# A Coding of Real Null Four-Momenta into World-Sheet

Authors: David B. Fairlie
Journal: Advances in Mathematical Physics
 (2009)

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## Abstract

The results of minimizing the action for string-like systems on a
simply connected world sheet are shown to encode the Cartesian components
of real null momentum four-vectors into coordinates on the
world sheet. This identification arises consistently from different approaches
to the problem.


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## Body

## 1. Recapitulation

This paper is based
upon an old unpublished article by
Fairlie and  Roberts
[1], which  dates back
to circa 1972, on a model for amplitudes suggested by string theory, or rather
the dual-resonance model as it was then called. (The results in this paper are
recorded in the Ph.D. thesis of Roberts [2].) I was so overwhelmed by the evident truth of the
famous paper
of Goddard et al. quantising the bosonic
string in 26 dimensions [3], which I regard as one of the classic papers in string
theory, that I never submitted this paper for publication. However since
recently, there has appeared an article by Sommerfield and  Thorn [4], the 4th
section of which is
closely related to the model presented in [1], it may be an appropriate time to give these ideas an
airing. Also other recent developments have given rise to an interpretation of
maximally violating helicity (MVH) amplitudes in
Yang Mills theory in terms of topological string amplitudes [5]; the connection between null
four-vectors and Koba-Nielsen variables which is at
the heart of [1] may
not be entirely coincidental. The intention was to construct a viable amplitude
for particles living in a strictly four-dimensional space-time, and with zero
mass instead of the tachyon ground state which bedevilled the bosonic string dual
resonance model. One of the features of tractable models of physical processes
which has come to be more appreciated in the intervening years is that there is
frequently a mismatch between what is tractable mathematically and what one
should like to have; for example, the potential integrability of N→∞ supersymmetric Yang Mills as against the
intractability of QCD, or the Sine Gordon model which displays both solitons
and Lorentz invariance, at the cost of working in two dimensions. Here a feature
analogous to self-dual Yang-Mills theory, which possesses instantons in a space
of even signature, is present; I have realised that the theory presented is
more mathematically compelling in a space of signature (2,2),
though a Lorentzian interpretation is by no means ruled out. This will be
discussed later in relation to the work of Gross and Mende on high energy
scattering [6]. The
starting point is the famous Koba-Nielsen formula, which gives an elegant
expression for the N point tree amplitude for N particles with incoming momenta piμ for the ground state of open strings [7],(1.1)A(s,t)t=∫−∞∞∏1NdzkdVabcθ(zi−zi+1)∏j>i(zi−zj)−2α′pi⋅pj,(1.2)dVabc=dzadzbdzc(zb−za)(zc−za)(za−zc).(This
integration measure is introduced as a consequence of conformal invariance; to
account for the property that the real axis along which the integration is
performed is invariant under transformations of the Möbius group, provided that α′(piμ)2=−1.) This means invariance under the
mapping:(1.3)z′↦az+bcz+d,ad−bc=1.It has been shown that this formula arises as a
contribution to string scattering from a simply connected world sheet, thanks
to the properties of conformal invariance. Another way of writing (1.1) is as an
exponential:(1.4)∫∞∞exp(∑i<j−2α′pi⋅pjlog(zi−zj))dzkdVabc.
The exponent in the integrand may
be interpreted as the (Euclidean) contribution to the action where the momenta
enter the upper half-plane at designated points zk which are then integrated over to give the
contribution to the path integral for the amplitude arising from a simply
connected world sheet. One of the chief deficiencies in (1.1) is the tachyon
condition, namely, that piμ is light like. This requirement follows from
the invariance under mappings which preserve the upper-half complex plane. The
radical idea behind [1] was to give the formula for the amplitude a different
interpretation; do not integrate, but instead determine the coordinates zi by minimising the integrand; this is
tantamount, in the second version to use the method of steepest descents. The
equations to be satisfied are, setting α′=1,(1.5)∑jpi⋅pj(zi−zj)=0.These equations may be seen to
be satisfied, provided that we are in a 4-dimensional space with signature (2,2) with null four-momenta pjμ,
and the coordinates zj (here on the real line) are given
by(1.6)zj=pj0+pj1pj2−pj3=pj2+pj3pj0−pj1;(pj0)2+(pj3)2−(pj1)2−(pj2)2=0.This works
because(1.7)(zi−zj)=pi⋅pj+pi0pj1−pi1pj0+pi3pj2−pi2pj3(pi2−pi3)(pj0−pj1),(zi−zj)=−pi⋅pj+pi0pj1−pi1pj0−pi3pj2−pi2pj3(pi0−pi1)(pj2−pj3).The second equation is obtained
by using the alternative expression of (zi,zj).
By subtracting and rationalising, we
have(1.8)(pi2−pi3)(pj0−pj1)−(pi0−pi1)(pj2−pj3)=2pi⋅pj(zi−zj).Summing over all particle
positions zj except zj=zi and invoking the conservation of momentum, ∑pjμ=0,
we see that (1.5) is satisfied. If instead of the real line, the integration in
(1.1) is performed over the boundary of the unit disc, the points on the boundary
where the momenta enter may be parametrised by(1.9)zj=pj0+ipj3pj1+ipj2=pj1−ipj2pj0−pj3;(pj0)2+(pj3)2−(pj1)2−(pj2)2=0.There is a Möbius transformation
(1.3) which connects the two representations, for the plane and the
disc,(1.10)zdisc=i+zplanei−zplane.Indeed complex Möbius
transformations on zi are equivalent to SU(2,2) transformations on piμ.


