eip·Q
(−)(z)eipq̂e+2α
′p̂·p log zeip·Q
(+)(z) (59)
and compute the following vacuum expectation value:
〈0, 0|
V (zi, pi)|0, 0〉 (60)
It can be easily computed using the Baker-Haussdorf relation
eAeB = eBeAe[A,B] (61)
that is valid if the commutator, as in our case, [A,B] is a c-number. In our
case the commutation relations to be used are:
[Q(+)(z), Q(−)(w)] = −2α′ log
and the second one in Eq. (51). Using them one gets:
V (z; p)V (w; k) =: V (z; p)V (w; k) : (z − w)2α
′p·k (63)
16 Paolo Di Vecchia
〈0, 0|
V (zi, pi)|0, 0〉 =
(zi − zj)2α
′pi·pj (2π)dδ(d)(
pi) (64)
where the normal ordering requires that all creation operators be put on the
left of the annihilation one and the momentum operator p̂ be put on the right
of the position operator q̂. This means that
(2π)dδ(d)(
pi)BN =
1 dziθ(zi − zi+1)
dVabc
(zi − zi+1)α0−1
× 〈0, 0|
V (zi, pi)|0, 0〉 (65)
By choosing the three variables za, zb and zc as in Eq. (39) we can rewrite the
previous equation as follows:
(2π)dδ(d)(
pi)BN =
θ(zi − zi+1)×
(zi − zi+1)α0−1
〈0, p1|
V (zi; pi)|0, pN 〉 (66)
where we have taken z2 = 1 and we have defined (α0 ≡ α′p2i ; i = 1 . . .N) :
V (zN ; pN)|0, 0〉 ≡ |0; pN〉 ; 〈0; 0| lim
z2α01 V (z1; p1) = 〈0, p1| (67)
Before proceeding to factorize the N -point amplitude let us study the prop-
erties under the projective group of the operators that we have introduced.
We have already seen that the projective group leaves the integrand of the
Koba-Nielsen representation of the N -point amplitude invariant. The projec-
tive group has three generators L0, L1 and L−1 corresponding respectively to
dilatations, inversions and translations. Assuming that the Fubini-Veneziano
fields Q(z) transforms as a field with weight 0 (as a scalar) we can immedi-
ately write the commutation relations that Q(z) must satisfy. This means in
fact that, under a projective transformation, Q(z) transforms as follows:
Q(z) → QT (z) = Q
αz + β
γz + δ
; αδ − βγ = 1 (68)
Expanding for small values of the parameters we get:
QT (z) = Q(z) + (ǫ1 + ǫ2z + ǫ3z
dQ(z)
+ o(ǫ2) (69)
The birth of string theory 17
This means that the three generators of the projective group must satisfy the
following commutation relations with Q(z):
[L0, Q(z)] = z
; [L−1, Q(z)] =
; [L1, Q(z)] = z
They are given by the following expressions in terms of the harmonic oscilla-
tors:
L0 = α
′p̂2 +
na†n · an ; L1 =
2α′p̂ · a1 +
n(n+ 1)an+1 · a†n (71)
L−1 = L
2α′p̂ · a†1 +
n(n+ 1)a
n+1 · an (72)
They annihilate the vacuum
L0|0, 0〉 = L1|0, 0〉 = L−1|0, 0〉 = 0 (73)
that is therefore called the projective invariant vacuum, and satisfy the algebra
that is called Gliozzi algebra [24]9:
[L0, L1] = −L1 ; [L0, L−1] = L−1 ; [L1, L−1] = 2L0 (74)
The vertex operator with momentum p is a projective field with weight equal
to α0 = α
′p2. It transforms in fact as follows under the projective group:
[Ln, V (z, p)] = z
n+1 dV (z, p)
+ α0(n+ 1)z
nV (z, p) ; n = 0,±1 (75)
or in finite form as follows:
UV (z, p)U−1 =
(γz + δ)2α0
αz + β
γz + δ
where U is the generator of an arbitrary finite projective transformation.
Since U leaves the vacuum invariant, by using Eq. (76) it is easy to show
that:
〈0, 0|
V (z′i, p)|0, 0〉 =
(γzi + δ)
2α0〈0, 0|
V (zi, p)|0, 0〉 (77)
that together with the following equation:
(z′i − z′i+1)α0−1 =
(zi − zi+1)α0−1
(γzi + δ)
−2α0(78)
9 See also Ref. [25].
18 Paolo Di Vecchia
implies that the integrand of the N -point amplitude in Eq. (65) is invariant
under projective transformations.
We are now ready to factorize the N -point amplitude and find the spec-
trum of mesons.
From Eq.s (75) and (76) it is easy to derive the transformation of the
vertex operator under a finite dilatation:
zL0V (1, p)z−L0 = V (z, p)zα0 (79)
Changing the integration variables as follows:
; i = 2, 3 . . .N − 2 ; det
= z3z4 . . . zN−2 (80)
where the last term is the jacobian of the trasformation from zi to xi, we get
from Eq.(66) the following expression:
AN ≡ 〈0, p1|V (1, p2)DV (1, p3) . . .DV (1, pN−1)|0, pN〉 (81)
where the propagator D is equal to:
dxxL0−1−α0(1− x)α0−1 = Γ (L0 − α0)Γ (α0)
Γ (L0)
AN = (2π)
dδ(d)
BN (83)
The factorization properties of the amplitude can be studied by inserting in
the channel (1,M) or equivalently in the channel (M +1, N) described by the
Mandelstam variable
s = −(p1 + p2 + . . . pM )2 = −(pM+1 + pM+2 . . .+ pN )2 ≡ −P 2 (84)
the complete set of states given in Eq. (55):
〈p(1,M)|λ, P 〉〈λ, P |D|µ, P 〉〈µ, P |p(M+1,N)〉 (85)
where
〈p(1,M)| = 〈0, p1|V (1, p2)DV (1, p3) . . . V (1, pM ) (86)
|p(M+1,N)〉 = V (1, pM+1)D . . . V (1, pN−1)|pN , 0〉 (87)
Introducing the quantity:
The birth of string theory 19
na†n · an (88)
it is possible to rewrite
〈λ, P |D|µ, P 〉 =
〈λ, P |
(−1)m
α0 − 1
R+m− α(s)
|µ, P 〉 (89)
where s is the variable defined in Eq. (84). Using this equation we can rewrite
Eq. (85) as follows
〈p(1,M)|λ, P 〉
〈λ, P |
(−1)m
α0 − 1
R+m− α(s)
|µ, P 〉〈µ, P |p(M+1,N)〉(90)
This expression shows that amplitude AN has a pole in the channel (1,M)
when α(s) is equal to an integer n ≥ 0 and the states |λ〉 that contribute to
its residue are those satisfying the relation:
R|λ〉 = (n−m)|λ〉 ; m = 0, 1 . . . n (91)
The number of independent states |λ〉 contributing to the residue gives the
degeneracy of states for each level n.
Because of manifest relativistic invariance the space spanned by the com-
plete system of states in Eq. (55) contains states with negative norm corre-
sponding to those states having an odd number of oscillators with timelike
directions (see Eq. (51)). This is not consistent in a quantum theory where
the states of a system must span a positive definite Hilbert space. This means
that there must exist a number of relations satisfied by the external states that
decouple a number of states leaving with a positive definite Hilbert space. In
order to find these relations we rewrite the state in Eq. (87) going back to the
Koba-Nielsen variables:
|p(1,M)〉 =
dziθ(zi − zi+1)]
(zi − zi+1)α0−1×
× V (1, p1)V (z2, p2) . . . V (zM−1, pM−1)|0, pM 〉 (92)
Let us consider the operator U(α) that generate the projective transformation
that leaves the points z = 0, 1 invariant:
1− α(z − 1) = z + α(z
2 − z) + o(α2) (93)
From the transformation properties of the vertex operators in Eq. (76) it
is easy to see that the previous transformation leaves the state in Eq. (92)
invariant:
20 Paolo Di Vecchia
U(α)|p(1,M)〉 = |p(1,M)〉 (94)
This means that the generator of the previous transformation annihilates the
state in Eq. (92):
W1|p(1,M)〉 = 0 ; W1 = L1 − L0 (95)
The explicit form of W1 follows from the infinitesimal form of the transforma-
tion in Eq. (93). This condition that is of the same kind of the relations that
on shell amplitudes with the emission of photons satisfy as a consequence of
gauge invariance, implies that the residue at the pole in Eq. (90) can be fac-
torized with a smaller number of states. It turns out, however, that a detailed
analysis of the spectrum shows that negative norm states are still present.
This can be qualitatively understood as follows. Due to the Lorentz metric
we have a negative norm component for each oscillator. In order to be able
to decouple all negative norm states we need to have a gauge condition of
the type as in Eq. (95) for each oscillator. But the number of oscillators is
infinite and, therefore, we need an infinite number of conditions of the type
as in Eq. (95). It was found in Ref. [6] that, if we take α0 = 1, then one can
easily construct an infinite number of operators that leave the state in Eq.
(92) invariant. In the next section we will concentrate on this case.
4 The case α0 = 1
If we take α0 = 1 many of the formulae given in the previous section simplify.
