Harmonic bilocal fields generated by globally
conformal invariant scalar fields
Nikolay M. Nikolov1,3,
Karl-Henning Rehren2,
Ivan Todorov1,3
September 23, 2021
1 Institute for Nuclear Research and Nuclear Energy,
Tsarigradsko Chaussee 72, BG-1784 Sofia, Bulgaria
2 Institut für Theoretische Physik, Universität Göttingen,
Friedrich-Hund-Platz 1, D-37077 Göttingen, Germany
3 Abdus Salam International Centre for Theoretical Physics,
Strada Costiera 11, I–34014 Trieste, Italy
Abstract
The twist two contribution in the operator product expansion of φ1(x1)
φ2(x2) for a pair of globally conformal invariant, scalar fields of equal scal-
ing dimension d in four space–time dimensions is a field V1(x1, x2) which is
harmonic in both variables. It is demonstrated that the Huygens bilocality
of V1 can be equivalently characterized by a “single–pole property” concern-
ing the pole structure of the (rational) correlation functions involving the
product φ1(x1) φ2(x2). This property is established for the dimension d = 2
of φ1, φ2. As an application we prove that any system of GCI scalar fields
of conformal dimension 2 (in four space–time dimensions) can be presented
as a (possibly infinite) superposition of products of free massless fields.
Subject classification: PACS 2003: 11.10.-z. 03.70.+k, MSC 2000: 81T10
1 Introduction
Global Conformal Invariance (GCI) of Minkowski space Wightman fields yields
rationality of correlation functions [14]. This result opens the way for a non-
perturbative construction and analysis of GCI models for higher dimensional
Quantum Field Theory (QFT), by exploring further implications of the Wight-
man axioms.
By choosing the axiomatic approach, we avoid any bias about the possible
origin of the model, because we aim at a broadest possible perspective. On
the other hand, the assumption of GCI limits the analysis to a class of theories
http://arxiv.org/abs/0704.1960v4
Harmonic Bilocal Fields 2
that can be parameterized by its (generating) field content and finitely many
coefficients for each correlation function (see Sect. 2). As anomalous dimensions
under the assumption of GCI are forced to be integral, there is no perturbative
approach within this setting, but it is conceivable that a theory with a contin-
uous coupling parameter may exhibit GCI at discrete values (that appear as
renormalization group fixed points). An example of this type is provided by the
Thirring model: it is locally conformal invariant for any value of the coupling
constant g and becomes GCI for positive integer g2 [5].
Previous axiomatic treatments of conformal QFT were focussed on the rep-
resentation theory and harmonic analysis of the conformal group [6, 10] as tools
for the Operator Product Expansion (OPE). The general projective realization
of conformal symmetry in QFT was already emphasized in [15, 16] and found to
constitute a (partial) organization of the OPE. GCI is complementary in that it
assumes true representations (trivial covering projection). A necessary condition
for this highly symmetric situation is the presence of infinitely many conserved
tensor currents (as we shall see in Sect. 3.3).
The first cases studied under the assumption of GCI were theories generated
by a scalar field φ(x) of (low) integral dimension d > 1. (The case d = 1
corresponds to a free massless field with a vanishing truncated 4-point function
wtr4 .) The cases 2 6 d 6 4, which give rise to non-zero w
4 were considered in
[12, 13, 11].1
The main purpose in these papers was to study the constraints for the 4-
point correlation (=Wightman) functions coming from theWightman (= Hilbert
space) positivity. This was achieved by using the conformal partial wave expan-
sion. An important technical tool in this expansion is the splitting of the OPE
into different twist contributions (see (2.10)). Each partial wave gives a non-
rational contribution to the complete rational 4-point function. It is therefore
remarkable that the sum of the leading, twist two, conformal partial waves (cor-
responding to the contributions of all conserved symmetric traceless tensors in
the OPE of basic fields) can be proven in certain cases to be a rational function.
This means that the twist two part in the OPE of two fields φ is convergent in
such cases to a bilocal field, V1(x1, x2), which is our first main result in the present
paper. Throughout, “bilocal” means Huygens (= space–like and time–like) lo-
cality with respect to both arguments. Proving bilocality exploits the bounds on
the poles due to Wightman positivity, and the conservation laws for twist two
tensors which imply that the bilocal fields are harmonic in both arguments.
Trivial examples of harmonic bilocal fields are given by bilinear free field
constructions of the form : ϕ(x1)ϕ(x2) :, : ψ̄(x1)γµ(x1 − x2)
µψ(x2) :, or (x1 −
µ(x1 − x2)
ν : Fµσ(x1)F
ν (x2) :. A major purpose of this paper is to explore
whether harmonic twist two fields can exist which are not of this form, and
whether they can be bilocal. Moreover, we show that the presence of a bilocal
1The last two references are chiefly concerned with the case d = 4 (in D = 4 space-time
dimensions) which appears to be of particular interest as corresponding to a (gauge invariant)
Lagrangian density. The intermediate case d = 3 is briefly surveyed in [18].
Harmonic Bilocal Fields 3
field V1 completely determines the structure of the theory in the case of a scaling
dimension d = 2. The first step towards the classification of d = 2 GCI fields
was made in [12] where the case of a unique scalar field was considered. Here we
extend our study to the most general case of a theory generated by an arbitrary
(countable) set of d = 2 scalar fields. Our second main result states that such
fields are always combinations of Wick products of free fields (and generalized
free fields).
The paper is organized as follows.
Section 2 contains a review of relevant results concerning the theory of GCI
scalar fields.
In Sect. 3 we study conditions for the existence of the harmonic bilocal field
V1(x1, x2). We prove that Huygens bilocality of V1(x1, x2) is equivalent to the sin-
gle pole property (SPP), Definition 3.1, which is a condition on the pole structure
of the leading singularities of the truncated correlation functions of φ1(x1)φ2(x2)
whose twist expansion starts with V1(x1, x2). This nontrivial condition qualifies
a premature announcement in [2] that Huygens bilocality is automatic.
Indeed, the SPP is trivially satisfied for all correlations of free field construc-
tions of harmonic fields with other (products of) free fields, due to the bilinear
structure of V1. Thus any violation of the SPP is a clear signal for a nontriv-
ial field content of the model. Moreover, the SPP will be proven from general
principles for an arbitrary system of d = 2 scalar fields (the case studied in [2]).
Yet, although the pole structure of U(x1, x2) turns out to be highly constrained
in general by the conservation laws of twist two tensor currents, the SPP does
not follow for fields of higher dimensions, as illustrated by a counter-example of
a 6-point function of d = 4 scalar fields involving double poles (Sect. 3.5).
The existence of V1(x1, x2) as a Huygens bilocal field in a theory of dimension
d = 2 fields allows to determine the truncated correlation functions up to a single
parameter in each of them. This is exploited in Sect. 4, where an associative
algebra structure of the OPE of d = 2 scalar fields and harmonic bilocal fields
is revealed. The free-field representation of these fields is inferred by solving an
associated moment problem.
2 Properties of GCI scalar fields
2.1 Structure of correlation functions and pole bounds
We assume throughout the validity of the Wightman axioms for a QFT on the
D = 4 flat Minkowski space–time M (except for asymptotic completeness) – see
[17]. Our results can be, in fact, generalized in a straightforward way to any
even space–time dimension D. The condition of GCI in the Minkowski space is
an additional symmetry condition on the correlation functions of the theory [14].
In the case of a scalar field φ(x), it asserts that the correlation functions of φ(x)
Harmonic Bilocal Fields 4
are invariant under the substitution
φ(x) 7→ det
, (2.1)
where x 7→ g(x) is any conformal transformation of the Minkowski space, ∂g
its Jacobi matrix and d > 0 is the scaling dimension of φ. An important point
is that the invariance of Wightman functions
∣∣φ(x1) · · · φ(xn)
under the
transformation (2.1) should be valid for all xk ∈ M in the domain of definition
of g (in the sense of distributions). It follows that d must be an integer in order
to ensure the singlevaluedness of the prefactor in (2.1). Thus, GCI implies that
only integral anomalous dimensions can occur.
The most important consequences of GCI in the case of scalar fields φk(x) of
dimensions dk are summarized as follows.
(a) Huygens Locality ([14, Theorem 4.1]). Fields commute for non light–like
separations. This has an algebraic version:
(x1 − x2)
φ1(x1), φ2(x2)
= 0 (2.2)
for a sufficiently large integer N .
(b) Rationality of Correlation Functions (cf. [14, Theorem 3.1]). The general
form of Wightman functions is:
∣∣φ1(x1) · · ·φn(xn)
{µjk}
C{µjk}
(ρjk)
µjk , (2.3)
where here and in what follows we set
ρjk := (xjk − i 0 e0)
2 = (xjk)
2 + i 0 x0jk , xjk := xj − xk ; (2.4)
the sum in Eq. (2.3) is over all configurations of integral powers {µjk = µkj}
subject to the following conditions:
j (6=k)
µjk = −dk, (2.5)
and pole bounds µjk > −
dj + dk
δdjdk − 1
. Equation (2.5) follows from
the conformal invariance under (2.1); the pole bounds express the absence of
non-unitary representations in the OPE of two fields [14, Lemma 4.3]. Under
these conditions the sum in (2.3) is always finite and there are a finite number
of free parameters for every n-point correlation function. We shall refer to the
form (2.3) as a Laurent polynomial in the variables ρjk.
2Writing correlation functions in terms of the conformally invariant cross ratios is particu-
larly useful to parameterize 4-point functions. A basis of cross ratios for an n-point function
is used in the proof of Lemma 3.6. The general systematics of the pole structure, however, is
more transparent in terms of the present variables.
Harmonic Bilocal Fields 5
(c) The truncated Wightman functions
∣∣φ1(x1) · · · φn(xn)
are of the
same form like (2.3) but with pole degrees µtr
bounded by
µtrjk > −
dj+dk
(2.6)
(cf. [14, Corollary 4.4]).
The cluster condition, expressing the uniqueness of the vacuum, requires that
if a non-empty proper subset of points xk among all xi (i = 1, . . . , n) is shifted by
t · a (a2 6= 0), then the truncated function must vanish in the limit t → ∞. For
the two-point clusters {xj , xk}, this condition is ensured by (2.6) in combination
with with (2.5). For higher clusters, it puts further constraints on the admissible
linear combinations of terms of the form (2.3). Note however, that because of
possible cancellations the individual terms need not vanish in the cluster limit.
The cluster condition will be used in establishing the single pole property for
d = 2.
2.2 Twist expansion of the OPE and bi–harmonicity of twist two
contribution
The most powerful tool provided by GCI is the explicit construction of the OPE
of local fields in the general (axiomatic) framework.
Let φ1(x) and φ2(x) be two GCI scalar fields of the same scaling dimension
d and consider the operator distribution
U(x1, x2) = (ρ12)
φ1(x1)φ2(x2)− 〈0|φ1(x1)φ2(x2)|0〉
. (2.7)
As a consequence of the pole bounds (2.6), U(x1, x2) is smooth in the difference
x12. This is to be understood in a weak sense for matrix elements of U between
bounded energy states. Obviously, U(x1, x2) is a Huygens bilocal field in the
sense that [
(x1 − x)
2(x2 − x)
U(x1, x2), ψ(x)
= 0 (2.8)
for every field ψ(x) that is Huygens local with respect to φk(x). Then, one
introduces the OPE of φ1(x1)φ2(x2) by the Taylor expansion of U in x12
U(x1, x2) =
µ1,...,µn=0
12 · · · x
µ1...µn
(x2) , (2.9)
where Xnµ1...µn(x2) are Huygens local fields. We can consider the series (2.9)
as a formal power series, or as a convergent series in terms of the analytically
continued correlation functions of U(x1, x2). We will consider at this point the
series (2.9) just as a formal series. (See also [1] for the general case of constructing
OPE via multilocal fields in the context of vertex algebras in higher dimensions.)
Since the prefactor in (2.7) transforms as a scalar density of conformal weight
(1−d, 1−d) then U(x1, x2) transforms as a conformal bilocal field of weight (1, 1).
Harmonic Bilocal Fields 6
Hence, the local fields Xnµ1...µn in (2.9) have scaling dimensions n+2 but are not,
in general, quasiprimary.3 One can pass to an expansion in quasiprimary fields by
subtracting from Xnµ1...µn derivatives of lower dimensional fields X
µ1...µn′
. The
resulting quasiprimary fields Okµ1...µℓ are traceless tensor fields of rank ℓ and
dimension k. The difference
k − ℓ (“dimension − rank”) (2.10)
is called twist of the tensor field Okµ1...µℓ . Unitarity implies that the twist is
non-negative [10], and by GCI, it should be an even integer. In this way one can
reorganize the OPE (2.9) as follows
U(x1, x2) = V1(x1, x2) + ρ12 V2(x1, x2) + (ρ12)
2 V3(x1, x2) + · · · , (2.11)
where Vκ(x1, x2) is the part of the OPE (2.9) containing only twist 2κ contri-
butions. Note that Eq. (2.11) contains also the information that the twist 2κ
contributions contain a factor (ρ12)
κ−1 (i.e. Vκ are “regular” at x1 = x2), which
is a nontrivial feature of this OPE (obtained by considering 3-point functions).
Thus, the expansion in twists can be viewed as a light-cone expansion of the
Since the twist decomposition of the fields is conformally invariant then each
Vκ will be behave, at least infinitesimally, as a scalar (κ, κ) density under con-
formal transformations.
Every Vκ is a complicated (formal) series in twist 2κ fields and their deriva-
tives:
Vκ(x1, x2) =
Kµ1...µℓκ (x12, ∂x2)O
µ1...µℓ
(x2) , (2.12)
where K
µ1...µℓ
κ (x12, ∂x2) are infinite formal power series in x12 with coefficients
that are differential operators in x2 acting on the quasiprimary fields O. The
important point here is that the series K
µ1...µℓ
κ (x12, ∂x2) can be fixed universally
for any (even generally) conformal QFT. This is due to the universality of con-
formal 3-point functions. The explicit form of K
µ1...µℓ
κ (x12, ∂x2) can be found in
[6, 7] (see also [13]).
Thus, we can at this point consider Vκ(x1, x2) only as generating series for
the twist 2κ contributions to the OPE of φ(x1)φ(x2) but we still do not know
whether these series would be convergent and even if they were, it would not be
evident whether they would give bilocal fields. In the next section we will see
that this is true for the leading, twist two part under certain conditions, which
are automatically fulfilled for d = 2.
The higher twist parts Vκ (κ > 1) are certainly not convergent to Huygens
bilocal fields, since their 4-point functions, computed in [13], are not rational.
3Quasiprimary fields transform irreducibly under conformal transformations.
Harmonic Bilocal Fields 7
The major difference between the twist two tensor fields and the higher twist
fields is that the former satisfy conservation laws:
∂xµ1O
µ1...µℓ
(x) = 0 (ℓ > 1) . (2.13)
This is a well known consequence of the conformal invariance of the 2-point func-
tion and the Reeh–Schlieder theorem. It includes, in particular, the conservation
laws of the currents and the stress–energy tensor. It turns out that V1(x1, x2)
encodes in a simple way this infinite system of equations.
Theorem 2.1. ([13]) The system of differential equations (2.13) is equivalent to
the harmonicity of V1(x1, x2) in both arguments (bi–harmonicity) as a formal
series, i.e.,
�x1V1(x1, x2) = 0 = �x2V1(x1, x2).
The proof is based on the explicit knowledge of the K series in (2.12) and it is
valid even if the theory is invariant under infinitesimal conformal transformations
only.
The separation of the twist two part in (2.11) amounts to a splitting of U of
the form
U(x1, x2) = V1(x1, x2) + ρ12 Ũ(x1, x2) . (2.14)
This splitting can be thought in terms of matrix elements of U(x1, x2) expanded
as a formal power series according to (2.9). It is unique by virtue of Theorem 2.1,
due to the following classical Lemma:
Lemma 2.2. ([3, 1]) Let u(x) be a formal power series in x ∈ C4 (or, CD) with
coefficients in a vector space V . Then there exist unique formal power series
v(x) and ũ(x) with coefficients in V such that
u(x) = v(x) + x2 ũ(x) (2.15)
and v(x) is harmonic in x (i.e., �x v(x) = 0). (2.15) is called the harmonic
decomposition of u(x) (in the variable x around x = 0), and the formal power
series v(x) is said to be the harmonic part of u(x).
3 Bilocality of twist two contribution to the OPE
Let us sketch our strategy for studying bilocality of V1(x1, x2).
The existence of the field V1(x1, x2) can be established by constructing its cor-
relation functions. On the other hand, every correlation function4
·V1(x1, x2) ·
4This short-hand notation stands for
˛φ3(x3) · · ·φk(xk) V1(x1, x2) φk+1(xk+1)
· · · φn(xn)
, here and in the sequel.
Harmonic Bilocal Fields 8
of V1 is obtained (originally, as a formal power series in x12) under the split-
ting (2.14). It thus appears as a harmonic decomposition of the corresponding
correlation function
· U(x1, x2) ·
of U :
· U(x1, x2) ·
· V1(x1, x2) ·
+ ρ12
· Ũ(x1, x2) ·
. (3.1)
Note that we should initially treat the left hand side of (3.1) also as a formal
power series in x12 in order to make the equality meaningful. It is important that
this series is always convergent as a Taylor expansion of a rational function in a
certain domain around x1 = x2 in M
, for the complexified Minkowski space
MC = M + iM , according to the standard analytic properties of Wightman
functions. We shall show in Sect. 3.1 that this implies the separate convergence
of both terms in the right hand side of (3.1). Hence, the key tool in constructing
V1 are the harmonic decompositions
F (x1, x2) = H(x1, x2) + ρ12 F̃ (x1, x2) (3.2)
of functions F (x1, x2) that are analytic in certain neighbourhoods of the diagonal
{x1 = x2}.
Recall that H in (3.2) is uniquely fixed as the harmonic part of F in x1 around
x2, due to Lemma 2.2. This is equivalent to the harmonicity �x1 H(x1, x2) =
0. On the other hand, according to Theorem 2.1 we have to consider also the
second harmonicity condition on H, �x2 H(x1, x2) = 0, i.e., H is the harmonic
part in x2 around x1. This leads to some “integrability” conditions for the initial
function F (x1, x2), which we study in Sect. 3.2.
Next, to characterize the Huygens bilocality of V1, we should have rationality
of its correlation functions
· V1(x1, x2) ·
, which is due to a straightforward
extension of the arguments of [14, Theorem 3.1]. But we have started with the
correlation functions of U , which are certainly rational. Hence, we should study
another condition on U , namely that its correlation functions have a rational
harmonic decomposition. We show in Sect. 3.3 that this is equivalent to a simple
condition on the correlation functions of U , which we call “Single Pole Property”
(SPP).
