Causality, Crossing and Analyticity in Conformal Field Theories
aa r X i v : . [ h e p - t h ] F e b February 5, 2021
Causality, Crossing and Analyticity in Conformal Field Theories
Jnanadeva Maharana Institute of PhysicsandNISERBhubaneswar - 751005, India
Abstract
Analyticity and crossing properties of four point function are investigated inconformal field theories in the frameworks of Wightman axioms. A Hermitianscalar conformal field, satisfying the Wightman axioms, is considered. Thecrucial role of microcausality in deriving analyticity domains is discussed anddomains of analyticity are presented. A pair of permuted Wightman functionsare envisaged. The crossing property is derived by appealing to the techniqueof analytic completion for the pair of permuted Wightman functions. Theoperator product expansion of a pair of scalar fields is studied and analyticityproperty of the matrix elements of composite fields, appearing in the operatorproduct expansion, is investigated. An integral representation is presented forthe commutator of composite fields where microcausality is a key ingredient.Three fundamental theorems of axiomatic local field theories; namely, PCTtheorem, the theorem proving equivalence between PCT theorem and weaklocal commutativity and the edge-of-the-wedge theorem are invoked to derivea conformal bootstrap equation rigorously. E-mail [email protected] . Introduction .The purpose of this article is to investigate the intimate relationships betweencausality, analyticity as well as crossing properties of four point functions in confor-mal field theories. The seminal paper of Mack and Salam [1] led to vigorous researchactivities in conformal field theory (CFT). The rapid progress of research during thatperiod following work of Mack and Salam [1] has been chronicled in several booksand review articles [2, 3, 4, 5, 6, 7, 8, 9].It is well known that wide class of physical systems exhibit scale invariance at thecritical points. The critical indices have universal character. Polyakov [10] had con-jectured that such theories might be invariant under the conformal symmetry whichencompasses the scale invariance. The conformal bootstrap program was initiatedby Migdal [11], by Ferrara and his collaborators [12, 13], by Mack and Todorov [14],and by Polyakov [15]. The proposed conformal bootstrap program synthesises twoimportant ingredients: (i) conformal invariance and (ii) operator produce expansion.Moreover, crossing symmetry plays a crucial role in the derivation of the consistencyconditions. We shall elaborate on some aspects of the latter and its relevance toour investigation in the sequel. The computation of the correlation functions in con-formal field theories is of paramount interest. A lot of attention has been focusedon the formal developments and to address important issues in conformal fiend the-ory itself. The bootstrap program is understood intuitively in the following sense[11, 12, 13, 14, 15]. The essential idea is that if we consider a 4-point correlationfunction with a given ordering of operators and another 4-point function where theoperator orderings are permuted then the two correlation functions are related ifcrossing symmetry is invoked. It becomes transparent when conformal partial waveexpansion is implemented in each channel. As a result, a set of consistency conditionsemerge from the requirements of bootstrap hypothesis. It is accepted that the two cor-relation functions are analytic continuation of each other. The interests in conformalbootstrap have been revived with renewed vigor in the recent past [7, 8, 9]. Causality,crossing and analyticity are three pillars of the conformal bootstrap paradigm.The bootstrap concept stems from the S-matrix approach [16, 17] to understandthe hadronic interactions phenomenologically in an era prior to the discovery of QCDas the theory of strong interactions. It will be useful to consider an illustrative exam-ple. A vast number of meson and baryon resonances were produced in high energyaccelerators. Let us consider meson-meson scatterings: (i) a + b → c + d and (ii) a + ¯ c → ¯ b + d . The first process corresponds to a direct channel reaction and thelatter is the crossed channel one. In the low energy region, in each channel, the twoprocesses are dominated by exchanges of resonances. Notice, however, that crossingsymmetry relates the two amplitudes. The allowed kinematical variables for each ofthe two reactions do not overlap. To state it more explicitly; for the direct channelprocess, s ≥ m and t ≤
0, whereas in case of crossed channel reaction, s ≤ t ≥ m . Here equal mass scattering is assumed, m being the mass of the external1article. In the fist case, s is c.m energy squared and t is momentum transfer squared;however, for crossed channel reaction the roles are reversed. If crossing is assumedthen a set of equations relating coupling constants and masses involved in two chan-nels get related as consistent conditions. Thus the bootstrap hypothesis was testedphenomenologically. It was a challenge to construct a crossing symmetric amplitudeand Veneziano [18] succeeded in presenting such an amplitude. The crossing symme-try was rigorously proved in the framework of axiomatic field theory by Bros, Epsteinand Glaser[19]. Let us elaborate this aspect for future reference. The microcausality,as one of the axioms of LSZ [20] formulation, was the main ingredients to demon-strate that the absorptive parts of s-channel and u-channel amplitudes of the 4-pointfunctions coincided in an unphysical domain of real kinematical variables. It wasproved rigorously from the theory of several complex variables for analytic functionsthat these two absorptive amplitudes are analytic continuation of each other. Theconformal bootstrap program was initiated to provide a field theoretic basis to thebootstrap paradigm of the phenomenological S-matrix philosophy. There have beenstimuli for further research in conformal bootstrap proposal due to its applicationsto variety of physical problems. The research in conformal field theories has spreadin diverse directions such as in a large class of gauge theories and supersymmetrictheories. Moreover, studies of structure of two dimensional conformal field has flour-ished and has influenced research in several directions. It is recognized that conformalsymmetry has played a key role in establishing the AdS ↔ CF T correspondencewhich has led to spectacular developments in our understanding of the relationshipsbetween string theory and quantum field theories [21]: a realization of the gauge-gravity duality.A very important attribute of the conformal field theory approach is that onecomputes the correlations functions of the product of field operators rather then theS-matrix elements. It is worthwhile to discuss this point. In the axiomatic fieldtheory approach [20, 22, 23, 24, 25, 26, 27, 28], it has been demonstrated that howthe analyticity of the scattering amplitude is intertwined with microcausality. Theanalyticity properties of amplitude are rigorously proved from axioms of general fieldtheories. Let us recall some of the important results which are derived from a set ofaxioms introduced by Lehmann, Symanzik and Zimmermann (LSZ)[20] who laid thefoundations of general field theories. The S-matrix elements are computed from thereduction technique of LSZ.Notice, however, that while computing the scattering amplitudes [20], the exter-nal particles are on the mass shell. The analyticity properties of the amplitude arereflected through the dispersion relations. The dispersion relations are proved fromfundamental principles such as Lorentz invariance and microcausality. There are twoadditional important ingredients in the LSZ formulation besides the stipulated pos-tulates: (i) The existence of (asymptotic) in and out fields so that a complete set of2perators can be constructed in terms of either in or out field. And (ii) it requiresthat there are massive particles in the theory. Therefore, the second assumption,intuitively implies the existence of short range forces in the theories under considera-tion. The analyticity and crossing symmetry of the S-matrix are proved in axiomaticframeworks for massive field theories. These results are derived in the linear pro-gram without invoking the the unitarity property of S-matrix which is a nonlinearconstraint. One of the most important accomplishments of the axiomatic approachis to study analyticity property of the amplitude in order to determine the domain ofholomorphy. Moreover, in order to identify the domain of analyticity of the amplitudein t , the momentum transferred squared, the Jost-Lehmann-Dyson [29, 30] represen-tation played a very crucial role. This paved the way to prove the fixed- t dispersionrelations. However, Bogoliubov [28] has developed an alternative technique to go offmass shell for the external particles and he independently proved fixed- t dispersionrelations for scattering amplitudes. The analyticity properties of the S-matrix areutilized to prove rigorous theorems, usually expressed as upper and lower bounds, onexperimentally measurable parameters. Therefore, if any of those bounds are violatedin high energy experiments then some of the axioms will be questioned. There is noevidence for violation of any of the bounds so far.There are formidable problems when the spectrum of a theory contains only mass-less particles. The pure Yang-Mills theory, Einstein’s theory of gravitation and someconformal field theories belong to this category. Our intuition guides us to the root ofthe problem. Notice that, in such theories, the presence of massless particles wouldlead to long range forces. Consequently, there are conceptual difficulties in definingthe asymptotic in and out states. Therefore, the LSZ technique is not suitable, whiledealing with some of the theories noted above. As we shall discuss later, conformalfield theories do not admit a discrete mass parameter. Therefore, the LSZ formalism,as presented in its original formulation, is not quite suitable in the context conformalfield theories. The Poincar´e group is contained in the conformal group, the enlargedsymmetry group. The theories are severely constrained when the conformal symme-try is enforced. One notable feature is that the states are not labelled by a ( mass ) ,unlike the Poincare invariant conventional field theories. We recall that the genera-tors of the spacetime translations, P µ , does not commute with the Casimir operatorsconstructed from the generators of conformal group. Whereas, in the context of con-ventional QFT formulations, P = P µ P µ is a Casimir. The other Casimir operatoris [23] W µ W µ where W µ = ǫ µνρτ P ν M ρτ , the Pauli-Luansky vector. The eigenvalue ofthe second Casimir is associated with the helicity of the particle. Therefore, a statevector is represented by its mass and helicity in quantum field theories. We shalldiscuss in the sequel, some aspects of the unitary irreducible representations of theconformal field theories which are quite different from those of the Poincar´e invariantquantum field theories. 3t is evident from the preceding discussions that computation of the scattering am-plitudes, as stipulated by LSZ for theories satisfying their axioms, cannot go throughin a straightforward manner in case of CFT. Thus it is not possible derive an ex-pression for the scattering operator rigorously using the LSZ reduction technique.Recently, however, an interesting development has provided evaluation of form factorand scattering amplitude from the perspectives of LSZ technique [31] by incorporat-ing a theorem from LSZ paper. They define form factors and scattering amplitudesin conformal field theories. These are the coefficient of singularity of the Fouriertransform of time-ordered correlation functions as the limit p i → p i standsfor four momenta of external legs. The form factor, F , is extracted from the fourpoint function. It is shown that F is crossing symmetric, analytic and it admits apartial wave expansion. We note that the authors work in Lorentzian signature met-ric. They obtain momentum space representation for the four point function from thetime ordered product of four field operators. Indeed, this investigation is an endeavorto address important issues such as analyticity and crossing symmetry from the per-spective of LSZ, however, they have circumvented the notion of explicitly introducing’in’, ’out’ fields as well as interacting fields in their formulation. Moreover, the ideaof interpolation, how interacting field field interpolates into ’in’ and ’out’ fields, is notutilized.We subscribe to the philosophy that it is desirable to compute correlation functionsin conformal field theories. As a consequence, we do not invoke the concept of scatter-ing operator in our approach. We shall unravel the relationship between analyticity,causality and crossing as we proceed. In recent times, the Euclidean formulation ofconformal field theory has been widely adopted. There are certain advantages in thisapproach while the position space description is utilized. The merits of Euclideanspace formulation and its power has been discussed in [7, 8, 9]. On the other handthe Minkowski space (Lorentzian signature metric) formulation has its own meritsas has been discussed in [32, 33]. We feel that the choice of the Lorentzian metricis more suitable for establishing relationship between causality and analyticity as isevident from the rigorous results derived in axiomatic field theories. As a motivation,we present an illustrative example which is a very simplified version of Toll’s work inthe context of dispersion relation [34].Let us consider a wave packet, ψ ( z, t ), moving with velocity c along the z -axis.The target is located at the point z = 0. The Fourier transform of ψ ( z, t ) is f ( ω ) andwe can write ψ ( z, t ) = 1 √ π Z + ∞∞ dωf ( ω ) e iω ( zc − t ) (1)The spherically symmetric scattered wave has the form S ( r, t ) = 1 r √ π Z + ∞−∞ ˜ S ( ω ) f ( ω ) e iω ( zc − t ) (2)4ere r is the radial distance from z = 0. We may invert eq. (1) to get f ( ω ) = 1 √ π Z + ∞−∞ dtψ (0 , t ) e iωt ) (3)We recognize that if the wave packet does not reach the origin, z = 0, then there willbe no observation of scattered wave; in other words if ψ (0 , t ) = 0 , for t < f ( ω ) is regular for Im ω >
0; i.e. it is regular on the upper half of the complex ω plane.For us, causality, in the present context, means no scattered wave would arrive at adistance r from the origin after an interval rc when the incident wave has not reached z = 0. S ( r, t ) = 0 , for ( ct − r ) < . (5)Let us take inverse Fourier transform of (2). We conclude that f ( ω ) ˜ S ( ω ) is analytic inthe upper half ω plane; i.e. it is analytic when Im ω >
0. Therefore, ˜ S ( ω ), the Fouriertransform of the amplitude, is analytic for Im ω >
0. This property is exploited towrite a dispersion relation. In the context of special relativity velocity of light, c ,is the limiting velocity and therefore, two points separated by spacelike distance arecausality disconnected. In the context of quantum theory, if there are two Hermi-tian operators located at spacelike distance we can make observations simultaneously.In relativistic quantum field theories the causality is stated to be [ O ( x ) , O ( x ′ )] =0 , if ( x − x ′ ) < O ( x ) and O ( x ′ ) are two local operators. The analyticityproperties of scattering amplitude crucially depend on the axiom of microcausality.The dispersion relations rest primarily on this axiom. In order to understand con-cepts such as causality and crossing intuitively it is desirable to work in a Minkowskispacetime manifold with a Lorentzian metric. Our adopted signature for the flatspace metric is g µν = diag (+ , − , − , − ).Moreover, we shall attempt to reveal how these three fundamental properties areintertwined in the Wightman formulation of field theories in the context of conformalfield theories. The necessary ingredients, to explore these aspects would be discussedat an appropriate juncture. The correlation functions are the vacuum expectation val-ues of field or local operators i.e. the Wightman functions. However, it must be bornein mind that not all conformal fields fulfill the requirements of Wightman axioms.We shall discuss this aspect in the later part of this section. Another advantage ofadopting the Wightman’s formulation is that the Wightman functions are boundaryvalues of analytic functions. It facilitates to establish relationship between causalityand analyticity and subsequently paves the way to an understanding of crossing once5e define analytic functions from the Wightman functions.The operator product expansion (OPE) has played a very important role in theadvancements of conformal field theories in recent years. Wilson [35] pioneered thetechnique of operator product expansion in QFT. His seminal works have profoundlyinfluenced research in diverse branches of physics. Wilson’s theory was further ad-vanced rigorously by Wilson and Zimmermann [36] in the frameworks of Wightmanaxioms [24]. It was shown that under certain conditions, the operator product expan-sion can be derived giving complete information on short distance behaviors. Theyprovided procedures to construct composite field operators from the perspectives ofWightman axioms, supplemented by extra hypothesis, relevant for operator productexpansions in QFT. It was argued, by Wilson and Zimmermann that , when theproducts of local operators are envisaged singularities invariably occur as separationbetween the two operators tends to vanish. These singularities appear as a conse-quence of relativistic invariance and positive definite metric in Hilbert space [37].Subsequently, Otterson and Zimmermann [38], concentrated on Wilson expansion forproduct of two scalar field operators: A ( x ) A ( x ). They demonstrated, following theapproach of [36], that this operator product could be used to define composite localoperators i.e. local with respect to ( x + x ) /