The N -point amplitude in Eq. (38) becomes:
1 dziθ(zi − zi+1)
dVabc
(zi − zj)2α
′pi·pj (96)
that can be rewritten in the operator formalism as follows:
(2π)4δ(
pi)BN =
1 dziθ(zi − zi+1)
dVabc
〈0, 0|
V (zi, pi)|0, 0〉 (97)
By choosing z1 = ∞, z2 = 1 and zN = 0 it becomes
(2π)4δ(
pi)BN =
θ(zi − zi+1)〈0, p1|
V (zi; pi)|0, pN 〉 (98)
The birth of string theory 21
where
V (zN ; pN )|0, 0〉 ≡ |0; pN 〉 ; 〈0; 0| lim
z21V (z1; p1) = 〈0, p1| (99)
Eq. (81) is as before, but now the propagator becomes:
dxxL0−2 =
L0 − 1
(100)
This means that Eq. (89) becomes:
〈λ, P |D|µ, P 〉 = 〈λ, P | 1
L0 − 1
|µ, P 〉 (101)
and Eq. (90) has the simpler form:
〈p(1,M)|λ, P 〉〈λ, P |
R − α(s)
|λ, P 〉〈λ, P |p(M+1,N)〉 (102)
BN has a pole in the channel (1,M) when α(s) is equal to an integer n ≥ 0 and
the states |λ〉 that contribute to its residue are those satisfying the relation:
R|λ〉 = n|λ〉 (103)
Their number gives the degeneracy of the states contributing to the pole at
α(s) = n. The N -point amplitude can be written as:
BN = 〈p(1,M)|D|p(M+1,N)〉 (104)
where
|p(1,M)〉 =
∫ M−1
[dziθ(zi − zi+1)]×
× V (1, p1)V (z2, p2) . . . V (zM−1, pM−1|0, pM 〉 (105)
Using Eq. (79) and changing variables from zi, i = 2 . . .M−1 to xi = zi+1zi , i =
1 . . .M − 2 with z1 = 1 we can rewrite the previous equation as follows:
|p(1,M)〉 = V (1, p1)DV (1, p2) . . .DV (1, pM−1)|0, pM 〉 (106)
where the propagator D is defined in Eq. (100).
We want now to show that the state in Eq.s (105) and (106) is not only
annihilated by the operator in Eq. (95), but, if α0 = 1 [6], by an infinite set
of operators whose lowest one is the one in Eq. (95). We will derive this by
using the formalism developed in Ref. [26] and we will follow closely their
derivation.
Starting from Eq.s (70) Fubini and Veneziano realized that the generators
of the projective group acting on a function of z are given by:
22 Paolo Di Vecchia
L0 = −z
; L−1 = −
; L1 = −z2
(107)
They generalized the previous generators to an arbitrary conformal transfor-
mation by introducing the following operators, called Virasoro operators:
Ln = −zn+1
(108)
that satisfy the algebra:
[Ln, Lm] = (n−m)Ln+m (109)
that does not contain the term with the central charge! They also showed that
the Virasoro operators satisfy the following commutation relations with the
vertex operator:
[Ln, V (z, p)] =
zn+1V (z, p)
(110)
More in general actually they define an operator Lf corresponding to an
arbitrary function f(ξ) and Lf = Ln if we choose f(ξ) = ξ
n. In this case the
commutation relation in Eq. (110) becomes:
[Lf , V (z, p)] =
(zf(z)V (z, p)) (111)
By introducing the variable:
ξf(ξ)
(112)
where A is an arbitrary constant, one can rewrite Eq. (111) in the following
form:
[Lf , zf(z)V (z, p)] =
(zf(z)V (z, p)) (113)
This implies that, under an arbitrary conformal transformation z → f(z),
generated by U = eαLf , the vertex operator transforms as:
eαLfV (z, p) zf(z) e−αLf = V (z′, p)z′f(z′) (114)
where the parameter α is given by:
ξf(ξ)
(115)
On the other hand, this equation implies:
zf(z)
z′f(z′)
(116)
The birth of string theory 23
that, inserted in Eq. (114), implies that the quantity V (z, p) dz is left invariant
by the transformation z → f(z):
eαLfV (z, p)dze−αLf = V (z′, p)dz′ (117)
Let us now act with the previous conformal transformation on the state in
Eq. (105). We get:
eαLf |p(1,M)〉 =
[dziθ(zi − zi+1)] eαLfV (1, p1)e−αLf×
×eαLfV (z2, p2)e−αLf . . . . . . eαLfV (zM−1, pM−1)e−αLf eαLf |0, pM 〉 =
θ(zi − zi+1)× eαLfV (1, p1)e−αLf×
× V (z′2, p2)dz′2 . . . V (z′M−1, pM−1)dz′M−1eαLf |0, pM 〉 (118)
where we have used Eq. (117). The previous transformation leaves the state
invariant if both z = 0 and z = 1 are fixed points of the conformal transfor-
mation. This happens if the denominator in Eq. (115) vanishes when ξ = 0, 1.
This requires the following conditions:
f(1) = 0 ; lim
ξf(ξ) = 0 (119)
Expanding ξ near the poinr ξ = 1 we can determine the relation between z
and z′ near z = z′ = 1. We get:
ze−αf
1− z + ze−αf ′(1)
(120)
and from it we can determine the conformal factor:
(1 − z + ze−αf ′(1))2
→ eαf
′(1) (121)
in the limit z → 1. Proceeding in the same near the point z = z′ = 0 we get:
zf(0)eαf(0)
f(0) + zf ′(0)(1− eαf(0)
→ zeαf(0) (122)
in the limit z → 0. This means that Eq. (118) becomes
eα(Lf−f
′(1)−f(0))|p(1,M)〉 = |p(1,M)〉 (123)
A choice of f that satisfies Eq.s (119) is the following:
24 Paolo Di Vecchia
f(ξ) = ξn − 1 (124)
that gives the following gauge operator:
Wn = Ln − L0 − (n− 1) (125)
that annihilates the state in Eq. (105):
Wn|p1...M 〉 = 0 ; n = 1 . . .∞ (126)
These are the Virasoro conditions found in Ref. [6]. There is one condition for
each negative norm oscillator and, therefore, in this case there is the possibility
that the physical subspace is positive definite. An alternative more direct
derivation of Eq. (126) can be obtained by acting with Wn on the state in Eq.
(106) and using the following identities:
WnV (1, p) = V (1, p)(Wn + n) ; (Wn + n)D = [L0 + n− 1]−1Wn (127)
The second equation is a consequence of the following equation:
L0 = xL0+nLn (128)
Eq.s (127) imply
WnV (1, p)D = V (1, p)[L0 + n− 1]−1Wn (129)
This shows that the operator Wn goes unchanged through all the product
of terms V D until it arrives in front of the term V (1, pM−1)|0, pM 〉. Going
through the vertex operator it becomes Ln − L0 + 1 that then annihilate the
state
(Ln − L0 + 1)|pM , 0〉 = 0 (130)
This proves Eq. (126).
Using the representation of the Virasoro operators given in Eq. (108) Fu-
bini and Veneziano showed that they satisfy the algebra given in eq. (109)
without the central charge. The presence of the central charge was recognized
by Joe Weis10 in 1970 and never published. Unlike Fubini and Veneziano [26]
he used the expression of the Ln operators in terms of the harmonic oscillators:
2α′np̂ · an +
m(n+m)an+m · am+
m(n−m)am−n · am ;n ≥ 0 Ln = L†n (131)
10 See noted added in proof in Ref. [26].
The birth of string theory 25
He got the following algebra:
[Ln, Lm] = (n−m)Ln+m +
n(n2 − 1)δn+m;0 (132)
where d is the dimension of the Minkowski space-time. We write here d for the
dimension of the Minkowski space, but we want to remind you that almost
everybody working in a model for mesons at that time took for granted that
the dimension of the space-time was d = 4. As far as I remember the first
paper where a dimension d 6= 4 was introduced was Ref. [27] where it was
shown that the unitarity violating cuts in the non-planar loop become poles
that were consistent with unitarity if d = 26.
In the last part of this section we will generalize the factorization procedure
to the Shapiro-Virasoro model whose N -point amplitude is given in Eq. (49).
In this case we must introduce two sets of harmonic oscillators commuting
with each other and only one set of zero modes satisfying the algebra [28] :
[anµ, a
mν ] = [ãnµ, ã
mν ] = ηµνδnm ; [q̂µ, p̂ν ] = iηµν (133)
In terms of them we can introduce the Fubini-Veneziano operator
Q(z, z̄) = q̂ − 2α′p̂ log(zz̄) + i
−n − a†nzn
ãnz̄
−n − ã†nz̄n
(134)
We can then introduce the vertex operator:
V (z, z̄; p) =: eip·Q(z,z̄) : (135)
and write the N -point amplitude in Eq. (95) in the following factorized form:
i=1 d
dVabc
V (zi, z̄i, pi))
|0〉 =
= (2π)4δ(4)(
i=1 d
dVabc
|zi − zj|α
′pi·pj (136)
where the radial ordered product is given by
V (zi, z̄i, pi))
V (zi, z̄i, pi))
θ(|zi| − |zi+1|) + . . . (137)
26 Paolo Di Vecchia
and the dots indicate a sum over all permutations of the vertex operators.
By fixing z1 = ∞, z2 = 1, zN = 0 we can rewrite the previous expression
as follows:
∫ N−1
d2zi〈0, p1|R
V (zi, z̄i, pi))
|0, pN〉 (138)
For the sake of simplicity let us consider the term corresponding to the per-
mutation 1, 2, . . .N . In this case the Koba-Nielsen variables are ordered in
such a way that |zi| ≥ |zi+1| for i = 1, . . .N −1. We can then use the formula:
V (zi, z̄i, pi)) = z
L̃0−1
i V (1, 1, pi)z
i (139)
and change variables:
; |wi| ≤ 1 (140)
to rewrite Eq. (138) as follows:
〈0, p1|V (1, 1, pi1)DV (1, 1, p2)D . . . V (1, 1, pN−1)|0, pN 〉 (141)
where
wL0−1w̄L̃0−1 =
L0 + L̃0 − 2
· sinπ(L0 − L̃0)
L0 − L̃0
(142)
We can now follow the same procedure for all permutations arriving at the
following expression:
〈0, p1|P [V (1, 1, p2)DV (1, 1, p3)D . . . V (1, 1, pN−1)]|0, pN〉 (143)
where P means a sum of all permutations of the particles.