In this way we establish in Sect. 3.4 that V1 always exists as a Huygens
bilocal field in the case of scalar fields of dimension d = 2. However, for higher
scaling dimensions one cannot anymore expect that V1 is Huygens bilocal in
general. This is illustrated by a counter-example, involving the 6-point function
of a system of d = 4 fields, given at the end of Sect. 3.5.
3.1 Convergence of harmonic decompositions
To analyze the existence of the harmonic decomposition of a convergent Taylor
series we use the complex integration techniques introduced in [1].
Let MC =M + iM be the complexification of Minkowski space, which in this
subsection is assumed to be D–dimensional, and E =
x : (i x0, x1, . . . , xD−1)
its Euclidean real submanifold, and SD−1 ⊂ E the unit sphere in E. We
Harmonic Bilocal Fields 9
denote by ‖·‖ the Hilbert norm related to the fixed coordinates in MC: ‖x‖
|x0|2 + · · · + |xD−1|2.
Let us also introduce for any r > 0 a real compact submanifold Mr of MC:
ζ ∈MC : ζ = r e
iθw, ϑ ∈ [0, π], w ∈ SD−1
(3.3)
(note that ϑ ∈ [π, 2π] gives another parameterization of Mr). Then there is an
integral representation for the harmonic part of a convergent Taylor series.
Lemma 3.1. (cf. [1, Sect. 3.3 and Appendix A]) Let u(x) be a complex formal
power series that is absolutely convergent in the ball ‖x‖ < r, for some r > 0, to
an analytic function U(x). Then the harmonic part v(x) of u(x) (around x = 0),
which is provided by Lemma 2.2, is absolutely convergent for
|x2|+ 2 r ‖x‖ < r2. (3.4)
The analytic function V (x) that is the sum of the formal power series v(x) has
the following integral representation:
V (x) =
(z− x)2
U(z) , V1 =
= iπ|SD−1|, (3.5)
where r′ < r, |x2| + 2 r′ ‖x‖ < r′2, and the (complex) integration measure
is obtained by the restriction of the complex volume form dDz (= dz0∧
· · · ∧dzD−1) on MC (∼= CD) to the real D–dimensional submanifold Mr′ (3.3),
r′ > 0.
Proof. Consider the Taylor expansion in x of the function
1 − x
(z − x)2
and write it in the form (cf. [1, Sect. 3.3])
(z− x)2
(z2)−
−ℓHℓ(z, x), Hℓ(z, x) =
hℓµ(z)hℓµ(x), (3.6)
where {hℓµ(u)} is an orthonormal basis of harmonic homogeneous polynomials
of degree ℓ on the sphere SD−1. This expansion is convergent for
∣∣+ 2
∣∣z · x
∣∣ (3.7)
since its left–hand side is related to the generating function for Hℓ:
1− λ2 x2 y2
(1− 2λ x · y + λ2 x2 y2)
λℓHℓ(x, y) , (3.8)
the expansion (3.8) being convergent for λ 6 1 if |x2y2|+2|x ·y| < 1. Then if we
fix r′ < r and z varies on Mr′ , a sufficient condition for (3.7) is |x
2| + 2 r′ ‖x‖ <
r′2 (since sup
w∈ SD−1
|w · x| = ‖x‖).
Harmonic Bilocal Fields 10
On the other hand, writing u(z) =
k=0 uk(z), where uk are homogeneous
polynomials of degree k, we get by the absolute convergence of u(z) the relation
(valid for |x2| + 2 r′ ‖x‖ < r′2)
(z− x)2
U(z) =
k,ℓ=0
(z2)−
−ℓHℓ(x, z)uk(z) .
(3.9)
Noting next that in the parameterization (3.3) of Mr′ we have d
= i r′D
eiD ϑ dϑ ∧ dσ(w), where dσ(w) is the volume form on the unit sphere, we obtain
for the right hand side of (3.9):
k,ℓ=0
eiϑ(k−ℓ)
dσ(w)
|SD−1|
Hℓ(x,w)uk(w) .
Now if we write, according to Lemma 2.2, uk(z) =
ck,j,µ′ (z
2)j hk−2j,µ′(z)
then we get by the orthonormality of hℓ,µ(w)
k,ℓ=0
δℓ,k−2j
eiϑ(k−ℓ) ck,j,µ hk−2j,µ(x)
ck,0,µ hk,µ(x) = v(x) .
The latter proves both: the convergence of v(x) in the domain (3.4) (since r′ < r
was arbitrary) and the integral representation (3.5). �
As an application of this result we will prove now
Proposition 3.2. For all n and k, and for all local fields φj (j = 3, . . . , n) the
Taylor series
∣∣φ3(x3) · · · φk(xk) V1(x1, x2) φk+1(xk+1) · · · φn(xn)
(3.10)
in x12 converge absolutely in the domain
‖x12‖+
‖x12‖2 +
∣∣x212
‖x2j‖+
‖x2j‖2 +
∣∣x22j
∣∣x22j
∣∣ ∀ j (3.11)
(j = 3, . . . , n). They all are real analytic and independent of k for mutually
nonisotropic points.
Proof. Let
Fk(x12, x23, . . . , x2n)
∣∣φ3(x3) · · ·φk(xk) U(x1, x2) φk+1(xk+1) · · · φn(xn)
(3.12)
Harmonic Bilocal Fields 11
be the correlation functions, analytically continued in x12.
As Fk, which is a rational function, depends on x := x12 via a sum of products
of powers
(x− x2j)
it has a convergent expansion in x for
∣∣+ 2
∣∣x · x2j
∣∣x22j
∣∣ . (3.13)
If we want Fk to have a convergent Taylor expansion for ‖x‖ < r we get the
following sufficient condition
∣∣x22j
∣∣− 2 r ‖x2j‖. (3.14)
By Lemma 3.1 we conclude that the series (3.10) is convergent for
|x212|+ 2 r ‖x12‖ < r
2. (3.15)
Combining both (sufficient) conditions (3.14) and (3.15) for r we find that they
are compatible if ‖x12‖ +
‖x12‖2 +
∣∣x212
‖x2j‖2 +
∣∣x22j
∣∣ − ‖x2j‖, which is
equivalent to (3.11). �
Note that one can also prove a similar convergence property for the correla-
tion functions of several V1.
Remark 3.1. The domain of convergence of (3.10) should be Lorentz invariant.
Hence, (3.10) are convergent in the smallest Lorentz invariant set containing the
domain (3.11). Such a set is determined by the values of the invariants x212, x
and x12 · x2j and it turns out to be the set
∣∣x212
∣∣ 12 ∣∣x22j
∣∣ 12 6
∣∣x12 · x2j
(∣∣x22j
∣∣ 12 −
∣∣x212
∣∣ 12 )2
or equivalently
√∣∣x212
∣∣∣∣x22j
∣∣x12 · x2j
∣∣2 <
(∣∣x22j
∣∣ 12 −
∣∣x212
∣∣ 12 )2
. (3.16)
Outside the domain of convergence (3.16), the correlations of V1(x1, x2) have
to be defined by analytic continuation. When the correlations are rational, V1
is Huygens bilocal, but the counter-example presented in Sect. 3.5 shows that
rationality is not automatic. Then, it is not even obvious that the continuations
are single–valued within the tube of analyticity required by the spectrum con-
dition, i.e., that V1 exists as a distribution in all of M × M . Nontrivial case
studies, however, show that at least for xk space–like to both x1 and x2, the
continuation is single–valued and preserves the independence on the position k
in (3.10) where V1(x1, x2) is inserted. This leads us to conjecture
Conjecture 3.3. The twist two field V1(x1, x2), whose correlations are defined
as the analytic continuations of the harmonic parts of those of U(x1, x2), exists
and is bilocal in the ordinary sense, i.e., it commutes with φ(x) and V1(x, x
′) if
x and x′ are space–like to x1 and x2.
We hope to return to this conjecture elsewhere (see also the Note added in
proof). Note that the argument that locality implies Huygens locality [14] does
not pass to bilocal fields.
Harmonic Bilocal Fields 12
3.2 Consequences of bi–harmonicity
Now our objective is to find the harmonic decomposition of the rational functions
F (x1, x2) that depend on x1 and x2 through the intervals ρik = (xi−xk)
2, i = 1, 2,
k = 3, . . . , n, for some additional points x3, . . . , xn. The F ’s, as correlation
functions of U(x1, x2), have the form
F (x1, x2) =
(ρ12)
q Fq(x1, x2) ≡
(ρ12)
{ρik}{i,k}6={1,2}
, (3.17)
Fq(x1, x2) =
{µ1i},{µ2i}
Cq,{µ1j},{µ2j}
(ρ1j)
(ρ2i)
µ2j , (3.18)
where M ∈ N and µ1j , µ2j (j = 3, . . . , n) are integers > −d such that
j>3 µ1j
j>3 µ2j = −1 − q, and the coefficients Cq,{µ1j},{µ2j} may depend on ρjk
(j, k > 3).
IfH is the harmonic part of F in x12, then the leading part F0 (of order (ρ12)
is also the leading part of H. We shall now proceed to show that bi–harmonicity
of H (Theorem 2.1), together with the first principles of QFT including GCI,
implies strong constraints on F0.
Proposition 3.4. Let F0(x1, x2) be as in (3.18), and let H(x1, x2) be its har-
monic part with respect to x1 around x2. Then H is also harmonic with respect
to x2, if and only if F0 satisfies the differential equation
(E1D2 − E2D1)F0 = 0, (3.19)
where E1 =
i=3 ρ2i∂1i (with ∂jk = ∂kj =
), D1 =
36j<k6n ρjk∂1j∂1k,
and similarly for E2 and D2, exchanging 1 ↔ 2.
Proof. By Proposition 3.2 (see also Remark 3.1) we can consider H as a function
in the 2n − 3 variables ρ1i, ρ2i (i > 3) and ρ12, analytic in some domain that
includes ρ12 = 0.
Expanding H =
q(ρ12)
qHq/q!, the functions Hq are homogeneous of degree
−1 − q in both sets of variables ρ1i and ρ2i, and H0 = F0. To impose the
harmonicity with respect to the variable x1, we use the identity [11, App. C]
�x1F = −4
26i<j6n
ρij ∂1i∂1j F
ρij =(xi−xj)2
, (3.20)
valid for homogeneous functions of ρ1i of degree −1, to express the wave operator
�x1 as a differential operator with respect to the set of variables ρ1i (i > 2). This
yields the recursive system of differential equations
E1Hq+1 = −D1Hq. (3.21)
Performing the same steps with respect to the variable x2, one obtains
E2Hq+1 = −D2Hq. (3.22)
Harmonic Bilocal Fields 13
Eq. (3.19) then arises as the integrability condition for the pair of inhomogeneous
differential equations for H1 (putting q = 0), observing that E2E1 − E1E2 =∑
ρ1i∂1i −
ρ2i∂2i vanishes on H1 by homogeneity.
Conversely, if (3.19) is fulfilled, thenH1 exists and satisfies (D1E2−D2E1)H1 =
−(D1D2 − D2D1)H0 = 0 because D1 and D2 commute. But this is equivalent
to (D2E1 − D1E2)H1 = 0, which is in turn the integrability condition for the
existence of H2, and so on. It follows that bi–harmonicity imposes no further
conditions on the leading function H0 = F0. �
The differential equation (3.19) imposes the following constraints on the lead-
ing part F0 of the rational correlation function F (3.17):
Corollary 3.5. Assume that the function F0 as in (3.18) satisfies the differential
equation (3.19). Then
(i) If F0 contains a “double pole” of the form (ρ1i)
µ1i(ρ1j)
µ1j with i 6= j and µ1i
and µ1j both negative, then its coefficients must be regular in ρ2k (k 6= i, j).
(ii) F0 cannot contain a “triple pole” of the form (ρ1i)
µ1i(ρ1j)
µ1j (ρ1k)
µ1k with
i, j, k all different and µ1i, µ1j , µ1k all negative.
The same hold true, exchanging 1 ↔ 2.
Proof. Pick any variable, say ρ2k, and decompose F0 =
r>−p(ρ2k)
rfr as a Lau-
rent polynomial in ρ2k. The differential equation (3.19) turns into the recursive
system
ρij∂1i∂1j −
i,j 6=k
ρ2iρkj∂1i∂2j
 r · fr = Xrfr−1 + Y fr
of differential equations for the functions fr which are Laurent polynomials in the
remaining variables. The precise form of the polynomial differential operators
Xr and Y does not matter. Assume the lowest power −p of ρ2k to be negative.
For r = −p, the right-hand-side vanishes. Because the term ρij∂1i∂1j on the
left-hand-side would produce a singularity that cannot be cancelled by any other
term, f−p cannot have a “double pole” in any pair of variables ρ1i, ρ1j with i 6= j
and i, j 6= k. This property passes recursively to all fr with r < 0, because also
the right-hand-side never can contain such a pole. This implies that a double
pole in a pair of variables ρ1i, ρ1j with i 6= j cannot multiply a term that is
singular in ρ2k unless k = i or k = j, proving (i).
If the coefficient of the double pole were singular in ρ1k, k 6= i, j, then the
resulting double pole in the pair ρ1i, ρ1k resp. ρ1j , ρ1k would imply regularity
also in ρ2j resp. ρ2i. Hence the coefficient of a triple pole must be regular in all
ρ2m, which contradicts the total homogeneity −1 of F0 in these variables. This
proves the statement (ii). �
3.3 A necessary and sufficient condition for Huygens bilocality
Definition 3.1. (“Single Pole Property”, SPP) Let f(x1, . . . , xn) be a Laurent
polynomial in the variables ρij , i.e., regarded as a function of x1 only, it is a
Harmonic Bilocal Fields 14
finite linear combination of functions of the form
(ρ1j)
µ1j ≡
(x1 − xj)
, (3.23)
where µ1j (j > 2) are integers and the coefficients may depend on the parameters
ρjk (j, k > 2). Then f is said to satisfy the single pole property with respect to
x1 if it contains no terms for which there are j 6= k (j, k > 2) such that both µ1,j
and µ1,k are negative.
The significance of SPP stems from the fact that the harmonic parts H of
F0, i.e., the correlation functions of V1, are again Laurent polynomials if and
only if F0 satisfies the SPP. Namely, if H is a harmonic Laurent polynomial, the
same argument as in [11, Lemma C.1] (using the representation (3.20) of the
wave operator) shows that H fulfils the SPP with respect to x1, and so does F0,
because it is the leading part of order (ρ12)
0 of H. The converse is an immediate
consequence of Lemma 3.6 (allowing for a relabelling and multiple counting of
the points x3, . . . , xn, which are not required to be distinct).
Lemma 3.6. Let n > 4. Every finite linear combination of monomials of the
gn(x1) =
i=4 ρ1i
(ρ13)
i=4(x1 − xi)
[(x1 − x3)2]n−2
(3.24)
has a rational harmonic decomposition in x1 around x2
gn(x1) = hn(x1) + (x1 − x2)
2 · g̃n(x1) (3.25)
i.e., hn is harmonic with respect to x1 and g̃n is regular at x1 = x2, and both
hn and g̃n are rational. More precisely, (ρ13)
n−2(ρ23)
n−3hn is a homogeneous
polynomial of total degree 2(n − 3) in the variables {ρij : 1 6 i < j}, which is
separately homogoneous of degree n− 3 in the variables {ρ1i : i > 2} and in the
variables {ρ12, ρ2i : i > 3}.
Proof. It is convenient to introduce the variables
ρ1iρ23
ρ13ρ2i
, si =
ρ12ρ3i
ρ13ρ2i
, uij =
ρ12ρ23ρij
ρ13ρ2iρ2j
(4 6 i < j 6 n). (3.26)
We claim that hn(x1) is of the form
hn(x1) =
fn(ti, si, uij)
, (3.27)
where fn are polynomials of degree n − 3 such that fn(ti, si = 0, uij = 0) =∏n
i=4 ti. Because all si and uij contain a factor ρ12, these properties ensure that
g̃n given by (gn − hn)/ρ12 is regular in ρ12.
Harmonic Bilocal Fields 15
Using again the identity (3.20) for the wave operator, and transforming this
into a differential operator with respect to the set of variables (3.26), we find
�x1 hn(x1) = −4
(ρ13)
·Dfn(ti, si, uij), (3.28)
where D is the differential operator
D = (1 + t∂t + s∂s + u∂u)(s∂t + s∂s + u∂u)− (s∂s + u∂u)∂t − u∂t∂t (3.29)
with shorthand notations for degree-preserving operators
t∂t =
ti∂ti , s∂t =
si∂ti , s∂s =
si∂si , u∂u =
46i<j6n
uij∂uij
and degree-lowering operators
∂ti , u∂t∂t =
46i<j6n
uij∂ti∂tj .
To solve the condition Dfn = 0 for harmonicity, we make an ansatz
fn(ti, si, uij) =
(sk, ukl) ·
i∈N\K
(ti − si),
where N ≡ {4, . . . , n}, g
are polynomials in the variables sk, ukl (k, l ∈ K)
only, and g
= 1. Then the harmonicity condition Dfn = 0 is equivalent to the
recursive system
(n− 2− |K|+∆)∆ g
K\{k}
k,l∈K,k<l
(ukl − sk − sl) g
K\{k,l}
where |K| is the number of elements of the set K and the differential operator
∆ = s∂s + u∂u measures the total polynomial degree r in sk and ukl. Since one
can divide by (n − 2− |K|+ r)r if r > 0, there is a unique polynomial solution
such that g
(sk = 0, ukl = 0) = 0 (K 6= ∅), and g
is of order 6 |K|. So fn
is of order n − 3. (Explicitly, the first three functions are f3 = 1, f4 = t4 − s4
and f5 = (t4 − s4)(t5 − s5) +
(u45 − s4 − s5).) An inspection of the recursion
also shows that all possible factors ρ2i in the denominators of the arguments of
fn cancel with the factors in the prefactor in (3.27), thus hn can have poles only
in ρ13 and ρ23 of the specified maximal degree. This proves the Lemma. �
The upshot of the previous discussion is a necessary and sufficient condition
for the Huygens bilocality of V1 which directly refers to the local correlation
functions of the theory:
Harmonic Bilocal Fields 16
Theorem 3.7. The field V1(x1, x2) weakly converges on bounded energy states
to a Huygens bilocal field which is conformal of weight (1, 1), if and only if the
leading parts F0 of the Laurent polynomials F (3.17) satisfy the “single pole
property” (Def. 3.1) with respect to both x1 and x2. In this case, the formal
series H converge to Laurent polynomials in (xi − xj)
2 subject to the same pole
bounds, specified in Theorem 2.1, as F .
Proof. We know already that if V1 is a Huygens bilocal field, then its correlation
functions H are Laurent polynomials of the form (2.3), and that this implies the
SPP for F0 with respect to x1 and x2. Conversely, if the SPP holds for F0 with
respect to x1 and x2, then H are Laurent polynomials by Lemma 3.6, and hence
V1 is relatively Huygens bilocal with respect to the fields φi. Since the general
argument [4] that relative locality implies local commutativity of a field with
itself refers only to local fields, we want to give an explicit argument for the case
at hand.
All the previous remains true when in (3.10) or (3.17) a product of fields
φk(xk)k+1φ(xk+1) is replaced by U(xk, xk+1). By assumption, and because U
is bilocal, the contributions of order (ρk,k+1)
0 to the correlation functions of
U(xk, xk+1) fulfil the SPP with respect to xk and xk+1. By Lemma 3.6, this
property is preserved upon the passage to the harmonic parts with respect to
x1 and x2. One may therefore continue in the same way with xk, xk+1, and
eventually find that all mixed correlation functions of φ’s and V1’s converge to
rational functions. By this convergence we conclude that all products of φ’s and
V1’s converge on the vacuum, and this then defines V1 as a Huygens bilocal field,
since its matrix elements will satisfy Huygens locality.
The conformal properties of V1 follow from the preservation of the homo-
geneity and the pole degrees in the harmonic decomposition, as guaranteed by
Lemma 3.6. �
For n = 4 points, the SPP is trivially satisfied because of homogeneity. Hence
the 4-point function
∣∣V ∗1 V1
is always rational. It follows that its expansion
in (transcendental) partial waves [11] cannot terminate. This means that (unless
V1 = 0 in which case there is not even a stress-energy tensor) a GCI QFT
necessarily contains infinitely many conserved tensor fields of arbitrarily high
spin.
3.4 The case of dimension 2
Let us consider now the case of scalar fields φk of dimension 2. We claim that in
this case, Corollary 3.5 in combination with the cluster condition is sufficient to
establish the SPP, Definition 3.1. Hence we conclude by Theorem 3.7 that the
twist two harmonic fields V1(x1, x2) are indeed Huygens bilocal fields.
To prove our claim, we use that by (2.6), µij > −1, hence the SPP is equiv-
alent to the statement that there can be no term contributing to
∣∣φ1(x1) · · ·
φn(xn)
, for which there is i with more than two µij negative (j 6= i). Thus
assume that there is a term with, say, µ12 = µ13 = µ14 = −1. It constitutes a
Harmonic Bilocal Fields 17
double pole for each of the three harmonic fields V1(x1, xj) (j = 2, 3, 4). Then by
homogeneity (2.5), there must be more poles in xj (j = 2, 3, 4), but these cannot
be of the form ρjk with k > 4 by Corollary 3.5. Hence (up to permutations of
2, 3, 4) µ23 = µ24 = −1, µ34 = 0. Again by homogeneity (2.5), the dependence
on x1, . . . , x4 must be given by a linear combination of terms
ρ1kρ4ℓ
ρ12ρ13ρ14ρ23ρ24
(3.30)
with k, ℓ > 4. Applying the cluster limit (Sect. 2.1) to the points x1, x2, x3, x4
in (3.30), the limit diverges ∼ t4. This behavior is tamed to ∼ t2 by anti–
symmetrization in k, ℓ, but it cannot be cancelled by any other terms. Hence
the assumption leads to a contradiction.
This proves the SPP if the generating scalar fields have dimension d = 2.
3.5 A d = 4 6-point function violating the SPP
We proceed with an example of 6-point function violating the SPP in the case of
two d = 4 GCI scalar fields Li(x) such that the bilocal field U(x1, x2) obtained
from L1(x1)L2(x2) has a non-zero skew–symmetric part. Let L be any linear
combination of L1 and L2.
The following admissible contribution to the truncated part of the 6-point
function 〈0|U(x1, x2)L(x3)L(x4)U(x5, x6)|0〉 clearly violates the SPP:
F0(x1, x2) = A12A56
ρ15ρ26ρ34 − 2ρ15ρ23ρ46 − 2ρ15ρ24ρ36
ρ13ρ14ρ23ρ24 · ρ34 · ρ35ρ45ρ36ρ46
, (3.31)
where Aij stands for the antisymmetrization in the arguments xi, xj . It is ad-
missible as a truncated 6-point structure because (ρ12ρ56)
−3F0 obeys all the pole
bounds of Sect. 2 for a correlation 〈0|L1(x1)L2(x2)L(x3)L(x4)L1(x5) L2(x6)|0〉
of six fields of dimension d = 4.
On the other hand, F0 satisfies the differential equation
(E1D2 − E2D1)F0(x1, x2) = 0 (3.32)
(and similar in the variables x5 and x6), ensuring that F0 is the leading part of a
bi–harmonic function, analytic in a neighborhood of x1 = x2 and x5 = x6, repre-
senting a contribution to the twist two 6-point function 〈0|V1(x1, x2)L(x3)L(x4)
V1(x5, x6)|0〉, of which F0 is the leading part. This function cannot be a Laurent
polynomial in the ρij by our general argument that the leading part of a bi–
harmonic Laurent polynomial cannot satisfy the SPP. Hence the twist two field
V1(x1, x2) cannot be Huygens bilocal.
The resulting contribution to the conserved local current 4-point function
〈0|Jµ(x1)L(x3)L(x4)Jν(x5)|0〉
tr is obtained through Jµ(x) = i(∂xµ−∂yµ) V1(x, y)|x=y.
It also satisfies the pertinent pole bounds. This structure is rational as it should,
because only the leading part F0 contributes. In fact, while the 6-point structure
involving the harmonic field cannot be reproduced by free fields due to its double
Harmonic Bilocal Fields 18
pole, the resulting 4-point structure does arise as one of the three independent
connected structures contributing to 4-point functions involving two Dirac cur-
rents : ψ̄aγ
µψb : and two Yukawa scalars ϕ : ψ̄cψd : (allowing for internal flavours
a, b, . . . ).
4 The theory of GCI scalar fields of scaling dimension
d = 2
The scaling dimension d = 2 is the minimal dimension of a GCI scalar field for
which one could expect the existence of nonfree models. It turns out however,
that in this case the fields can be constructed as composite fields of free, or
generalized free, fields. Namely, we will establish the following result.
Theorem 4.1. Let {Φm(x)}
m=1 be a system of real GCI scalar fields of scaling
dimension d = 2. Then it can be realized by a system of generalized free fields
{ψm(x)} and a system of independent real massless free fields {ϕm(x)}, acting
on a possibly larger Hilbert space, as follows:
Φm(x) =
αm,j ψj(x) +
j,k=1
βm,j,k :ϕj(x)ϕk(x) : , (4.1)
where αm,j and βm,j,k = βm,k,j are real constants such that
α2m,j < ∞ and
j,k=1
m,j,k
< ∞. Here, we assume the normalizations 〈0|ϕj(x1) ϕk(x2) |0〉 =
δjk (ρ12)
−1, 〈0|ψj(x1) ψk(x2) |0〉 = δjk (ρ12)
The proof of Theorem 4.1 is given at the end of Sect. 4.2. The main reason
for this result is the fact that in the d = 2 case the harmonic bilocal fields exist
and furthermore, they are Lie fields. This was originally recognized in [12], [2]
under the assumption that there is a unique field φ of dimension 2. We are
extending here the result to an arbitrary system of d = 2 GCI scalar fields.
If we assume the existence of a stress-energy tensor as a Wightman field5,
the generalized free fields must be absent in (4.1), and the number of free fields
must be finite. In this case, the iterated OPE generates in particular the bilocal
field 1
i :ϕi(x)ϕi(y):. As this field has no other positive-energy representation
than those occurring in the Fock space [2], nontrivial possibilities for correlations
between non-free fields and the fields (4.1) are strongly limited.
4.1 Structure of the correlation functions
We consider a GCI QFT generated by a set of hermitean (real) scalar fields.
We denote by F the real vector space of all GCI real scalar fields of scaling
5A stress-energy tensor always exists as a quadratic form between states generated by the
fields Φm from the vacuum [8].
Harmonic Bilocal Fields 19
dimension 2 in the theory. (Note that the space F may be larger than the linear
span of the original system of d = 2 fields of Theorem 4.1.) We shall find in this
section the explicit form of the correlation functions of the fields from F .
Theorem 4.2. Let φ1(x), . . . , φn(x) ∈ F then their truncated n-point functions
have the form
〈0|φ1(x1) · · · φn(xn)|0〉
c(n)(φσ1 , . . . , φσn)
ρσ1σ2 · · · ρσnσ1
(4.2)
where c(n) are multilinear functionals c(n) : F⊗n → R with the inversion and
cyclic symmetries c(n)(φ1, . . . , φn) = c
(n)(φn, . . . , φ1) = c
(n)(φn, φ1, . . . , φn−1).
Before we prove the theorem, let us first illustrate it on the example of the
free field realization (4.1). In this case one finds
Φm1 ,Φm2
αm1,jαm2,j +
j,k=1
βm1,j,k βm2,j,k
αm1,jαm2,j + Tr βm1βm2 ,
Φm1 , . . . ,Φmn
= Tr βm1 · · · βmn for n > 2 (4.3)
where βm =
βm,j,k
Proof of Theorem 4.2. We first recall the general form (2.3) of the truncated
correlation function with pole bounds (2.6) that read in this case: µtrjk > −1.
The argument in Sect. 3.4 shows that the nonzero contributing terms in Eq. (2.3)
have for every j = 1, . . . , n exactly two negative µtrjk or µ
kj for some k = k1, k2
different from j.
The nonzero terms are therefore products of “disjoint cyclic products of prop-
agators” of the form 1/ρk1k2ρk2k3 · · · ρkr−1krρkrk1 . But cycles of length r < n are
in conflict with the cluster condition (Sect. 2). We conclude that 〈0|φ1(x1) · · ·
φn(xn)|0〉
tr is a linear combination of terms like those in (4.2) with some co-
efficients cσ(φ1, . . . , φn) depending on the permutations σ ∈ Sn and on the
fields φj (multilinearly). Locality, i.e. 〈0|φ1(x1) · · · φn(xn)|0〉
tr = 〈0|φσ′1(xσ
) · · ·
φσ′n(xσ′n)|0〉
tr, then implies cσ′σ(φ1, . . . , φn) = cσ(φσ′1
, . . . , φσ′n) (σ, σ
′ ∈ Sn), so
that cσ(φ1, . . . , φn) = c
(n)(φσ1 , . . . , φσn) for some c
(n) : F⊗n → R. The equali-
ties c(n)(φ1, . . . , φn) = c
(n)(φn, . . . , φ1) = c
(n)(φn, φ1, . . . , φn−1) are again due to
locality. �
As we already know by the general results of the previous section, the har-
monic bilocal field exist in the case of fields of dimension d = 2. Moreover, the
knowledge of the correlation functions of the d = 2 fields allows us to find the
form of the correlation functions of the resulting bilocal fields. This yields an
algebraic structure in the space of real (local and bilocal) scalar fields, which we
proceed to display.
Harmonic Bilocal Fields 20
Let us introduce together with the space F of d = 2 fields also the real vector
space V of all real harmonic bilocal fields. We shall consider F and V as built
starting from our original system of d = 2 fields {Φm} of Theorem 4.1, by the
following constructions.
(a) If φ1(x), φ2(x) ∈ F then introducing the bilocal (1, 1)–field U(x1, x2) =
φ1(x1)φ2(x2)− 〈0|φ1(x1)φ2(x2)|0〉
in accord with Eq. (2.7), we consider its
harmonic decomposition U(x, y) = V1(x, y) + (x−y)
2 Ũ(x, y). We denote V1(x, y)
by φ1 ∗ φ2; this defines a bilinear map F ⊗ F
(b) If now v(x, y) ∈ V then vt(x, y) := v(y, x) also belongs to V and γ(v)
is a field from F .
(c) If v(x, y), v′(x, y) ∈ V then there is a harmonic bilocal field
(v∗v′)
:= w-lim
x′ → y′
x′−y′
x, x′
y′, y
−〈0|v
x, x′
y′, y
. (4.4)
The existence of the above weak limit (i.e., a limit within correlation functions)
will be established below together with the independence of x′ = y′ and the
regularity of the resulting field for (x− y)2 = 0.
(d) If v(x, y) ∈ V and φ(x) ∈ F then we can construct the following bilocal
field belonging to V:
(v ∗ φ)
:= w-lim
x′ → y
x′ − y
x, x′
− 〈0|v
x, x′
, (4.5)
where again the existence of the limit and the regularity for (x− y)2 = 0 will be
established later.
One can define similarly a product φ ∗ v ∈ V, but it would then be expressed
as: (vt ∗ φ)t.
To summarize, we have three bilinear maps: F ⊗ F
→V, V ⊗ V
→V, V ⊗
→V, and two linear ones: V
→V, V
→F . Applying these maps we construct
F and V inductively, starting from our original system of d = 2 fields, given
in Theorem 4.1, and at each step of this inductive procedure, we establish the
existence of the above limits in (c) and (d). In fact, we shall establish this
together with the structure of the truncated correlation functions for the fields
in F and V.6
Before we state the inductive result it is convenient to introduce the vector
space
Â = F × V (4.6)
6Since we shall use the notion of truncated correlation functions also for bilocal fields let
us briefly recall it. If B1, . . . , Bn are some smeared (multi)local fields then their truncated
correlation functions are recursively defined by: 〈0|B1 · · ·Bn|0〉 =
∪̇P = {1,...,n}
{j1,...,jk} ∈P
〈0|Bj1 · · ·Bjk |0〉
tr (the sum being over all partitions P of {1, . . . , n})
Harmonic Bilocal Fields 21
and endow it with the following bilinear operation
(φ1, v1) ∗ (φ2, v2) :=
0, φ1 ∗ φ2 + v1 ∗ v2 + v1 ∗ φ2 + (v
2 ∗ φ1)
, (4.7)
and with the transposition
(φ, v)t := (φ, vt) . (4.8)
The spaces F and V will be considered as subspaces in Â. Thus, the new
operation ∗ in Â combines the above listed three operations. We shall see later
that Â is actually an associative algebra under the product (4.7). We note
that the transposition t (4.8) is an antiinvolution with respect to the product:
(q1 ∗ q2)
t = qt2 ∗ q
1, for every q1, q2 ∈ Â.
Proposition 4.3. There exist multilinear functionals
c(N) : Â⊗N → R (4.9)
such that if we take elements q1, . . . , qn+m ∈ Â : qk := vk
xk[0], xk[1]
∈ V, where
[ε] stands for a Z/2Z–value and k = 1, . . . , n, and qk := φk−n
∈ F for
k = n+ 1, . . . , n +m, then the truncated correlation functions can be written in
the following form:
〈0|v1
x1[0], x1[1]
· · · vn
xn[0], xn[1]
· · ·φm
|0〉tr
2(n+m)
σ∈Sn+m
(ε1,...,εn)∈ (Z/2Z)
Kσ,ε Tσ,ε
x1[0], . . . , xn[1], xn+1, . . . , xn+m
. (4.10)
Here: Kσ,ε are coefficients given by Kσ,ε := c
(n+m)
[εσ1 ]
σ1 , . . . , q
[εσn+m ]
, where
we set εn+1 = · · · = εn+m = 0, and q
[0] := q, q[1] := qt (for q ∈ Â); the terms
Tσ,ε are the following cyclic products of intervals
Tσ,ε =
xσn+m − xσ1[ε1]
xσk[1+εk] − xσk+1[εk+1]
xσn[1+εn] − xσn+1
xσn+k − xσn+k+1
. (4.11)
It follows by Eq. (4.10) that the limits in the steps (c) and (d) above are well
defined.
Before the proof let us make some remarks. First, we used the same notation
c(n) as in Theorem 4.2 since the above multilinear functionals are obviously an
extension of the previous, i.e., Eq. (4.10) reduces to Eq. (4.2) for m = 0. Let us
also give an example for Eq. (4.10) with n = m = 1:
〈0|v(x1, x2)φ(x3)|0〉 =
c(2)(v, φ)
ρ23 ρ31
+ c(2)(vt, φ)
ρ13 ρ32
+ c(2)(φ, v)
ρ31 ρ23
+ c(2)(φ, vt)
ρ32 ρ13
. (4.12)
Harmonic Bilocal Fields 22
As one can see, c(n) (as well as c(n) of Theorem 4.2) possess a cyclic and an
inversion symmetry:
q1, . . . , qn
= c(n)
qn, q1 . . . , qn−1
= c(n)
qtn, . . . , q
. (4.13)
This is the reason for choosing the prefactors in Eqs. (4.2) and (4.10) (the inverse
of the orders of the symmetry groups).
Proof of Proposition 4.3. According to our preliminary remarks it is enough to
prove that Eq. (4.10) is consistent with the operations F ⊗ F
→V, V ⊗ V
V ⊗ F
→V and V
Starting with F ⊗ F
→V one should prove that any truncated correlation
function
·φ1(x1)φ2(x2)·
given by Eq. (4.10) yields a harmonic decomposition:
·φ1(x1)φ2(x2) ·
·(φ1∗φ2)
x1, x2
+ρ12R(x1, x2), with a correlation
function
· (φ1 ∗ φ2)
x1, x2
given by Eq. (4.10) and a rational function R
regular at ρ12 = 0. This gives us relations of the type
c(n+2)(q1, . . . , φ1, φ2, . . . , qn) = c
(n+1)(q1, . . . , φ1 ∗ φ2, . . . , qn) . (4.14)
Next, having correlation functions of type
· v1(x1, x2)v2(x3, x4) ·
v(x1, x2)φ(x3) ·
of the form (4.10), one verifies that the limits (4.4) and (4.5)
exist within these correlation functions, and they yield expressions for
· (v1 ∗
x1, x4
· (v ∗ φ)
x1, x3
consistent with (4.10). As a result we
obtain again relations between the c’s:
c(n+2)(q1, . . . , v1, v2, . . . , qn) = c
(n+1)(q1, . . . , v1 ∗ v2, . . . , qn) ,
c(n+2)(q1, . . . , v, φ, . . . , qn) = c
(n+1)(q1, . . . , v ∗ φ, . . . , qn) . (4.15)
Finally, one verifies that setting x1 = x2 in
· v(x1, x2) ·
we obtain the
correlation functions
· γ(v)
with the relation
c(n+1)(q1, . . . , (v + v
t), . . . , qn) = 2 c
(n+1)(q1, . . . , γ(v), . . . , qn) . (4.16)
This completes the proof of Proposition 4.3 as well as the proof that the products
V ⊗ V
→V and V ⊗ F
→V are well defined. �
4.2 Associative algebra structure of the OPE
Note that Eqs. (4.14), (4.15) read (under (4.7))
q1, . . . , qk, qk+1, . . . , qn
= c(n−1)
q1, . . . , qk ∗ qk+1, . . . , qn
. (4.17)
This implies that the bilinear operation ∗ on Â is an associative product.