2. The OPE also depends, in addition,on a vector, ζ µ , which is proportional to distance ( x − x ) µ between A ( x ) and A ( x ).They investigated locality and analyticity properties of appropriate matrix elementsfrom the Wightman axiom point of view. Their conclusions are based on rigoroustechniques and are quite robust.We focus our attentions to study the consequences of locality and analyticity inconformal field theories and it will be carried out in the light of the works of [36] and[38]. Moreover, the procedures of [38] is appropriately adopted for the conformal fieldtheories. In the context of CFT, the OPE imposes severe constraints on the structuralframeworks of the theories. Furthermore, it enables us to extract important conclu-sions without resorting to any specific model i.e. without introducing a Lagrangiandensity or an action. We are aware that the issue of convergence of OPE in conformalfield theory is quite pertinent. The derivation of the conformal bootstrap equation isintimately connected with the convergence of conformal partial wave expansion. Itis worth while to note that, in the context of conformal field theories, supplementarypostulates are needed in addition to the Wightman axioms. Microcausality is a car-dinal principle of axiomatic local field theories. Therefore, for a theory with threeimportant ingredients such as (i) an enlarged symmetry like the conformal invari-ance, (ii) microcausality and (iii) the power of operator product expansion; some ofthe important and most general attributes are extracted. For example, the two pointfunction and the three point function of a CFT are determined up to multiplicativeconstant factors. Furthermore, very important characteristics and structure of fourpoint functions are understood from these ingredients.6he crossing symmetry, in the context of conformal field theories in conjunctionwith the power of operator product expansion, is generally discussed in the coordinatespace description. Let us consider the three point function for a scalar field where thefield operators are located at spacetime points x , x and x and is defined as follows. W ( x , x , x ) = < | φ ( x ) φ ( x ) φ ( x ) | > . If we consider a permuted field configura-tion then W ( x , x , x ) = < | φ ( x ) φ ( x ) φ ( x ) | > ; the two correlation functions areequal i.e. W ( x , x , x ) = W ( x , x , x ) when ( x − x ) < x µ and x µ separation is spacelike. The essential point to note is that it is necessary to provethat W ( x , x , x ) and W ( x , x , x ) are analytic continuation of each other. Similararguments can be invoked for the four point functions as well as for n -point functionswhile discussing crossing.We hold the view that the study of the analyticity properties of the correlationfunctions in CFT is best accomplished in the Wightman’s formulation. The Wight-man functions are considered to be boundary values of analytic functions of severalcomplex variables. Let us consider an n-point Wightman function W n ( x , x , ....x n ) = < | φ ( x ) φ ( x ) .......φ ( x n ) | > . It is the boundary value of an analytic function. If wepermute location of field operators then another corresponding Wightman functionwould be defined which would be boundary value of another analytic function. Ourtask, in the context of CFT, is to investigate analyticity properties of four pointWightman functions. Consider the four point function W = < | φ ( x ) φ ( x ) φ ( x ) φ ( x ) | > .It is crossing symmetric if we compute W using standard OPE technique as long asall four points are separated by spacelike distances, modulo the issues related to theconvergence of the operator product expansions. Indeed, the consistency requirementsfor the four point amplitude, computed by contraction of pair of field (for differentconfigurations), leads to the so called conformal bootstrap program (see section 3 forthe work in this direction). We would like to address the following question: how farwe can proceed within an axiomatic framework to infer about crossing and analyticityproperties of the correlation functions of conformal field theories?Our goal is modest. We intend to study only a class of four dimensional confor-mal field theories which respect Wightman axioms. The generators of the conformalgroup are: (i) the ten generators of the Poincar´e group i.e. spacetime translations andLorentz transformations respectively denoted by P µ and M µν . (ii) The five additionalgenerators are: the dilatation operator, D, ( associated with the scale transforma-tion) and the four generators corresponding to special conformal transformations aredenoted by K µ . Thus there are altogether fifteen generators in D = 4. The scalarfield, φ ( x ), transforms as[ P µ , φ ( x )] = i∂ µ φ ( x ) , [ M µν , φ ( x )] = i ( x µ ∂ ν − x ν ∂ µ ) φ ( x ) (6)[ D, φ ( x )] = ( d + x ν ∂ ν ) φ ( x ) , [ K µ , φ ( x )] = i ( x ∂ µ − x µ x ν x ν − x µ d ) φ ( x ) (7)7here d is the scale dimension of the field φ ( x ). The above transformation rules aremodified appropriately when we consider tensor fields. The fifteen generators of theconformal group satisfy the algebra of SO (4 , C = 12 M µν M µν − K µ P µ − iD − D (8) C = − (cid:18) W µ K µ + K µ W µ (cid:19) − ǫ µνρτ M µν M ρτ (9) C = 14 (cid:20) K µ K µ P ν P ν − K µ M µν M νρ P ρ − K µ M µν P ν ( D + 6 i )+ 34 ( M µν M µν ) + 116 ( ǫ µνρτ M µν M ρτ ) + M µν M µν ( D + 8 iD − C − − D − iD +80 D + 128 iD + 36 C − iC D − C D (cid:21) (10)where W µ = ǫ µνρτ P ν M ρτ and we remind that P µ does not commute with C , C and C . The expressions for the Casimir operators for four dimensional Minkowskispace, as given above, are according to the notations and conventions of Fradkin andPlachek [5, 39].Let us discuss Wightman approach in the context of conformal field theories. Theanalyticity properties structure of conformal field theories was systematically inves-tigated by L¨uscher and Mack [40] extensively. We shall discuss some of the salientaspects in the next section. Let ˜ φ ( p ) denote the Fourier transform of φ ( x ). Noticethat the Fourier transform, ˜ φ ( p ), of every conformal field does not satisfy spectralitycondition of Wightman axioms [22]; i.e. p ≥ p ≥
0. A special type of con-formal fields fulfill the spectrality condition. These are designated as nonderivativefields [41]. They satisfy the constraint [ φ (0) , K µ ] = 0. Note that x -dependence of thefield can be accomplished through the transformation: φ ( x ) = e − iP.x φ (0) e iP.x . Weshall consider the nonderivative conformal fields, denoted by φ ( x ), throughout thisinvestigation. A state belonging to a unitary irreducible representation is specified byappropriate quantum numbers of the covering group of SO (2 , SO (4 ,
2) to be SU (2 , i)The vacuum : The vacuum is denoted as | > . The vacuum is conformally invariantand therefore, the generators of the conformal group annihilate the vacuum. ( ii) The Hilbert Space : A radically new approach is invoked in the study of conformalfield theories (CFT). The structure of the Hilbert space, in the case of conformal fieldtheories, is different which would be alluded to later [5].We discuss briefly the correspondence between local operators and states. Thisconcept plays a crucial role in understanding of the structure of the Hilbert space inconformal field theories. The state may be constructed either in the momentum spacerepresentation or in the coordinate representation as well. Consider a local operator, O ( x ), acting on the vacuum. The operator ↔ state correspondence is O ( x ) | > = |O ( x ) > (11)In fact we could consider the operator at x = 0, i.e. O (0) and generate x -dependenceby a translation operation. The momentum space state vector is |O ( p ) > = Z d xe ip.x |O ( x ) > (12)Therefore, the vacuum expectation values of product of operators can be evaluatedeither for those in p -space representations or those in x -space representations. In theframework of conformal field theories, we have a set of fields. One does not envisage amodel or a Lagrangian density with a single field or a set of fields while investigatinggeneral structure of CFT. Thus the Hilbert space is constructed from such a set offields: { Φ m } : Φ ( x ) , Φ ( x ) , ....... (13)Each field, Φ m ( x ), carries a scale dimension, d m , and might be endowed with its owntensor structure; moreover, it might be characterized with internal quantum numbers.The product of two conformal fields is expanded in terms of complete set of local fieldswith C-number coefficients. Therefore, we need an infinite set of local fields whichbelong to unitary irreducible representation of the conformal group. Generically weexpress the product of a pair asΦ( x )Φ( x ) = ∞ X m =0 A m Z C m ( x : x , x )Φ m ( x ) d x (14)where { Φ m ( x ) } are the set of fields belonging to irreducible representation of theconformal group and C m ( x ; x , x ), the C-number functions, which have singular be-havior in the short distance limit. A m are a set of constants, may be interpreted ascoupling constants and these are determined from the dynamical inputs of the spe-cific theory under considerations. We may associate a state vector with each of thefields in { Φ m ( x ) } from the hypothesis of state ↔ operator correspondence alluded to9lready. A state vector is created when a field operator acts on the conformal vacuumas noted earlier. Thus a state in the Hilbert space is defined | Φ m > = Φ m | > (15)The full Hilbert space is decomposed into direct sum of mutually orthogonal spacessince two normalized states | Φ m > and | Φ n > with respective scale dimensions d m and d n are orthogonal i.e. < Φ m | Φ n > = δ m,n . In fact each of the states belongingto unitary irreducible representations of the conformal group constitute subvectorspaces. Therefore, the full Hilbert space, H , decomposes into direct sums as H = ⊕H χ (16)where H χ is the subspace where the complete set of state vectors are created by com-plete set of fields Φ χ which belong to an irreducible representation of the conformalgroup. Here χ stands, collectively, for all the quantum numbers that characterize theirreducible representation. A noteworthy feature is the closure of the algebra in thesense that the product of two fields can expanded in terms fields of conformal fieldtheory: symbolically; Φ m Φ n ∼ P l Φ l .The choice of Lorentzian signature to study the properties of conformal field the-ories has certain advantages as has been noted in the early phase of the research inCFT [1, 2, 11, 15, 5]. Moreover, the Wightman function as VEV of the product of fieldoperators and their variances in the form of the VEV of R-products or A-product offields have been utilized for good reasons. For example, Polyakov [15] had emphasizedfor adopting R-product and A-product of a pair of fields to evaluate correspondingWightman function. Its importance was noted while computing the discontinuity offour point function in connection with the conformal bootstrap equations.Let us consider the three point Wightman function for scalar fields with scaledimension d . Our interest lies in the study of analyticity properties. Therefore,we adopt the iǫ prescription for all the Wightman functions in what follows: ( seemore details in subsequent next section when we complexify the coordinates). Aftersuitably implementing the conformal transformations on each of the scalar fields, thethree point Wightman function assumes the following form. W ( x , x , x ) = < | φ ( x ) φ ( x ) φ ( x ) | > = g ( x − iǫx ) − d/ ( x − iǫx ) − d/ ( x − iǫx ) − d/ (17)The above expression is presented for the scalar field, φ ( x ); however, the expressionfor three point function of three arbitrary field is already known in the literature.Here g is interpreted a the coupling constant.10hen fields are separated by spacelike distances i.e. ( x − x ) <
0, ( x − x ) < x − x ) < W ( x , x , x ) = W ( x , x , x ) = W ( x , x , x ) as noted earlier; expressed ex-plicitly < | φ ( x ) φ ( x ) φ ( x ) | > = < | φ ( x ) φ ( x ) φ ( x ) | > = < | φ ( x ) φ ( x ) φ ( x ) | > (18)This is a consequence of the axiom of microcausality. Indeed, the equation relatingdifferent 3-point function (18) for the permutations of the field operators is the cross-ing relation for vertex functions. In order to recognize the importance of the studyof analyticity, let us examine the expressions for a pair of three point functions. Inthe case of the Wightman functions these are boundary values of analytic functions(see later). Consider the expressions for W and W W = g ( x − iǫx ) − d/ ( x − iǫx ) − d/ ( x − iǫx ) − d/ (19)and W = g ( x − iǫx ) − d/ ( x − iǫx ) − d/ ( x − iǫx ) − d/ (20)We note that W and W do not necessarily coincide when x , x and x aretimelike. Let us compare each term of (19) and (20) on the r.h.s and especially fo-cus attention on the second term of (19) and the third term of (20). The former is( x − iǫ ( x − x ) ) − d/ whereas the latter is ( x + iǫ ( x ) ) − d/ . Therefore, whenwe approach the real axis in the complex plane we must approach it from oppositedirections. Thus in order to provide a proof of crossing we have to appeal to the themethod of analytic completion. In recent years, the study of the analyticity proper-ties in CFT have attracted attentions [46, 47, 48, 49, 50, 32, 33].It is more interesting to consider the a pair of 4-point Wightman function, withthe iǫ prescription, which have the following form in the convention of [9] W ( x , x , x , x ) = < | φ ( x ) φ ( x ) φ ( x ) φ ( x ) | > = [ 1(˜ z − iǫ ˜ z )(˜ z − iǫ ˜ z ) ] d F ( Z , Z ) (21)for the case when we have only one scalar field, φ ( x ), with scale dimension d ; and˜ z = x − x , ˜ z = x − x , ¯ z = x − x , ˜ z = x − x , ˜ z = x − x , ˜ z = x − x .Here Z and Z are the cross ratios Z = (˜ z − iǫ ˜ z )(¯ z − iǫ ˜ z )(˜ z − iǫ ˜ z )(˜ z − iǫ ˜ z ) (22)and Z = (˜ z − iǫ ˜ z )(˜ z − iǫ ˜ z )(˜ z − iǫ ˜ z )(˜ z − iǫ ˜ z ) (23)11 is a function which depends on cross ratios and its form is determined by themodel under considerations. The discussions of crossing operation is to be treatedwith care. For example, in order to establish crossing, it is desirable to show how W ( x , x , x , x ) is analytically continued to W ( x , x , x , x ) . They coincide forreal x and x when ( x − x ) < A ( x ) , B ( x ) and C ( x ). This proposal was suitably modified by Streater [57] toimplement the technique of analytic completion for a pair of Wightman functions. Weadopted Streater’s prescription to prove crossing for a pair of three point functionsby in CFT applying method of analytic completion. We recall, if we permute a pairof field to obtain a new three point function from a given configuration then, it wasshown that, the two vertex functions are analytic continuation of each other. Oneremark is quite pertinent at this stage. The Wightman functions are boundary values12f analytic functions of several complex variables of complexified coordinates as willbe defined in the next section. The important point is these analytic functions aredefined over a domain known as extended tubes (see Section 2 for details). Thereis a Wightman function defined for a given ordering of fields. There a Wightmanfunction for each permutation of the field ordering. Moreover, corresponding to eachWightman function there is an analytic function, defined over an extended tube andthe Wightman function is its boundary value. The domain of holomorphy is the unionof the extended tubes on which the permuted Wightman functions are defined. Itis argued that the set of analytic functions so obtained are analytic continuation ofeach other. Notice that Wightman function, defined as vacuum expectation valueof product of field operators is not related to another Wightman function obtainedfrom permutation of the fields. Thus it might not be possible, in our approach, toidentify the envelope of holomorphy when the union of domain of analyticities of allpermuted Wightman functions taken. The reason is that the envelope of holomorphymight be larger than the union of the domains of analyticities. We shall elaborateon this aspect in Section 2. We carry out a detailed study, adopting an argument ofJost which provides a relationship between the retarded three point function and thethree point Wightman function. This result is appropriately tailored for conformalfield theory.Section 3 deals with operator product expansion in conformal field theories adopt-ing the formalism of Wilson and Zimmermann. The prescription laid down by Otter-son and Zimmerman [38] for operator product expansion of two field operator in theframeworks of Wilson is quite rigorous. We employ the procedure for operator productexpansion of nonderivative conformal fields. The matrix elements of the compositefield are investigated to examine consequences of microcausality. It is shown howa theorem, analogous to the Jost-Lehmann-Dyson theorem can be proved. Further-more, the analyticity property of the matrix elements are derived to prove crossingfor the four point function. In the later part of third section (in the subsection), weappeal to PCT theorem to derive conformal bootstrap equation in a novel way. ThePCT operation, applied to a four point Wightman function, transforms into anotherWightman function. These two four point functions are equal if PCT is a symmetry.The equivalence between PCT theorem and weak local commutativity (WLC) playsa very important role in the derivation of the equation. First, the conformal partialwave (CPW) expansion technique is employed to the each of the four point equationswhich are related by WLC. The two functions coincide at the Jost point for real val-ues of spacetime coordinates when their separation is spacelike. As we shall explainin Section 3, the two Wightman functions are boundary values of analytic functionsand they are holomorphic as complex valued functions. We argue that these two fourpoint functions are analytic continuation of each other. Thus, we present a rigorousderivation of the conformal bootstrap equation. In a short communication [58], wehave reported part of this result. The summary of our work and conclusions are13ncorporated in Section 4.