If we want to consider the factorization of the amplitude on the pole at
s = −(p1 + . . . pM )2 we get only the following contribution:
〈p(1...M)|D|p(M+1...N)〉 (144)
where
|p(M+1...N)〉 = P [V (1, 1, pM+1)D . . . V (1, 1, pN−1]|0, pN 〉 (145)
〈p(1...M)| = 〈0, p1|P [V (1, 1, p2)D . . . V (1, 1, pM)] (146)
The amplitude is factorized by introducing a complete set of states and rewrit-
ing Eq. (141) as follows:
The birth of string theory 27
〈p1...M |λ, λ̃〉
2π〈λ, λ̃|δL0,L̃0|λ, λ̃〉
L0 + L̃0 − 2
〈λ, λ̃|p(M+1,...N)〉 (147)
By writing
p̂2 +R ; L̃0 =
p̂2 + R̃ (148)
na†n · an ; R̃ =
nã†n · ãn (149)
we can rewrite Eq. (147) as follows
〈p1...M |λ, λ̃〉
2π〈λ, λ̃|δR,R̃|λ, λ̃〉
R + R̃− α(s)
〈λ, λ̃|p(M+1,...N)〉 (150)
We see that the amplitude for the Shapiro-Virasoro model has simple poles
only for even integer values of αSV (s) = 2 +
s = 2n ≥ 0 and the residue at
the poles factorizes in a sum with a finite number of terms. Notice that the
Regge trajectory of the Shapiro-Virasoro model has double intercept and half
slope of that of the generalized Veneziano model.
5 Physical states and their vertex operators
In the previous section, we have seen that the residue at the poles of the N -
point amplitudes factorizes in a sum of a finite number of terms. We have also
seen that some of these terms, due to the Lorentz metric, correspond to states
with negative norm. We have also derived a number of ”Ward identities” given
in Eq. (126) that imply that some of the terms of the residue decouple. The
question to be answered now is: Is the space spanned by the physical states
a positive norm Hilbert space? In order to answer this question we need first
to find the conditions that characterize the on shell physical states |λ, P 〉
and then to determine which are the states that contribute to the residue
of the pole at α(s = −P 2) = n. In other words, we have to find a way of
characterizing the physical states and of eliminating the spurious states that
decouple in Eq. (102) as a consequence of Eq.s (126). A state |λ.P 〉 contributes
at the residue of the pole in Eq.(102) for α(s = −P 2) = n if it is on shell,
namely if it satisfies the following equations:
R|λ, P 〉 = n|λ, P 〉 ; α(−P 2) = 1− α′P 2 = n (151)
that can be written in a unique equation:
28 Paolo Di Vecchia
(L0 − 1)|λ, P 〉 = 0 (152)
Because of Eq. (126) we also know that a state of the type:
|s, P 〉 = W †m|µ, P 〉 (153)
is not going to contribute to the residue of the pole. We call it a spurious or
unphysical state. We start constructing the subspace of spurious states that
are on shell at the level n. Let us consider the set of orthogonal states |µ, P 〉
such that
R|µ, P 〉 = nµ|µ, P 〉 ; L0|µ, P 〉 = (1−m)|µ, P 〉 ; 1− α′P 2 = n (154)
where
m = n+ nµ (155)
In terms of these states we can construct the most general spurious state that
is on shell at the level n. It is given by
|s, P 〉 = W †m|µ, P 〉 ; (L0 − 1)|s, P 〉 = 0 (156)
per any positive integer m. Using Eq. (154), eq. (156) becomes:
|s, P 〉 = L†m|µ, P 〉 (157)
where |µ, P 〉 is an arbitrary state satisfying Eq.s (154).
A physical state |λ, P 〉 is defined as the one that is orthogonal to all spuri-
ous states appearing at a certain level n. This means that it must satisfy the
following equation:
〈λ.P |L†ℓ |µ, P 〉 = 0 (158)
for any state |µ, P 〉 satisfying Eq.s (154). In conclusion, the on shell physical
states at the level n are characterized by the fact that they satisfy the following
conditions:
Lm|λ, P 〉 = (L0 − 1)|λ, P 〉 = 0 ; 1− α′P 2 = n (159)
These conditions characterizing the physical subspace were first found by Del
Giudice and Di Vecchia [28] where the analysis described here was done.
In order to find the physical subspace one starts writing the most general
on shell state contributing to the residue of the pole at level n in Eq. (154).
Then one imposes Eq.s (159) and determines the states that span the physical
subspace. Actually, among these states one finds also a set of zero norm states
that are physical and spurious at the same time. Those states are of the form
given in Eq. (157), but also satisfy Eq.s (159). It is easy to see that they are
not really physical because they are not contributing to the residue of the pole
The birth of string theory 29
at the level n. This follows from the form of the unit operator given in the
space of the physical states by:
norm 6=0
|λ, P 〉〈λ, P |+
[|λ0, P 〉〈µ0, P |+ |µ0, P 〉〈λ0, P |] (160)
where |λ0, P 〉 is a zero norm physical and spurious state and |µ0, P 〉 its con-
jugate state. A conjugate state of a zero norm state is obtained by changing
the sign of the oscillators with timelike direction. Since |λ0, P 〉 is a spurious
state when we insert the unit operator, given in Eq. (160), in Eq. (102) we
see that the zero norm states never contribute to the residue because their
contribution is annihilated either from the state 〈p(1,M)| or from the state
|p(M+1,N)〉. In conclusion, the physical subspace contains only the states in
the first term in the r.h.s. of Eq. (160).
Let us analyze the first two excited levels. The first excited level corre-
sponds to a massless gauge field. It is spanned by the states ǫµa
1µ|0, P 〉. In
this case the only condition that we must impose is:
1µ|0, P 〉 = 0 =⇒ P · ǫ = 0 (161)
Choosing a frame of reference where the momentum of the photon is given by
Pµ ≡ (P, 0....0, P ) , Eq. (161) implies that the only physical states are:
1i |0, P 〉+ ǫ(a
1;0 − a
1;d−1)|0, P 〉 ; i = 1 . . . d− 2 (162)
where ǫi and ǫ are arbitrary parameters. The state in Eq. (162) is the most
general state of the level N = 1 satisfying the conditions in Eq. (159). The
first state in eq. (162) has positive norm, while the second one has zero norm
that is orthogonal to all other physical states since it can be written as follows:
1;0 − a
1;D−1)|0, P 〉 = L
1|0, P 〉 (163)
in the frame of reference where Pµ ≡ (P, ...0, P ). Because of the previous
property it is decoupled from the physical states together with its conjugate:
1,0 + a
1,d−1)|0, P 〉 (164)
In conclusion, we are left only with the transverse d− 2 states corresponding
to the physical degrees of freedom of a massless spin 1 state. At the next level
n = 2 the most general state is given by:
[αµνa
1,ν + β
2,µ]|0, P 〉 (165)
If we work in the center of mass frame where Pµ = (M,0) we get the following
most general physical state:
|Phys >= αij [a†1,ia
1,j −
(d− 1)
1,k]|0, P 〉+
30 Paolo Di Vecchia
+βi[a
2,i + a
1,i]|0, P >〉+
1,i +
1,0 − 2a
|0, P 〉 (166)
where the indices i, j run over the d− 1 space components. The first term in
(166) corresponds to a spin 2 in (d− 1) dimensional space and has a positive
norm being made with space indices. The second term has zero norm and is
orthogonal to the other physical states since it can be written as L+1 a
1,i|0, P 〉.
Therefore it must be eliminated from the physical spectrum together with its
conjugate, as explained above. Finally, the last state in (166) is spinless and
has a norm given by:
2(d− 1)(26− d) (167)
If d < 26 it corresponds to a physical spin zero particle with positive norm. If
d > 26 it is a ghost. Finally, if d = 26 it has a zero norm and is also orthogonal
to the other physical states since it can be written in the form:
2 + 3L
1 )|0 > (168)
It does not belong, therefore, to the physical spectrum. The analysis of this
level was done in Ref. [29] with d = 4. This did not allow the authors of
Ref. [29] to see that there was a critical dimension.
The analysis of the physical states can be easily extended [28] to the
Shapiro-Virasoro model. In this case the physical conditions given in Eq. (159)
for the open string, become [28]:
Lm|λ, λ̃〉 = L̃m|λ, λ̃〉 = (L0 − 1)|λ, λ̃〉 = (L̃0 − 1)|λ, λ̃〉 = 0 (169)
for any positive integer m. It can be easily seen from the previous equations
that the lowest state of the Shapiro-Virasoro model is the vacuum |0a, 0ã, p〉
corresponding to a tachyon with mass α′p2 = 4, while the next level described
by the state a
1ν |0a, 0ã, p〉 contains massless states corresponding to the
graviton, a dilaton and a two-index antisymmetric tensor Bµν .
Having characterized the physical subspace one can go on and construct
a N -point scattering amplitude involving arbitrary physical states. This was
done by Campagna, Fubini, Napolitano and Sciuto [30] where the vertex oper-
ator for an arbitrary physical state was constructed in analogy with what has
been done for the ground tachyonic state. They associated to each physical
state |α, P 〉 a vertex operator Vα(z, P ) that is a conformal field with conformal
dimension equal to 1:
[Ln, Vα(z, p)] =
zn+1Vα(z, p)
(170)
and reproduces the corresponding state acting on the vacuum as follows:
Vα(z; p)|0, 0〉 ≡ |α; p〉 ; 〈0; 0| lim
z2Vα(z; p) = 〈α, p| (171)
The birth of string theory 31
It satisfies, in addition, the hermiticity relation:
V †α (z, P ) = Vα(
,−P )(−1)α(−P
2) (172)
An excited vertex that will play an important role in the next section is the
one associated to the massless gauge field. It is given by:
Vǫ(z, k) ≡ ǫ ·
dQ(z)
eik·Q(z) ; k · ǫ = k2 = 0 (173)
Because of the last two conditions in Eq. (173) the normal order is not neces-
sary. It is convenient to give the expression of
dQ(z)
in terms of the harmonic
oscillators:
P (z) ≡ dQ(z)
−n−1 (174)
It is a conformal field with conformal dimension equal to 1. The rescaled
oscillators αn are given by:
nan ; α−n =
na†n ; n > 0 ; α0 =
2α′p̂ (175)
In terms of the vertex operators previously introduced the most general
amplitude involving arbitrary physical states is given by [30]:
(2π)4δ(
1 dziθ(zi − zi+1)
dVabc
〈0, 0|
Vαi(zi, pi)|0, 0〉(176)
In the case of the Shapiro-Virasoro model the tachyon vertex operator is
given in Eq. (135). By rewriting Eq. (134) as follows:
Q(z, z̄) = Q(z) + Q̃(z̄) (177)
where
Q(z) =
q̂ − 2α′p̂ log(z) + i
−n − a†nzn
(178)
Q̃(z̄) =
q̂ − 2α′p̂ log(z̄) + i
ãnz̄
−n − ã†nz̄n
(179)
we can write the tachyon vertex operator in the following way:
V (z, z̄, p) =: eip·Q(z)eip·Q̃(z̄) : (180)
32 Paolo Di Vecchia
This shows that the vertex operator corresponding to the tachyon of the
Shapiro-Virasoro model can be written as the product of two vertex oper-
ators corresponding each to the tachyon of the generalized Veneziano model.