Indeed, consider the element q :=
q1 ∗q2
∗q3−q1∗
q2 ∗q3
for q1, q2, q3 ∈ Â.
By (4.7) q is a bilocal field. Equation (4.17) implies that all c’s in which q enters
vanish and hence, by Eq. (4.10) q has zero correlation functions with all other
Harmonic Bilocal Fields 23
fields, including itself. But then this (bilocal) field is zero by the Reeh–Schlieder
theorem, since its action on the vacuum will be identically zero.
Thus, introducing the cartesian product Â (4.6) was not only convenient for
combining three types of bilinear operations in one but also as a compact expres-
sion for the associativity (Eqs. (4.14), (4.15)). However, Â carries a redundant
information due to the following relation:
−γ(v),
(v + vt)
∗ q = 0 = q ∗
−γ(v),
(v + vt)
(4.18)
for every v ∈ V and q ∈ Â. To prove (4.18) we point out first that it is equivalent
to the identities v ∗ φ = γ(v) ∗ φ and v′ ∗ v = v′ ∗ γ(v) for v = vt ∈ V and any
φ ∈ F , v′ ∈ V. These identities can be established again first for the c’s, and
then proceeding by using the Reeh–Schlieder theorem, as in the above proof of
associativity.
Hence, the redundancy in Â is because we can identify symmetric bilocal
fields v = vt ∈ V with their restrictions to the diagonal, γ(v) ∈ F , and this
is compatible with the product ∗. Let us point out that the restriction of the
map γ to the t–invariant subspace Vs := {v ∈ V : v = v
t} is an injection into
F . The latter follows from a simple analysis of the 4-point functions of v and
the Reeh–Schlieder theorem: if v(x, y) = v(y, x) and 〈0|v(x, x)v(y, y)|0〉 = 0
then 〈0|v(x, x′)v(y, y′)|0〉 = 0. In this way we see that we can identify in Â the
symmetric harmonic bilocal fields v = vt with their restriction on the diagonal
γ(v) ∈ F .
Formally, the above considerations can be summarized in the following ab-
stract way. Let us introduce the quotient
A := Â
−γ(v),
(v + vt)
: v ∈ V
. (4.19)
It is an associative algebra according to Eq. (4.18). The involution t : Â → Â
can be transferred to an involution on the quotient (4.19) and we denote it by
t as well. The spaces F and V are mapped into A by the natural compositions
F → Â → A and V → Â → A. The injectivity of γ on Vs implies that the maps
F → A and V → A so defined are actually injections. Hence, we shall treat F
and V also as subspaces of A. Furthermore, A becomes a direct sum of vector
spaces
A = F ⊕ Va , (4.20)
q ∈ A : qt = q
= F ⊇ Vs
v ∈ V : vt = v
q ∈ A : qt = −q
= Va :=
v ∈ V : vt = −v
Hence, the t–symmetric elements of A are identified with the d = 2 local fields,
while the t–antisymmetric elements of A, with the antisymmetric, harmonic
bilocal (1, 1) fields. (Neither F nor Va are subalgebras of A.)
To summarize, the associative algebra A is obtained from Â by identifying
the space Vs of symmetric bilocal fields with its image γ
⊆ F .
Harmonic Bilocal Fields 24
For simplicity we will denote the equivalence class in A of an element q ∈ Â
again by q. Also note that the c’s can be transferred as well, to multilinear
functionals on A, since the kernel of the quotient (4.19) is contained in the
kernel of each c(n) by (4.16). We shall use the same notation c(n) also for the
multilinear functional c(n) on A.
Example 4.1. Let us illustrate the above algebraic structures on the simplest
example of a QFT generated by a pair of d = 2 GCI fields Φ1 and Φ2 given
by normal a pair of two mutually commuting free massless fields ϕj : Φ1(x) =
: ϕ21(x) : − : ϕ
2(x) :
and Φ2(x) = ϕ1(x)ϕ2(x). Their OPE algebra involves
a set of four independent harmonic bilocal fields Vjk(x1, x2) := :ϕj(x1)ϕk(x2) :
(j, k = 1, 2), which satisfy
Vjk(x1, x2)
= Vkj(x1, x2) = Vjk(x2, x1). For in-
stance, we have Φ1 ∗ Φ2 = V12 − V21.
7 Also note that Φ1 = γ(V1) for V1(x1, x2)
= :ϕ1(x1)ϕ1(x2) : − :ϕ2(x1)ϕ2(x2) :, etc.
By the associativity and Eq. (4.17) we have
q1, . . . , qn
= c(2)
q1 ∗ · · · ∗ qn−1, qn
(4.21)
for q1, . . . , qn ∈ A. Let us consider now c
(2) and define the following symmetric
bilinear form on A: 〈
q1, q2
:= c(2)
qt1, q2
. (4.22)
First note that F and Va are orthogonal with respect to this bilinear form: this is
due to the fact that there is no nonzero three point conformally invariant scalar
function of weights (2, 1, 1), which is antisymmetric in the second and third
arguments. Next, we claim that (4.22) is strictly positive definite. This is a
straightforward consequence of the Wightman positivity and the Reeh–Schlieder
theorem (one should consider separately the positivity on F and Va). In partic-
ular, (4.22) is nondegenerate. By Eqs. (4.13) and (4.17) we have:
q1 ∗ q2, q3
q2, q
1 ∗ q3
(4.23)
for all q1, q2, q3 ∈ A.
Let us introduce now an additional splitting of F . Denote by F0 the kernel
of the product, i.e.,
F0 :=
ψ ∈ F : ψ ∗ q = 0 ∀q ∈ A
ψ ∈ F : q ∗ ψ = 0 ∀q ∈ A
(4.24)
(the second equality is due to the identity φ ∗ q = (qt ∗ φ)t). Let F1 be the
orthogonal complement in F of F0 with respect to the scalar product (4.22):
F1 :=
φ ∈ F :
= 0 ∀ψ ∈ F0
. (4.25)
The meaning of fields belonging to F0 becomes immediately clear if we note
that c(n) for n > 3 are zero if one of the arguments belongs to F0 (this is due to
7i.e., in the OPE Φ1(x1)Φ2(x2) there appears the antisymmetric bilocal field V12(x1, x2) −
V21(x1, x2) that involves only odd rank conserved tensor currents in its expansion in local fields
Harmonic Bilocal Fields 25
Eq. (4.21)). Hence, all their truncated functions higher than two point are zero,
i.e., the fields belonging to F0 are generalized free d = 2 fields. Furthermore,
these fields commute with all other fields from F1 and Va ≡ A
(1): this is because
of the vanishing of c(2)(ψ, q) if ψ ∈ F0 and q ∈ F1 ⊕ Va, as well as of all
c(n+1)(ψ, q1, . . . , qn) for n > 2 if ψ ∈ F0 and q1, . . . , qn ∈ A (by (4.21) and
(4.24)).
Clearly, F1⊕Va is a subalgebra of A: this follows from Eq. (4.23) with q3 ∈ F0
along with the definitions (4.24) and (4.25). Let us denote it by
B := F1 ⊕ Va . (4.26)
We are now ready to state the main step towards the proof of Theorem 4.1.
Proposition 4.4. There is a homomorphism ι from the associative algebra B
into the algebra of Hilbert–Schmidt operators over some real separable Hilbert
space, such that
q1, . . . , qn
· · · ι
, (4.27)
and ι
are symmetric operators while ι
are antisymmetric.
We shall give the proof of this proposition in the subsequent subsection. The
main reason leading to it is that B becomes a real Hilbert algebra with an integral
trace on it. Here we proceed to show how Theorem 4.1 can be proven by using
the above results.
Proof of Theorem 4.1. Let Φm = Φ
m + Φ
m be the decomposition of each field
Φm according to the splitting F = F0 ⊕ F1. Take an orthonormal basis ψm in
F0 and let Φ
αm,j ψj , and βm =
βm,j,k
be the symmetric matrix
corresponding to the Hilbert–Schmidt operator ι
(m = 1, 2, . . . ). Then
Eqs. (4.3) and (4.27) show that the constants αm,j and βm,j,k so defined satisfy
the conditions of Theorem 4.1. �
Remark 4.1. In general, we have F1 % Vs. This is because the elements of
F1 correspond, by Proposition 4.4, to Hilbert–Schmidt symmetric operators and
on the other hand, the elements of V are obtained, according to the inductive
construction of Sect. 4.1, as products of elements of F and will, hence, correspond
to trace class operators.
4.3 Completion of the proofs
It remains to prove Proposition 4.4. We start with an inequality of Cauchy–
Schwartz type.
Lemma 4.5. Let q1, q2 ∈ A be such that each of them belongs either to F or to
Va. Then we have
q1 ∗ q2, q1 ∗ q2
q1 ∗ q1, q1 ∗ q1
q2 ∗ q2, q2 ∗ q2
. (4.28)
Harmonic Bilocal Fields 26
Proof. Consider
q1∗q1+λ q2∗q2, q1∗q1+λ q2∗q2
> 0 and use that
q1∗q1, q2∗q2
q1 ∗ q2, q1 ∗ q2
if each of q1, q2 belongs either to F or to Va. �
The space B (4.26) is a real pre–Hilbert space with a scalar product given by
(4.22). It is also invariant under the action of t (actually the eigenspaces of t are
F1 and Va). The left action of B on itself gives us an algebra homomorphism
ι : B → LinR B (4.29)
of B into the algebra of all operators over B. Moreover, the elements of F are
mapped into symmetric operators and the elements of Va, into antisymmetric
(this is due to (4.23)).
Lemma 4.6. Every element of B is mapped into a Hilbert–Schmidt operator.
Proof. Since B is generated by F1 (according to the inductive construction of F
and V in Sect. 4.1) it is enough to show this for the elements of F1.
Let φ ∈ F1 and consider the commutative subalgebra Bφ of B generated by
φ. The algebra Bφ is freely generated by φ, i.e., is isomorphic to the algebra
λR[λ] of polynomials in a single variable λ (↔ φ), since φ belongs to the orthog-
onal complement of F0 (4.24). For a p(λ) ∈ λR[λ] we shall denote by φ[p] the
corresponding element of Bφ. In particular,
φ[p1] ∗ φ[p2] = φ[p1p2]. (4.30)
Setting
φ∗(n+1) := φ∗n ∗ φ, c
:= c(2)
φ∗n, φ
φ∗n, φ
(4.31)
(φ∗1 := φ, n > 1) we obtain a positive definite functional over the algebra λ2R[λ]
∼= φ ∗ Bφ (due to Eq. (4.23) and the positivity of
(4.22)).
Then, by the Hamburger theorem about the classical moment problem ([9,
Chap. 12, Sect. 8]) we conclude that there exists a bounded positive Borel mea-
sure dµ
on R, such that
λ2 p (λ)
dµ(λ) (4.32)
for every p(λ) ∈ R[λ]. Using this we can extend the fields φ[p](x) to φ[f ](x)
for Borel measurable functions f having compact support with respect to µ
in R\{0}. The latter can be done in the following way. Fix ε ∈ (0, 1) and
let g1, . . . , gn be Schwartz test functions on M . By Theorem 4.2 the correla-
tors 〈0|φ[p1][g1] · · ·φ
[pn][gn]|0〉 depend polynomially on c
φ[pk1 ], . . . , φ
[pkj ]
pk1 · · · pkj
for all {k1, . . . , kj} ⊆ {1, . . . , n}. But for every ε ∈ (0, 1) there
exists a norm
‖q‖ε = Aε sup
|λ|6ε
qk(λ)
∣∣∣ + Bε
R \ (−ε, ε)
∣∣qk(λ)
∣∣ dµ(λ) (4.33)
Harmonic Bilocal Fields 27
on λ2R[λ] ∋ q(λ), where Aε and Bε are some positive constants, such that for
every q1, . . . , qm ∈ λ
2R[λ]
∣∣∣ c
q1(λ) · · · qm(λ)
]∣∣∣ 6
∣∣qk(λ)
dµ(λ)
‖qk‖ε .
Hence,
∣∣〈0|φ[p1][g1] · · ·φ[pn][gn]|0〉
∣∣ 6 C
‖pk‖ε ‖gk‖S for some constant C
and Schwartz norm ‖·‖S (not depending on pk and gk). Since for every ε ∈
(0, 1) the Banach space L1
R\{(−ε, ε)}, µ
is contained in the completion of
λ2R[λ] with respect to the norms (4.33), we can extend the linear functional
c[p(λ)] as well as the correlators 〈0|φ[p1][g1] · · · φ
[pn][gn]|0〉 to a functional c[f(λ)]
and correlators 〈0|φ[f1][g1] · · ·φ
[fn][gn]|0〉 defined for Borel functions f, f1, . . . , fn
compactly supported with respect to µ in R\{0}. Thus, we can extend the fields
φ[p] by extending their correlators.
By the continuity we also have for arbitrary Borel functions f, fk, compactly
supported in R\{0}:
φ[f1] ∗ φ[f2] = φ[f1f2], c(n)
φ[f1], . . . , φ[fn]
f1 · · · fn
f ] =
dµ(λ) (4.34)
(cp. (4.32)), and c(n) determine the correlation functions of φ[fk] as in Theo-
rem 4.2.
In particular, for every characteristic function χS of a compact subset S ⊂
R\{0} we have φ[χS ]∗φ[χS ] = φ[χS ]. Hence, for such a d = 2 field we will have that
all its truncated correlation functions are given by (4.2) with all normalization
constants c(n) equal to one and the same value c(2)
φ[χs], φ[χs]
. Then, as shown in
[12, Theorem 5.1], Wightman positivity requires this value to be a non-negative
integer, i.e.,
φ[χS ], φ[χS ]
dµ(λ)
∈ {0, 1, 2, . . . } (4.35)
(it is zero iff φ[χS ] = 0). Hence, the restriction of the measure dµ(λ)/λ2 to R\{0}
is a (possibly infinite) sum of atom measures of integral masses, each supported
at some γk ∈ R\{0} for k = 1, . . . , N (and N could be infinity). In particular,
the measure µ is supported in a bounded subset of R.
By Lemma 4.5 we can define ι(φ[f ]) as a closable operator on B if f is a Borel
measurable function with compact support in R\{0}. It follows then that the
projectors ι(φ[χS ]), for a compact S ⊆ R\{0}, provide a spectral decomposition
for ι(φ) (in fact, ι(φ[f ]) = f
). Thus, ι(φ) has discrete spectrum with eigen-
values γk (k ∈ N), each of a multiplicity given by the integer c
χ{γk} , φ
χ{γk}
Then ι(φ) is a Hilbert–Schmidt operator since
γ2k c
χ{γk} , φ
χ{γk}
dµ(λ)
R\{0}
dµ(λ) <∞
Harmonic Bilocal Fields 28
(µ being a bounded measure). �
The completion of the proof of Proposition 4.4 is provided now by the fol-
lowing corollary.
Corollary 4.7. For every q1, q2 ∈ B one has c
q1, q2
ι(q1)ι(q2)
Proof. If q1 = q2 ∈ F1 this follows from the proof of Lemma 4.6 and hence, by
a polarization, for any q1, q2 ∈ F1. The general case can be obtained by using
the facts that B is generated by F1 and c
(2) has the symmetry c(2)(q1 ∗ q2, q3) =
c(2)(q1, q2 ∗ q3). �
5 Discussion. Open problems
The main result of Sect. 4, the (generalized) free field representation of a system
{φa} of GCI scalar fields of conformal dimension d = 2 (Theorem 4.1), is obtained
by revealing and exploiting a rich algebraic structure in the space F × V of all
d = 2 real scalar fields and of all harmonic bilocal fields of dimension (1, 1).
However, this structure is mainly due to the fact that we are in the case of
lower scaling dimension: there is only one possible singular structure in the
OPE (after truncating the vacuum part). One can try to establish such a result
in spaces of spin–tensor bilocal fields (of dimension
) satisfying
linear (first order) conformally invariant differential equations (that again imply
harmonicity). If these equations together with the corresponding pole bounds
imply such singularities in the OPE, which can be “split” one would be able to
prove the validity of free field realizations in such more general theories, too.
One may also attempt to study models, say in a theory of a system of scalar
fields of dimension d = 4, without leaving the realm of scalar bilocal harmonic
fields V1 (of dimension (1, 1)). In [11] there have been found examples of 6–point
functions of harmonic bilocal fields, which do not have free field realizations.
However, our experience with the d = 2 case shows that in order to complete the
model (including the check of Wightman positivity for all correlation functions)
it is crucial to describe the OPE in terms of some simple algebraic structure
(e.g., associative, or Lie algebras).
On the other hand going beyond bilocal V1’s is a true signal of nontriviality
of a GCI model. Our analysis of Sect. 3 shows that this can be characterized
by a simple property of the correlation functions: the violation of the single
pole property (of Sect. 3.3). From this point of view a further exploration of
the example of Sect. 3.5 within a QFT involving currents appears particularly
attractive.
Note added in proof. In [19], we have determined the biharmonic function
whose leading part is given by Eq. (3.31). It involves dilogarothmic functions,
whose arguments are algebraic functions of conformal cross ratios. This exem-
plifies the violation of Huygens bilocality for the biharmonic fields, Theorem 3.7.
Harmonic Bilocal Fields 29
Yet, in support of Conjecture 3.3, it is shown that the structure of the cuts is in
a nontrivial manner consistent with ordinary bilocality.
Acknowledgements.
We thank Yassen Stanev for an enlightening discussion.
This work was started while N.N. and I.T. were visiting the Institut für
Theoretische Physik der Universität Göttingen as an Alexander von Humboldt
research fellow and an AvH awardee, respectively. It was continued during the
stay of N.N. at the Albert Einstein Institute for Gravitational Physics in Potsdam
and of I.T. at the Theory Group of the Physics Department of CERN. The paper
was completed during the visit of N.N. and I.T. to the High Energy Section
of the I.C.T.P. in Trieste, and of K.-H.R. at the Erwin Schrödinger Institute
in Vienna. We thank all these institutions for their hospitality and support.
N.N. and I.T. were partially supported by the Research Training Network of
the European Commission under contract MRTN-CT-2004-00514 and by the
Bulgarian National Council for Scientific Research under contract PH-1406.