2. The Wightman Functions in Conformal Field Theory .We study the crossing and analyticity properties of the four point function of a con-formal field, φ ( x ), in this section. The crossing symmetry relates two permutedWightman functions. We have noted, in Section 1, the motivations for the studyof crossing in CFT. It was pointed out that two Wightman functions, where a pairof fields get interchanged, coincide when the corresponding pair of spacetime coor-dinates are separated by spacelike distance. The conformal bootstrap equation isderived under such a condition. This is not the full story. The proof of crossingin QFT, from the axiomatic stand point, demands more. It is necessary to provethat the pair of functions have analytic continuations to their domain of holomorphyalthough the pair coincide at certain spacetime point. Therefore, the domain of holo-morphy of each of the functions have to be identified. This is the first task. The roleof OPE in CFT has been emphasized already. In turn, as was noted in the previoussection, the Hilbert space structure of CFT is different from that of the conventional(axiomatic) fields theories. We discuss this aspect very briefly in the sequel. We haveemphasized the relationship between microcausality and analyticity. A simple way tobring out their intimate relationship is to consider crossing property of the three pointWightman function. This problem has been studied recently [55]. We briefly recallessence of this work, later in this section and incorporate our new results. It will seta background to study relation between microcausality and analyticity properties ofthe four point function. Moreover, we invoke the arguments of [55] to prove crossingfor four point function. .The Wightman functions [22] are vacuum expectation values of field operators. Theyare envisaged as boundary values of analytic function of several complex variables.Wightman [59] has argued that if all the vacuum expectation values < | φ ( x ) φ ( x ) ...φ ( x n ) | > for all the permutations of φ ( x ) ....φ ( x n ) are given then the operator φ ( x ) is deter-mined. Our intent is to consider the Wightman functions for the scalar conformalfield φ ( x ) and study their analyticity properties and crossing symmetry. The axiomsare: A1 . Invariance of the theory under proper inhomogeneous Lorentz group. A2 . The existence of vacuum. There exists a Hilbert space, H , spanned by thephysical state vectors. The states have nonnegative energy spectrum i.e. p ≥ p ≥
0. There exists a vacuum, | > , the lowest energy state such that if P µ is theenergy momentum operator, P µ | > = 0. The vacuum is unique and stable. A3 . There exist field operators which are tempered. In other words, vacuum expec-tation values of operators are tempered distributions in the Schwartzian sense. A4 . Local commutativity. Expressed in another way; local operators commute (for14osons) or anticommute (for fermions) when they are separated by spacelike distance.For bosonic local operators,[ O ( x ) , O ( x ′ )] = 0 , if ( x − x ′ ) < W n ( x , x , ...x n ) is W n ( x , x , ...x n ) = < | φ ( x ) φ ( x ) ...φ ( x n ) | > (25)We know that these vacuum expectation values are not ordinary functions but aredistributions. They are to be interpreted as linear functionals as defined below W [ f ] = Z d x ..d x n W n ( x , x , ...x n ) f ( x , x ...x n ) (26)Thus we assign a complex number with the introduction of f ( x , x ..x n ) in the abovethe functional. Moreover, f ( x , x ...x n ) are infinitely differentiable functions and theyvanish outside a bounded domain of the four dimensional spacetime manifold. Notethat fields are operator valued distributions. Consequently, the spacetime average ofoperators are interpreted as observables. Thus operators of the form φ [ f ] = Z d xφ ( x ) f ( x ) (27)are meaningful. It is assumed that φ [ f ] is defined as the class of all infinitelydifferentiable functions of compact support in spacetime. In this light, note that W φ [ f ] = < | φ [ f ] | > is a linear functional with respect to f . More refined statementscan be made by introducing a sequence of test functions { f n ( x ) } and by specifyingconvergence properties [25]. In the optics of the preceding discussions the n-pointfunction W n ( x , x , ...x n ) = < | φ ( x ) φ ( x ) ...φ ( x n ) | > is a distribution in each of thevariables x i W n [ f , f , ....f n ] = < | φ ( f ) φ ( f ) .....φ ( f n ) | > (28)We can give a precise interpretation to W n ( x , x , ...x n ) in terms of the infinitelydifferentiable test functions { f n } . Now on, when we deal with Wightman functions W n ( x , x , ...x n ), it is understood that they have interpretations as alluded to above.As consequence of translational invariance of the theory; we conclude that W n ( x , x , ....x n )depends on the difference of coordinates W n ( x , x , ....x n ) = W n ( y , y ....y n − ) (29)where y i = x i − x i +1 . Moreover, W n ( y , y ....y n − ) are invariant under inhomogeneousLorentz transformations; for a real orthochronous Lorentz transformation W n ( y , y ....y n − ) = W n (Λ r y , Λ r y .... Λ r y n − ) (30)15here Λ r is a real Lorentz transformations, det Λ r = 1. Local commutativity:
It follows from axiom ( A
4) that i.e. [ φ ( x ) , φ ( x ′ )] = 0 if( x − x ′ ) <
0. Consequently, W n ( x , x , ....x j , x j +1 , ....x n ) = W n ( x , x , ....x j +1 , x j , ....x n ) , if ( x j − x j +1 ) < f W ( p , p , ...p n − ) denote the Fourier transform of W n ( y , y ....y n − ) then W n ( y , y ....y n − ) = Z d p d p ...d p n − e − i P n − j =1 p j .y j ˜ W n ( p , p ...p n − ) (32)From the temperedness property we know that W n -functions have at most polynomialgrowth at infinity. It is required that spectrum of the physical states must havetimelike four momenta and positive energy. Therefore, from the stability of vacuumand support condition f W n ( p , p ...p n − ) = 0 , unless p i ≥ , p i ≥ , i = 1 , ...n − . (33)The analyticity structure of W n ( x , x , ...x n ), in the coordinate space is related tothe support properties of the Fourier transformed Wightman functions, f W n , in themomentum space.The function W n ( ξ , ξ ...ξ n − ) of complex variables ξ µj = y µj − iη µj , j = 1 , , ...n − W n ( y , y ....y n − ).The set of complex variables { ξ µi } are defined as follows: the real pair { y µi , η µi } are suchthat η µi ∈ V + , i.e. η i ≥ , η i ≥
0. Thus { η j } is in the forward lightcone; moreover, −∞ < y iµ < + ∞ . This is the definition of forward tube T n − . The primitive domainis now identified. Wightman functions, the distributions, are boundary values ofanalytic functions i.e. W n ( y , y ...y n − ) = lim { η j → } W n ( ξ , ξ , ...ξ n − ) (34)Note that W n ( ξ , ξ ...ξ n − ) are also invariant under real orthochronous Lorentz trans-formations; det Λ r = 1. W n ( ξ , ξ ...ξ n − ) = W n (Λ r ξ , Λ r ξ ... Λ r ξ n − ) (35)Λ r is real proper Lorentz transformation. Moreover, according to Hall and Wight-man [60], if W n ( ξ , ξ ...ξ n − ) is analytic in the tube, T n − , and is invariant underreal orthochronous Lorentz transformations then W n ( ξ , ξ ...ξ n − ) is invariant undercomplex Lorentz transformations where ( ξ , ξ , ...ξ n − ) → (Λ ξ , Λ ξ , ... Λ ξ n − ); notethat ( ξ , ξ , ...ξ n − ) ∈ T n − . Here Λ ∈ SL + (2 C ), det Λ = 1. The set of points(Λ ξ , Λ ξ , ... Λ ξ n − ), for arbitrary Λ ∈ SL + (2 C ), define the extended tube T ′ n − . Theenlargement of domain of holomorphy is achieved through this procedure. Important16oint to note is that T n − does not contain the real points of { ξ j } . The extendedtube contains real points { y i } . The axioms: Lorentz invariance, uniqueness of vac-uum, stability of vacuum and local commutativity lead to the following assertion.The function W n ( ξ , ξ ...ξ n − ) exists and is analytic in the domain specified above.It is also single valued. It is analytic in the domain which is union of the permutedextended tubes.The Wightman functions, W n ( { y i } ) are analytic functions of real variables { y i } when all of them are spacelike [60, 61]. The result of Hall and Wightman are veryimportant for our purpose. We have noted earlier that when we define W n ( { ξ i } ) inthe extended tube they are analytic functions of complex four vectors { ξ µi } . Halland Wightman proved that if W n ( ξ , ξ , ...ξ n − ) is analytic in the four vector vari-ables { ξ µi } which is invariant under the Lorentz transformations then it is an analyticfunction of scalar products of those complex four vectors. The statement, intuitively,seems to be reasonable; however, the proof is quite formidable when mathematicalrigor is enforced. There are two significant implications of this theorem in the con-text of our work. First, the number of variables that the analytic function dependson is reduced considerably. For example, the three point function, W ( x , x , x ) de-pends on two four vectors, y µ = ( x − x ) µ and y µ = ( x − x ) µ , from translationinvariance of the theory. Thus it is a function of eight complex four vectors. Inthe light of Hall-Wightman theorem, now we know that W ( ξ , ξ ) depends on theinvariants z ij = ξ i .ξ j , i, j = 1 ,
2. In order to bring out the essence of the secondimplication let us recall the following facts. Notice that z ij is a complex symmetricmatrix ( ξ k = y k − iη k , k = 1 ,
2) in the forward light cone and η j is in the interiorof the lightcone. This set { z ij } is a domain of analyticity of an invariant function.Moreover, the Wightman function is the boundary value when η j →
0. It has beenproved by Hall and Wightman [60] that there are set of points { ξ i } on the boundaryof the tube with following properties. These vectors can be used to construct matrices ξ i .ξ j which lie in the interior. Moreover, they have argued that an invariant analyticfunction in the tube will not admit an arbitrary invariant distribution as boundaryvalue. It has also been proved that the boundary value is an analytic function of realvariables ξ i .ξ j , i, j = 1 , , ...n −
1, where η j = 0, in a certain domain. The analyticfunction is uniquely determined once its values are known in some subdomain of theboundary of the tube. These remarks might look as if they are out of context to beinterjected at this juncture. The importance is intimately related to the Jost theoremwhich we shall encounter in the discussion of crossing.The next question to ask is where do the real points of T ′ n − reside and what aretheir attributes? Jost proved the following: [62] The real points { y µ , y µ , ...y µn − } liein the extended tube, if and only if, the convex hull of { y µ , y µ , ...y µn − } only containsspacelike points. To elaborate a little bit, the convex hull of points { y µ , y µ , ...y µn − } is the set of all four vectors of the form λ y µ + λ y µ + ...λ n − y µn − where the set { λ i } λ i ≥ P n − i λ i = 1. Therefore, the real points of theextended tube are the ones for which if we take an arbitrary convex linear combina-tion ( P n − i =1 λ i y µi , λ i ≥ , P n − i λ i = 1), are always spacelike i.e. ( P n − i =1 λ i y µi ) < λ i ≥ , P n − i λ i = 1). This is the Jost point. The following remarks are pertinentin order to appreciate the importance of the theorem of Jost. We recognize that thedetermination of Jost point implies the existence of a domain consisting of points y J µ , y J µ , ..y Jn − µ where { y Jkµ } are real and spacelike. These points lie in the interior ofthe extended tube T ′ n − . However, they reside on the boundary of the forward tube, T n − . The Wightman function W n ( y , y , ...y n − ) is an analytic function of the set ofvariables { y jµ } at the Jost points. We can expand W n in a convergent power seriesin these variables. We argue that if we know the Wightman function in the neigh-borhood of y J µ , y J µ ...y Jn − µ , the Jost points, then the Wightman function is uniquelydetermined for the set of variables y µ , y µ , ...y n − µ of its argument. We shall utilizethis fact in the sequel. Moreover, their importance is realized in the study of crossingand in the context PCT theorem. This is a very powerful result. We have been dis-cussing crossing symmetry. We have alluded to the fact that the permuted Wightmanfunctions coincide at spacelike point. This theorem has far reaching consequences aswould be evident in the later part of this section as well as in the next section.The importance of Wightman formulation in conformal field theories has been rec-ognized long ago cite [2, 15, 40, 41, 45, 5]. Fradkin and Palchik [5] have presented avery comprehensive exposition in their book. Subsequently, Mack [41] has rigorouslyinvestigated the convergence of operator product expansion in conformal field theoriesutilizing the Wightman formulation. We have noted, in the last section, that not allconformal field theories satisfy the spectrality condition of Wightman axioms. All therepresentations of SU (2 , { φ i } be a st of confor-mal fields of dimensions d i and transform as finite dimensional representations of theLorentz group, L = SL (2 C ). Let U ≈ SU (2) the rotation subgroup of L ; ¯ U stands forset all finite dimensional irreducible representations, denoted by l . Therefore, a repre-sentation, characterizing the field, would be collectively denoted by χ = ( l, d ). As anillustration, consider the unitary irreducible representations of SU (2 , l , which is a finite dimensional representation of the Lorentz group SL (2 C ).Suppose the highest weight representation of l is (2 j , j ) (note that 2 j and 2 j arenonnegative integers). Now, χ = ( l, d ) and d min = 2 j + 2 j + 2 for nonzero j and j .In addition the field might be endowed with internal quantum numbers. However,all along, we consider only a single Hermitian nonderivative conformal scalar field, φ .18he Wightman axioms are respected by φ ( x ). We had discussed the structure of theHilbert space and we recall that it decomposes into several disjoint subspaces. H = X χ ⊕H χ (36)Here H χ stands for subspaces where each vector is characterized by χ which are thequantum number belonging to the irreducible representations of the covering groupalluded to earlier. It follows from the algebra in OPE that even if we start with asingle field, φ ( x ), the operator product expansion of a pair of this field needs infiniteset of composite operator of nonderivative type[41]. Therefore, we need the states as-sociated with them when we construct the full Hilbert space, H , which has subspaces H χ .Now let us consider two four point Wightman functions for the conformal field the-ory: W ( x , x , x , x ) = < | φ ( x ) φ ( x ) φ ( x )) φ ( x ) | > and W ( x , x , x , x ) = < | φ ( x ) φ ( x ) φ ( x ) φ ( x ) | > . Where would they coincide? We associate a tube(see the precise definition given earlier in this section) with respect to the first W ( x , x , x , x ) and another tube with the second W ( x , x , x , x ). They are con-tinued to one another as regular functions in the union of two extended tubes asso-ciated with each of the functions. Note that the Jost point has a real neighborhoodin the extended tube. Consider, for the case at hand, f ( ξ , ξ , ξ ) and f ( ξ ′ , ξ ′ , ξ ′ )which are analytic in their corresponding extended tubes, T ′ . To remind the reader,from translational invariance argument each 4-point Wightman function depends onlyon three coordinates, say y , y , y and y ′ , y ′ , y ′ and we can define corresponding ex-tended tubes. Let these two functions coincide for a real neighborhood in the extendedtube f ( y ′′ , y ′′ , y ′′ ) = f ( y ′′ , y ′′ , y ′′ ) (37)for y ′′ , y ′′ , y ′′ in a real neighborhood of a Jost point. The essential conclusion of theJost theorem is f ( ξ , ξ , ξ ) = f ( ξ , ξ , ξ ) (38)This is the edge-of-the-wedge theorem [63, 64, 65] and the implications of the theoremin the context of conformal bootstrap equation will be discussed in Section 3. Thistheorem was first proved, in the context of dispersion relations from LSZ axiomaticfield theory [62]. We recapitulate a few points for the sake of motivations and his-torical considerations . If we consider a four point scattering amplitude, it has aright hand cut and a left hand cut. The right hand cut originates from the direct, s -channel, reaction; whereas, the left hand cut arises from the crossed channel, the u -channel, process. The discontinuities across the cuts are related to the absorptiveparts of the respective amplitudes. The edge-of-the-wedge theorem proves that thetwo absorptive amplitudes are analytic continuation of each other [63, 64]. Thus cross-ing was proved from analyticity property of the amplitude and a dispersion relation19ould be written down. We recall that the theorem was proved for S-matrix elementswhere external particles are on the mass shell and furthermore, equations of motionwere utilized in obtaining matrix elements of source current commutators. The samearguments may be extended to the cases of retarded and advanced commutators ofcurrents. In the context of conformal field theory, we deal with Wightman functions.Therefore, a different route has to be chosen when we intend to prove analyticity andcrossing in the present context. The edge-of-the-wedge theorem has been proved forWightman functions by Epstein [65]. Subsequently, the holomorphicity and envelopof holomorphy were studied by Streater [66, 68] and Tomozawa [67]. The precedingstatements in the context of (38) is a qualitative and intuitive argument about theedge-of-the-wedge theorem. We shall discuss this aspect in more details in the nextsection. .Our objective is to discuss analyticity and crossing properties of the four point Wight-man function for Hermitian scalar nonderivative conformal field. The three pointWightman function is W ( x , x , x ) = < | φ ( x ) φ ( x ) φ ( x ) | > (39)A crossed three point function, for example, is W ( x , x , x ) and the two coincidewhen ( x − x ) < W ( x , x , x ) depends on two variables, y = x − x and y = x − x , as a consequence of translational invariance. The function ofinterest is W ( x , x , x ) − W ( x , x , x ). The recent progress made in this directionby us is presented in the sequel. The above difference, W ( x , x , x ) − W ( x , x , x ),may be expressed as commutator < | φ ( x )[ φ ( x ) , φ ( x )] | > and it vanishes when( x − x ) <
0. The goal is to study its analyticity property which is intimatelyconnected with crossing. We adopted a variance of the representation, due to Dyson[56], of the double commutator of three scalar fields: < | [ C ( x ) , [ B ( x ) , A ( x )]] | > where A, B, C are the three scalar fields. In fact Dyson’s technique together with thearguments of Streater [57] were suitably adopted by us to obtain a representation for20he VEV of our interest. Define F ( y , y ) = (cid:18) W ( x , x , x ) − W ( x , x , x ) (cid:19) (40)Note that F ( y , y ) = 0 for y < F ( p, q ), of F ( y , y ) has a representation [57, 55]˜ F ( p, q ) = Z Ψ ( p, u, s ) δ (( u − q ) − s ) ǫ (( u − q ) ) d uds (41)This is a generalized version of the Jost-Lehmann-Dyson representation [29, 30].The function Ψ ( p, u, s ) has following properties: It vanishes unless the hyperbolain the q -space i.e. ( u − q ) = s lies in the union of two domains characterized as q ∈ V + ∪ ( p − q ) ∈ V + . It was concluded that Ψ ( p, u, s ) = 0 except when the condi-tions p, u ∈ ( u ∈ V + ∩ ( p − u ) ∈ V + ) are fulfilled.It is necessary to identify the extended tubes for the two Wightman functions: (i) W ( ξ , ξ ) is regular in the extended tube T ′ ( ξ , ξ ). (ii) Similarly, W ( ξ + ξ , − ξ )is regular in the extended tube T ′ ( ξ + ξ , − ξ ). If f W ( p, q ) and f W ′ ( p, q ) denote theFourier transforms of the Wightman functions W ( ξ , ξ ) and W ( ξ + ξ , − ξ ) respec-tively; note that the latter corresponds to the crossed 3-point Wightman function.The point to note is that W ( ξ , ξ ) = W ( ξ + ξ , − ξ ) in a domain where they areregular since these are Jost points i.e. they corresponds to real points separated byspacelike distance. We conclude that they analytically continue to one another in thedomain T ′ = T ′ ( y , y ) ∪ T ′ ( y + y , − y ) (42)Let us denote the Fourier transforms of W ( y , y ) and W ( y + y , − y ) respectivelyby f W ( p, q ) and f W ′ ( p, q ). We can read off the support properties to be˜ W ( p, q ) = 0 , unless p > , p > , and q > , q > W ′ ( p, q ) = 0 , unless p > , p > p − q ) > , ( p − q ) > φ ( x ). Note that y = ( x − x ) , y = ( x − x ) and x − x = y + y . In termsof ξ and ξ , it assumes the form, W ( ξ , ξ ) = const . (cid:20) ξ ξ ( ξ + ξ ) (cid:21) d/ (44)Let us now consider the scale transformation: ξ i → λξ i . Then W ( ξ , ξ ) → W ( λξ , λξ ) = [( 1 λ ) ] d (cid:20) ξ ξ ( ξ + ξ ) (cid:21) d/ = [( 1 λ ) ] d W ( ξ , ξ ) (45)21t the Jost point, for the real coordinates satisfy y < , y <
0. Moreover, the de-nominator of (44) is analytic whenever ξ , ξ , and ( ξ + ξ ) are negative for real ξ i .Note that if real ξ , real ξ are spacelike so is the sum real ( ξ + ξ ). Thus, expressedin terms of ξ and ξ , the two three point functions W x , x , x ) and W ( x , x , x )coincide since both can be considered as boundary values of corresponding analyticfunctions at the Jost point. These two functions are analytic functions in the realenvironment for spacelike separated points. It follows from the Jost theorem and theedge-of-the-wedge theorem that they are analytic continuation of each other. Thusthe crossing is established.The preceding discussions lead to the conclusion that the two Wightman func-tions are boundary values of analytic functions with known support properties. Thuscrossing is proved for a pair of Wightman functions such that one is obtained from theother from interchange of a pair of fields. Let us invoke the Hall-Wightman theorem[60] to discuss the analytic continuation. The three point function W ( x , x , x ) de-pends of two variables y i , i = 1 ,
2. It is boundary value of an analytic function of twocomplex variables, ξ i , i = 1 ,
2. Thus it depends on eight four-vectors. On the otherhand, it follows from the Hall-Wightman theorem that W is a function of Lorentzinvariant variables constructed from ξ and ξ : it depends on three complex variables: z jk = ξ µj ξ µk , j, k = 1 ,
2; expressed explicitly; z = ξ , z = ξ and z = ξ .ξ . TheJost points are real and spacelike; moreover they belong to T ′ . If the Jost points aredenoted by v i , i = 1 , v i <
0, and v + λv is also a Jost point with 0 < λ < ξ i , i = 1 , ξ + λξ ) < , ξ , ξ real (46)We derive a relationship among the variables z jk . Reality of λ implies z > √ z z (47)Obviously the two Wightman functions W ( x , x , x ) and W ( x , x , x ) are equal atthe Jost point, The corresponding extended tube for W ( x , x , x ) is T ′ ( ξ + ξ , − ξ ).Invoking Jost’s theorem, we conclude that the two Wightman functions are analyticcontinuation of each other. We have shown that the crossing hold for Wightman func-tions while considered pairwise. Therefore, all the six permuted three point Wightmanfunctions are analytic continuations of each other when one pair is considered at atime. It is important to note, however, that the envelope of holomorphy could be amuch larger domain. We were contented to prove crossing in this simple approach.It is worth while to dwell upon the analyticity properties of the three point func-tion in the momentum space representation of three point function. The analyticityproperties in momentum space representation of n-point functions have been inves-tigated in the past in the frameworks of axiomatic field theories [69, 70, 71, 72].22he retarded functions, R -functions, defined in the coordinate space are endowedwith retardedness and causal properties. It has been demonstrated that the Fouriertransform of the VEV of R-Products are boundary values of analytic functions ofcomplexified momenta. There is a close analogy between the three point Wightmanfunction and the retarded three point function from the analysis of Jost [73] whichhas been further studied by Brown [74].We study the analyticity properties of three point function of conformal fieldtheory in the momentum space. Recently, there has been quite a bit of interest in un-derstanding momentum space descriptions of correlation functions in conformal fieldtheories [75, 76, 77, 78, 79, 80, 32, 33, 81, 82, 83, 84, 85, 86, 55]. In particular, oneof our interests is the study of the analyticity properties of three point function inthe momentum space description [32, 33, 55]. We were motivated to undertake thisstudy by the recent two papers [32, 33]. We know, from Jost theorem, that Wightmanfunctions are analytic in the spacelike regions i.e. when the coordinate separationsare spacelike. Moreover, the Fourier transform of a Wightman function for spacelikecoordinate differences would have conjugate momenta lying in the spacelike region.Let us define the vacuum expectation value of R -product of n scalar field to be r n ( y , y , ...y n − ) where y i = x i − x i +1 . Therefore, for a single type of real scalar field, φ ( x ) r n ( y , y , ..y n − ) = < | R φ ( x ) φ ( x ) ...φ ( x n − ) | > (48)The Green function is G n ( p , p , ...p n − ) = Z d y d y ...d y n − e i P n − j =1 p j .y j r n ( y , y , ...y n − ) (49)The next step is to define an analytic function of complexified momentum variables: p µj → ( p µj + iq µj ) , j = 1 , , ...n −
1. The ( n −
1) real four vectors ( p µj , q µj ) are such that q j ∈ V + and p µj is unrestricted. The Fourier transform G n ( k , k , ..k n − ) = Z d y d y ..d y n − e i P n − j =1 k j .y j r n ( y , y , ...y n − ) (50)has good convergence property. Now the Green function G n ( p , p , ..p n − ) is boundaryvalue of an analytic function G n ( p , p , ..p n − ) = lim { q i → } G ( k , k , ...k n − ) (51)Ruelle [70] has proved analog of Jost theorem in the momentum space. Since { q µi } ∈ V + , we may choose a coordinate frame such that q µi = ( q i , ) and there are norestrictions on p µi . A simplified version of Ruelle’s theorem can be expressed asfollows: The function G ( k i , k i ) can have singularities if q i = 0 and p µi is not spacelike.23his theorem is valuable for us in what follows.The case of three point function will be taken up now in the light of the observationsof Jost [73]. We deal with the R -product where the product is of a single conformalscalar field, φ ( x ) is defined to be R φ ( x ) φ ( x ) φ ( x ) = θ ( x − x ) θ ( x − x )[[ φ ( x ) , φ ( x )] , φ ( x )]+ θ ( x − x ) θ ( x − x )[[ φ ( x ) , φ ( x )] , φ ( x )] (52)Let us consider the implications of causality on the first double commutator on the r.h.s of (52), suppressing the presence of the two θ -functions,[[ φ ( x ) , φ ( x )] , φ ( x )] = 0 for ( x − x ) < , or [( x − x ) < and ( x − x ) < y = x − x and y = x − x , we may express these constraints in termsof y i -variables. Similar constraints follow for the second terms of (52). We concen-trate on the three point function < | φ ( x ) φ ( x ) φ ( x ) | ) > to study momentum spaceanalyticity properties. It depends on two variables y and y and we introduce com-plexified coordinates ξ , ξ . Moreover, W ( ξ , ξ ) = R dp dp e − i ( ξ .y + ξ .p ) f W ( p , p )and p , p ∈ V + . We also know from works of Ruelle [70] that, in the coordinatespace description, we may go to a frame where η i = ( η i , , , , i = 1 ,
2; this is per-mitted since η i ∈ V + . Note that −∞ < y i < + ∞ , i = 1 ,
2, the real part of ξ i (weremind the reader that ξ j = y j − iη j , j = 1 ,
2) . We also know that, for real ξ i ,each of the three point functions W ( { ξ i } (appearing in the R -product) are analyticwhenever { y i } ∈ T ′ and thus are spacelike, from the Jost theorem. We may adoptanother theorem of Ruelle [70] for this case; W ( η , η , y , y ) can only have singular-ities if two of its arguments ( η , y ) and ( η , y ) are such that η = η and y µ − y µ is not spacelike. In what follows, we shall consider a three point Wightman function W ( ξ , ξ ) taking the clue from analysis of Jost [73]. Let us consider the momentumspace representation of three point function f W ( p , p ) = Z d y d y e i ( p .y + p .y ) W ( y , y ) (54)Now if we want this Fourier transform to be convergent. We compexify the momentumvariables (as before): ( p , p ) → ( k = p + iq , k = p + iq ). Thus the tube, e T isdefined in the momentum space and it is the primitive domain; moreover, the integral(54) is convergent since q and q ∈ V + . The momentum-space 3-point function isthe boundary value of an analytic function: f W ( k , k ) f W ( p , p ) = lim { q i }→ f W ( k , k ) (55)Assuming that Ruelle’s theorem is valid for f W ( k , k ) we may invoke the momentumspace theorem [70]. Thus lim { q i }→ f W ( k , k ) is analytic when real momenta p µ and p µ lie in the spacelike region. We draw attention to a few points. Let us consider the24ifference R φ ( x ) φ ( x ) φ ( x ) − A φ ( x ) φ ( x ) φ ( x ) where the A -product is defines suchthat ( x i − x j ) is replaced by ( x j − x i ) . Now consider the VEV of this difference anddenote the VEV of R -product as r ( x, x , x ) and correspondingly VEV of A -productas a ( x, x , x ) r ( x, x , x ) − a ( x, x , x ) = < | [[ φ ( x ) , φ ( x )] , phi ( x )] | > (56)We arrive at above expression using the properties of θ -functions and also the factthat θ ( x ) + θ ( − x ) = 1. This is the double commutator we have encountered before.If we open up the double commutator it will be sum of four three point functions.We also know the constraints due to microcausality. Note that there are altogethersix permuted Wightman functions and they are different. Let us consider the mo-mentum space functions. Then we complexify the momenta and define the analyticfunctions in these complex variables. Now we appeal to Hall-Wightman theorem [60]for these functions. Obviously, we have function which depends on Lorentz invariants˜ z ij = k i .k j , i, j = 1 , ,