Analogously the vertex operator corresponding to an arbitrary physical
state of the Shapiro-Virasoro model can always be written as a product of
two vertex operators of the generalized Veneziano model:
Vα,β(z, z̄, p) = Vα(z,
)Vβ(z̄,
) (181)
The first one contains only the oscillators αn, while the second one only the
oscillators α̃n. They both contain only half of the total momentum p and
the same zero modes p̂ and q̂. The two vertex operators of the generalized
Veneziano model are both conformal fields with conformal dimension equal
to 1. If they correspond to physical states at the level 2n, they satisfy the
following relation (n = ñ):
+ n = 1 (182)
They lie on the following Regge trajectory:
p2 ≡ αSV (−p2) = 2n (183)
as we have already seen by factorizing the amplitude in Eq. (150).
6 The DDF states and absence of ghosts
In the previous section we have derived the equations that characterize the
physical states and their corresponding vertex operators. In this section we
will explicitly construct an infinite number of orthonormal physical states with
positive norm.
The starting point is the DDF operator introduced by Del Giudice, Di Vec-
chia and Fubini [31] and defined in terms of the vertex operator corresponding
to the massless gauge field introduced in eq. (173):
Ai,n =
i Pµ(z)e
ik·Q(z) (184)
where the index i runs over the d−2 transverse directions, that are orthogonal
to the momentum k. We have also taken
= 1. Because of the log z term
appearing in the zero mode part of the exponential, the integral in Eq. (184),
that is performed around the origin z = 0, is well defined only if we constrain
the momentum of the state, on which Ai,n acts, to satisfy the relation:
2α′p · k = n (185)
The birth of string theory 33
where n is a non-vanishing integer.
The operator in Eq. (184) will generate physical states because it com-
mutes with the gauge operators Lm:
[Lm, An;i] = 0 (186)
since the vertex operator transforms as a primary field with conformal dimen-
sion equal to 1 as it follows from Eq. (170).
On the other hand it also satisfies the algebra of the harmonic oscillator
as we are now going to show. From Eq. (184) we get:
[An,i, Am,j] = −
dzǫi · P (z)eik·Q(ζ)ǫj · P (ζ)eik
′ ·Q(ζ) (187)
where
2α′p · k = n ; 2α′p · k′ = m (188)
and k and k′ are supposed to be in the same direction, namely
kµ = nk̂µ ; k
µ = mk̂µ (189)
2α′p · k̂ = 1 (190)
Finally the polarizations are normalized as:
ǫi · ǫj = δij (191)
Since k̂ · ǫi = k̂ · ǫj = k̂2 = 0 a singularity for z = ζ can appear only from the
contraction of the two terms P (ζ) and P ((z) that is given by:
〈0, 0|ǫi · P (z)ǫj · P (ζ)|0, 0〉 = −
2α′δij
(z − ζ)2
(192)
Inserting it in Eq. (187) we get:
[An,i, Am,j ] = δij in
dζk̂ · P (ζ)e−i(n+m))k̂·Q(ζ) =
= inδijδn+m;0
dζk̂ · P (ζ) (193)
where we have used the fact that the integrand is a total derivative and
therefore one gets a vanishing contribution unless n + m = 0. If n + m = 0
from Eq.s (174) and (190) we get:
[An,i, Am,j ] = nδijδn+m;0 ; i, j = 1 . . . d− 2 (194)
34 Paolo Di Vecchia
Eq. (194) shows that the DDF operators satisfy the harmonic oscillator alge-
In terms of this infinite set of transverse oscillators we can construct an
orthonormal set of states:
|i1, N1; i2, N2; . . . im, Nm〉 =
Aik,−Nk√
|0, p〉 (195)
where λh is the multiplicity of the operator Aih,−Nh in the product in Eq.
(195) and the momentum of the state in Eq. (195) is given by
P = p+
k̂Ni (196)
They were constructed in four dimensions where they were not a complete
system of states 11 and it took some time to realize that in fact they were a
complete system of states if d = 26 [32, 33] 12. Brower [32] and Goddard and
Thorn [33] showed also that the dual resonance model was ghost free for any
dimension d ≤ 26. In d = 26 this follows from the fact that the DDF operators
obviously span a positive definite Hilbert space (See Eq. (194)). For d < 26
there are extra states called Brower states [32]. The first of these states is
the last state in Eq. (166) that becomes a zero norm state for d = 26. But
also for d < 26 there is no negative norm state among the physical states.
The proof of the no-ghost theorem in the case α0 = 1 is a very important
step because it shows that the dual resonance model constructed generalizing
the four-point Veneziano formula, is a fully consistent quantum-relativistic
theory! This is not quite true because, when the intercept α0 = 1, the lowest
state of the spectrum corresponding to the pole in the N -point amplitude for
α(s) = 0, is a tachyon with mass m2 = − 1
. A lot of effort was then made
to construct a model without tachyon and with a meson spectrum consistent
with the experimental data. The only reasonably consistent models that came
out from these attempts, were the Neveu-Schwarz [7] for mesons and the
Ramond model [8] for fermions that only later were recognized to be part of a
unique model that nowadays is called the Neveu-Schwarz-Ramond model. But
this model was not really more consistent than the original dual resonance
11 Because of this Fubini did not want to publish our result, but then he went to a
meeting in Israel in spring 1971 giving a talk on our work where he found that
the audience was very interested in our result and when he came back to MIT we
decided to publish our result.
12 I still remember Charles Thorn coming into my office at Cern and telling me:
Paolo, do you know that your DDF states are complete if d = 26? I quickly redid
the analysis done in Ref. [29] with an arbitrary value of the space-time dimension
obtaining Eq.s (166) and (167) that show that the spinless state at the level
α(s) = 2 is decoupled if d = 26. I strongly regretted not to have used an arbitrary
space-time dimension d in the analysis of Ref. [29] .
The birth of string theory 35
model because it still had a tachyon with mass m2 = − 1
. The tachyon
was eliminated from the spectrum only in 1976 through the GSO projection
proposed by Gliozzi, Scherk and Olive [34].
Having realized that, at least for the critical value of the space-time dimen-
sion d = 26, the physical states are described by the DDF states having only
d− 2 = 24 independent components, open the way to Brink and Nielsen [35]
to compute the value α0 = 1 of the Regge trajectory with a very physical ar-
gument. They related the intercept of the Regge trajectory to the zero point
energy of a system with an infinite number of oscillators having only d − 2
independent components:
α0 = −
n (197)
This quantity is obviously infinite and, in order to make sense of it, they in-
troduced a cutoff on the frequencies of the harmonic oscillators obtaining an
infinite term that they eliminated by renormalizing the speed of light and a
finite universal constant term that gave the intercept of the Regge trajectory.
Instead of following their original approach we discuss here an alternative ap-
proach due to Gliozzi [36] that uses the ζ-function regularization. He rewrites
Eq. (197) as follows:
α0 = −
n = −
n−s = −
ζR(−1) = 1 (198)
where in the last equation we have used the identity ζR(−1) = − 112 and we
have put d = 26. Since the Shapiro-Virasoro model has two sets of trans-
verse harmonic oscillators it is obvious that its intercept is twice that of the
generalized Veneziano model.
Using the rules discussed in the previous section we can construct the
vertex operator corresponding to the state in Eq. (195). It is given by:
V(i;Ni)(z, P ) =
dziǫi · P (zi)eiNik̂·Q(zi) : eip·Q(z) : (199)
where the integral on the variable zi is evaluated along a curve of the complex
plane zi containing the point z. The singularity of the integrand for zi = z is
a pole provided that the following condition is satisfied.
2α′p · k̂ = 1 (200)
The last vertex in Eq. (199) is the vertex operator corresponding to the ground
tachyonic state given in Eq. (59) with α′p2 = 1.
Using the general form of the vertex one can compute the three-point
amplitude involving three arbitrary DDF vertex operators. This calculation
36 Paolo Di Vecchia
has been performed in Ref. [37] and since the vertex operators are conformal
fields with dimension equal to 1 one gets:
〈0, 0|V
(z1, P1)V(i(2)
(z2, P2)V(i(3)
(z3, P3)|0, 0〉 =
(z1 − z2)(z1 − z3)(z2 − z3)
(201)
where the explicit form of the coefficient C123 is given by:
C123 = 1〈0, 0|2〈0, 0|3〈0, 0|e
r.s=1
n,m=1
−n;iN
−m;i+
−n;i×
× eτ0
(α′Π2r−1)|N (1)k1 , i
〉1|N (2)k2 , i
〉2|N (3)k3 , i
〉3 (202)
where
N rsnm = −N rnNsm
nmα1α2α3
nαs +mαr
; N rn =
Γ (−nαr+1
αrn!Γ (1− nαr+1αr − n)
(203)
Π = Pr+1αr − Prαr+1 ; r = 1, 2, 3 (204)
Π is independent on the value of r chosen as a consequence of the equations:
Pr = 0 (205)
7 The zero slope limit
In the introduction we have seen that the dual resonance model has been
constructed using rules that are different from those used in field theory.
For instance, we have seen that planar duality implies that the amplitude
corresponding to a certain duality diagram, contains poles in both s and t
channels, while the amplitude corresponding to a Feynman diagram in field
theory contains only a pole in one of the two channels. Furthermore, the
scattering amplitude in the dual resonance model contains an infinite number
of resonant states that, at high energy, average out to give Regge behaviour.