## 2. Alternative Approach

Consider a
two-dimensional surface embedded in a four-dimensional space and take as
parametric representation of the surface the
four-vectorsXμ(σ,τ) where σ and τ are intrinsic coordinates on the surface with
metric:(2.1)ds2=Edσ2+2Fdσdτ+Gdτ2,where(2.2)E=(∂Xμ∂σ)2,F=(∂Xμ∂σ)(∂Xμ∂τ),G=(∂Xμ∂τ)2(see [1]). The Nambu-Goto Lagrangian
describing the dynamics of the field Xμ(σ,τ) is a measure of the area of the world sheet
and is the reparameterisation invariant form(2.3)ℒ=α′∬EG−F2dσdτ.On the other hand, it is well
known that there exists a transformation to a coordinate system of so-called
isometric coordinates in which the Lagrangian takes the simple quadratic
form(2.4)ℒ′=∬((∂Xμ∂σ)2+(∂Xμ∂τ)2)dσdτ,which is invariant only under
the subset of reparameterisations of the variables (σ,τ) which are conformal, that is, those
transformations which satisfy the Cauchy-Riemann equations. It is well known
that in the coordinate system where σ and τ are isometric parameters defined by E=G;F=0.
In this frame, the Euler equation minimising (2.3) becomes linear and is
just(2.5)∇2Xμ=0.
The
conditions for an isometric coordinate system may be written in the
following form due to Weierstrass(2.6)(∂ζμ∂z)2=E−G+2iF=0,where Xμ is the real part of ζμ in view of the fact that (2.5) is satisfied
provided that ζ is an analytic function of z=σ+iτ.
The Weierstrass' condition shows that conformal mappings of coordinate systems
preserve the isometric property. We can make a link with the Virasoro
conditions for closed strings [8, 9] by noting that this is in fact the gauge condition of the
model; writing(2.7)(∂ζμ∂z)2=∑−∞∞Lnzn=0.This is too stringent to demand
as an operator equation. Instead, we require that the matrix elements of (2.7)
should vanish for all z, that is,(2.8)〈ψ†|Ln|ψ〉=0,∀n≥0.This is satisfied provided that Ln=Ln†=0. These conditions are the familiar Virasoro
conditions for closed strings with zero mass ground states. A typical solution
of (2.5) with a finite number of singularities is given by(2.9)ζμ=∑i=1i=npiμlog(z−zi).
By
applying the Weierstrass condition, we have(2.10)∑i,jpi⋅pj(z−zi)(z−zj)=∑i,jpi⋅pj(1(z−zi)(zj−zi)−1(z−zj)(zj−zi))=0.This has to be true for all z,
which evidently requires that ∑i,jpi⋅pj=0 and, with conservation of four-momentum,
also requires the same conditions ∑j(pi⋅pj/(zi−zj))=0 as before. In the case of the four-point
function, the solution of these conditions (1.5) may be readily solved in terms
of the cross-ratio λ to give(2.11)λ=(zi−z2)(z3−z4)(z1−z3)(z4−z2)=p1⋅p2p1⋅p3=st,where(2.12)s=(p1μ+p2μ)2,t=(p1μ+p3μ)2,u=(p1μ+p4μ)2,s+t+u=0.This
result, in terms of the cross-ratio, is independent of the metric, so also
works in a Lorentz metric with signature (3,1).
The resulting amplitude A(s,t,u) with s+t+u=0 may be evaluated to give(2.13)A(s,t,u)=(−s)−α′s(−t)−α′t(−u)α′u.As s→∞ at fixed  tA(s,t,u)→t−α′ss−α′t,
that is, it exhibits Regge asymptotic behaviour. The subject of asymptotic
behaviour of high energy string amplitudes was examined to all orders sometime
afterwards by Gross and  Mende [6] who found the same connection
(2.11) between the
cross-ratio and the Mandelstam variables.


## 3. Lorentz Signature

As has been
remarked, the minimisation condition in the case of the four-point function may
be solved in terms of cross-ratios. This suggests that the conditions may be
solved directly in terms of the variableszj whatever the metric
is. This is indeed the case; for real
four-momenta, the solution may be expressed
as(3.1)zj=pj0+pj3pj1−ipj2=pji+ipj2pj0−pj3,(pj0)2−(pj1)2−(pj2)2−(pj3)2=0.The difference is that in the case of signature (2,2),
the four-momenta may be parametrised as pj0=rcosh(θj),pj1=rsinh(θj),pj2=rcosh(ϕj),pj3=rsinh(ϕj),
which imply that
(3.2)zj=exp(θj+ϕj),so the variables lie on the real
line. Alternatively, a trigonometric parameterisation may be employed, in which
case zj=exp(iθj+iϕj).
However in the case of signature (1,3),
the parameterisation is mixed;  pj0=rcosh(θj),pj3=rsinh(θj),pj1=rcos(ϕj),pj2=rsinh(ϕj),
which imply that zj=iexp(θj+iϕj),
so there is no obvious integration contour for (1.1).