3. We also know k µ + k µ + k µ = 0 due to translational invarianceand the complexified momenta are conjugate to x, x , x . The point to note is thatnow we can define a function which is analytic in the union of the extended tubesassociated with the three point analytic functions. If we go to boundary values ineach of the real valued momenta the corresponding Fourier transformed Wightmanfunctions are different. We see the power of analytic completion in this simple exam-ple.It is intuitively quite appealing. If a Wightman function is defined for space-like coordinate, in the coordinate space representation, the conjugate momenta arealso spacelike when we take the Fourier transform and study analyticity. Our intu-ition guides that if those coordinate points correspond to Jost point, we expect thatthe conjugate momenta would be spacelike. Moreover, the momentum space Greenfunction will be analytic in momentum variables in the real environment of theseJost points (momentum space Jost points). It is quite safe to conjecture that ouraforementioned arguments regarding analyticity of f W ( k , k ) is valid. Let us startfrom the momentum space Jost point where f W ( k , k ) is analytic for Re ( k , k ).Moreover, f W ( k , k ) is analytic in the momentum space extended tube, ˜ T ′ . If weextend the argument of Jost, for momentum space three point function, f W ( k , k ),it is analytic for real spacelike p , p in their neighborhood. Let us implement aninfinitesimal complex Lorentz transformation on these real momenta . The momenta p , p will assume complex values and arguments of f W ( p , p ) will be complex (de-note them as k and k ). The resulting f W ( k , k ) would be analytically continuedin ˜ T ′ . Therefore, the function is analytic in the domain belonging to ˜ T ′ . We can saymore. Note that W ( ξ , ξ ) is a tempered distribution. The Fourier transform of atempered distribution is also a tempered distribution Froissart [87]. Consequently, itis bounded as is defined for a distribution i.e. the boundedness is to be understoodin the sense that f W ( k , k ) is a distribution (see Froissart [87] for detail discussion).25e examine the implications of these general arguments in what follows. Re-cently, the properties of three point function in the Lorentzian metric description hasdrawn attentions [32, 33]. Baurista and Godazgar [32] have systematically investi-gated the three point function in momentum space for the Euclidean metric as wellas for the Lorentzian signature metrics. They derive an expression for the three pointfunction as an integral which includes product of Bessel function as well as modifiedBessel functions. Their work revealed several interesting analyticity properties of theWightman function. Gillioz [33] studied the properties of three point function in themomentum space with Lorentzian signature metric. He employed operator productexpansion for momentum space operators and derived Ward identities. One advantageof this approach is that he constructed states, utilizing state ↔ correspondence, inthe momentum space in order to compute three point function. He has demonstratedthat the three point function for scalars is expanded in a double hypergeometric series.Moreover, he explored its behavior in various kinematical limits. Our objective is tostudy the analyticity properties in the momentum space in the light of the precedingdiscussions.Let us consider the tree point function given by (44). It is bounded and analyticfor spacelike { ξ i } , i = 1 ,
2. The Fourier transform of the coordinate space threepoint function has been investigated in detail in [32, 33]. Recall that the Fouriertransform of W ( x , x , x ) would depend on three momentum variable, however,the total momentum conservation implies that only two of the three momentumvariables are independent. It follows from Lorentz invariance that the three pointfunction f W ( p , p , p ) depends on the Lorentz invariants constructed from p µi , i =1 , ,
3. Furthermore, conformal invariance imposes restrictions on its structure. Werecall the results of Gillioz [33] for our purpose. The expression for the three pointfunction assumes the following generic form (for D = 4 in our metric convention) < | φ ( p ) φ ( p ) φ ( p ) | > = θ ( − p ) θ ( p ) δ (4) ( p + p + p )( p ) (3 d − / F ( (cid:18) p p , p p (cid:19) (57)where the energy momentum conserving δ -function implies p + p + p = 0. Moreover, F is a function of ratios of squares of momenta. Eventually, F gets related to Appell’s F generalized hypergeometric function of two variables and we refer to [33] for details.A few comments are in order at this stage: (i) Following Ruelle, we argue thatthe momenta can be complexified as p µi → k µi = ( p µi + iq µi ) to define an analyticallycontinued three point function in the complex momentum variable. We identify theprimitive domain to be ˜ T and the corresponding extended tube as ˜ T ′ . (ii) It followsfrom momentum conservation that p = ( p + p ) and the dimensionless ratio of themomenta, appearing as arguments of the function F go over to p p → p ( p + p ) and26 p → p ( p + p ) . The function, F ( (cid:18) p ( p + p ) , p ( p + p ) (cid:19) is conformally invariant as is evi-dent from the structure. (iii) Now consider p and p to be spacelike and so is the sum p + p . Furthermore, as p and p tend to asymptotic values, with p p → constant, inthe spacelike region, then ratios p ( p + p ) and p ( p + p ) tend to constants. Therefore, inthis region, the function, F ( (cid:18) p ( p + p ) , p ( p + p ) (cid:19) , will tend to constant. Moreover, wenote that the three point function is analytic for spacelike momenta and in its a realneighborhood belonging to T ′ . Furthermore, this function admits analytic continu-ation to the extended tube. (iv) It is evident that the three point function dependson Lorentz invariant variables and their ratios. This is analog of the Hall-Wightmantheorem [60] for three point function of our conformal field theory. The functionof the complex variables k i is a tempered distribution as the coordinate space threepoint function is a tempered distribution. .We proceed to study analyticity of four point Wightman function following the pre-scription discussed above. The four point function W ( x , x , x , x ) = < | φ ( x ) φ ( x ) φ ( x ) φ ( x ) | > (58)will be considered along with another permuted Wightman function. We shall pro-ceed to investigate analyticity and crossing adopting the spirit adopted for three pointfunctions. The translational invariance implies that W ( x , x , x , x ) = W ( y , y , y )where y i = x i − x i +1 . Let us recall the expression for the four point function (21),(22) and (23). In the light of the preceding discussion for the three point function, weexamine the analyticity of W ( x , x , x , x ) for real values of the coordinates ξ , ξ , ξ and when the separations are spacelike. Recall that four vectors, y i , i = 1 , ,
3, are allspacelike and their linear combinations with positive coefficients are also spacelike.It follows from the definition of Z and Z that these are ratio of y µi ’s and the ratio ispositive as long as each vector is spacelike. Moreover, Z and Z are scale invariant.The denominator appearing as prefactor of F ( Z , Z ) is also positive. We invoke theJost theorem and once again, the edge-of-the-wedge theorem when we consider thetwo four point functions W ( x , x , x , x ) and W ( x , x , x , x ) and argue that theyare analytic continuation of each other. Thus crossing is demonstrated from this per-spective.We intend to study the support properties from the Fourier transform of thefour point function. We aim at deriving an integral representation for the function,analogous to the Jost-Lehmann-Dyson representation as our next step. Thus f W ( p , p , p ) = Z d y d y d y e i P j =1 p j .y j W ( y , y , y ) (59)27ote, from the spectral condition that f W ( p , p , p ) = 0 , unless p j ∈ V + , j = 1 , , f W ( p , p , p ), is also adistribution in the momentum variables. Note that f W ( p , p , p ) is a boundary valueof a holomorphic function as W ( y , y , y ) is. We argue that W ( ξ , ξ , ξ ) , recall( ξ i = y i − iη i ) has a Laplace transform W ( ξ , ξ , ξ ) = Z d p d p d p e − i P p j . ( y j − iη j ) f W ( p , p , p ) (61)It is holomorphic in the upper half-plane (note that η i ∈ V + ensures convergence).Moreover, the derivatives dW ( { ξ j } ) dξ j exists and does not depend on direction. We re-mind that the function is holomorphic in the forward tube and it is Laplace transformof a distribution. The distributions f W ( p , p , p ) vanish for p µi / ∈ V + . The purposeof interjecting these remarks is to convey that our next steps are based on the knownanalyticity properties of the Fourier transformed four point function.We have introduced three complex variables and we have defined the forward tube.Thus W ( y , y , y ) is boundary value of analytic function where the complex coordi-nates are in T ( ξ , ξ , ξ ). Next, the extended tube T ′ is obtained by implementingorthochronous complex Lorentz transformations SL + (2 C ) on { ξ j } . In order to pro-ceed further, we define the permuted Wightman function where the locations of thetwo fields φ ( x ) and φ ( x ) are interchanged (recall the case of three point function). W ′ ( x , x , x , x ) = < | φ ( x ) φ ( x ) φ ( x ) φ ( x ) | > (62)The new complexified coordinates are ξ ′ = ξ , ξ ′ = ξ + ξ , ξ ′ = − ξ , expressed interms of { ξ j } . The associated forward tube is e T ( ξ , ξ + ξ , − ξ ). We can obtainthe extended tube e T ′ once we have constructed e T ( ξ , ξ + ξ , − ξ ). The supportproperties of the Fourier transform f W ′ ( p , p , p ) of W ′ ( x , x , x , x ) are f W ′ ( p , p , p ) = 0 , for p ∈ V + , p ∈ V + , p − p ∈ V + , and p / ∈ V + (63)which can be inferred from the expression for the Fourier transform. We now definea function e F ( p , p , p ) = f W ( p , p , p ) − f W ′ ( p , p , p )= f W = 0 , if p ∈ V + , p ∈ V + , p ∈ V + , and − f W ′ ( p , p , p ) = 0 , if p ∈ V + , p ∈ V + , p − p ∈ V + , and p / ∈ V + (64)We consider the two functions W ( ξ , ξ , ξ ) and W ′ ( ξ , ξ + ξ , − ξ ) to be analyticcontinuation of each other. The corresponding domain is the union of two extendedtubes T ′ = T ′ ( ξ , ξ , ξ ) ∪ ˜ T ′ ( ξ , ξ + ξ , − ξ ) (65)28e define the momentum space function e F ( p , p , p ) = Z < | φ ( x ) φ ( x )[ φ ( x ) , φ ( x )] | > e i ( p .y + p .y ) e + ip .y d y d y d y (66)Notice that, from microcausality, Z d p e − ip .y e F ( p , p , p ) = 0 if y < e F ( p , p , p ), following the arguments of [68] e F ( p , p , p ) = Z ¯Φ( u, p , v, s, κ ) δ [( p − u ) − s ] ǫ ([( p − u ] ) δ [( p − v ) − κ ] ǫ ([ p − v ] ) d ud vdsdk (68)The function ¯Φ( u, p , v, s, κ ) satisfies following properties: (i) Note that¯Φ( u, p , v, s, k ) = 0 , unless ( p − u ) > , p > , ( p − v ) > p − u ) = s and ( p − v ) = k defines a pair of hyperboloidswith four momenta satisfying above constraints. The function ¯Φ( u, p , v, s, k ) van-ishes unless these conditions are fulfilled. This is analog of the Jost-Lehmann-Dysonrepresentation and was considered by Streater [68, 66].Remark: We have considered two permuted Wightman functions and it is arguedthat their domain of holomorphy is the union two extended tube T ′ ( ξ , ξ , ξ ) and e T ( ξ , ξ + ξ , − ξ ) ′ . Let us construct the Lorentz invariant variables from the set { ξ i } for the application of Hall-Wightman theorem. z ij = ξ i .ξ j , i, j = 1 , ,
3. Weremind the reader that the three point Wightman function has six permutations ofthe ordering of the field. Similarly, the four point function has twelve permutations.The determination of the envelope of holomorphy for this case is a formidable task.The problem for this case has not been analyzed in so much of details as has beendone for three point functions by K¨alle´n and Wightman [51]. We recall that Lorentzinvariance and local commutativity are the two principles which are instrumental tostudy the general structures. In the case of three point function (the VEV ) can becontinued to a function regular in a certain domain M in the space of scalar productsof { ξ i } . We have seen that local commutativity plays an important role to identify theunion of domains which are formed when we consider permuted field configurations.There have been several attempts to [52, 53, 23] to implement similar prescriptionfor the four point function and identify the domain M and they have achieved somesuccess; however, the detail analysis carried out for the three point function [51] hasnot been accomplished for the four point function. We shall be contented with theanalysis for a pair of functions as envisaged above. We shall consider a simplifiedsituation guided by our previous attempt [55]. We consider two spacelike vectors on M which is a subspace of M . A third spacelike vector is considered which lies29n M but not in M . Then we determine the Jost point. This procedure enablesus to determine points of M similar to the condition derived for three point func-tions. Now there will be more inequalities (see 47) compared to the case of W ( ξ , ξ ).We begin with the set of complexified four vectors { ξ µi } , i = 1 , , M . Let us consider a submanifold M in M . Consider, twospacelike vectors which lie in M . Their linear combination, with a positive realcoefficient, is also a spacelike vector. Now choose a spacelike vector in the complimentof M . We should be in a position to obtain constraints on these three real spacelikevectors which would be similar to (47). For the case at hand, we have to considerthe decomposition of M into M ⊕ M ; M lies in the space compliment to M , forthree different configurations as discussed below.We go through the following steps:(i) Let ξ be a real spacelike vector in M which is compliment of corresponding M i.e. M = M ⊕ M .(ii) ξ and ξ are two spacelike vectors which are lying in M . Thus ξ + λξ alsospacelike where λ is positive and real. Let us consider the properties of the threevectors in three different configurations.Case (a): Notice that ξ . ( ξ + λξ ) < ξ < ξ ( ξ + λξ ) < q [ ξ ( ξ + λξ ) ] > λ >
0. We conclude that that[ ξ . ( ξ + λξ )] < ξ ( ξ + λξ ) (71)Note that the r.h.s of the above equation is positive. As before, define the Lorentzinvariant Hall-Wightman variables z ij = ξ i .ξ j , i, j = 1 , ,
3. We choose { ξ i } to bereal in the light of the preceding discussion to derive relations among z ij from theconstraint that λ >
0. When expressed in terms of z ij , The inequality (71) translateto z z − z + 2 λ ( z z − z z ) + λ ( z z − z ) > λ be positive and real.Instead of obtaining such constraints in a case by case basis let us consider the othertwo cases. We choose two spacelike vectors in a subspace M and another one whichlies in its compliment. The other two case are(b) ξ . ( ξ + λξ ) < ξ and ξ lie in an M and ξ in its compliment, M .(c) ξ . ( ξ + λξ ) <
0. Here we have another permuted scenario for { ξ , ξ , ξ } .We shall obtain two more equations from the case (b) and (c) which will beanalogous to (72). We can derived the set of constraints in a more efficient and30legant manner. Define the matrix Z = z z z z z z z z z (73)Note that Z -matrix is symmetric, Z T = Z as was introduced by Hall and Wightman.Let us define a matrix f M ( f M ) ij = (det Z )( Z − ) ij (74)The constraint equations arising from the requirement that λ > f M . Notice that we have to solve for the analog of equa-tions like (72) to derive the requisite constraints in terms of the matrix elements of f M . Moreover, there will be altogether three equations. The conditions are givenbelow:( a ′ ) We start with the condition ξ . ( ξ + λξ ) <
0. Positivity of λ leads to followinginequality: f M > f M f M .( b ′ ) For the case ξ . ( ξ + λξ ) <
0. Correspondingly the condition is : f M > f M f M .( c ′ ) For the third case i.e. ξ . ( ξ + λξ ) <
0; the condition becomes: f M > f M f M .Notice that same procedure can be adopted to derive constraints for the real valuesof the the complexified variables, ξ ′ , ξ ′ , ξ ′ ; ( ξ ′ = ξ + ξ , ξ ′ = − ξ , ξ ′ = ξ + ξ ).Moreover, W ( x , x , x , x ) and W ′ ( x , x , x , x ) are equal for ( x − x ) < ξ ) <
0. We have proved crossing for the pairof Wightman functions and identified the analyticity regions.Let us recall the structure of the four point function for the conformal field theory,expressed in terms of the three ξ -variables from (21), (22) and (23) W ( ξ , ξ , ξ ) = (cid:20) ξ ξ (cid:21) d ˜ F ( ˜ Z , ˜ Z ) (75)Now ˜ Z and ˜ Z are cross ratios of variables { ξ i } .˜ Z = ξ ξ ( ξ + ξ ) ( ξ + ξ ) (76)and ˜ Z = ( ξ + ξ + ξ ) ξ ( ξ + ξ ) ( ξ + ξ ) (77)Notice also that the cross ratios are conformally invariant. The prefactor of thefunction of cross ratio transforms according to the known rule. Thus W ( λξ , λξ , λξ ) = ( λ ) − d W ( ξ , ξ , ξ ) (78)31oreover, when all real set of { ξ i } are spacelike, i.e. (real ξ ) < , i = 1 , , W isanalytic, according to Jost theorem. Therefore, the crossing will be valid in this re-gion. We are not in a position to make more accurate statements since the functionaldependence of cross ratios are not known unless we appeal a specific model.