Also this property is not observed in field theory. The question that was
natural to ask, was then: is there any relation between the dual resonance
model and field theory? It turned out, to the surprise of many, that the dual
resonance model was not in contradiction with field theory, but was instead
an extension of a certain number of field theories. We will see that the limit in
The birth of string theory 37
which a field theory is obtained from the dual resonance model corresponds
to taking the slope of the Regge trajectory α′ to zero.
Let us consider the scattering amplitude of four ground state particles in
Eq. (1) that we rewrite here with the correct normalization factor:
A(s, t, u) = C0N
0 (A(s, t) +A(s, u) +A(t, u)) (206)
where
2g(2α′)
4 (207)
is the correct normalization factor for each external leg, g is the dimensionless
open string coupling constant that we have constantly ignored in the previous
sections and C0 is determined by the following relation:
′ = 1 (208)
that is obtained by requiring the factorization of the amplitude at the pole
corresponding to the ground state particle whose mass is given in Eq. (21).
Using Eq. (21) in order to rewrite the intercept of the Regge trajectory in
terms of the mass of the ground state particle m2 and the following relation
satisfied by the Γ - function:
Γ (1 + z) = zΓ (z) (209)
we can easily perform the limit for α′ → 0 of A(s, t) obtaining:
A(s, t) =
m2 − s
m2 − s
(210)
Performing the same limit on the other two planar amplitudes we get the
following expression for the total amplitude in Eq. (206):
A(s, t, u) =
2g(2α′)
(α′)2
m2 − s
m2 − s
m2 − u
(211)
By introducing the coupling constant:
g3 = 4g(2α
4 (212)
Eq. (211) becomes
A(s, t, u) = g23
m2 − s
m2 − s
m2 − u
(213)
that is equal to the sum of the tree diagrams for the scattering of four particles
with mass m of Φ3 theory with coupling constant equal to g3. We have shown
that, by keeping g3 fixed in the limit α
′ → 0, the scattering amplitude of four
38 Paolo Di Vecchia
ground state particles of the dual resonance model is equal to the tree diagrams
of Φ3 theory. This proof can be extended to the scattering of N ground state
particles recovering also in this case the tree diagrams of Φ3 theory. It is also
valid for loop diagrams that we will discuss in the next section. In conclusion,
the dual resonance model reduces in the zero slope limit to Φ3 theory. The
proof that we have presented here is due to J. Scherk [38] 13
A more interesting case to study is the one with intercept α0 = 1. We will
see that, in this case, one will obtain the tree diagrams of Yang-Mills theory,
as shown by Neveu and Scherk [40] 14.
Let us consider the three-point amplitude involving three massless gauge
particles described by the vertex operator in Eq. (173). It is given by the sum
of two planar diagrams. The first one corresponding to the ordering (123) is
given by:
3Tr (λa1λa2λa3)
〈0, 0|Vǫ1(z1, p1)Vǫ2(z2, p2)Vǫ3(z3, p3)|0, 0〉
[(z1 − z2)(z2 − z3)(z1 − z3)]−1
(214)
Using momentum conservation p1+ p2+ p3 = 0 and the mass shell conditions
p2i = pi · ǫi = 0 one can rewrite the previous equation as follows:
0Tr(λ
a1λa2λa3)
× [(ǫ1 · ǫ2)(p1 · ǫ3) + (ǫ1 · ǫ3)(p3 · ǫ2) + (ǫ2 · ǫ3)(p2 · ǫ1)] (215)
The second contribution comes from the ordering 132 that can be obtained
from the previous one by the substitution
Tr(λa1λa2λa3) → −Tr(λa1λa3λa2) (216)
Summing the two contributions one gets
oTr(λ
a1 [λa2 , λa3 ])
× [(ǫ1 · ǫ2)(p1 · ǫ3) + (ǫ1 · ǫ3)(p3 · ǫ2) + (ǫ2 · ǫ3)(p2 · ǫ1)] (217)
The factor
N0 = 2g(2α
′)(d−2)/4 (218)
is the correct normalization factor for each vertex operator if we normalize
the generators of the Chan-Paton group as follows:
δij (219)
13 See also Ref. [39].
14 See also Ref. [41].
The birth of string theory 39
It is related to C0 through the relation
′ = 2 (220)
g is the dimensionless open string coupling constant. Notice that Eq.s (218)
and (220) differ from Eq.s (207) and (208) because of the presence of the
Chan-Paton factors that we did not include in the case of Φ3 theory.
By using the commutation relations:
[λa, λb] = ifabcλc (221)
and the previous normalization factors we get for the three-gluon amplitude:
igYMf
a1a2a3 [(ǫ1 · ǫ2)((p1 − p2) · ǫ3 +
+(ǫ1 · ǫ3)((p3 − p1) · ǫ2) + (ǫ2 · ǫ3)((p2 − p3) · ǫ1)] (222)
that is equal to the 3-gluon vertex that one obtains from the Yang-Mills action
LYM = −
F aαβF
a , F
αβ = ∂αA
β − ∂βAaα + gYMfabcAbαAcβ (223)
where
gYM = 2g(2α
4 (224)
The previous procedure can be extended to the scattering of N gluons finding
the same result that one gets from the tree diagrams of Yang-Mills theory.
In the next section, we will discuss the loop diagrams. Also, in this case one
finds that the h-loop diagrams involving N external gluons reproduces in the
zero slope limit the sum of the h-loop diagrams with N external gluons of
Yang-Mills theory.
We conclude this section mentioning that one can also take the zero slope
limit of a scattering amplitude involving three and four gravitons obtaining
agreement with what one gets from the Einstein Lagrangian of general rela-
tivity. This has been shown by Yoneya [43].
8 Loop diagrams
The N -point amplitude previously constructed satisfies all the axioms of S-
matrix theory except unitarity because its only singularities are simple poles
corresponding to zero width resonances lying on the real axis of the Mandel-
stam variables and does not contain the various cuts required by unitarity [1].
15 The determination of the previous normalization factors can be found in the
Appendix of Ref. [42].
40 Paolo Di Vecchia
In order to eliminate this problem it was proposed already in the early days of
dual theories to assume, in analogy with what happens for instance in pertur-
bative field theory, that the N -point amplitude was only the lowest order (the
tree diagram) of a perturbative expansion and, in order to implement unitar-
ity, it was necessary to include loop diagrams. Then, the one-loop diagrams
were constructed from the propagator and vertices that we have introduced
in the previous sections [44]. The planar one-loop amplitude with M external
particles was computed by starting from a (M + 2)-point tree amplitude and
then by sewing two external legs together after the insertion of a propagator
D given in Eq. (100). In this way one gets:
(2α′)d/2(2π)d
〈P, λ|V (1, p1)DV (1, p2) . . . V (1, pN)D|P, λ〉 (225)
where the sum over λ corresponds to the trace in the space of the harmonic os-
cillators and the integral in ddP corresponds to integrate over the momentum
circulating in the loop. The previous expression for the one-loop amplitude
cannot be quite correct because all states of the space generated by the oscil-
lators in Eq. (51) are circulating in the loop, while we know that we should
include only the physical ones. This was achieved first by cancelling by hand
the time and one of the space components of the harmonic oscillators reducing
the degrees of freedom of each oscillator from d to d − 2 as suggested by the
DDF operators at least for d = 26. This procedure was then shown to be cor-
rect by Brink and Olive [45]. They constructed the operator that projects over
the physical states and, by inserting it in the loop, showed that the reduction
of the degrees of freedom of the oscillators from d to d− 2 was indeed correct.
This was, at that time, the only procedure available to let only the physical
states circulate in the loop because the BRST procedure was discovered a bit
later also in the framework of the gauge field theories!
To be more explicit let us compute the trace in Eq. (225) adding also the
Chan-Paton factor. We get:
(2π)dδ(d)
NTr(λa1 . . . λaM )
(8π2α′)d/2
τd/2+1
[f1(k)]
12 (2π)M×
dνM−1 . . .
dν2 τ
eG(νji)
]2α′pi·pj
; k ≡ e−πτ(226)
where νji ≡ νj − νi,
G(ν) = log
ie−πν
2τ Θ1(iντ |iτ)
f31 (k)
; f1(k) = k
(1− k2n) (227)
The birth of string theory 41
Θ1(ν|iτ) = −2k1/4 sinπν
1− e2iπνk2n
1− e−2iπνk2n
(1− k2n)(228)
Finally the normalization factor N0 is given in Eq. (218). We have performed
the calculation for an arbitrary value of the space-time dimension d. However,
in this way one gets also the extra factor of k
12 appearing in the first line
of Eq. (226) that implies that our calculation is actually only consistent if
d = 26. In fact, the presence of this factor does not allow one to rewrite the
amplitude, originally obtained in the Reggeon sector, in the Pomeron sector
as explained below. In the following we neglect this extra factor, implicitly
assuming that d = 26, but, on the other hand, still keeping an arbitrary d.
Using the relations:
f1(k) =
tf1(q) ; Θ1(iντ |iτ) = iΘ1(ν|it)t1/2eπν
2/t (229)
where t = 1
and q ≡ e−πt, we can rewrite the one-loop planar diagram in the
Pomeron channel. We get:
(2π)dδ(d)
NTr(λa1 . . . λaM )
(8π2α′)d/2
dt[f1(q)]
2−d(2π)M×
dνM−1 . . .
−Θ1(νji|it)
f31 (q)
]2α′pi·pj
(230)
Notice that, by factorizing the planar loop in the Pomeron channel, one con-
structed for the first time what we now call the boundary state [46] 16. This
can be easily seen in the way that we are now going to describe. First of all,
notice that the last quantity in Eq. (230) can be written as follows:
Θ1(νji|it)
f31 (q)
]2α′pi·pj
−2 sin(πνji)
1− q2ne2πiνji
1− q2ne−2πiνji
(1− q2n)2
]2α′pi·pj
(231)
This equation can be rewritten as follows:
〈p = 0|q2R
i=1 : e
ipi·Q(e2iπνi ) : |p = 0〉
Tr (〈p = 0|q2N |p = 0〉)
; R =
na†n · an (232)
16 See also the first paper in Ref. [47].
42 Paolo Di Vecchia
where the trace is taken only over the non-zero modes and momentum con-
servation has been used. It must also be stressed that the normal ordering of
the vertex operators in the previous equation is such that the zero modes are
taken to be both in the same exponential instead of being ordered as in Eq.