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http://arxiv.org/abs/0711.0628
	Introduction
	Properties of GCI scalar fields
	Structure of correlation functions and pole bounds
	Twist expansion of the OPE and bi–harmonicity of twist two contribution
	Bilocality of twist two contribution to the OPE
	Convergence of harmonic decompositions
	Consequences of bi–harmonicity
	A necessary and sufficient condition for Huygens bilocality
	The case of dimension 2
	A d=4 6-point function violating the SPP
	The theory of GCI scalar fields of scaling dimension d=2
	Structure of the correlation functions
	Associative algebra structure of the OPE
	Completion of the proofs
	Discussion. Open problems

  The twist two contribution in the operator product expansion of phi_1(x_1)
phi_2(x_2) for a pair of globally conformal invariant, scalar fields of equal
scaling dimension d in four space-time dimensions is a field V_1(x_1,x_2) which
is harmonic in both variables. It is demonstrated that the Huygens bilocality
of V_1 can be equivalently characterized by a "single-pole property" concerning
the pole structure of the (rational) correlation functions involving the
product phi_1(x_1) phi_2(x_2). This property is established for the dimension
d=2 of phi_1, phi_2. As an application we prove that any system of GCI scalar
fields of conformal dimension 2 (in four space-time dimensions) can be
presented as a (possibly infinite) superposition of products of free massless
fields.

Introduction
Global Conformal Invariance (GCI) of Minkowski space Wightman fields yields
rationality of correlation functions [14]. This result opens the way for a non-
perturbative construction and analysis of GCI models for higher dimensional
Quantum Field Theory (QFT), by exploring further implications of the Wight-
man axioms.
By choosing the axiomatic approach, we avoid any bias about the possible
origin of the model, because we aim at a broadest possible perspective. On
the other hand, the assumption of GCI limits the analysis to a class of theories
http://arxiv.org/abs/0704.1960v4
Harmonic Bilocal Fields 2
that can be parameterized by its (generating) field content and finitely many
coefficients for each correlation function (see Sect. 2). As anomalous dimensions
under the assumption of GCI are forced to be integral, there is no perturbative
approach within this setting, but it is conceivable that a theory with a contin-
uous coupling parameter may exhibit GCI at discrete values (that appear as
renormalization group fixed points). An example of this type is provided by the
Thirring model: it is locally conformal invariant for any value of the coupling
constant g and becomes GCI for positive integer g2 [5].
Previous axiomatic treatments of conformal QFT were focussed on the rep-
resentation theory and harmonic analysis of the conformal group [6, 10] as tools
for the Operator Product Expansion (OPE). The general projective realization
of conformal symmetry in QFT was already emphasized in [15, 16] and found to
constitute a (partial) organization of the OPE. GCI is complementary in that it
assumes true representations (trivial covering projection). A necessary condition
for this highly symmetric situation is the presence of infinitely many conserved
tensor currents (as we shall see in Sect. 3.3).
The first cases studied under the assumption of GCI were theories generated
by a scalar field φ(x) of (low) integral dimension d > 1. (The case d = 1
corresponds to a free massless field with a vanishing truncated 4-point function
wtr4 .) The cases 2 6 d 6 4, which give rise to non-zero w
4 were considered in
[12, 13, 11].1
The main purpose in these papers was to study the constraints for the 4-
point correlation (=Wightman) functions coming from theWightman (= Hilbert
space) positivity. This was achieved by using the conformal partial wave expan-
sion. An important technical tool in this expansion is the splitting of the OPE
into different twist contributions (see (2.10)). Each partial wave gives a non-
rational contribution to the complete rational 4-point function. It is therefore
remarkable that the sum of the leading, twist two, conformal partial waves (cor-
responding to the contributions of all conserved symmetric traceless tensors in
the OPE of basic fields) can be proven in certain cases to be a rational function.
This means that the twist two part in the OPE of two fields φ is convergent in
such cases to a bilocal field, V1(x1, x2), which is our first main result in the present
paper. Throughout, “bilocal” means Huygens (= space–like and time–like) lo-
cality with respect to both arguments. Proving bilocality exploits the bounds on
the poles due to Wightman positivity, and the conservation laws for twist two
tensors which imply that the bilocal fields are harmonic in both arguments.
Trivial examples of harmonic bilocal fields are given by bilinear free field
constructions of the form : ϕ(x1)ϕ(x2) :, : ψ̄(x1)γµ(x1 − x2)
µψ(x2) :, or (x1 −
µ(x1 − x2)
ν : Fµσ(x1)F
ν (x2) :. A major purpose of this paper is to explore
whether harmonic twist two fields can exist which are not of this form, and
whether they can be bilocal. Moreover, we show that the presence of a bilocal
1The last two references are chiefly concerned with the case d = 4 (in D = 4 space-time
dimensions) which appears to be of particular interest as corresponding to a (gauge invariant)
Lagrangian density. The intermediate case d = 3 is briefly surveyed in [18].
Harmonic Bilocal Fields 3
field V1 completely determines the structure of the theory in the case of a scaling
dimension d = 2. The first step towards the classification of d = 2 GCI fields
was made in [12] where the case of a unique scalar field was considered. Here we
extend our study to the most general case of a theory generated by an arbitrary
(countable) set of d = 2 scalar fields. Our second main result states that such
fields are always combinations of Wick products of free fields (and generalized
free fields).
The paper is organized as follows.
Section 2 contains a review of relevant results concerning the theory of GCI
scalar fields.
In Sect. 3 we study conditions for the existence of the harmonic bilocal field
V1(x1, x2). We prove that Huygens bilocality of V1(x1, x2) is equivalent to the sin-
gle pole property (SPP), Definition 3.1, which is a condition on the pole structure
of the leading singularities of the truncated correlation functions of φ1(x1)φ2(x2)
whose twist expansion starts with V1(x1, x2). This nontrivial condition qualifies
a premature announcement in [2] that Huygens bilocality is automatic.
Indeed, the SPP is trivially satisfied for all correlations of free field construc-
tions of harmonic fields with other (products of) free fields, due to the bilinear
structure of V1. Thus any violation of the SPP is a clear signal for a nontriv-
ial field content of the model. Moreover, the SPP will be proven from general
principles for an arbitrary system of d = 2 scalar fields (the case studied in [2]).
Yet, although the pole structure of U(x1, x2) turns out to be highly constrained
in general by the conservation laws of twist two tensor currents, the SPP does
not follow for fields of higher dimensions, as illustrated by a counter-example of
a 6-point function of d = 4 scalar fields involving double poles (Sect. 3.5).
The existence of V1(x1, x2) as a Huygens bilocal field in a theory of dimension
d = 2 fields allows to determine the truncated correlation functions up to a single
parameter in each of them. This is exploited in Sect. 4, where an associative
algebra structure of the OPE of d = 2 scalar fields and harmonic bilocal fields
is revealed. The free-field representation of these fields is inferred by solving an
associated moment problem.
2 Properties of GCI scalar fields
2.1 Structure of correlation functions and pole bounds
We assume throughout the validity of the Wightman axioms for a QFT on the
D = 4 flat Minkowski space–time M (except for asymptotic completeness) – see
[17]. Our results can be, in fact, generalized in a straightforward way to any
even space–time dimension D. The condition of GCI in the Minkowski space is
an additional symmetry condition on the correlation functions of the theory [14].
In the case of a scalar field φ(x), it asserts that the correlation functions of φ(x)
Harmonic Bilocal Fields 4
are invariant under the substitution
φ(x) 7→ det
, (2.1)
where x 7→ g(x) is any conformal transformation of the Minkowski space, ∂g
its Jacobi matrix and d > 0 is the scaling dimension of φ. An important point
is that the invariance of Wightman functions
∣∣φ(x1) · · · φ(xn)
under the
transformation (2.1) should be valid for all xk ∈ M in the domain of definition
of g (in the sense of distributions). It follows that d must be an integer in order
to ensure the singlevaluedness of the prefactor in (2.1). Thus, GCI implies that
only integral anomalous dimensions can occur.
The most important consequences of GCI in the case of scalar fields φk(x) of
dimensions dk are summarized as follows.
(a) Huygens Locality ([14, Theorem 4.1]). Fields commute for non light–like
separations. This has an algebraic version:
(x1 − x2)
φ1(x1), φ2(x2)
= 0 (2.2)
for a sufficiently large integer N .
(b) Rationality of Correlation Functions (cf. [14, Theorem 3.1]). The general
form of Wightman functions is:
∣∣φ1(x1) · · ·φn(xn)
{µjk}
C{µjk}
(ρjk)
µjk , (2.3)
where here and in what follows we set
ρjk := (xjk − i 0 e0)
2 = (xjk)
2 + i 0 x0jk , xjk := xj − xk ; (2.4)
the sum in Eq. (2.3) is over all configurations of integral powers {µjk = µkj}
subject to the following conditions:
j (6=k)
µjk = −dk, (2.5)
and pole bounds µjk > −
dj + dk
δdjdk − 1
. Equation (2.5) follows from
the conformal invariance under (2.1); the pole bounds express the absence of
non-unitary representations in the OPE of two fields [14, Lemma 4.3]. Under
these conditions the sum in (2.3) is always finite and there are a finite number
of free parameters for every n-point correlation function. We shall refer to the
form (2.3) as a Laurent polynomial in the variables ρjk.
2Writing correlation functions in terms of the conformally invariant cross ratios is particu-
larly useful to parameterize 4-point functions. A basis of cross ratios for an n-point function
is used in the proof of Lemma 3.6. The general systematics of the pole structure, however, is
more transparent in terms of the present variables.
Harmonic Bilocal Fields 5
(c) The truncated Wightman functions
∣∣φ1(x1) · · · φn(xn)
are of the
same form like (2.3) but with pole degrees µtr
bounded by
µtrjk > −
dj+dk
(2.6)
(cf. [14, Corollary 4.4]).
The cluster condition, expressing the uniqueness of the vacuum, requires that
if a non-empty proper subset of points xk among all xi (i = 1, . . . , n) is shifted by
t · a (a2 6= 0), then the truncated function must vanish in the limit t → ∞. For
the two-point clusters {xj , xk}, this condition is ensured by (2.6) in combination
with with (2.5). For higher clusters, it puts further constraints on the admissible
linear combinations of terms of the form (2.3). Note however, that because of
possible cancellations the individual terms need not vanish in the cluster limit.
The cluster condition will be used in establishing the single pole property for
d = 2.
2.2 Twist expansion of the OPE and bi–harmonicity of twist two
contribution
The most powerful tool provided by GCI is the explicit construction of the OPE
of local fields in the general (axiomatic) framework.
Let φ1(x) and φ2(x) be two GCI scalar fields of the same scaling dimension
d and consider the operator distribution
U(x1, x2) = (ρ12)
φ1(x1)φ2(x2)− 〈0|φ1(x1)φ2(x2)|0〉
. (2.7)
As a consequence of the pole bounds (2.6), U(x1, x2) is smooth in the difference
x12. This is to be understood in a weak sense for matrix elements of U between
bounded energy states. Obviously, U(x1, x2) is a Huygens bilocal field in the
sense that [
(x1 − x)
2(x2 − x)
U(x1, x2), ψ(x)
= 0 (2.8)
for every field ψ(x) that is Huygens local with respect to φk(x). Then, one
introduces the OPE of φ1(x1)φ2(x2) by the Taylor expansion of U in x12
U(x1, x2) =
µ1,...,µn=0
12 · · · x
µ1...µn
(x2) , (2.9)
where Xnµ1...µn(x2) are Huygens local fields. We can consider the series (2.9)
as a formal power series, or as a convergent series in terms of the analytically
continued correlation functions of U(x1, x2). We will consider at this point the
series (2.9) just as a formal series. (See also [1] for the general case of constructing
OPE via multilocal fields in the context of vertex algebras in higher dimensions.)
Since the prefactor in (2.7) transforms as a scalar density of conformal weight
(1−d, 1−d) then U(x1, x2) transforms as a conformal bilocal field of weight (1, 1).
Harmonic Bilocal Fields 6
Hence, the local fields Xnµ1...µn in (2.9) have scaling dimensions n+2 but are not,
in general, quasiprimary.3 One can pass to an expansion in quasiprimary fields by
subtracting from Xnµ1...µn derivatives of lower dimensional fields X
µ1...µn′
. The
resulting quasiprimary fields Okµ1...µℓ are traceless tensor fields of rank ℓ and
dimension k. The difference
k − ℓ (“dimension − rank”) (2.10)
is called twist of the tensor field Okµ1...µℓ . Unitarity implies that the twist is
non-negative [10], and by GCI, it should be an even integer. In this way one can
reorganize the OPE (2.9) as follows
U(x1, x2) = V1(x1, x2) + ρ12 V2(x1, x2) + (ρ12)
2 V3(x1, x2) + · · · , (2.11)
where Vκ(x1, x2) is the part of the OPE (2.9) containing only twist 2κ contri-
butions. Note that Eq. (2.11) contains also the information that the twist 2κ
contributions contain a factor (ρ12)
κ−1 (i.e. Vκ are “regular” at x1 = x2), which
is a nontrivial feature of this OPE (obtained by considering 3-point functions).
Thus, the expansion in twists can be viewed as a light-cone expansion of the
Since the twist decomposition of the fields is conformally invariant then each
Vκ will be behave, at least infinitesimally, as a scalar (κ, κ) density under con-
formal transformations.
Every Vκ is a complicated (formal) series in twist 2κ fields and their deriva-
tives:
Vκ(x1, x2) =
Kµ1...µℓκ (x12, ∂x2)O
µ1...µℓ
(x2) , (2.12)
where K
µ1...µℓ
κ (x12, ∂x2) are infinite formal power series in x12 with coefficients
that are differential operators in x2 acting on the quasiprimary fields O. The
important point here is that the series K
µ1...µℓ
κ (x12, ∂x2) can be fixed universally
for any (even generally) conformal QFT. This is due to the universality of con-
formal 3-point functions. The explicit form of K
µ1...µℓ
κ (x12, ∂x2) can be found in
[6, 7] (see also [13]).
Thus, we can at this point consider Vκ(x1, x2) only as generating series for
the twist 2κ contributions to the OPE of φ(x1)φ(x2) but we still do not know
whether these series would be convergent and even if they were, it would not be
evident whether they would give bilocal fields. In the next section we will see
that this is true for the leading, twist two part under certain conditions, which
are automatically fulfilled for d = 2.
The higher twist parts Vκ (κ > 1) are certainly not convergent to Huygens
bilocal fields, since their 4-point functions, computed in [13], are not rational.
3Quasiprimary fields transform irreducibly under conformal transformations.
Harmonic Bilocal Fields 7
The major difference between the twist two tensor fields and the higher twist
fields is that the former satisfy conservation laws:
∂xµ1O
µ1...µℓ
(x) = 0 (ℓ > 1) . (2.13)
This is a well known consequence of the conformal invariance of the 2-point func-
tion and the Reeh–Schlieder theorem. It includes, in particular, the conservation
laws of the currents and the stress–energy tensor. It turns out that V1(x1, x2)
encodes in a simple way this infinite system of equations.
Theorem 2.1. ([13]) The system of differential equations (2.13) is equivalent to
the harmonicity of V1(x1, x2) in both arguments (bi–harmonicity) as a formal
series, i.e.,
�x1V1(x1, x2) = 0 = �x2V1(x1, x2).
The proof is based on the explicit knowledge of the K series in (2.12) and it is
valid even if the theory is invariant under infinitesimal conformal transformations
only.
The separation of the twist two part in (2.11) amounts to a splitting of U of
the form
U(x1, x2) = V1(x1, x2) + ρ12 Ũ(x1, x2) . (2.14)
This splitting can be thought in terms of matrix elements of U(x1, x2) expanded
as a formal power series according to (2.9). It is unique by virtue of Theorem 2.1,
due to the following classical Lemma:
Lemma 2.2. ([3, 1]) Let u(x) be a formal power series in x ∈ C4 (or, CD) with
coefficients in a vector space V . Then there exist unique formal power series
v(x) and ũ(x) with coefficients in V such that
u(x) = v(x) + x2 ũ(x) (2.15)
and v(x) is harmonic in x (i.e., �x v(x) = 0). (2.15) is called the harmonic
decomposition of u(x) (in the variable x around x = 0), and the formal power
series v(x) is said to be the harmonic part of u(x).
3 Bilocality of twist two contribution to the OPE
Let us sketch our strategy for studying bilocality of V1(x1, x2).
The existence of the field V1(x1, x2) can be established by constructing its cor-
relation functions. On the other hand, every correlation function4
·V1(x1, x2) ·
4This short-hand notation stands for
˛φ3(x3) · · ·φk(xk) V1(x1, x2) φk+1(xk+1)
· · · φn(xn)
, here and in the sequel.
Harmonic Bilocal Fields 8
of V1 is obtained (originally, as a formal power series in x12) under the split-
ting (2.14). It thus appears as a harmonic decomposition of the corresponding
correlation function
· U(x1, x2) ·
of U :
· U(x1, x2) ·
· V1(x1, x2) ·
+ ρ12
· Ũ(x1, x2) ·
. (3.1)
Note that we should initially treat the left hand side of (3.1) also as a formal
power series in x12 in order to make the equality meaningful. It is important that
this series is always convergent as a Taylor expansion of a rational function in a
certain domain around x1 = x2 in M
, for the complexified Minkowski space
MC = M + iM , according to the standard analytic properties of Wightman
functions. We shall show in Sect. 3.1 that this implies the separate convergence
of both terms in the right hand side of (3.1). Hence, the key tool in constructing
V1 are the harmonic decompositions
F (x1, x2) = H(x1, x2) + ρ12 F̃ (x1, x2) (3.2)
of functions F (x1, x2) that are analytic in certain neighbourhoods of the diagonal
{x1 = x2}.
Recall that H in (3.2) is uniquely fixed as the harmonic part of F in x1 around
x2, due to Lemma 2.2. This is equivalent to the harmonicity �x1 H(x1, x2) =
0. On the other hand, according to Theorem 2.1 we have to consider also the
second harmonicity condition on H, �x2 H(x1, x2) = 0, i.e., H is the harmonic
part in x2 around x1. This leads to some “integrability” conditions for the initial
function F (x1, x2), which we study in Sect. 3.2.
Next, to characterize the Huygens bilocality of V1, we should have rationality
of its correlation functions
· V1(x1, x2) ·
, which is due to a straightforward
extension of the arguments of [14, Theorem 3.1]. But we have started with the
correlation functions of U , which are certainly rational. Hence, we should study
another condition on U , namely that its correlation functions have a rational
harmonic decomposition. We show in Sect. 3.3 that this is equivalent to a simple
condition on the correlation functions of U , which we call “Single Pole Property”
(SPP).
In this way we establish in Sect. 3.4 that V1 always exists as a Huygens
bilocal field in the case of scalar fields of dimension d = 2. However, for higher
scaling dimensions one cannot anymore expect that V1 is Huygens bilocal in
general. This is illustrated by a counter-example, involving the 6-point function
of a system of d = 4 fields, given at the end of Sect. 3.5.