3. Analyticity and Causality in Operator Product Expansions and Con-formal Bootstrap Equation .In this section we discuss analyticity and causality properties of the four point Wight-man functions in the context of conformal bootstrap equation. First, we focus at-tention on operator product expansion of a pair of conformal fields. The proposal ofOtterson and Zimmermann [38] is appropriately modified in the context of CFT. Theirinvestigation is based on the work of Wilson and Zimmermann [36] who rigorouslystudied operator expansion in QFT from the Wightman axioms. The analyticityproperty of the matrix elements of composite operators appearing in OPE are in-vestigated. Mack [41] had initiated the study of the properties OPE in CFT fromWightman axiom view point. When we invoke microcausality for the commutator oftwo operator product expansions, the matrix element of the (difference) of two com-posite operators are constrained. Therefore, the analog of the Jost-Lehmann-Dysonrepresentation can be derived.The second part of this section is devoted to derive bootstrap equation in a novelway through the PCT theorem. The equivalence between PCT theorem and weaklocal commutativity (WLC) is invoked to related two four point functions. Theconformal partial wave expansion is implemented on each the four point Wightmanfunctions to derive the bootstrap equation. We draw attention to following points tohighlight our approach. The first point to note is that the two four point functionscoincide at the Jost point. It should be noted that the corresponding Fourier trans-formed Wightman functions depend on conjugate momenta which are spacelike andwe term them as ’unphysical’. Moreover, each four point function is boundary valueof an analytic function. These Wightman functions are analytic in real neighborhoodof Jost point. They are analytic in corresponding extended tubes. Thus it will bedemonstrated, with chain of arguments, that the two four point functions are analyticcontinuation of each other. The second point is the following. The conformal boot-strap equation hold at real points where coordinates have spacelike separations. Aswill be shown later, the bootstrap equations can be interpreted as boundary values ofanalytic functions defined over extended tubes. We feel that this proof is quite novelin the sense that the power of PCT theorem and its equivalence with WLC enablesus to derive the bootstrap equation in CFT rigorously. .32et us recapitulate the Wilson’s operator product expansion proposal for the productof two scalar fields A ( x ) A ( x ) as envisaged in [38], based on the works of Wilsonand Zimmermann [36]. In the case of scalar fields A ( x ) A ( x ) = k X j =1 f j ( ρ ) C j ( x, ζ , ρ ) + R ( x, ζ , ρ ) (79)where x µ = x µ + ρζ µ , x µ = x µ − ρζ µ , ρ > R ( x, ζ , ρ ) stands for the remainder ofthe series collectively. The series can be so organized that coefficients would satisfylim ρ → f j +1 ( ρ ) f j ( ρ ) = 0 , lim ρ → R f j ( ρ ) = 0 (80)Note that { f j ( ρ ) } are C-numbers and they become singular as ρ →
0. The operators, C j , depend on the vector x µ which is identified as the center of mass point. It alsodepends on another vector ζ µ which is proportional to the distance between the twooperators A ( x ) and A ( x ). We note that the ζ -dependence is connected with thedirectional dependence of the set of operators { C j } . It becomes obvious from therelations ζ µ = κ µ √ κ , ρ = √ κ (81)We can reexpress (79) as A ( x + κ ) A ( x − κ ) = k X j =1 f j ( √ κ ) C j ( x, κ √ κ ) + R (82)The scalar fields, A ( x ) and A ( x ) respect the Wightman axioms. The operators C j ( x, ζ ) are local in x for a given ζ . Otterson and Zimmermann [38] have investi-gated the relationship between causality and analyticity rigorously for OPE of twoscalar fields.Our intent is to investigate relationship between causality and analyticity in con-formal field theory in the frameworks of Wightman axioms and adopt the formalismintroduced by [38]. We remind that not all conformal field theories fulfill the require-ments of Wightman axioms. The Hermitian scalar nonderivative field, φ ( x ), respectsWightman axioms. Its Fourier transform, ˜ φ ( p ), satisfies the spectrality condition i.e. p ∈ V + . We consider a single conformal scalar field, φ ( x ). The operator productexpansion is φ ( x ) φ ( x ) = X χ X j f χj ( ρ ) C χj ( x, ζ ) (83)Mack [41] has investigated the convergence of the operator product expansion rig-orously for nonderivative scalar conformal field theory. The coefficients { f χj ( ρ ) } are33 -number functions which become singular as ρ →
0. For the nonderivative field, φ ( x ), in the OPE, the complete set of local operators, { C χj } , are also nonderivativeoperators [41] (see definition in Section 1). The sum over χ goes over all unitaryirreducible finite dimensional representations of the covering group of the conformalgroup, SO (4 , SU (2 , χ = [ l, δ ]; l corre-sponds to finite dimensional irreducible representation of SL (2 , C ); it is the ’Lorentzspin’ and δ ≥ δ min , real dimension. OPE, as noted in section 1, requires infinite set ofoperators belonging to irreducible representations of the covering group. Therefore,a Hilbert space is associated each of these local fields characterized by χ . Thus thefull Hilbert space, H , is decomposed as H = P χ ⊕H χ as discussed already. It willsuffice to consider a series expansion for a given χ (i.e. fixed χ ) in order to investigatecausality and analyticity properties. We pick up a generic term in the double seriesexpansion (83) and define X j f χj ( ρ ) C χj ( x, ζ ) = X j ˜ f j ˜ C j ( x, ζ ) , for a given χ (84)Note that the r.h.s of the OPE, (83), is a double sum. It suffices for us to analyzeproperties of ( ?? ) in the present context. Therefore, we investigate the causality andanalyticity properties of a single sum over j in (84) i.e. look at the summed over j term for a given χ . The conclusion drawn from here will hold for each of termof (83) in the sum over χ of the double sum. Here ˜ f j ( ρ )‘ and ˜ C j ( x, ζ ) stand for f χj ( ρ ) and C χj ( x, ζ ) respectively so that we do not carry the index χ everywhere. Theseries is so organized, as was adopted in [38], for a given sector of χ , that coefficientsof the operator ˜ C j ( x, ζ ) satisfy a condition analogous to (80); i.e.lim ρ → ˜ f j +1 ( ρ )˜ f j ( ρ ) = 0 (85)We define P j ( x, ζ , ρ ) = ˜ f j ( ρ ) ˜ C j ( x, ζ ) (86)following the clue from Otterson and Zimmermann [38]. Consequently,˜ C j ( x, ζ ) = lim ρ → P j ( x, ζ , ρ )˜ f j ( ρ ) (87)Define Fourier transform of ˜ C j ( x, ζ ) as¯ C j ( x, u ) = 12 π Z d ζ e iζ.u ˜ C j ( x, ζ ) (88)Let us envisage two state vectors | p > and | q > in the Hilbert space H χ for a given χ in the momentum space representation. Thus P µ | p > = p µ | p > and P µ | q > = q µ | q > .Now consider the matrix element which satisfies < p | ¯ C k ( x, u ) | q > = 0 , unless u ∈ V + (89)34n fact if < φ ( p ) | and | ψ ( q ) > are arbitrary states which are superposition of mo-mentum states the matrix element < φ ( p ) | ¯ C k ( x, u ) | ψ ( q ) > = 0 , unless u ∈ V + . Themicrocausality constraint is[ φ ( x ) , φ ( x )] = 0 , if ( x − x ) < C j ( x, ζ ) = ˜ C j ( x, − ζ ) , for ζ < F j ( x, ζ ) = < φ ( p ) | ( ˜ C j ( x, ζ ) − ˜ C j ( x, − ζ )) | ψ ( q ) > = 0 , for ζ < F j ( x, u ) = < φ ( p ) | ( ¯ C j ( x, u ) − ¯ C j ( x, − u )) | ψ ( q ) > = 0 , if u < F j ( u ) satisfies all the conditions of Jost-Lehmann-Dyson theorem [29, 30].It admits the representation˜ F j ( u ) = Z ds Z d u ′ Σ ( x, u − u ′ , s ) (94)The function Σ ( x, u − u ′ , s ) vanishes unless the hyperboloid ( u − u ′ ) = s lies in theregion u ≥
0; otherwise it is arbitrary. We recall that ˜ C j ( x, ζ ) is local in x for everyfixed ζ . Moreover, from the Fourier transform of ˜ C j ( x, ζ ), we know that u ∈ V + which is conjugate to ζ .It is necessary to discuss further the analyticity properties in the present context.Let us complexify ζ , i.e. define ˜ ζ = ζ − iα, α ∈ V + . We employ the argumentsdeveloped in the previous section to consider the complex ζ -plane. This defines theforward tube, T ( ˜ ζ ). Thus, ˜ C j ( x, ζ ) is boundary value of an analytic function G j ( x, ˜ ζ )such that ˜ C j ( x, ζ ) = lim α → G j ( x, ζ − iα ) (95)We obtain the corresponding extended tube T ′ ( x, ˜ ζ ) by implementing orthochronouscomplex Lorentz transformations, SL + (2 C ), on the points of the forward tube T ( ¯ ζ ).Moreover, G j ( x, ˜ ζ ) = 0 for ˜ ζ < , ˜ ζ real (96)Note that the real points are separated by spacelike distance.Therefore, ˜ C j ( x, ζ ) and˜ C j ( x, − ζ ) are analytic continuation of each other.We may interpret this conclusion as a proof of crossing in the following sense. (i) The35wo local operators ˜ C j ( x, ζ ) and ˜ C j ( x, − ζ ) coincide at the Jost point. (ii) Now we con-sider the difference of two Wightman functions W ( x , x , x , x ) − W ′ ( x , x , x , x )which is < | φ ( x ) φ ( x ) φ ( x ) φ ( x ) | > − < | φ ( x ) φ ( x ) φ ( x ) φ ( x ) | > (97)Once we implement OPE for the fields φ ( x ) φ ( x ) and φ ( x ) φ ( x ) we arrive at theJLD representation and finally see that two functions coincide for spacelike separatedpoints. .We proceed to derive bootstrap equation by invoking PCT theorem and its equiva-lence with weak local commutativity as alluded to in the beginning of this section. Letus briefly recapitulate important aspects of PCT theorem for motivations. The PCTtheorem is very profound. The theorem, known as Pauli-L¨uder theorem, was provedby Pauli [88] and by L¨uder [89, 90]. Furthermore, when the theorem was proved byPauli and L¨uder, they had not considered the possibilities of the violation of discretesymmetries: P, C and T. Moreover, in their proof, they considered Lagrangian fieldtheories. The violation of parity in weak interactions was proposed by Yang andLee [91] later. The parity violation was experimentally observed very soon by Wuand collaborators [92]. Therefore, with hindsight, we may say that the proof of Pauli-L¨uder PCT theorem did not consider the most general case at that juncture. Jost [62]proved PCT theorem rigorously for axiomatic local field theories and established theequivalence of the theorem with weak local commutativity of Wightman functions.Dyson [93] as well as Ruelle [94] have further investigated consequences of WLC andanalyticity properties of Wightman functions. Greenberg [95] has proved that viola-tion of PCT theorem by Wightman functions implies violation of Lorentz invarianceof the local field theory. One of the consequences of the PCT theorem is that themasses of particle and antiparticle be equal. The best experimental test is from the K and ¯ K mass difference [96]: − × − GeV < m K − m ¯ K < +4 × − GeV .It is obvious why so much of premium is placed on CPT theorem. Let us considerthe four point Wightman function to derive the conformal bootstrap equation fromthe PCT theorem.The discrete spacetime transformations: parity, P, and time reversal, T, havefollowing properties. Under P, ( t, x ) → ( t, − x ), whereas under time reversal, T,( t, x ) → ( − t, x ). All the additive quantum numbers characterizing a field reverse theirsigns under C. Moreover, T is an antilinear operator. The combined operator Θ = P CT