## 4. Minimal Surface Interpretation

Further insight
may be gained by a parameterisation of minimal surfaces embedded in
four-dimensional Euclidean space, originally due to
Eisenhart [10], but rediscovered by Shaw
[11] and quoted in
[1, 12]. It is given
by(4.1)X0=Re(f(z)−zf′(z)+g(z)′),X3=Im(f(z)−zf′(z)−g(z)′),X1=Re(g(z)−zg′(z)+f(z)′),X2=Im(zg′(z)−g(z)+f(z)′),where a prime denotes the
derivative with respect to the argument. Suppose we seek a parameterisation
where Xμ+aμ is the real part of ζμ=∑ipiμGi(z),
and aμ is an arbitrary origin. Then, thanks to the
linearity of the above equations, we can split f(z) and g(z) into sums of independent components, that is, f(z)=∑fi(z),g(z)=∑gi(z),
and deduce, up to shifts of origin,(4.2)(pi0+ipi3)Gi(z)=2(fi(z)−zfi′(z)+gi′(z)),(pi1−ipi2)Gi(z)=2(gi(z)−zgi′(z)+fi′(z)),(pi1+ipi2)Gi(z)=2(gi(z)−gi′(z)+fi′(z))*,(pi0−ipi3)Gi(z)=2(fi(z)−zfi′(z)+zg′(z))*.If one postulates, using the
same relations as obtained before,
then
(4.3)zi=pi0+ipi3pi1−ipi2=pi1+ipi2pi0−ipi3.These equations possess basic
solutions of the form(4.4)2fi(z)=(a−b)ln(z+1)(z+1)2(z1+1)+(a+b)ln(z−1)(z−1)2(z1−1)+(z1a+b)ln(z−z1)(z−z1)(1−z12)−a,2gi(z)=(b−a)ln(z+1)(z+1)2(z1+1)+(a+b)ln(z−1)(z−1)2(z1−1)+(z1b+a)ln(z−z1)(z−z1)(1−z12)−b,Gi=c(−1+ln(z−z1)),where a=(pi0+ipi3)c and b=z1(pi1−ipi2)c,
and c is a real parameter. If the real
parameterisation (1.8) is employed, then the zi lie on the real axis(4.5)Xμ=∑ipiμlog(z−zi)=∑πpiμΘ(z−zi)forzontherealaxis.As z moves from +∞ to −∞,Xμ  jumps by πpiμ at the point zi,
so the skew polygon formed by the partial sums of momenta (closed on account of
momentum conservation) is mapped into intervals on the real line.


## 5. Conclusion

The principal
message of this paper is to draw attention to the link between the Cartesian
components of real null four-momenta in four-dimensional flat space and complex
variables on a simply connected world sheet, associated with a minimal surface,
or a form of string evolution. The set of four momenta are also required to sum
to zero, that is, momentum is conserved in the system. Various aspects leading
to this identification are explored. The minimisation of the Koba-Nielsen
integrand, the consequence of the Weierstrass condition upon a linear
combination of elementary solutions to the free equations of motion, to
guarantee a minimal surface solution, and the direct determination of this
class of minimal surface solution from the Eisenhart parameterisation are all
shown to entail the same identification of a complex variable in terms of the
components of a null four-momentum. In a space of even signature(2,2);
in one representation, the complex variables lie on the real line; in another
on a circle; in the case of odd signature (Lorentz metric), there is no specific
curve on which the variables lie. SL(2,C) transformations of the complex variable
implement homogeneous Lorentz transformations upon the momentum.In our original paper, as is standard practice, the
optimistic anticipation of further development of these ideas was raised, but
it must be admitted that neither author has been able to add anything
substantially new in the intervening 35 years! However, as T.S. Eliot has said,
“A poem may have meanings which are hidden from its author.” It may be that the
further examination of solutions to the four-dimensional minimal surface
equations originally proposed by Eisenhart will be fruitful. The ideas of this
paper seem rooted in four dimensions; the parameterisation of classical string
solutions proposed in [12, 13] based upon the division algebras may contain the clue
to extend the connection between momenta and world sheet coordinates to 10
dimensions. The recent paper of Sommerfield and  Thorn [4] extends their ideas to AdS
space-time, and the picture of world sheets bounded by a closed polygon of null
lines which is presented therein and is also contained in [14] is essentially the same as that in
Section 4 of the
present paper. In addition, the treatment of high energy string amplitudes by
Gross and  Mende [6]
extends some aspects of this analysis to multiply connected world sheets.In this spirit, this revised and rewritten version of
[1] is offered in the
hope that some deeper connection between momentum space and the world sheet
will be discovered.



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