(59). By bringing all annihilation operators on the left of the creation ones,
from the expression in Eq. (232) one gets (zi ≡ e2πiνi):
(2π)dδ(d)
(−2 sinπνji)2α
′pi·pj×
n=1 Tr
n·ane
2α′pj ·
znj e
2α′pi· an√
Tr (〈p = 0|q2N |p = 0〉)
(233)
The trace can be computed by using the completeness relation involving co-
herent states |f〉 = efa† |0〉:
e−|f |
|f〉〈f | = 1 (234)
Inserting the previous identity operator in Eq. (233) one gets after some cal-
culation:
(2π)dδ(d)
(−2 sinπνji)2α
′pi·pj×
i.j=1
−2α′pi·pje2πinνji q
n(1−q2n) (235)
Expanding the denominator in the last exponent and performing the sum over
n one gets:
(2π)dδ(d)
(−2 sinπνji)2α
′pi·pj×
2α′pi·pj
log(1−e2πiνji q2(m+1)) (236)
that is equal to the last line of Eq. (231) apart from the δ-function for mo-
mentum conservation. In conclusion, we have shown that Eq.s (231) and (232)
are equal.
Using Eq. (231) we can rewrite Eq. (230) as follows:
NNM0 Tr(λ
a1 . . . λaM )
(8π2α′)d/2
dt[f1(q)]
2−d(2πi)M
dνM−1 . . .
The birth of string theory 43
. . .
λ〈p = 0, λ|q2R
i=1 : e
ipi·Q(e2iπνi ) : |p = 0, λ〉
λ〈p = 0, λ|q2N |p = 0, λ〉
(237)
where the sum over any state |λ〉 corresponds to taking the trace over the
non-zero modes. If d = 26 we can rewrite Eq. (237) in a simpler form:
NNM0 Tr(λ
a1 . . . λaM )
(8π2α′)d/2
dt (2πi)M
dνM−1 . . .
〈p = 0, λ|q2R−2
: eipi·Q(e
2iπνi ) : |p = 0, λ〉 (238)
The previous equation contains the factor
dtq2R−2 that is like the propa-
gator of the Shapiro-Virasoro model, but with only one set of oscillators as
in the generalized Veneziano model. In the following we will rewrite it com-
pletely with the formalism of the Shapiro-Virasoro model. This can be done
by introducing the Pomeron propagator:
dt q2N−2 =
D̂ ; D̂ ≡ α
zL0−1z̄L̃0−1; |z| ≡ q = e−πt(239)
and rewriting the planar loop in the following compact form:
〈B0|D̂|BM 〉 ; |B0〉 ≡
n |p = 0, 0a, 0ã〉 (240)
where |B0〉 is the boundary state without any Reggeon on it,
Td−1 =
2(d−10)/4
α′)−d/2−1 (241)
and |BM 〉 is instead the one with M Reggeons given by:
|BM 〉 = NM0 Tr(λa1 . . . λaM )(2πi)M
dνM−1 . . .
: eipi·Q(e
2iπνi ) : |B0〉 (242)
We want to stress once more that the normal ordering in the previous equa-
tion is defined by taking the zero modes in the same exponential. Both the
boundary states and the propagator are now states of the Shapiro-Virasoro
model. This means that we have rewritten the one-loop planar diagram, where
the states of the generalized Veneziano model circulate in the loop, as a tree
44 Paolo Di Vecchia
diagram of the Shapiro-Virasoro model involving two boundary states and a
propagator. This is what nowadays is called open/closed string duality.
Besides the one-loop planar diagram in Eq. (225), that is nowadays called
the annulus diagram, also the non-planar and the non-orientable diagrams
were constructed and studied. In particular the non-planar one, that is ob-
tained as the planar one in Eq. (225) but with two propagators multiplied
with the twist operator
Ω = eL−1(−1)R , (243)
had unitarity violating cuts that disappeared [27] if the dimension of the
space-time d = 26, leaving behind additional pole singularities. The explicit
form of the non-planar loop can be obtained following the same steps done
for the planar loop. One gets for the non-planar loop the following amplitude:
〈BR|D̂|BM 〉 (244)
where now both boundary states contain, respectively, R and M Reggeon
states. The additional poles found in the non-planar loop were called Pomerons
because they occur in the Pomeron sector, that today is called the closed string
channel, to distinguish them from the Reggeons that instead occur in the
Reggeon sector, that today is called the open string sector of the planar and
non-planar loop diagrams. At that time in fact, the states of the generalized
Veneziano models were called Reggeons, while the additional ones appearing
in the non-planar loop were called Pomerons. The Reggeons correspond nowa-
days to open string states, while the Pomerons to closed string states. These
things are obvious now, but at that time it took a while to show that the
additional states appearing in the Pomeron sector have to be identified with
those of the Shapiro-Virasoro model. The proof that the spectrum was the
same came rather early. This was obtained by factorizing the non-planar dia-
gram in the Pomeron channel [46] as we have done in Eq. (244). It was found
that the states of the Pomeron channel lie on a linear Regge trajectory that
has double intercept and half slope of the one of the Reggeons. This follows
immediately from the propagator D̂ in Eq. (239) that has poles for values of
the momentum of the Pomeron exchanged given by:
p2 = 2n (245)
that are exactly the values of the masses of the states of the Shapiro-Virasoro
model [48], while the Reggeon propagator in Eq. (100) has poles for values of
momentum equal to:
1− α′p2 = n (246)
However, it was still not clear that the Pomeron states interact among them-
selves as the states of the Shapiro-Virasoro model. To show this it was first
The birth of string theory 45
necessary to construct tree amplitudes containing both states of the general-
ized Veneziano model and of the Shapiro-Virasoro model [49]. They reduced
to the amplitudes of the generalized Veneziano (Shapiro-Virasoro) model if
we have only external states of the generalized Veneziano (Shapiro-Virasoro)
model. Those amplitudes are called today disk amplitudes containing both
open and closed string states. They were constructed [49] by using for the
Reggeon states the vertex operators that we have discussed in Sect. (5) in-
volving one set of harmonic oscillators and for the Pomeron states the vertex
operators given in Eq. (181) that we rewrite here:
Vα,β(z, z̄, p) = Vα(z,
)Vβ(z̄,
) (247)
because now both component vertices contain the same set of harmonic os-
cillators as in the generalized Veneziano model. Furthermore, each of the two
vertices is separately normal ordered, but their product is nor normal ordered.
The amplitude involving both kinds of states is then constructed by taking
the product of all vertices between the projective invariant vacuum and inte-
grating the Reggeons on the real axis in an ordered way and the Pomerons in
the upper half plane, as one does for a disk amplitude.
We have mentioned above that the two vertices are separately normal
ordered, but their product is not normal ordered. When we normal order
them we get, for instance for the tachyon of the Pomeron sector, a factor
(z − z̄)α′p2/2 that describes the Reggeon-Pomeron transition. This implies a
direct coupling [51] between the U(1) part of gauge field and the two-index
antisymmetric field Bµν , called Kalb-Ramond field [50], of the Pomeron sector,
that makes the gauge field massive [51].
It was then shown that, by factorizing the non-planal loop in the Pomeron
channel, one reproduced the scattering amplitude containing one state of
the Shapiro-Virasoro and a number of states of the generalized Veneziano
model [52]. If we have also external states belonging to the generalized
Shapiro-Virasoro model, then by factorizing the non-planar one loop ampli-
tude in the pure Pomeron channel, one would obtain the tree amplitudes of
the Shapiro-Virasoro model [52].
All this implies that the generalized Veneziano model and the Shapiro-
Virasoro model are not two independent models, but they are part of the
same and unique model. In fact, if one started with the generalized Veneziano
model and added loop diagrams to implement unitarity, one found the ap-
pearence in the non-planar loop of additional states that had the same mass
and interaction of those of the Shapiro-Virasoro model.
The planar diagram, written in Eq. (230) in the closed string channel, is
divergent for large values of t. This divergence was recognized to be due to
exchange, in the Pomeron channel, of the tachyon of the Shapiro-Virasoro
model and of the dilaton [47]. They correspond, respectively, to the first two
terms of the expansion:
[f1(q)]
−24 = e2πt + 24 +O
e−2πt
(248)
46 Paolo Di Vecchia
The first one could be cancelled by an analytic continuation, while the second
one could be eliminated through a renormalization of the slope of the Regge
trajectory α′ [47].
We conclude the discussion of the one-loop diagrams by mentioning
that the one-loop diagram for the Shapiro-Virasoro model was computed by
Shapiro [53] who also found that the integrand was modular invariant.
The computation of multiloop diagrams requires a more advanced tech-
nology that was also developed in the early days of the dual resonance model
few years before the discovery of its connection to string theory. In order to
compute multiloop diagrams one needs first to construct an object that was
called the N -Reggeon vertex and that has the properties of containing N sets
of harmonic oscillators, one for each external leg, and is such that, when we
saturate it with N physical states, we get the corresponding N -point ampli-
tude. In the following we will discuss how to determine the N -Reggeon vertex.
The first step toward the N -Reggeon vertex is the Sciuto-Della Selva-
Saito [54] vertex that includes two sets of harmonic oscillators that we denote
with the indices 1 and 2. It is equal to:
VSDS = 2〈x = 0, 0| : exp
dzX ′2(z) ·X1(1− z)
: (249)
where X is the quantity that we have called Q in Eq. (57) and the prime
denotes a derivative with respect to z. It satisfies the important property
of giving the vertex operator Vα(z = 1) of an arbitrary state |α〉 when we
saturate it with the corresponding state:
VSDS |α〉2 = Vα(z = 1) (250)
A shortcoming of this vertex is that it is not invariant under a cyclic permu-
tation of the three legs. A cyclic symmetric vertex has been constructed by
Caneschi, Schwimmer and Veneziano [55] by inserting the twist operator in
Eq. (243). But the 3-Reggeon vertex is not enough if we want to compute an
arbitrary multiloop amplitude. We must generalize it to an arbitrary number
of external legs. Such a vertex, that can be obtained from the one in Eq. (249)
with a very direct procedure, or that can also be obtained by sewing together
three-Reggeon vertices, has been written in its final form by Lovelace [56] 17.