For local field theories, if it is invariant under proper Lorentz transformation,the existence of Θ can be proved. Consider a complex scalar field, Φ( x ). The action36f the operator is Θ Φ( x ) Θ − = η Φ( − t, − x ) † , | η | = 1 (98) η , satisfying the constraint, is a phase and it is so chosen that the theory is PCTinvariant. A Wightman function of complex scalar field Φ( x ) transforms as followsunder Θ < | Φ( x )Φ( x ) ... Φ( x n ) | > → < | Φ( − x )Φ( − x ) ... Φ( − x n ) | > ∗ = < | Φ( − x n )Φ( − x n − ) ... Φ( − x ) | > (99)We deal with a real scalar conformal field, φ ( x ). We recall that the four point function, W ( x , x , x , x ) = < | φ ( x ) φ ( x ) φ ( x ) φ ( x ) | > which transforms as follows underthe PCT operation < | φ ( x ) φ ( x ) φ ( x ) φ ( x ) | > → < | φ ( − x ) φ ( − x ) φ ( − x ) φ ( − x ) | > (100)If the theory is invariant under PCT symmetry then < | φ ( x ) φ ( x ) φ ( x ) φ ( x ) | > = < | φ ( − x ) φ ( − x ) φ ( − x ) φ ( − x ) | > (101)We have noted, in the previous section, that W ( x , x , x , x ) depends on differenceof the coordinates, y j = x j − x j +1 due to translational invariance. Recall our defini-tion of complexified coordinates, ξ j = y j − iη j , η j ∈ V + . We argue, as before, that W ( { y j } ) , j = 1 , , W ( { ξ j } ) , j = 1 , , { ξ j } and ξ j ∈ T , the forward tube. We remind that W ( { ξ j } ) isalso invariant under orthochronous complex Lorentz transformation SL + (2 C ). Theset of points of ξ j ∈ T which are generated under arbitrary complex Lorentz trans-formations, Λ ∈ SL + (2 C ) define an extended tube T ′ . In other words, the points { Λ ξ j } obtained from { ξ j } belong to the extended tube. Moreover, there is a singlevalued continuation of W ( { ξ j } to the extended tube [60]. We have emphasized beforethat T contains only the complex points of the forward tube. On the other hand T ′ contains the real points, { y j } , as well. These points are spacelike; the Jost points.The Jost points are spacetime points in which all convex combinations of successivedifferences are spacelike. A Jost point is an ordered set ( x , x , x , x ). The Jost theorem [62] : A real point of ξ , ξ , ξ lies in the extended tube , T ′ , if andonly if all real four vectors of the form P λ j ξ µj , λ j ≥ , P λ j > are spacelike i.e.( P λ j ξ µj ) < , λ j ≥ , P λ j > The necessary and sufficient condition is thatall the real points of T ′ are spacelike. The equivalence between PCT theorem and WLC:If the PCT theorem holds for all x , x , x , x then for every x , x , x , x such thateach of the y j = x j − x j +1 is a Jost point. The WLC condition leading to < | φ ( x ) φ ( x ) , φ ( x ) φ ( x ) | > = < | φ ( x ) φ ( x ) , φ ( x ) φ ( x ) | > (102)37s satisfied. The converse statement of the Jost theorem is paraphrased as if WLCholds in a real neighborhood of (102), a Jost point, then the PCT condition (101) isvalid everywhere . Moreover, WLC implies validity of PCT symmetry for the confor-mal scalar. We are in a position to address the conformal bootstrap proposal in thepresent perspective and go through the following steps. I . The validity of the PCT theorem is assumed for conformal field theory. Moreover,we know that W ( ξ , ξ , ξ ) is a holomorphic function and (102) holds in the extendedtube T ′ . Furthermore, the four point function is boundary value of an analytic func-tion lim η j → W ( ξ , ξ , ξ ) = W ( y , y , y ) (103)We also know from [60] that W ( ξ , ξ , ξ ) is invariant under proper complex Lorentztransformations, SL + (2 C ): { ξ i } → Λ { ξ i } , ξ i ∈ T ′ . Let us choose a Λ such that theset of complex four vectors ξ µi → − ξ µi . Consequently, W ( ξ , ξ , ξ ) = W ( − ξ , − ξ , − ξ ) (104) II . We recall that the r.h.s. of the equation (101) is a statement of PCT invarianceof the four point function. It is also boundary value of an analytic function.lim η j → W ( ξ , ξ , ξ ) = W ( y , y , y ) = < | φ ( − x ) φ ( − x ) φ ( − x ) φ ( − x ) | > (105) III . Now consider the difference of two four point functions: W ( ξ , ξ , ξ ) −W ( ξ , ξ , ξ ).This is holomorphic in the domain T ′ . We know that this difference vanishes forRe ξ i , i = 1 , , C and C , in R n for generality. Let the functions f ( z ) and f ( z ), with z = x + iy (note x, y carry no indices and they have no relationship with { y µi } definedabove) satisfy following properties: (i) The two functions, f ( z ) and f ( z ), are definedand analytic in the intersection of the tube over C α , α = 1 ,
2; that is z : Im z ∈ C α and also analytic in a certain neighborhood of z = 0. (ii) When y tends to 0 frominside C α the two functions f ( x + iy ) and f ( x + iy ) tend to distributions T ( x ) and T ( x ) respectively; in D N where N is certain real neighborhood of the point z = 0.Recall that the Wightman function, a distribution, is boundary value of the analyticfunction. And (iii) T = T . Then f ( z ) and f ( z ) have a common analytic exten-sion f ( z ) on the intersection of the neighborhoods of z = 0 and the convex closure of C ∪ C . Consider a situation where C and C are completely opposite i.e. C ∩ ( − C )contains an open cone, then f ( z ) is analytic in a neighborhood of z = 0. It follows, inour context, that if the coordinate differences, x i − x i +1 , i = 1 , , , { y i } , i = 1 , , W ( ξ , ξ , ξ ) = W ( ξ , ξ , ξ ) (106)The converse of the above statement is the following. It is a consequence of Hall andWightman theorem [60] that if (106) holds good in an arbitrary neighborhood of T ′ it also holds good in the extended tube. Moreover, if it is also valid for passing intothe boundary in the tube T then we recover the condition of PCT invariance (101).In the historical context, note that the first proof of the edge-of-the-wedge theorem[63] was presented to prove dispersion relations for pion-nucleon scattering. Theamplitude was obtained by adopting the LSZ [20] reduction technique. Subsequently,the matrix element of causal commutator of the source currents was envisaged. Thusthe equations of motion were implicitly used. For the present case we are dealingwith Wightman functions and the edge-of-the-wedge theorem is invoked to prove(106). Thus the conclusion is that PCT invariance is equivalent to WLC in CFT. Itfollows from equations (104) and (106) that W ( ξ , ξ , ξ ) = W ( − ξ , − ξ , − ξ ) (107) Remark : Let us try to pass to the boundary in the above equation for any set of y , y , y in (107). A problem arises. We shall not be able to obtain a relation be-tween the two functions (in the above equation) for any set of { y i } for the followingreason. On the l.h.s. ξ , ξ , ξ approach the real vectors which are in V + . Notethat the real vectors of − ξ , − ξ , − ξ would be in V − . The equality holds for forRe ( ξ , ξ , ξ ) and Re ( − ξ , − ξ , − ξ ) when we are at the Jost point.Notice the important fact: at the real point of holomorphy, at the Jost point, wehave the following relation W ( ξ , ξ , ξ ) = < | φ ( x ) φ ( x ) φ ( x ) φ ( x ) | > = W ( − ξ , − ξ , − ξ ) = < | φ ( x ) φ ( x ) φ ( x ) φ ( x ) | > (108)This equation has important implications for the conformal bootstrap proposal.Let us proceed to envisage the four point Wightman function < | φ ( x ) φ ( x ) φ ( x ) φ ( x ) | > .We employ the conformal partial wave expansion. Introduce a complete set of states, {| Ψ > } , between the product of the two pair of operators: φ ( x ) φ ( x ) and φ ( x ) φ ( x ).The states {| Ψ > } belong to the full Hilbert space, H . Therefore, all irreducible rep-resentations of the conformal group are included as ’intermediate’ states. Now W = < | φ ( x ) φ ( x ) φ ( x ) φ ( x ) | > = X | Ψ > < | φ ( x ) φ ( x ) | Ψ >< Ψ | φ ( x ) φ ( x ) | > (109)The sum P | Ψ > also includes integration over coordinates as is the custom whenwe insert complete set of intermediate states. We resort to the state ↔ operator39orrespondence and then interpret < | φ ( x ) φ ( x ) | Ψ > as a three point function < | φ ( x ) φ ( x ) ˆΨ | > . Let us identify | Ψ > = ˆΨ | > ; ˆΨ represents the complete setof operator belonging to irreducible representations of the conformal group. Noticethat the second matrix element of the r.h.s. of the above equation is another threepoint function and < Ψ | = < | ˆ¯Ψ; ˆ¯Ψ is the adjoint of ˆΨ. We may reexpress (109) as < | φ ( x ) φ ( x ) φ ( x ) φ ( x ) | > = X Ψ , ˆΨ X αβ λ αφ φ W αβφ φ ˆΨ ¯ˆΨ φ φ λ βφ φ (110)where λ αφ φ and λ βφ φ can be read off from the above equation. Moreover, W αβφ φ ˆΨ ¯ˆΨ φ φ is the conformal partial waves (CPW) [13, 7, 8, 9]. The sum over complete set ofoperators include all the allowed irreducible representations of the conformal groupsuch as Lorentz spin, scale dimensions etc. We have discussed the analyticity and itsclose relationship with causality in the previous section in the Wightman’s formulationfor CFT. Let us briefly recall the analyticity properties of the commutator of a pair ofcomposite operators that appear in OPE. We consider the CPW expansion of anotherfamiliar four point function < | φ ( x ) φ ( x ) φ ( x ) φ ( x ) | > = X ˆΨ , ¯ˆΨ X αβ λ αφ φ W αβφ φ ˆΨ ¯ˆΨ φ φ λ βφ φ (111)The two expressions for Wightman functions (110) and (111) are equal at those Jostpoints and these are conformal bootstrap conditions [13, 15, 7, 8, 9].We remind that the Wightman functions, satisfying (110) and (111) are analyticfunctions in the extended tubes. Let us invoke Jost’s theorem [62] and Dyson’sarguments [93] for the proof of analyticity of the Wightman functions for real { ξ i } and when the point corresponds to Jost point. It follows from WLC condition that X ˆΨ X αβ λ αφ φ W αβφ φ ˆΨ ¯ˆΨ φ φ λ βφ φ = X ˆΨ , ¯ˆΨ X αβ λ αφ φ W αβφ φ ˆΨ ¯ˆΨ φ φ λ βφ φ (112)This is the conformal bootstrap equation. We have alluded to the importance andsignificance of this equation. Let us deliberate on a few points. Notice that the equa-tion holds when points are separated by spacelike distance. If we considered Fouriertransform of the two Wightman functions, W ( ξ , ξ , ξ ) and W ( ξ , ξ , ξ ) the conju-gate momenta are spacelike. Therefore, the two functions coincide in the region wheremomenta are spacelike. This is the situation in case of the absorptive amplitudes of s and u channels of a scattering amplitude derived from the LSZ reductions. Theabsorptive parts of the two amplitude are equal when momenta are real and lie inthe unphysical region. It was necessary to prove that the two absorptive parts areanalytic continuation of each other. The situation is the same here. It is necessary toidentify the domain of analyticity of each of the Wightman functions and argue thatthey are analytic continuation of each other. We have proved this through the chainof arguments presented earlier. We have persuasively argued in the previous sectionthat a pair of Wightman functions are holomorphic in the union of their extended40ubes. We think that the PCT theorem together with its equivalence of WLC pro-vide a very strong basis to derive the conformal bootstrap equation. It is needless toemphasize the crucial role played by the edge-of-the-wedge theorem. Remark:
The above bootstrap condition is not specifically valid for a nonderiva-tive scalar conformal field theory. If we consider four point Wightman function, fornonderivative conformal fields, which belong to finite dimensional irreducible repre-sentation of conformal group then the above proof will go through with appropriatemodifications. Now the corresponding Wightman function will carry tensor indices asthe fields would transform according to the representations of SL (2 C ) ⊗ SL (2 C ) andthey will carry their conformal dimensions [5]. Thus the W will transform covariantlyunder SL (2 C ) ⊗ SL (2 C ). The preceding arguments will essentially go through withadequate technical modifications only. Consequently, the analyticity properties andbootstrap equations will continue to hold for the general four point functions as longas the fields are of nonderivative types. Therefore, we conclude that the two resultingfour point functions, in general, will be analytic continuation of each other.To briefly summarize the contents of this section: (i) We considered OPE of a pairof nonderivative scalar conformal fields and expanded them in a set of composite (alsononderivative) conformal fields. Next, we considered OPE for the commutator of thepair of fields and noted that the commutator vanishes when they are separated byspacelike distance. Then we argue that the matrix element of the Fourier transformsof the difference of the two composite field operators enjoy certain support properties.Consequently, a representation, for the matrix elements could derived which is analo-gous to Jost-Lehmann-Dyson representation. The analyticity properties of the matrixelements are discussed. The connections with bootstrap equation are presented.In the second part, we invoked PCT theorem to aim at derivation of the boot-strap equation. The equivalence of weak local commutativity (WLC) with the PCTtheorem plays a crucial role in arriving at the bootstrap equation. We went througha number of steps. It is important, in our view, to note that although the two fourpoint Wightman functions coincide at the Jost point, one has to demonstrate thatthe two four point functions are analytic functions. It is necessary to identify theirdomain of holomorphy and then invoke the edge-of-the-wedge-theorem to show thatthese two functions are analytic continuation of each other.