3.1 Convergence of harmonic decompositions
To analyze the existence of the harmonic decomposition of a convergent Taylor
series we use the complex integration techniques introduced in [1].
Let MC =M + iM be the complexification of Minkowski space, which in this
subsection is assumed to be D–dimensional, and E =
x : (i x0, x1, . . . , xD−1)
its Euclidean real submanifold, and SD−1 ⊂ E the unit sphere in E. We
Harmonic Bilocal Fields 9
denote by ‖·‖ the Hilbert norm related to the fixed coordinates in MC: ‖x‖
|x0|2 + · · · + |xD−1|2.
Let us also introduce for any r > 0 a real compact submanifold Mr of MC:
ζ ∈MC : ζ = r e
iθw, ϑ ∈ [0, π], w ∈ SD−1
(3.3)
(note that ϑ ∈ [π, 2π] gives another parameterization of Mr). Then there is an
integral representation for the harmonic part of a convergent Taylor series.
Lemma 3.1. (cf. [1, Sect. 3.3 and Appendix A]) Let u(x) be a complex formal
power series that is absolutely convergent in the ball ‖x‖ < r, for some r > 0, to
an analytic function U(x). Then the harmonic part v(x) of u(x) (around x = 0),
which is provided by Lemma 2.2, is absolutely convergent for
|x2|+ 2 r ‖x‖ < r2. (3.4)
The analytic function V (x) that is the sum of the formal power series v(x) has
the following integral representation:
V (x) =
(z− x)2
U(z) , V1 =
= iπ|SD−1|, (3.5)
where r′ < r, |x2| + 2 r′ ‖x‖ < r′2, and the (complex) integration measure
is obtained by the restriction of the complex volume form dDz (= dz0∧
· · · ∧dzD−1) on MC (∼= CD) to the real D–dimensional submanifold Mr′ (3.3),
r′ > 0.
Proof. Consider the Taylor expansion in x of the function
1 − x
(z − x)2
and write it in the form (cf. [1, Sect. 3.3])
(z− x)2
(z2)−
−ℓHℓ(z, x), Hℓ(z, x) =
hℓµ(z)hℓµ(x), (3.6)
where {hℓµ(u)} is an orthonormal basis of harmonic homogeneous polynomials
of degree ℓ on the sphere SD−1. This expansion is convergent for
∣∣+ 2
∣∣z · x
∣∣ (3.7)
since its left–hand side is related to the generating function for Hℓ:
1− λ2 x2 y2
(1− 2λ x · y + λ2 x2 y2)
λℓHℓ(x, y) , (3.8)
the expansion (3.8) being convergent for λ 6 1 if |x2y2|+2|x ·y| < 1. Then if we
fix r′ < r and z varies on Mr′ , a sufficient condition for (3.7) is |x
2| + 2 r′ ‖x‖ <
r′2 (since sup
w∈ SD−1
|w · x| = ‖x‖).
Harmonic Bilocal Fields 10
On the other hand, writing u(z) =
k=0 uk(z), where uk are homogeneous
polynomials of degree k, we get by the absolute convergence of u(z) the relation
(valid for |x2| + 2 r′ ‖x‖ < r′2)
(z− x)2
U(z) =
k,ℓ=0
(z2)−
−ℓHℓ(x, z)uk(z) .
(3.9)
Noting next that in the parameterization (3.3) of Mr′ we have d
= i r′D
eiD ϑ dϑ ∧ dσ(w), where dσ(w) is the volume form on the unit sphere, we obtain
for the right hand side of (3.9):
k,ℓ=0
eiϑ(k−ℓ)
dσ(w)
|SD−1|
Hℓ(x,w)uk(w) .
Now if we write, according to Lemma 2.2, uk(z) =
ck,j,µ′ (z
2)j hk−2j,µ′(z)
then we get by the orthonormality of hℓ,µ(w)
k,ℓ=0
δℓ,k−2j
eiϑ(k−ℓ) ck,j,µ hk−2j,µ(x)
ck,0,µ hk,µ(x) = v(x) .
The latter proves both: the convergence of v(x) in the domain (3.4) (since r′ < r
was arbitrary) and the integral representation (3.5). �
As an application of this result we will prove now
Proposition 3.2. For all n and k, and for all local fields φj (j = 3, . . . , n) the
Taylor series
∣∣φ3(x3) · · · φk(xk) V1(x1, x2) φk+1(xk+1) · · · φn(xn)
(3.10)
in x12 converge absolutely in the domain
‖x12‖+
‖x12‖2 +
∣∣x212
‖x2j‖+
‖x2j‖2 +
∣∣x22j
∣∣x22j
∣∣ ∀ j (3.11)
(j = 3, . . . , n). They all are real analytic and independent of k for mutually
nonisotropic points.
Proof. Let
Fk(x12, x23, . . . , x2n)
∣∣φ3(x3) · · ·φk(xk) U(x1, x2) φk+1(xk+1) · · · φn(xn)
(3.12)
Harmonic Bilocal Fields 11
be the correlation functions, analytically continued in x12.
As Fk, which is a rational function, depends on x := x12 via a sum of products
of powers
(x− x2j)
it has a convergent expansion in x for
∣∣+ 2
∣∣x · x2j
∣∣x22j
∣∣ . (3.13)
If we want Fk to have a convergent Taylor expansion for ‖x‖ < r we get the
following sufficient condition
∣∣x22j
∣∣− 2 r ‖x2j‖. (3.14)
By Lemma 3.1 we conclude that the series (3.10) is convergent for
|x212|+ 2 r ‖x12‖ < r
2. (3.15)
Combining both (sufficient) conditions (3.14) and (3.15) for r we find that they
are compatible if ‖x12‖ +
‖x12‖2 +
∣∣x212
‖x2j‖2 +
∣∣x22j
∣∣ − ‖x2j‖, which is
equivalent to (3.11). �
Note that one can also prove a similar convergence property for the correla-
tion functions of several V1.
Remark 3.1. The domain of convergence of (3.10) should be Lorentz invariant.
Hence, (3.10) are convergent in the smallest Lorentz invariant set containing the
domain (3.11). Such a set is determined by the values of the invariants x212, x
and x12 · x2j and it turns out to be the set
∣∣x212
∣∣ 12 ∣∣x22j
∣∣ 12 6
∣∣x12 · x2j
(∣∣x22j
∣∣ 12 −
∣∣x212
∣∣ 12 )2
or equivalently
√∣∣x212
∣∣∣∣x22j
∣∣x12 · x2j
∣∣2 <
(∣∣x22j
∣∣ 12 −
∣∣x212
∣∣ 12 )2
. (3.16)
Outside the domain of convergence (3.16), the correlations of V1(x1, x2) have
to be defined by analytic continuation. When the correlations are rational, V1
is Huygens bilocal, but the counter-example presented in Sect. 3.5 shows that
rationality is not automatic. Then, it is not even obvious that the continuations
are single–valued within the tube of analyticity required by the spectrum con-
dition, i.e., that V1 exists as a distribution in all of M × M . Nontrivial case
studies, however, show that at least for xk space–like to both x1 and x2, the
continuation is single–valued and preserves the independence on the position k
in (3.10) where V1(x1, x2) is inserted. This leads us to conjecture
Conjecture 3.3. The twist two field V1(x1, x2), whose correlations are defined
as the analytic continuations of the harmonic parts of those of U(x1, x2), exists
and is bilocal in the ordinary sense, i.e., it commutes with φ(x) and V1(x, x
′) if
x and x′ are space–like to x1 and x2.
We hope to return to this conjecture elsewhere (see also the Note added in
proof). Note that the argument that locality implies Huygens locality [14] does
not pass to bilocal fields.
Harmonic Bilocal Fields 12
3.2 Consequences of bi–harmonicity
Now our objective is to find the harmonic decomposition of the rational functions
F (x1, x2) that depend on x1 and x2 through the intervals ρik = (xi−xk)
2, i = 1, 2,
k = 3, . . . , n, for some additional points x3, . . . , xn. The F ’s, as correlation
functions of U(x1, x2), have the form
F (x1, x2) =
(ρ12)
q Fq(x1, x2) ≡
(ρ12)
{ρik}{i,k}6={1,2}
, (3.17)
Fq(x1, x2) =
{µ1i},{µ2i}
Cq,{µ1j},{µ2j}
(ρ1j)
(ρ2i)
µ2j , (3.18)
where M ∈ N and µ1j , µ2j (j = 3, . . . , n) are integers > −d such that
j>3 µ1j
j>3 µ2j = −1 − q, and the coefficients Cq,{µ1j},{µ2j} may depend on ρjk
(j, k > 3).
IfH is the harmonic part of F in x12, then the leading part F0 (of order (ρ12)
is also the leading part of H. We shall now proceed to show that bi–harmonicity
of H (Theorem 2.1), together with the first principles of QFT including GCI,
implies strong constraints on F0.
Proposition 3.4. Let F0(x1, x2) be as in (3.18), and let H(x1, x2) be its har-
monic part with respect to x1 around x2. Then H is also harmonic with respect
to x2, if and only if F0 satisfies the differential equation
(E1D2 − E2D1)F0 = 0, (3.19)
where E1 =
i=3 ρ2i∂1i (with ∂jk = ∂kj =
), D1 =
36j<k6n ρjk∂1j∂1k,
and similarly for E2 and D2, exchanging 1 ↔ 2.
Proof. By Proposition 3.2 (see also Remark 3.1) we can consider H as a function
in the 2n − 3 variables ρ1i, ρ2i (i > 3) and ρ12, analytic in some domain that
includes ρ12 = 0.
Expanding H =
q(ρ12)
qHq/q!, the functions Hq are homogeneous of degree
−1 − q in both sets of variables ρ1i and ρ2i, and H0 = F0. To impose the
harmonicity with respect to the variable x1, we use the identity [11, App. C]
�x1F = −4
26i<j6n
ρij ∂1i∂1j F
ρij =(xi−xj)2
, (3.20)
valid for homogeneous functions of ρ1i of degree −1, to express the wave operator
�x1 as a differential operator with respect to the set of variables ρ1i (i > 2). This
yields the recursive system of differential equations
E1Hq+1 = −D1Hq. (3.21)
Performing the same steps with respect to the variable x2, one obtains
E2Hq+1 = −D2Hq. (3.22)
Harmonic Bilocal Fields 13
Eq. (3.19) then arises as the integrability condition for the pair of inhomogeneous
differential equations for H1 (putting q = 0), observing that E2E1 − E1E2 =∑
ρ1i∂1i −
ρ2i∂2i vanishes on H1 by homogeneity.
Conversely, if (3.19) is fulfilled, thenH1 exists and satisfies (D1E2−D2E1)H1 =
−(D1D2 − D2D1)H0 = 0 because D1 and D2 commute. But this is equivalent
to (D2E1 − D1E2)H1 = 0, which is in turn the integrability condition for the
existence of H2, and so on. It follows that bi–harmonicity imposes no further
conditions on the leading function H0 = F0. �
The differential equation (3.19) imposes the following constraints on the lead-
ing part F0 of the rational correlation function F (3.17):
Corollary 3.5. Assume that the function F0 as in (3.18) satisfies the differential
equation (3.19). Then
(i) If F0 contains a “double pole” of the form (ρ1i)
µ1i(ρ1j)
µ1j with i 6= j and µ1i
and µ1j both negative, then its coefficients must be regular in ρ2k (k 6= i, j).
(ii) F0 cannot contain a “triple pole” of the form (ρ1i)
µ1i(ρ1j)
µ1j (ρ1k)
µ1k with
i, j, k all different and µ1i, µ1j , µ1k all negative.
The same hold true, exchanging 1 ↔ 2.
Proof. Pick any variable, say ρ2k, and decompose F0 =
r>−p(ρ2k)
rfr as a Lau-
rent polynomial in ρ2k. The differential equation (3.19) turns into the recursive
system
ρij∂1i∂1j −
i,j 6=k
ρ2iρkj∂1i∂2j
 r · fr = Xrfr−1 + Y fr
of differential equations for the functions fr which are Laurent polynomials in the
remaining variables. The precise form of the polynomial differential operators
Xr and Y does not matter. Assume the lowest power −p of ρ2k to be negative.
For r = −p, the right-hand-side vanishes. Because the term ρij∂1i∂1j on the
left-hand-side would produce a singularity that cannot be cancelled by any other
term, f−p cannot have a “double pole” in any pair of variables ρ1i, ρ1j with i 6= j
and i, j 6= k. This property passes recursively to all fr with r < 0, because also
the right-hand-side never can contain such a pole. This implies that a double
pole in a pair of variables ρ1i, ρ1j with i 6= j cannot multiply a term that is
singular in ρ2k unless k = i or k = j, proving (i).
If the coefficient of the double pole were singular in ρ1k, k 6= i, j, then the
resulting double pole in the pair ρ1i, ρ1k resp. ρ1j , ρ1k would imply regularity
also in ρ2j resp. ρ2i. Hence the coefficient of a triple pole must be regular in all
ρ2m, which contradicts the total homogeneity −1 of F0 in these variables. This
proves the statement (ii). �
3.3 A necessary and sufficient condition for Huygens bilocality
Definition 3.1. (“Single Pole Property”, SPP) Let f(x1, . . . , xn) be a Laurent
polynomial in the variables ρij , i.e., regarded as a function of x1 only, it is a
Harmonic Bilocal Fields 14
finite linear combination of functions of the form
(ρ1j)
µ1j ≡
(x1 − xj)
, (3.23)
where µ1j (j > 2) are integers and the coefficients may depend on the parameters
ρjk (j, k > 2). Then f is said to satisfy the single pole property with respect to
x1 if it contains no terms for which there are j 6= k (j, k > 2) such that both µ1,j
and µ1,k are negative.
The significance of SPP stems from the fact that the harmonic parts H of
F0, i.e., the correlation functions of V1, are again Laurent polynomials if and
only if F0 satisfies the SPP. Namely, if H is a harmonic Laurent polynomial, the
same argument as in [11, Lemma C.1] (using the representation (3.20) of the
wave operator) shows that H fulfils the SPP with respect to x1, and so does F0,
because it is the leading part of order (ρ12)
0 of H. The converse is an immediate
consequence of Lemma 3.6 (allowing for a relabelling and multiple counting of
the points x3, . . . , xn, which are not required to be distinct).
Lemma 3.6. Let n > 4. Every finite linear combination of monomials of the
gn(x1) =
i=4 ρ1i
(ρ13)
i=4(x1 − xi)
[(x1 − x3)2]n−2
(3.24)
has a rational harmonic decomposition in x1 around x2
gn(x1) = hn(x1) + (x1 − x2)
2 · g̃n(x1) (3.25)
i.e., hn is harmonic with respect to x1 and g̃n is regular at x1 = x2, and both
hn and g̃n are rational. More precisely, (ρ13)
n−2(ρ23)
n−3hn is a homogeneous
polynomial of total degree 2(n − 3) in the variables {ρij : 1 6 i < j}, which is
separately homogoneous of degree n− 3 in the variables {ρ1i : i > 2} and in the
variables {ρ12, ρ2i : i > 3}.
Proof. It is convenient to introduce the variables
ρ1iρ23
ρ13ρ2i
, si =
ρ12ρ3i
ρ13ρ2i
, uij =
ρ12ρ23ρij
ρ13ρ2iρ2j
(4 6 i < j 6 n). (3.26)
We claim that hn(x1) is of the form
hn(x1) =
fn(ti, si, uij)
, (3.27)
where fn are polynomials of degree n − 3 such that fn(ti, si = 0, uij = 0) =∏n
i=4 ti. Because all si and uij contain a factor ρ12, these properties ensure that
g̃n given by (gn − hn)/ρ12 is regular in ρ12.
Harmonic Bilocal Fields 15
Using again the identity (3.20) for the wave operator, and transforming this
into a differential operator with respect to the set of variables (3.26), we find
�x1 hn(x1) = −4
(ρ13)
·Dfn(ti, si, uij), (3.28)
where D is the differential operator
D = (1 + t∂t + s∂s + u∂u)(s∂t + s∂s + u∂u)− (s∂s + u∂u)∂t − u∂t∂t (3.29)
with shorthand notations for degree-preserving operators
t∂t =
ti∂ti , s∂t =
si∂ti , s∂s =
si∂si , u∂u =
46i<j6n
uij∂uij
and degree-lowering operators
∂ti , u∂t∂t =
46i<j6n
uij∂ti∂tj .
To solve the condition Dfn = 0 for harmonicity, we make an ansatz
fn(ti, si, uij) =
(sk, ukl) ·
i∈N\K
(ti − si),
where N ≡ {4, . . . , n}, g
are polynomials in the variables sk, ukl (k, l ∈ K)
only, and g
= 1. Then the harmonicity condition Dfn = 0 is equivalent to the
recursive system
(n− 2− |K|+∆)∆ g
K\{k}
k,l∈K,k<l
(ukl − sk − sl) g
K\{k,l}
where |K| is the number of elements of the set K and the differential operator
∆ = s∂s + u∂u measures the total polynomial degree r in sk and ukl. Since one
can divide by (n − 2− |K|+ r)r if r > 0, there is a unique polynomial solution
such that g
(sk = 0, ukl = 0) = 0 (K 6= ∅), and g
is of order 6 |K|. So fn
is of order n − 3. (Explicitly, the first three functions are f3 = 1, f4 = t4 − s4
and f5 = (t4 − s4)(t5 − s5) +
(u45 − s4 − s5).) An inspection of the recursion
also shows that all possible factors ρ2i in the denominators of the arguments of
fn cancel with the factors in the prefactor in (3.27), thus hn can have poles only
in ρ13 and ρ23 of the specified maximal degree. This proves the Lemma. �
The upshot of the previous discussion is a necessary and sufficient condition
for the Huygens bilocality of V1 which directly refers to the local correlation
functions of the theory:
Harmonic Bilocal Fields 16
Theorem 3.7. The field V1(x1, x2) weakly converges on bounded energy states
to a Huygens bilocal field which is conformal of weight (1, 1), if and only if the
leading parts F0 of the Laurent polynomials F (3.17) satisfy the “single pole
property” (Def. 3.1) with respect to both x1 and x2. In this case, the formal
series H converge to Laurent polynomials in (xi − xj)
2 subject to the same pole
bounds, specified in Theorem 2.1, as F .
Proof. We know already that if V1 is a Huygens bilocal field, then its correlation
functions H are Laurent polynomials of the form (2.3), and that this implies the
SPP for F0 with respect to x1 and x2. Conversely, if the SPP holds for F0 with
respect to x1 and x2, then H are Laurent polynomials by Lemma 3.6, and hence
V1 is relatively Huygens bilocal with respect to the fields φi. Since the general
argument [4] that relative locality implies local commutativity of a field with
itself refers only to local fields, we want to give an explicit argument for the case
at hand.