Here we do not derive it, but we give directly its expression written in Ref. [56]:
VN,0 =
i=1 dzi
dVabc
i=1[V
i (0)]
[i<x = 0, Oa|] δ(
i,j=1
n,m=0
a(i)n Dnm(ΓV
i Vj) a
(251)
17 See also Ref. [57]. Earlier papers on the N-Reggeon can be found in Ref.s [58].
The birth of string theory 47
where a
0 ≡ αi0 =
2α′p̂i is the momentum of particle i and the infinite
matrix:
Dnm(γ) =
∂mz [γ(z)]
n|z=0 ; n,m = 1.. : D00(γ) = − log |
AD −BC
Dn0 =
)n ; D0n =
)n ; γ(z) =
Az +B
Cz +D
(252)
is a ”representation” of the projective group corresponding to the conformal
weight ∆ = 0, that satisfies the eqs.:
Dnm(γ1γ2) =
Dnl(γ1)Dlm(γ2) +Dn0(γ1)δ0m +D0m(γ2)δn0 (253)
Dnm(γ) = Dmn(Γγ
−1Γ ) Γ (z) =
(254)
Finally Vi is a projective transformation that maps 0, 1 and ∞ into zi−1, zi
and zi+1.
The previous vertex can be written in a more elegant form as follows:
VN,0 =
i=1 dzi
dVabc
i=1[V
i (0)]
[i<x = 0, Oa|] δ(
dz∂X(i)(z)p̂i logV
i (z)
i,j=1
dy∂X(i)(z) log[Vi(z)− Vj(y)]∂X(j)(y)
(255)
where the quantities X(i) are what we called Q, namely the Fubini-Veneziano
field, in the previous sections. The N -Reggeon vertex that satisfies the impor-
tant property of giving the scattering amplitude of N physical particle when
we saturate it with their corresponding states, is the fundamental object for
computing the multiloop amplitudes. In fact, if we want to compute a M -loop
amplitude withN external states, we need to start from the (N+2M)-Reggeon
vertex and then we have to sew the M pairs together after having inserted a
propagator D. In this way we obtain an amplitude that is not only integrated
over the punctures zi (i = 1 . . . N) of the N external states, but also over
the additional 3h− 3 moduli corresponding to the punctures variables of the
48 Paolo Di Vecchia
states that we sew together and the integration variable of the M propaga-
tors. h is the number of loops. The multiloop amplitudes have been obtained
in this way already in 1970 [59, 60, 61] and, through the sewing procedure,
one obtained functions, as the period matrix, the abelian differentials, the
prime form, etc., that are well defined on Riemann surface! The only thing
that was missing, was the correct measure of integrations over the 3h−3 vari-
ables because it was technically not possible to let only the physical states to
circulate in the loops. This problem was solved only much later [62, 63] when
a BRST invariant formulation of string theory and the light-cone functional
integral could be used for computing multiloops. They are two very different
approaches that, however, gave the same result. For the sake of completeness
we write here the planar h-loop amplitude involving M tachyons:
M (p1, . . . , pM ) = N
h Tr(λa1 · · ·λaM ) Ch
2gs (2α
(d−2)/4
[dm]Mh
G(h)(zi, zj)
V ′i (0)V
j (0)
2α′pi·pj
, (256)
where Nh Tr(λa1 · · ·λaM ) is the appropriate U(N) Chan-Paton factor, g is
the dimensionless open string coupling constant, Ch is a normalization factor
given by
(2π)dh
g2h−2s
(2α′)d/2
, (257)
and G(h) is the h-loop bosonic Green function
G(h)(zi, zj) = logE(h)(zi, zj)−
ωµ (2πImτµν)
ων , (258)
with E(h)(zi, zj) being the prime form, ω
µ (µ = 1, . . . , h) the abelian differen-
tials and τµν the period matrix. All these objects, as well as the measure on
moduli space [dm]Mh , can be explicitly written in the Schottky parametrization
of the Riemann surface, and their expressions for arbitrary h can be found for
example in Ref. [64]. It is given by
[dm]Mh =
dVabc
V ′i (0)
dkµ dξµ dηµ
k2µ (ξµ − ηµ)2
(1− kµ)2
(259)
× [det (−iτµν)]−d/2
(1− knα)−d
(1− knα)2
where kµ are the multipliers, ξµ and ηµ are the fixed points of the generators
of the Schottky group,
The birth of string theory 49
9 From dual models to string theory
The approach presented in the previous sections is a real bottom-up ap-
proach. The experimental data were the driving force in the construction of
the Veneziano model and of its generalization to N external legs. The rest of
the work that we have described above consisted in deriving its properties. The
result is, except for a tachyon, a fully consistent quantum-relativistic model
that was a source of fascination for those who worked in the field. Although
the model grew out of S-matrix theory where the scattering amplitude is the
only observable object, while the action or the Lagrangian have not a central
role, some people nevertheless started to investigate what was the underly-
ing microscopic structure that gave rise to such a consistent and beautiful
model. It turned out, as we know today, that this underlying structure is that
of a quantum-relativistic string. However, the process of connecting the dual
resonance model (actually two of them the generalized Veneziano and the
Shapiro-Virasoro model) to string theory took several years from the origi-
nal idea to a complete and convincing proof of the conjecture. The original
conjecture was independently formulated by Nambu [20, 65], Nielsen [66] and
Susskind [21] 18. If we look at it in retrospective, it was at that time a fantastic
idea that shows the enormous physical intuition of those who formulated it.
On the other hand, it took several years to digest it before one was able to
derive from it all the deep features of the dual resonance model. Because of
this, the idea that the underlying structure was that of a relativistic string,
did not really influence most of the research in the field up to 1973. Let me
try to explain why.
A common feature of the work of Ref.s [20, 66, 21] is the suggestion that
the infinite number of oscillators, that one got through the factorization of
the dual resonance model, naturally comes out from a two-dimensional free
Lagrangian for the coordinate Xµ(τ, σ) of a one-dimensional string, that is an
obvious generalization of the Lagrangian that one writes for the coordinate
Xµ(τ) of a pointlike object in the proper-time gauge:
=⇒ L ∼
(260)
Being this theory conformal invariant the Virasoro operators were also con-
structed together with their algebra. In this very first formulation, however,
the Virasoro generators Ln were just the generators associated to the confor-
mal symmetry of the string world-sheet Lagrangian given in Eq. (260) as in
any conformal field theory. It was not clear at all why they should imply the
gauge conditions found by Virasoro or, in modern terms, why they should be
zero classically. The basic ingredient to solve this problem was provided by
Nambu [65] and Goto [68] who wrote the non-linear Lagrangian proportional
18 See also Ref. [67].
50 Paolo Di Vecchia
to the area spanned by the string in the external target space. They proceeded
in analogy with the point particle and wrote the following action:
−dσµνdσµν (261)
where
dσµν =
dζα ∧ dζβ = ∂Xµ
ǫαβdσdτ (262)
Xµ(σ, τ) is the string coordinate and ζ
0 = τ and ζ1 = σ are the coordinates of
the string worldsheet. ǫαβ is an antisymmetric tensor with ǫ01 = 1. Inserting
eq. (262) in (261) and fixing the proportionality constant one gets the Nambu-
Goto action [65, 68]:
S = −cT
(Ẋ ·X ′)2 − Ẋ2X ′2 (263)
where
Ẋµ ≡ ∂X
µ ≡ ∂X
(264)
and T ≡ 1
is the string tension, that replaces the mass appearing in the
case of a point particle. In this formulation, the string Lagrangian is invariant
under any reparametrization of the world-sheet coordinates σ and τ and not
only under the conformal transformations. This, in fact, implies that the two-
dimensional world-sheet energy-momentum tensor of the string is actually zero
as we will show later on. But it took still a few years to connect the Nambu-
Goto action to the properties of the dual resonance model. In the meantime
an analogue model was formulated [69] that reproduced the tree and loop
amplitudes of the generalized Veneziano model. This approach anticipated
by several years the path integral derivation of dual amplitudes. It was very
closely related to the functional integral formulation of Ref.s [70].
However, one needed to wait until 1973 with the paper of Goddard,
Goldstone, Rebbi and Thorn [71], where the Nambu-Goto action was cor-
rectly treated, all its consequences were derived and it became completely
clear that the structure underlying the dual resonance model was that of a
quantum-relativistic string. The equation of motion for the string were de-
rived from the action in Eq. (263) by imposing δS = 0 for variations such
that δXµ(τi) = δX
µ(τf ) = 0. One gets:
∂X ′µ
δXµ +
∂X ′µ
δXµ|σ=πσ=0
(265)
where L is the Lagrangian in Eq. (263). Since δXµ is arbitrary, from eq. (265)
one gets the Euler-Lagrange equation of motion
The birth of string theory 51
∂X ′µ
= 0 (266)
and the boundary conditions
∂X ′µ
= 0 or δXµ = 0 at σ = 0, π (267)
for an open string and
Xµ(τ, 0) = Xµ(τ, π) (268)
for a closed string. In the case of an open string, the first kind of boundary
condition in Eq.(267) corresponds to Neumann boundary conditions, while
the second one to Dirichlet boundary conditions. Only the Neumann bound-
ary conditions preserve the translation invariance of the theory and, there-
fore, they were mostly used in the early days of string theory. It must be
stressed, however, that Dirichlet boundary conditions were already discussed
and used in the early days of string theory for constructing models with off-
shell states [72].