All the previous remains true when in (3.10) or (3.17) a product of fields
φk(xk)k+1φ(xk+1) is replaced by U(xk, xk+1). By assumption, and because U
is bilocal, the contributions of order (ρk,k+1)
0 to the correlation functions of
U(xk, xk+1) fulfil the SPP with respect to xk and xk+1. By Lemma 3.6, this
property is preserved upon the passage to the harmonic parts with respect to
x1 and x2. One may therefore continue in the same way with xk, xk+1, and
eventually find that all mixed correlation functions of φ’s and V1’s converge to
rational functions. By this convergence we conclude that all products of φ’s and
V1’s converge on the vacuum, and this then defines V1 as a Huygens bilocal field,
since its matrix elements will satisfy Huygens locality.
The conformal properties of V1 follow from the preservation of the homo-
geneity and the pole degrees in the harmonic decomposition, as guaranteed by
Lemma 3.6. �
For n = 4 points, the SPP is trivially satisfied because of homogeneity. Hence
the 4-point function
∣∣V ∗1 V1
is always rational. It follows that its expansion
in (transcendental) partial waves [11] cannot terminate. This means that (unless
V1 = 0 in which case there is not even a stress-energy tensor) a GCI QFT
necessarily contains infinitely many conserved tensor fields of arbitrarily high
spin.
3.4 The case of dimension 2
Let us consider now the case of scalar fields φk of dimension 2. We claim that in
this case, Corollary 3.5 in combination with the cluster condition is sufficient to
establish the SPP, Definition 3.1. Hence we conclude by Theorem 3.7 that the
twist two harmonic fields V1(x1, x2) are indeed Huygens bilocal fields.
To prove our claim, we use that by (2.6), µij > −1, hence the SPP is equiv-
alent to the statement that there can be no term contributing to
∣∣φ1(x1) · · ·
φn(xn)
, for which there is i with more than two µij negative (j 6= i). Thus
assume that there is a term with, say, µ12 = µ13 = µ14 = −1. It constitutes a
Harmonic Bilocal Fields 17
double pole for each of the three harmonic fields V1(x1, xj) (j = 2, 3, 4). Then by
homogeneity (2.5), there must be more poles in xj (j = 2, 3, 4), but these cannot
be of the form ρjk with k > 4 by Corollary 3.5. Hence (up to permutations of
2, 3, 4) µ23 = µ24 = −1, µ34 = 0. Again by homogeneity (2.5), the dependence
on x1, . . . , x4 must be given by a linear combination of terms
ρ1kρ4ℓ
ρ12ρ13ρ14ρ23ρ24
(3.30)
with k, ℓ > 4. Applying the cluster limit (Sect. 2.1) to the points x1, x2, x3, x4
in (3.30), the limit diverges ∼ t4. This behavior is tamed to ∼ t2 by anti–
symmetrization in k, ℓ, but it cannot be cancelled by any other terms. Hence
the assumption leads to a contradiction.
This proves the SPP if the generating scalar fields have dimension d = 2.
3.5 A d = 4 6-point function violating the SPP
We proceed with an example of 6-point function violating the SPP in the case of
two d = 4 GCI scalar fields Li(x) such that the bilocal field U(x1, x2) obtained
from L1(x1)L2(x2) has a non-zero skew–symmetric part. Let L be any linear
combination of L1 and L2.
The following admissible contribution to the truncated part of the 6-point
function 〈0|U(x1, x2)L(x3)L(x4)U(x5, x6)|0〉 clearly violates the SPP:
F0(x1, x2) = A12A56
ρ15ρ26ρ34 − 2ρ15ρ23ρ46 − 2ρ15ρ24ρ36
ρ13ρ14ρ23ρ24 · ρ34 · ρ35ρ45ρ36ρ46
, (3.31)
where Aij stands for the antisymmetrization in the arguments xi, xj . It is ad-
missible as a truncated 6-point structure because (ρ12ρ56)
−3F0 obeys all the pole
bounds of Sect. 2 for a correlation 〈0|L1(x1)L2(x2)L(x3)L(x4)L1(x5) L2(x6)|0〉
of six fields of dimension d = 4.
On the other hand, F0 satisfies the differential equation
(E1D2 − E2D1)F0(x1, x2) = 0 (3.32)
(and similar in the variables x5 and x6), ensuring that F0 is the leading part of a
bi–harmonic function, analytic in a neighborhood of x1 = x2 and x5 = x6, repre-
senting a contribution to the twist two 6-point function 〈0|V1(x1, x2)L(x3)L(x4)
V1(x5, x6)|0〉, of which F0 is the leading part. This function cannot be a Laurent
polynomial in the ρij by our general argument that the leading part of a bi–
harmonic Laurent polynomial cannot satisfy the SPP. Hence the twist two field
V1(x1, x2) cannot be Huygens bilocal.
The resulting contribution to the conserved local current 4-point function
〈0|Jµ(x1)L(x3)L(x4)Jν(x5)|0〉
tr is obtained through Jµ(x) = i(∂xµ−∂yµ) V1(x, y)|x=y.
It also satisfies the pertinent pole bounds. This structure is rational as it should,
because only the leading part F0 contributes. In fact, while the 6-point structure
involving the harmonic field cannot be reproduced by free fields due to its double
Harmonic Bilocal Fields 18
pole, the resulting 4-point structure does arise as one of the three independent
connected structures contributing to 4-point functions involving two Dirac cur-
rents : ψ̄aγ
µψb : and two Yukawa scalars ϕ : ψ̄cψd : (allowing for internal flavours
a, b, . . . ).
4 The theory of GCI scalar fields of scaling dimension
d = 2
The scaling dimension d = 2 is the minimal dimension of a GCI scalar field for
which one could expect the existence of nonfree models. It turns out however,
that in this case the fields can be constructed as composite fields of free, or
generalized free, fields. Namely, we will establish the following result.
Theorem 4.1. Let {Φm(x)}
m=1 be a system of real GCI scalar fields of scaling
dimension d = 2. Then it can be realized by a system of generalized free fields
{ψm(x)} and a system of independent real massless free fields {ϕm(x)}, acting
on a possibly larger Hilbert space, as follows:
Φm(x) =
αm,j ψj(x) +
j,k=1
βm,j,k :ϕj(x)ϕk(x) : , (4.1)
where αm,j and βm,j,k = βm,k,j are real constants such that
α2m,j < ∞ and
j,k=1
m,j,k
< ∞. Here, we assume the normalizations 〈0|ϕj(x1) ϕk(x2) |0〉 =
δjk (ρ12)
−1, 〈0|ψj(x1) ψk(x2) |0〉 = δjk (ρ12)
The proof of Theorem 4.1 is given at the end of Sect. 4.2. The main reason
for this result is the fact that in the d = 2 case the harmonic bilocal fields exist
and furthermore, they are Lie fields. This was originally recognized in [12], [2]
under the assumption that there is a unique field φ of dimension 2. We are
extending here the result to an arbitrary system of d = 2 GCI scalar fields.
If we assume the existence of a stress-energy tensor as a Wightman field5,
the generalized free fields must be absent in (4.1), and the number of free fields
must be finite. In this case, the iterated OPE generates in particular the bilocal
field 1
i :ϕi(x)ϕi(y):. As this field has no other positive-energy representation
than those occurring in the Fock space [2], nontrivial possibilities for correlations
between non-free fields and the fields (4.1) are strongly limited.
4.1 Structure of the correlation functions
We consider a GCI QFT generated by a set of hermitean (real) scalar fields.
We denote by F the real vector space of all GCI real scalar fields of scaling
5A stress-energy tensor always exists as a quadratic form between states generated by the
fields Φm from the vacuum [8].
Harmonic Bilocal Fields 19
dimension 2 in the theory. (Note that the space F may be larger than the linear
span of the original system of d = 2 fields of Theorem 4.1.) We shall find in this
section the explicit form of the correlation functions of the fields from F .
Theorem 4.2. Let φ1(x), . . . , φn(x) ∈ F then their truncated n-point functions
have the form
〈0|φ1(x1) · · · φn(xn)|0〉
c(n)(φσ1 , . . . , φσn)
ρσ1σ2 · · · ρσnσ1
(4.2)
where c(n) are multilinear functionals c(n) : F⊗n → R with the inversion and
cyclic symmetries c(n)(φ1, . . . , φn) = c
(n)(φn, . . . , φ1) = c
(n)(φn, φ1, . . . , φn−1).
Before we prove the theorem, let us first illustrate it on the example of the
free field realization (4.1). In this case one finds
Φm1 ,Φm2
αm1,jαm2,j +
j,k=1
βm1,j,k βm2,j,k
αm1,jαm2,j + Tr βm1βm2 ,
Φm1 , . . . ,Φmn
= Tr βm1 · · · βmn for n > 2 (4.3)
where βm =
βm,j,k
Proof of Theorem 4.2. We first recall the general form (2.3) of the truncated
correlation function with pole bounds (2.6) that read in this case: µtrjk > −1.
The argument in Sect. 3.4 shows that the nonzero contributing terms in Eq. (2.3)
have for every j = 1, . . . , n exactly two negative µtrjk or µ
kj for some k = k1, k2
different from j.
The nonzero terms are therefore products of “disjoint cyclic products of prop-
agators” of the form 1/ρk1k2ρk2k3 · · · ρkr−1krρkrk1 . But cycles of length r < n are
in conflict with the cluster condition (Sect. 2). We conclude that 〈0|φ1(x1) · · ·
φn(xn)|0〉
tr is a linear combination of terms like those in (4.2) with some co-
efficients cσ(φ1, . . . , φn) depending on the permutations σ ∈ Sn and on the
fields φj (multilinearly). Locality, i.e. 〈0|φ1(x1) · · · φn(xn)|0〉
tr = 〈0|φσ′1(xσ
) · · ·
φσ′n(xσ′n)|0〉
tr, then implies cσ′σ(φ1, . . . , φn) = cσ(φσ′1
, . . . , φσ′n) (σ, σ
′ ∈ Sn), so
that cσ(φ1, . . . , φn) = c
(n)(φσ1 , . . . , φσn) for some c
(n) : F⊗n → R. The equali-
ties c(n)(φ1, . . . , φn) = c
(n)(φn, . . . , φ1) = c
(n)(φn, φ1, . . . , φn−1) are again due to
locality. �
As we already know by the general results of the previous section, the har-
monic bilocal field exist in the case of fields of dimension d = 2. Moreover, the
knowledge of the correlation functions of the d = 2 fields allows us to find the
form of the correlation functions of the resulting bilocal fields. This yields an
algebraic structure in the space of real (local and bilocal) scalar fields, which we
proceed to display.
Harmonic Bilocal Fields 20
Let us introduce together with the space F of d = 2 fields also the real vector
space V of all real harmonic bilocal fields. We shall consider F and V as built
starting from our original system of d = 2 fields {Φm} of Theorem 4.1, by the
following constructions.
(a) If φ1(x), φ2(x) ∈ F then introducing the bilocal (1, 1)–field U(x1, x2) =
φ1(x1)φ2(x2)− 〈0|φ1(x1)φ2(x2)|0〉
in accord with Eq. (2.7), we consider its
harmonic decomposition U(x, y) = V1(x, y) + (x−y)
2 Ũ(x, y). We denote V1(x, y)
by φ1 ∗ φ2; this defines a bilinear map F ⊗ F
(b) If now v(x, y) ∈ V then vt(x, y) := v(y, x) also belongs to V and γ(v)
is a field from F .
(c) If v(x, y), v′(x, y) ∈ V then there is a harmonic bilocal field
(v∗v′)
:= w-lim
x′ → y′
x′−y′
x, x′
y′, y
−〈0|v
x, x′
y′, y
. (4.4)
The existence of the above weak limit (i.e., a limit within correlation functions)
will be established below together with the independence of x′ = y′ and the
regularity of the resulting field for (x− y)2 = 0.
(d) If v(x, y) ∈ V and φ(x) ∈ F then we can construct the following bilocal
field belonging to V:
(v ∗ φ)
:= w-lim
x′ → y
x′ − y
x, x′
− 〈0|v
x, x′
, (4.5)
where again the existence of the limit and the regularity for (x− y)2 = 0 will be
established later.
One can define similarly a product φ ∗ v ∈ V, but it would then be expressed
as: (vt ∗ φ)t.
To summarize, we have three bilinear maps: F ⊗ F
→V, V ⊗ V
→V, V ⊗
→V, and two linear ones: V
→V, V
→F . Applying these maps we construct
F and V inductively, starting from our original system of d = 2 fields, given
in Theorem 4.1, and at each step of this inductive procedure, we establish the
existence of the above limits in (c) and (d). In fact, we shall establish this
together with the structure of the truncated correlation functions for the fields
in F and V.6
Before we state the inductive result it is convenient to introduce the vector
space
Â = F × V (4.6)
6Since we shall use the notion of truncated correlation functions also for bilocal fields let
us briefly recall it. If B1, . . . , Bn are some smeared (multi)local fields then their truncated
correlation functions are recursively defined by: 〈0|B1 · · ·Bn|0〉 =
∪̇P = {1,...,n}
{j1,...,jk} ∈P
〈0|Bj1 · · ·Bjk |0〉
tr (the sum being over all partitions P of {1, . . . , n})
Harmonic Bilocal Fields 21
and endow it with the following bilinear operation
(φ1, v1) ∗ (φ2, v2) :=
0, φ1 ∗ φ2 + v1 ∗ v2 + v1 ∗ φ2 + (v
2 ∗ φ1)
, (4.7)
and with the transposition
(φ, v)t := (φ, vt) . (4.8)
The spaces F and V will be considered as subspaces in Â. Thus, the new
operation ∗ in Â combines the above listed three operations. We shall see later
that Â is actually an associative algebra under the product (4.7). We note
that the transposition t (4.8) is an antiinvolution with respect to the product:
(q1 ∗ q2)
t = qt2 ∗ q
1, for every q1, q2 ∈ Â.
Proposition 4.3. There exist multilinear functionals
c(N) : Â⊗N → R (4.9)
such that if we take elements q1, . . . , qn+m ∈ Â : qk := vk
xk[0], xk[1]
∈ V, where
[ε] stands for a Z/2Z–value and k = 1, . . . , n, and qk := φk−n
∈ F for
k = n+ 1, . . . , n +m, then the truncated correlation functions can be written in
the following form:
〈0|v1
x1[0], x1[1]
· · · vn
xn[0], xn[1]
· · ·φm
|0〉tr
2(n+m)
σ∈Sn+m
(ε1,...,εn)∈ (Z/2Z)
Kσ,ε Tσ,ε
x1[0], . . . , xn[1], xn+1, . . . , xn+m
. (4.10)
Here: Kσ,ε are coefficients given by Kσ,ε := c
(n+m)
[εσ1 ]
σ1 , . . . , q
[εσn+m ]
, where
we set εn+1 = · · · = εn+m = 0, and q
[0] := q, q[1] := qt (for q ∈ Â); the terms
Tσ,ε are the following cyclic products of intervals
Tσ,ε =
xσn+m − xσ1[ε1]
xσk[1+εk] − xσk+1[εk+1]
xσn[1+εn] − xσn+1
xσn+k − xσn+k+1
. (4.11)
It follows by Eq. (4.10) that the limits in the steps (c) and (d) above are well
defined.
Before the proof let us make some remarks. First, we used the same notation
c(n) as in Theorem 4.2 since the above multilinear functionals are obviously an
extension of the previous, i.e., Eq. (4.10) reduces to Eq. (4.2) for m = 0. Let us
also give an example for Eq. (4.10) with n = m = 1:
〈0|v(x1, x2)φ(x3)|0〉 =
c(2)(v, φ)
ρ23 ρ31
+ c(2)(vt, φ)
ρ13 ρ32
+ c(2)(φ, v)
ρ31 ρ23
+ c(2)(φ, vt)
ρ32 ρ13
. (4.12)
Harmonic Bilocal Fields 22
As one can see, c(n) (as well as c(n) of Theorem 4.2) possess a cyclic and an
inversion symmetry:
q1, . . . , qn
= c(n)
qn, q1 . . . , qn−1
= c(n)
qtn, . . . , q
. (4.13)
This is the reason for choosing the prefactors in Eqs. (4.2) and (4.10) (the inverse
of the orders of the symmetry groups).
Proof of Proposition 4.3. According to our preliminary remarks it is enough to
prove that Eq. (4.10) is consistent with the operations F ⊗ F
→V, V ⊗ V
V ⊗ F
→V and V
Starting with F ⊗ F
→V one should prove that any truncated correlation
function
·φ1(x1)φ2(x2)·
given by Eq. (4.10) yields a harmonic decomposition:
·φ1(x1)φ2(x2) ·
·(φ1∗φ2)
x1, x2
+ρ12R(x1, x2), with a correlation
function
· (φ1 ∗ φ2)
x1, x2
given by Eq. (4.10) and a rational function R
regular at ρ12 = 0. This gives us relations of the type
c(n+2)(q1, . . . , φ1, φ2, . . . , qn) = c
(n+1)(q1, . . . , φ1 ∗ φ2, . . . , qn) . (4.14)
Next, having correlation functions of type
· v1(x1, x2)v2(x3, x4) ·
v(x1, x2)φ(x3) ·
of the form (4.10), one verifies that the limits (4.4) and (4.5)
exist within these correlation functions, and they yield expressions for
· (v1 ∗
x1, x4
· (v ∗ φ)
x1, x3
consistent with (4.10). As a result we
obtain again relations between the c’s:
c(n+2)(q1, . . . , v1, v2, . . . , qn) = c
(n+1)(q1, . . . , v1 ∗ v2, . . . , qn) ,
c(n+2)(q1, . . . , v, φ, . . . , qn) = c
(n+1)(q1, . . . , v ∗ φ, . . . , qn) . (4.15)
Finally, one verifies that setting x1 = x2 in
· v(x1, x2) ·
we obtain the
correlation functions
· γ(v)
with the relation
c(n+1)(q1, . . . , (v + v
t), . . . , qn) = 2 c
(n+1)(q1, . . . , γ(v), . . . , qn) . (4.16)
This completes the proof of Proposition 4.3 as well as the proof that the products
V ⊗ V
→V and V ⊗ F
→V are well defined. �
4.2 Associative algebra structure of the OPE
Note that Eqs. (4.14), (4.15) read (under (4.7))
q1, . . . , qk, qk+1, . . . , qn
= c(n−1)
q1, . . . , qk ∗ qk+1, . . . , qn
. (4.17)
This implies that the bilinear operation ∗ on Â is an associative product.
Indeed, consider the element q :=
q1 ∗q2
∗q3−q1∗
q2 ∗q3
for q1, q2, q3 ∈ Â.
By (4.7) q is a bilocal field. Equation (4.17) implies that all c’s in which q enters
vanish and hence, by Eq. (4.10) q has zero correlation functions with all other
Harmonic Bilocal Fields 23
fields, including itself. But then this (bilocal) field is zero by the Reeh–Schlieder
theorem, since its action on the vacuum will be identically zero.
Thus, introducing the cartesian product Â (4.6) was not only convenient for
combining three types of bilinear operations in one but also as a compact expres-
sion for the associativity (Eqs. (4.14), (4.15)). However, Â carries a redundant
information due to the following relation:
−γ(v),
(v + vt)
∗ q = 0 = q ∗
−γ(v),
(v + vt)
(4.18)
for every v ∈ V and q ∈ Â. To prove (4.18) we point out first that it is equivalent
to the identities v ∗ φ = γ(v) ∗ φ and v′ ∗ v = v′ ∗ γ(v) for v = vt ∈ V and any
φ ∈ F , v′ ∈ V. These identities can be established again first for the c’s, and
then proceeding by using the Reeh–Schlieder theorem, as in the above proof of
associativity.
Hence, the redundancy in Â is because we can identify symmetric bilocal
fields v = vt ∈ V with their restrictions to the diagonal, γ(v) ∈ F , and this
is compatible with the product ∗. Let us point out that the restriction of the
map γ to the t–invariant subspace Vs := {v ∈ V : v = v
t} is an injection into
F . The latter follows from a simple analysis of the 4-point functions of v and
the Reeh–Schlieder theorem: if v(x, y) = v(y, x) and 〈0|v(x, x)v(y, y)|0〉 = 0
then 〈0|v(x, x′)v(y, y′)|0〉 = 0. In this way we see that we can identify in Â the
symmetric harmonic bilocal fields v = vt with their restriction on the diagonal
γ(v) ∈ F .
Formally, the above considerations can be summarized in the following ab-
stract way. Let us introduce the quotient
A := Â
−γ(v),
(v + vt)
: v ∈ V
. (4.19)
It is an associative algebra according to Eq. (4.18). The involution t : Â → Â
can be transferred to an involution on the quotient (4.19) and we denote it by
t as well. The spaces F and V are mapped into A by the natural compositions
F → Â → A and V → Â → A. The injectivity of γ on Vs implies that the maps
F → A and V → A so defined are actually injections. Hence, we shall treat F
and V also as subspaces of A. Furthermore, A becomes a direct sum of vector
spaces
A = F ⊕ Va , (4.20)
q ∈ A : qt = q
= F ⊇ Vs
v ∈ V : vt = v
q ∈ A : qt = −q
= Va :=
v ∈ V : vt = −v
Hence, the t–symmetric elements of A are identified with the d = 2 local fields,
while the t–antisymmetric elements of A, with the antisymmetric, harmonic
bilocal (1, 1) fields. (Neither F nor Va are subalgebras of A.)
To summarize, the associative algebra A is obtained from Â by identifying
the space Vs of symmetric bilocal fields with its image γ
⊆ F .
Harmonic Bilocal Fields 24
For simplicity we will denote the equivalence class in A of an element q ∈ Â
again by q. Also note that the c’s can be transferred as well, to multilinear
functionals on A, since the kernel of the quotient (4.19) is contained in the
kernel of each c(n) by (4.16). We shall use the same notation c(n) also for the
multilinear functional c(n) on A.
Example 4.1. Let us illustrate the above algebraic structures on the simplest
example of a QFT generated by a pair of d = 2 GCI fields Φ1 and Φ2 given
by normal a pair of two mutually commuting free massless fields ϕj : Φ1(x) =
: ϕ21(x) : − : ϕ
2(x) :
and Φ2(x) = ϕ1(x)ϕ2(x). Their OPE algebra involves
a set of four independent harmonic bilocal fields Vjk(x1, x2) := :ϕj(x1)ϕk(x2) :
(j, k = 1, 2), which satisfy
Vjk(x1, x2)
= Vkj(x1, x2) = Vjk(x2, x1). For in-
stance, we have Φ1 ∗ Φ2 = V12 − V21.
7 Also note that Φ1 = γ(V1) for V1(x1, x2)
= :ϕ1(x1)ϕ1(x2) : − :ϕ2(x1)ϕ2(x2) :, etc.
By the associativity and Eq. (4.17) we have
q1, . . . , qn
= c(2)
q1 ∗ · · · ∗ qn−1, qn
(4.21)
for q1, . . . , qn ∈ A. Let us consider now c
(2) and define the following symmetric
bilinear form on A: 〈
q1, q2
:= c(2)
qt1, q2
. (4.22)
First note that F and Va are orthogonal with respect to this bilinear form: this is
due to the fact that there is no nonzero three point conformally invariant scalar
function of weights (2, 1, 1), which is antisymmetric in the second and third
arguments. Next, we claim that (4.22) is strictly positive definite. This is a
straightforward consequence of the Wightman positivity and the Reeh–Schlieder
theorem (one should consider separately the positivity on F and Va). In partic-
ular, (4.22) is nondegenerate. By Eqs. (4.13) and (4.17) we have:
q1 ∗ q2, q3
q2, q
1 ∗ q3
(4.23)
for all q1, q2, q3 ∈ A.
Let us introduce now an additional splitting of F . Denote by F0 the kernel
of the product, i.e.,
F0 :=
ψ ∈ F : ψ ∗ q = 0 ∀q ∈ A
ψ ∈ F : q ∗ ψ = 0 ∀q ∈ A
(4.24)
(the second equality is due to the identity φ ∗ q = (qt ∗ φ)t). Let F1 be the
orthogonal complement in F of F0 with respect to the scalar product (4.22):
F1 :=
φ ∈ F :
= 0 ∀ψ ∈ F0
. (4.25)
The meaning of fields belonging to F0 becomes immediately clear if we note
that c(n) for n > 3 are zero if one of the arguments belongs to F0 (this is due to
7i.e., in the OPE Φ1(x1)Φ2(x2) there appears the antisymmetric bilocal field V12(x1, x2) −
V21(x1, x2) that involves only odd rank conserved tensor currents in its expansion in local fields
Harmonic Bilocal Fields 25
Eq. (4.21)). Hence, all their truncated functions higher than two point are zero,
i.e., the fields belonging to F0 are generalized free d = 2 fields. Furthermore,
these fields commute with all other fields from F1 and Va ≡ A
(1): this is because
of the vanishing of c(2)(ψ, q) if ψ ∈ F0 and q ∈ F1 ⊕ Va, as well as of all
c(n+1)(ψ, q1, . . . , qn) for n > 2 if ψ ∈ F0 and q1, . . . , qn ∈ A (by (4.21) and
(4.24)).
Clearly, F1⊕Va is a subalgebra of A: this follows from Eq. (4.23) with q3 ∈ F0
along with the definitions (4.24) and (4.25). Let us denote it by
B := F1 ⊕ Va . (4.26)
We are now ready to state the main step towards the proof of Theorem 4.1.
Proposition 4.4. There is a homomorphism ι from the associative algebra B
into the algebra of Hilbert–Schmidt operators over some real separable Hilbert
space, such that
q1, . . . , qn
· · · ι
, (4.27)
and ι
are symmetric operators while ι
are antisymmetric.
We shall give the proof of this proposition in the subsequent subsection. The
main reason leading to it is that B becomes a real Hilbert algebra with an integral
trace on it. Here we proceed to show how Theorem 4.1 can be proven by using
the above results.
Proof of Theorem 4.1. Let Φm = Φ
m + Φ
m be the decomposition of each field
Φm according to the splitting F = F0 ⊕ F1. Take an orthonormal basis ψm in
F0 and let Φ
αm,j ψj , and βm =
βm,j,k
be the symmetric matrix
corresponding to the Hilbert–Schmidt operator ι
(m = 1, 2, . . . ). Then
Eqs. (4.3) and (4.27) show that the constants αm,j and βm,j,k so defined satisfy
the conditions of Theorem 4.1. �
Remark 4.1. In general, we have F1 % Vs. This is because the elements of
F1 correspond, by Proposition 4.4, to Hilbert–Schmidt symmetric operators and
on the other hand, the elements of V are obtained, according to the inductive
construction of Sect. 4.1, as products of elements of F and will, hence, correspond
to trace class operators.
4.3 Completion of the proofs
It remains to prove Proposition 4.4. We start with an inequality of Cauchy–
Schwartz type.
Lemma 4.5. Let q1, q2 ∈ A be such that each of them belongs either to F or to
Va. Then we have
q1 ∗ q2, q1 ∗ q2
q1 ∗ q1, q1 ∗ q1
q2 ∗ q2, q2 ∗ q2
. (4.28)
Harmonic Bilocal Fields 26
Proof. Consider
q1∗q1+λ q2∗q2, q1∗q1+λ q2∗q2
> 0 and use that
q1∗q1, q2∗q2
q1 ∗ q2, q1 ∗ q2
if each of q1, q2 belongs either to F or to Va. �
The space B (4.26) is a real pre–Hilbert space with a scalar product given by
(4.22). It is also invariant under the action of t (actually the eigenspaces of t are
F1 and Va). The left action of B on itself gives us an algebra homomorphism
ι : B → LinR B (4.29)
of B into the algebra of all operators over B. Moreover, the elements of F are
mapped into symmetric operators and the elements of Va, into antisymmetric
(this is due to (4.23)).
Lemma 4.6. Every element of B is mapped into a Hilbert–Schmidt operator.
Proof. Since B is generated by F1 (according to the inductive construction of F
and V in Sect. 4.1) it is enough to show this for the elements of F1.
Let φ ∈ F1 and consider the commutative subalgebra Bφ of B generated by
φ. The algebra Bφ is freely generated by φ, i.e., is isomorphic to the algebra
λR[λ] of polynomials in a single variable λ (↔ φ), since φ belongs to the orthog-
onal complement of F0 (4.24). For a p(λ) ∈ λR[λ] we shall denote by φ[p] the
corresponding element of Bφ. In particular,
φ[p1] ∗ φ[p2] = φ[p1p2]. (4.30)
Setting
φ∗(n+1) := φ∗n ∗ φ, c
:= c(2)
φ∗n, φ
φ∗n, φ
(4.31)
(φ∗1 := φ, n > 1) we obtain a positive definite functional over the algebra λ2R[λ]
∼= φ ∗ Bφ (due to Eq. (4.23) and the positivity of
(4.22)).
Then, by the Hamburger theorem about the classical moment problem ([9,
Chap. 12, Sect. 8]) we conclude that there exists a bounded positive Borel mea-
sure dµ
on R, such that
λ2 p (λ)
dµ(λ) (4.32)
for every p(λ) ∈ R[λ]. Using this we can extend the fields φ[p](x) to φ[f ](x)
for Borel measurable functions f having compact support with respect to µ
in R\{0}. The latter can be done in the following way. Fix ε ∈ (0, 1) and
let g1, . . . , gn be Schwartz test functions on M . By Theorem 4.2 the correla-
tors 〈0|φ[p1][g1] · · ·φ
[pn][gn]|0〉 depend polynomially on c
φ[pk1 ], . . . , φ
[pkj ]
pk1 · · · pkj
for all {k1, . . . , kj} ⊆ {1, . . . , n}. But for every ε ∈ (0, 1) there
exists a norm
‖q‖ε = Aε sup
|λ|6ε
qk(λ)
∣∣∣ + Bε
R \ (−ε, ε)
∣∣qk(λ)
∣∣ dµ(λ) (4.33)
Harmonic Bilocal Fields 27
on λ2R[λ] ∋ q(λ), where Aε and Bε are some positive constants, such that for
every q1, . . . , qm ∈ λ
2R[λ]
∣∣∣ c
q1(λ) · · · qm(λ)
]∣∣∣ 6
∣∣qk(λ)
dµ(λ)
‖qk‖ε .
Hence,
∣∣〈0|φ[p1][g1] · · ·φ[pn][gn]|0〉
∣∣ 6 C
‖pk‖ε ‖gk‖S for some constant C
and Schwartz norm ‖·‖S (not depending on pk and gk). Since for every ε ∈
(0, 1) the Banach space L1
R\{(−ε, ε)}, µ
is contained in the completion of
λ2R[λ] with respect to the norms (4.33), we can extend the linear functional
c[p(λ)] as well as the correlators 〈0|φ[p1][g1] · · · φ
[pn][gn]|0〉 to a functional c[f(λ)]
and correlators 〈0|φ[f1][g1] · · ·φ
[fn][gn]|0〉 defined for Borel functions f, f1, . . . , fn
compactly supported with respect to µ in R\{0}. Thus, we can extend the fields
φ[p] by extending their correlators.
By the continuity we also have for arbitrary Borel functions f, fk, compactly
supported in R\{0}:
φ[f1] ∗ φ[f2] = φ[f1f2], c(n)
φ[f1], . . . , φ[fn]
f1 · · · fn
f ] =
dµ(λ) (4.34)
(cp. (4.32)), and c(n) determine the correlation functions of φ[fk] as in Theo-
rem 4.2.
In particular, for every characteristic function χS of a compact subset S ⊂
R\{0} we have φ[χS ]∗φ[χS ] = φ[χS ]. Hence, for such a d = 2 field we will have that
all its truncated correlation functions are given by (4.2) with all normalization
constants c(n) equal to one and the same value c(2)
φ[χs], φ[χs]
. Then, as shown in
[12, Theorem 5.1], Wightman positivity requires this value to be a non-negative
integer, i.e.,
φ[χS ], φ[χS ]
dµ(λ)
∈ {0, 1, 2, . . . } (4.35)
(it is zero iff φ[χS ] = 0). Hence, the restriction of the measure dµ(λ)/λ2 to R\{0}
is a (possibly infinite) sum of atom measures of integral masses, each supported
at some γk ∈ R\{0} for k = 1, . . . , N (and N could be infinity). In particular,
the measure µ is supported in a bounded subset of R.
By Lemma 4.5 we can define ι(φ[f ]) as a closable operator on B if f is a Borel
measurable function with compact support in R\{0}. It follows then that the
projectors ι(φ[χS ]), for a compact S ⊆ R\{0}, provide a spectral decomposition
for ι(φ) (in fact, ι(φ[f ]) = f
). Thus, ι(φ) has discrete spectrum with eigen-
values γk (k ∈ N), each of a multiplicity given by the integer c
χ{γk} , φ
χ{γk}
Then ι(φ) is a Hilbert–Schmidt operator since
γ2k c
χ{γk} , φ
χ{γk}
dµ(λ)
R\{0}
dµ(λ) <∞
Harmonic Bilocal Fields 28
(µ being a bounded measure). �
The completion of the proof of Proposition 4.4 is provided now by the fol-
lowing corollary.
Corollary 4.7. For every q1, q2 ∈ B one has c
q1, q2
ι(q1)ι(q2)
Proof. If q1 = q2 ∈ F1 this follows from the proof of Lemma 4.6 and hence, by
a polarization, for any q1, q2 ∈ F1. The general case can be obtained by using
the facts that B is generated by F1 and c
(2) has the symmetry c(2)(q1 ∗ q2, q3) =
c(2)(q1, q2 ∗ q3). �
5 Discussion. Open problems
The main result of Sect. 4, the (generalized) free field representation of a system
{φa} of GCI scalar fields of conformal dimension d = 2 (Theorem 4.1), is obtained
by revealing and exploiting a rich algebraic structure in the space F × V of all
d = 2 real scalar fields and of all harmonic bilocal fields of dimension (1, 1).
However, this structure is mainly due to the fact that we are in the case of
lower scaling dimension: there is only one possible singular structure in the
OPE (after truncating the vacuum part). One can try to establish such a result
in spaces of spin–tensor bilocal fields (of dimension
) satisfying
linear (first order) conformally invariant differential equations (that again imply
harmonicity). If these equations together with the corresponding pole bounds
imply such singularities in the OPE, which can be “split” one would be able to
prove the validity of free field realizations in such more general theories, too.
One may also attempt to study models, say in a theory of a system of scalar
fields of dimension d = 4, without leaving the realm of scalar bilocal harmonic
fields V1 (of dimension (1, 1)). In [11] there have been found examples of 6–point
functions of harmonic bilocal fields, which do not have free field realizations.
However, our experience with the d = 2 case shows that in order to complete the
model (including the check of Wightman positivity for all correlation functions)
it is crucial to describe the OPE in terms of some simple algebraic structure
(e.g., associative, or Lie algebras).
On the other hand going beyond bilocal V1’s is a true signal of nontriviality
of a GCI model. Our analysis of Sect. 3 shows that this can be characterized
by a simple property of the correlation functions: the violation of the single
pole property (of Sect. 3.3). From this point of view a further exploration of
the example of Sect. 3.5 within a QFT involving currents appears particularly
attractive.
Note added in proof. In [19], we have determined the biharmonic function
whose leading part is given by Eq. (3.31). It involves dilogarothmic functions,
whose arguments are algebraic functions of conformal cross ratios. This exem-
plifies the violation of Huygens bilocality for the biharmonic fields, Theorem 3.7.
Harmonic Bilocal Fields 29
Yet, in support of Conjecture 3.3, it is shown that the structure of the cuts is in
a nontrivial manner consistent with ordinary bilocality.
Acknowledgements.
We thank Yassen Stanev for an enlightening discussion.
This work was started while N.N. and I.T. were visiting the Institut für
Theoretische Physik der Universität Göttingen as an Alexander von Humboldt
research fellow and an AvH awardee, respectively. It was continued during the
stay of N.N. at the Albert Einstein Institute for Gravitational Physics in Potsdam
and of I.T. at the Theory Group of the Physics Department of CERN. The paper
was completed during the visit of N.N. and I.T. to the High Energy Section
of the I.C.T.P. in Trieste, and of K.-H.R. at the Erwin Schrödinger Institute
in Vienna. We thank all these institutions for their hospitality and support.
N.N. and I.T. were partially supported by the Research Training Network of
the European Commission under contract MRTN-CT-2004-00514 and by the
Bulgarian National Council for Scientific Research under contract PH-1406.
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	Introduction
	Properties of GCI scalar fields
	Structure of correlation functions and pole bounds
	Twist expansion of the OPE and bi–harmonicity of twist two contribution
	Bilocality of twist two contribution to the OPE
	Convergence of harmonic decompositions
	Consequences of bi–harmonicity
	A necessary and sufficient condition for Huygens bilocality
	The case of dimension 2
	A d=4 6-point function violating the SPP
	The theory of GCI scalar fields of scaling dimension d=2
	Structure of the correlation functions
	Associative algebra structure of the OPE
	Completion of the proofs
	Discussion. Open problems