From Eq. (263) one can compute the momentum density along the string:
≡ Pµ = cT
′2 −X ′µ(Ẋ ·X ′)
(Ẋ ·X ′)2 − Ẋ2X ′2
(269)
and obtain the following constraints between the dynamical variables Xµ and
c2T 2x′
+ P 2 = x′ · P = 0 (270)
They are a consequence of the reparametrization invariance of the string La-
grangian. Because of this one can choose the orthonormal gauge specified by
the conditions:
Ẋ2 +X ′
= Ẋ ·X ′ = 0 (271)
that nowadays is called conformal gauge. In this gauge eq. (269) becomes:
Pµ = cT Ẋµ
∂X ′µ
= −cTX ′µ (272)
and therefore the eq. of motion in eq.(266) becomes:
Ẍµ −X ′′µ = 0 (273)
while the boundary condition in eq.(267) becomes:
X ′µ(σ = 0, π) = 0 (274)
52 Paolo Di Vecchia
The most general solution of the eq. of motion and of the boundary conditions
can be written as follows:
Xµ(τ, σ) = qµ + 2α′pµτ + i
[aµne
−inτ − a+µn einτ ]
cosnσ√
(275)
for an open string and
Xµ(τ, σ) = qµ + 2α′pµτ +
[ãµne
−2in(τ+σ) − ã+µn e2in(τ+σ)]
[aµne
−2in(τ−σ) − a+µn e2in(τ−σ)]
(276)
for a closed string. This procedure really shows that, starting from the
Nambu-Goto action, one can choose a gauge (the orthonormal or confor-
mal gauge) where the equation of motion of the string becomes the two-
dimensional D’Alembert equation in Eq. (273). Furthermore, the invariance
under reparametrization of the Nambu-Goto action implies that the two-
dimensional energy-momentum tensor is identically zero at the classical level
(See Eq. (271)).
As the Lorentz gauge in QED the orthonormal gauge does not fix com-
pletely the gauge. We can still perform reparametrizations that leave in the
conformal gauge: they are conformal transformatiuons. Introducing the vari-
able z = eiτ the generators of the conformal transformations for the open
string can be written as follows:
dzzn+1
αn−m · αm = 0 (277)
where
αµn =
naµn if n > 0√
2α′pµ if n = 0√
na†µn if n < 0
(278)
They are zero as a consequence of Eq.s (270) that in the conformal gauge
become Eq.s (271). In the case of a closed string we get instead:
L̃n =
dzzn+1
= 0 (279)
dz̄z̄n+1
= 0 (280)
The birth of string theory 53
In terms of the harmonic oscillators introduced in eq. (276) we get
αm · αn−m = 0 ; L̃n =
α̃m · α̃n−m = 0 (281)
where for the non-zero modes we have used the convention in (278), while the
zero mode is given by:
0 = α̃
(282)
In conclusion, the fact that we have reparametrization invariance implies that
the Virasoro generators are classically identically zero. When we quantize the
theory one cannot and also does not need to impose that they are vanishing at
the operator level. They are imposed as conditions characterizing the physical
states.
〈Phys′|Ln|Phys〉 = 〈Phys′|(L0 − 1)|Phys〉 = 0 ; n 6= 0 (283)
These equations are satisfied if we require:
Ln|Phys >= (L0 − 1)|Phys >= 0 (284)
The extra factor −1 in the previous equations comes from the normal ordering
as explained in Eq. (198).
The authors of Ref. [71] further specified the gauge by fixing it completely.
They introduced the light-cone gauge specified by imposing the condition:
X+ = 2α′p+τ (285)
where
X0 ±Xd−1√
X0 ±Xd−1√
(286)
In this gauge the only physical degrees of freedom are the transverse ones.
In fact the components along the directions 0 and d − 1 can be expressed in
terms of the transverse ones by inserting Eq. (285) in the constraints in Eq.
(271) and getting:
Ẋ− =
4α′p+
(Ẋ2i +X
i ) X
2α′p+
Ẋi ·X ′i (287)
that up to a constant of integration determine completely X− as a function
of X i. In terms of oscillators we get
α+n = 0 ;
2α′α−n =
αin−mα
m n 6= 0 (288)
54 Paolo Di Vecchia
for an open string and
α+n = α̃
n = 0 n 6= 0 (289)
together with
2α′α−n =
αin−mα
2α′α̃−n =
α̃in−mα̃
m (290)
in the case of a closed string.
This shows that the physical states are described only by the transverse
oscillators having only d − 2 components. Those transverse oscillators corre-
spond to the transverse DDF operators that we have discussed in Section 6.
The authors of Ref. [71] also constructed the Lorentz generators only in terms
of the transverse oscillators and they showed that they satisfy the correct
Lorentz algebra only if the space-time dimension is d = 26. In this way the
spectrum of the dual resonance model was completely reproduced starting
from the Nambu-Goto action if d = 26! On the other hand, the choice of
d = 26 is a necessity if we want to keep Lorentz invariance!
Immediately after this, the interaction was also included either by adding
a term describing the interaction of the string with an external gauge field [73]
or by using a functional formalism [74, 75].
In the following we will give some detail only of the first approach for the
case of an open string. A way to describe the string interaction is by adding
to the free string action an additional term that describes the interaction of
the string with an external field.
SINT =
dDyΦL(y)JL(y) (291)
where ΦL(y) is the external field and JL is the current generated by the string.
The index L stands for possible Lorentz indices that are saturated in order to
have a Lorentz invariant action.
In the case of a point particle, such an interaction term will not give any
information on the self-interaction of a particle.
In the case of a string, instead, we will see that SINT will describe the
interaction among strings because the external fields that can consistently
interact with a string are only those that correspond to the various states of
the string, as it will become clear in the discussion below.
This is a consequence of the fact that, for the sake of consistency, we must
put the following restrictions on SINT :
• It must be a well defined operator in the space spanned by the string
oscillators.
The birth of string theory 55
• It must preserve the invariances of the free string theory. In particular, in
the ”conformal gauge” it must be conformal invariant.
• In the case of an open string, the interaction occurs at the end point of
a string (say at σ = 0). This follows from the fact that two open strings
interact attaching to each other at the end points.
The simplest scalar current generated by the motion of a string can be written
as follows
J(y) =
dσδ(σ)δ(d)[yµ − xµ(τ, σ)] (292)
where δ(σ) has been introduced because the interaction occurs at the end of
the string. For the sake of simplicity we omit to write a coupling constant g
in (292).
Inserting (292) in (291) and using for the scalar external field Φ(y) = eik·y
a plane wave, we get the following interaction:
SINT =
dτ : eik·X(τ,0) : (293)
where the normal ordering has been introduced in order to have a well defined
operator. The invariance of (293) under a conformal transformation τ → w(τ)
requires the following identity:
SINT =
dτ : eik·X(τ,0) : =
dw : eik·X(w,0) : (294)
or, in other words, that
: eik·X(τ,0) :=⇒ w′(τ) : eik·X(w,0) : (295)
This means that the integrand in Eq. (294) must be a conformal field with
conformal dimension equal to one and this happens only if α′k2 = 1. The
external field corresponds then to the tachyonic lowest state of the open string.
Another simple current generated by the string is given by:
Jµ(y) =
dσδ(σ)Ẋµ(τ, σ)δ
(d)(y −X(τ, σ)) (296)
Inserting (296) in (291) we get
SINT =
dτẊµ(τ, 0)ǫ
µeik·X(τ,0) (297)
if we use a plane wave for Φµ(y) = ǫµe
ik·y. The vertex operator in eq. (297)
is conformal invariant only if
k2 = ǫ · k = 0 (298)
56 Paolo Di Vecchia
and, therefore, the external vector must be the massless photon state of the
string. We can generalize this procedure to an arbitrary external field and
the result is that we can only use external fields that correspond to on shell
physical states of the string.
This procedure has been extended in Ref. [73] to the case of external
gravitons by introducing in the Nambu-Goto action a target space metric and
obtaining the vertex operator for the graviton that is a massless state in the
closed string theory. Remember that, at that time, this could have been done
only with the Nambu-Goto action because the σ-model action was introduced
only in 1976 first for the point particle [76] and then for the string [77]. As
in the case of the photon it turned out that the external field corresponding
to the graviton was required to be on shell. This condition is the precursor of
the equations of motion that one obtains from the σ-model action requiring
the vanishing of the β-function [78].
One can then compute the probability amplitude for the emission of a
number of string states corresponding to the various external fields, from an
initial string state to a final one. This amplitude gives precisely the N -point
amplitude that we discussed in the previous sections [73]. In particular, one
learns that, in the case of the open string, the Fubini-Veneziano field is just
the string coordinate computed at σ = 0:
Qµ(z) ≡ Xµ(z, σ = 0) ; z = eiτ (299)
In the case of a closed string we get instead:
Qµ(z, z̄) ≡ Xµ(z, z̄) ; z = e2i(τ−σ) , z̄ = e2i(τ+σ) (300)
Finally, let me mention that with the functional approach Mandelstam [74]
and Cremmer and Gervais [79] computed the interaction between three arbi-
trary physical string states and reproduced in this way the coupling of three
DDF states given in Eq. (202) and obtained in Ref. [37] by using the operator
formalism. At this point it was completely clear that the structure underlying
the generalized Veneziano model was that of an open relativistic string, while
that underlying the Shapiro-Virasoro model was that of a closed relativistic
string. Furthermore, these two theories are not independent because, if one
starts from an open string theory, one gets automatically closed strings by
loop corrections.
10 Conclusions
In this contribution, we have gone through the developments that led from
the construction of the dual resonance model to the bosonic string theory
trying as much as possible to include all the necessary technical details. This
is because we believe that they are not only important from an historical point
of view, but are also still part of the formalism that one uses today in many
The birth of string theory 57
string calculations. We have tried to be as complete and objective as possible,
but it could very well be that some of those who participated in the research
of these years, will not agree with some or even many of the statements we
made. We apologize to those we have forgotten to mention or we have not
mentioned as they would have liked.
Finally, after having gone through the developments of these years, my
thoughts go to Sergio Fubini who shared with me and Gabriele many of the
ideas described here and who is deeply missed, and to my friends from Flo-
rence, Naples and Turin for a pleasant collaboration in many papers discussed
here.
Acknowledgments
I thank R. Marotta and I. Pesando for a critical reading of the manuscript.
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	The birth of string theory
	Paolo Di Vecchia