4. Summary and Conclusions .
The objective of this investigation was to study analyticity and crossing propertiesof four point correlation functions of conformal field theory. We adopted Wightman’s41ormulation. It is well known that Wightman axioms are not respected by all con-formal field theories. Therefore, we considered a nonderivative real scalar field, φ ( x ),which has the desired properties [41]. We invoked the arguments that Wightmanfunctions are boundary values of analytic functions of several complex variables andfocused on the four point function. The primitive domain of analyticity of the cor-responding analytic function was identified to be the forward tube, T since the fourpoint function depends on three independent coordinates due to translational invari-ance. Next, the extended tube, T ′ was defined following the standard procedurewhere the function is analytic and is single valued. A permuted four point Wightmanfunction was considered. The two four point functions are related by crossing. As aprelude, we presented analyticity and crossing properties of the three point functionin some detail incorporating essential results of [55].Moreover, we have studied the analyticity properties of three point function in themomentum space representation. The work of Jost [73] was crucial. The R-productof three point function was shown to be related to Wightman functions. We utilizedthis fact to investigate analyticity of momentum space three point function.Next, we analyzed the crossing properties of four point function. In general, eachof the permuted Wightman function is boundary value of the corresponding analyticfunction. The domain of analyticity is union of the domain of holomorphy of fourpoint functions. However, the entire domain of holomorphy of the collection all theWightman function might be much larger. We have considered a pair of Wightmanfunctions at a time and then found the domain of holomorphicity. The Fourier trans-forms of the two four point functions were considered and we read off their supportproperties in the momentum space We adopted the prescriptions of [55] to deriverepresentation of the Fourier transform of the four point Wightman function. Thisrepresentation is similar to the Jost-Lehmann-Dyson representation derived for thematrix element of causal current commutators in field theory. Indeed, starting fromthis point, we get insights into crossing symmetry for a pair of permuted Wightmanfunction at a point where the two functions coincide. Next step was to derive the do-main of analyticity in the coordinate space. We depended on two important results.First, the Hall-Wightman theorem was invoked to argue that the analytic functiondepend on Lorentz invariants constructed from the complexified four vectors. Thesecond ingredient was to appeal to the Jost theorem. As far as we know, the envelopeof holomorphy have not been constructed for four point Wightman functions com-pletely in QFT [52, 53, 54] i.e. as exhaustively as for the three point function [51]. Ourprincipal goal is to establish crossing for four point functions. Therefore, it suffices,for our purpose, to consider a pair of Wightman functions and identify the domain ofholomorphy. Consequently, we can proceed to prove crossing for Wightman functionstaken pairwise. There was one more technical obstacle. In case of the three pointfunctions there were only two Hall-Wightman Lorentz invariant complex variables for42ll practical purposes (although there are three of them z , z , z ; and crossing wasproved [55]). Note that for the four point functions, the number of Hall-Wightmanvariables increase and the prescription of [55] does not go through in a straight forwardmanner. Therefore, we simplified the task a little bit and derive constraints on (real)Hall-Wightman invariants since we go to a domain where Jost theorem is applicable.Thus in Section 2, we established crossing for a pair of permuted Wightman functions.The third section was devoted to study analyticity properties of the matrix ele-ments of composite operators which arise in the OPE. In this section the microcausal-ity plays a crucial role in establishing the analyticity properties. The OPE of a pairof nonderivative conformal fields has been investigated by Mack [41]. In the OPE,the composite fields are also of nonderivative type and they belong to the irreduciblerepresentations of the conformal group. Consequently, they respect Wightman ax-ioms. We considered, OPE of the commutator of the two scalar fields. Thus weobtain difference of two composite fields as has ben demonstrated in section 3. Weconsidered the Fourier transforms and then took matrix elements of the compositeoperators. This matrix element is constrained from the microcausality arguments.The constraints are the same as those utilized to derive the Jost-Lehmann-Dysonrepresentation (JLD). We recall that, JLD representation was crucial to derive cross-ing in QFT. Moreover, we use the techniques developed in section 2 to study theanalyticity properties of these matrix elements. We presented a derivation of the con-formal bootstrap equation from a novel perspective. We invoked two very powerfultheorems of axiomatic field theory to accomplish our goal. We first appeal to thePCT theorem. PCT theorem is profound and is respected by all local field theorieswhich respect Wightman axioms. We noted that two Wightman functions are equalif they are PCT transform of each other. We utilized the equivalence of PCT theo-rem and weak local commutativity which was rigorously proved by Jost. Therefore,we were able to relate two Wightman functions at Jost point. Next, we presenteda series of steps to relate the two Wightman functions. The conformal partial waveexpansion technique was applied to arrive at the conformal bootstrap equation. It isnot adequately emphasized that the equality between two four point functions holdswhen a pair of coordinates, for their real vales, are separated by spacelike distance.It is essential to prove that the four point functions are analytic at that point. More-over, from the perspectives of Wightman axioms, it is required that we identify theirrespective extended domain of holomorphicity of the two functions in the complexdomain. Dyson [93] and Ruelle [94] have proved the analyticity of the Wightmanfunctions at Jost point in the context of WLC [62]. Furthermore, as has been arguedin section 3, the bootstrap equation is not special to the case of Wightman functionof four scalar field. Indeed, a four point functions can be defined as product of fourconformal fields belonging to irreducible representations of the conformal group, in ageneral scenario, as long as they satisfy Wightman axioms. Then the correspondingbootstrap equation holds. It is important to note that three fundamental theorems of43xiomatic local field theories such as PCT theorem, the theorem stating equivalencebetween PCT theorem and weak local commutativity and the edge-of-the-wedge the-orem, are invoked to derive the conformal bootstrap equation rigorously.We conclude the article with following remarks. It is tempting to suggest thatcrossing and analyticity of n-point function can be derived following the techniquesintroduced here. It must be noted that proof of crossing and analyticity for n-pointfunctions in axiomatic QFT is a formidable task. Bros [97] has comprehensively re-viewed the progress on crossing and analyticity properties of n-point amplitude inaxiomatic QFT at that juncture. Our understanding is that several issues have re-mained unresolved in this topic. Another avenue to explore, in order to derive crossingrelations for n-point functions in the context of CFT, is to follow a clue from QFT.We should look for a generalization of Jost-Lehmann-Dyson representation for then-point function in CFT. The JLD representation was the principal ingredient sincethe coincident region was identified through this technique. Thus the singularity freeregion was identified. There have been attempts, in the past, to obtain integral repre-sentations for the VEV of product of field operators and VEV of the commutators ofstring of field operators in the axiomatic QFT [66, 68]. Therefore, those results mightbe utilized to prove crossing for, at least, a pair of permuted Wightman functions.However, we have not established existence of JLD representation to n-point functionsin CFT. Therefore, it might be premature to speculate that problem of crossing andanalyticity could be solved in a straight forward manner in CFT. However, conformalsymmetry is very powerful. It is well known that two point and three point functionsget fixed in CFT up to constant coefficients. Moreover, conformal symmetry imposesconstraints on the structure of the n-point functions ( n > .Acknowledgments: I am thankful to Edward Witten for very valuable discussions onanalyticity properties of scattering amplitudes. I was influenced by his suggestionsthat the correlation functions of CFT have better prospects of unraveling the analytic-ity properties. The warm hospitality of Professor Witten at the Institute for AdvancedStudy, in 2017, is gratefully acknowledged. I would like to thank Hadi Godazgar forseveral useful remarks and for valuable suggestions to improve the manuscript. Witten has suggested that it might be worth while to investigate questions about analyticcontinuations and crossing properties of correlation functions of conformal field theories. [98] eferences
1. G. Mack and A. Salam, Ann. Phys. , 174 (1969).2. S. Ferrara, R. Gatto and A. F. Grillo, Springer Tracts in Mod. Phys. ,1(1973).3. G. Mack, Lecture Notes in Physics, ,Springer Berlin-Heidenberg-New York,1972. in Mod. Phys. ,1 (1973).4. E. S. Fradkin and M. Ya Palchik, Phys. Rep. C44 , 249 (1978).5. E. S. Fradkin and M. Ya Palchik, Conformal Field Theory in D-dimensions,Springer Science Business Media, Dordrecht, 1996.6. I. T. Todorov, M. C. Mintechev and V.R. Petkova, Conformal Invariance inQuantum Field Theory, Publications of Scuola Normale Superiore, BrirkhauserVerlag, 2007.7. D. Simon-Duffin, TASI Lectures 2015, ArXiv: 1602.07982[hep-th].8. J. Penedones, TASI Lecture 2016, ArXiv: 1608.04948[hep-th].9. D. Poland, S. Rychkov and A. Vichi, Rev. Mod. Phys. , 051002.2019 (2019).10. A. M. Polyakov, JETP Lett. , 381 (1970).11. A. A. Migdal, Phys. Lett. bf 37B, 98 (1971); ,386, (1971).12. S. Ferrara, A. F. Grillo and R. Gato, Annals of Phys. 76, 161,1973.13. . Ferrara, A. F. Grillo, R. Gatto and G. Parisi, Nuovo. Cim. A19 ,667 (1974).14. G. Mack and I. Todorov, Phys. Rev. D8 , 1764 (1973).15. A. M. Polyakov, Z. Eksp. Teor. Fiz, , 23, (1974).16. G. F. Chew, S-Matrix Theory of Strong Interactions, W.A. Benjamin Inc 1962.17. G. F. Chew ”Bootstrap: A scientic idea?” Science, , 762 (1968).18. G. Veneziano, Nuovo Cim. , 190 (1968).19. J. Bross, H. Epstein and V. Glaser, Nuovo. Cim., , 1265 (1964).20. H. J. Lehmann, K. Symanzik and W. Zimmermann, Nuovo Cim. , 205 (1955).21. J. Maldacena, Adv. Theor. Math. Phys. , 231 (1998).452. A. S. Wightman, Phys. Rev. , 860 (1956).23. S. S. Schweber, Introduction to Relativistic Quantum Field Theory, Harper andRow, New York, Evanston and London, 1961.24. R. F. Streater and A. S. Wightman PCT Spin Statistics and All That, Benjamin,New York, 1964.25. R. Jost, General Theory of Quantized Fields, American Mathematical Society,Providence, Rhode Island, 196526. R. Haag, Local Quantum Physics: Field Fields, Particles, Algebras, Springer,1996.27. C. Itzykson and J. -B. Zubber Quantum Field Theory, Dover Publications Mi-neola, New York, 2008.28. N. N. Bogolibov, A.A. Logunov, A.I. Oksak and I. T. Todorov, General Princi-ples of Quantum Field Theory, Klwer Academic Publisher, Dordrecht/Boston/NewYork/London, 1990.29. R. Jost and H. Lehmann, Nuovo Cimen. , 1598 (1957).30. F. J. Dyson, Phys. Rev. , 1460 (1958).31. G. Gillios, M. Meineri and, J. Penedones, A Scattering A scattering amplitudein conformal field theory, arXiv:2003.07361 [hep-th].32. T. Baurista and H. Godazgar, JHEP, 01, 142 (2020).33. M. Gillioz, Commun.Math.Phys. , 227 (2020).34. J. S. Toll, Phys. Rev. , 1760 (1956).35. K. G. Wilson, Phys. Rev. , 1499 (1969).36. K. G. Wilson and W. Zimmermann, Commun. in Math. Phys. , 87, (1972).37. H. J. Lehmann, Nuovo Cim. , 342 (1954).38. P. Otterson and W. Zimmermann, Commun. Math. Phys. , 107 (1972).39. E. S. Fradkin and M. Ya Palchik, Ann. Phys. , 174 (1969).40. M. L¨uscher and G. Mack, Commun. Math. Phys. , 203 (1975).41. G. Mack, Commun. in Math. Phys. , 155 (1977).42. T. Yao, J. Math. Phys. , 1731 (1967).463. T. Yao, J. Math. Phys. , 1615 (1968).44. T. Yao, J. Math. Phys. , 315 (1971).45. G. Mack, Commun. in Math. Phys. , 1 (1972).46. Z. Komargodski and A. Zhiboedov JHEP , 140 (2013.47. T. Hartman, S. Kundu and A. Tajdini, JHEP , 066 (2017).48. T. Hartman, S. Jain and S. Kundu, JHEP, 05, 099 (2016).49. M. S. Costa, T. Hansen and J. Penedones, JHEP, 10, 197 (2017).50. S. Caron-Huot, JHEP, 09, 078 (2017).51. G. K¨all¨en and A. S. Wighgtman, (Kgl. Dan. Mat.-fys. Skrifter, 1: No. 6(1958).52. D. Kleitman, Nucl. Phys. , 459 (1959).53. G. K¨all¨en Nucl. Phys. , 568 (1961).54. B. Geyer, V. A. Matveev, D. Robaschik and E. Wieczorek, Rep. Math. Phys. , 203 (1976).55. J. Maharana, Mod. Phys. Lett. A 35 , 2050186 (2020).56. F. J. Dyson, Phys. Rev. , 1717 (1958)57. R. F. Streater, Proc. R. Soc. , 39 (1960).58. J. Maharana, PCT Theorem, Wightman formulation and Conformal Bootstrap,ArXiv: 2102.0188; (to be published).59. A. S. Wightman, Lectures on Field Theory Summer School 1958, Varena Suppl.Nuovo Cimento , no1 (1959).60. D. Hall and A. S. Wightman, Del Kong. Danske Viden. Selska. , no. 5, 1957.61. D. Ruelle, Helvetica Phys. Act. , 135 (1959).62. R. Jost, Helvetica Phys. Act. , 409 (1957).63. H. J. Bremmermann, R. Oehme and J. G. Taylor, Phys. Rev. , 2178 (1958).64. J. G. Taylor, Ann. Phys. , 391 (1958).65. H. Epstein, J. Math. Phys. , 524 (1960).476. R. F. Streater, J. Math. Phys. , 256 (1962).67. Y. Tomozawa, J. Math. Phys. , 1240 (1963).68. R. F. Streater, Nuovo Cim. , 937 (1960).69. H. Araki and N. Burgoyne, Nuovo. Cim. , 342 (1960).70. D. Ruelle, Nuovo. Cim. , 358 (1961).71. H. Araki, Prog. Th. Phys. , 83 (1961).72. The positivity condition in momentum space CERN Preprint TH.980 (unpub-lished).73. R. Jost, Helvetica Phys. Acta , 263 (1958).74. W. S. Brown, J. Math. Phys. , 221 (1961).75. C. Coriano, L. Delle Rose, E. Mottola and M. Srino, JHEP , 011 (2013).76. A. Bzowski, P. McFadden and K.Skenderis, JHEP , 111 (2014).77. A. Bzowski, P. McFadden and K.Skenderis, ArXiv 1910.10162.78. C. Coriano and M. M. Maglio, JHEP , 107 (2019).79. H. Isono, T. Noumi and G. Shiu, JHEP , 136 (2018).80. H. Isono, T. Noumi and G. Shiu, JHEP , 183 (2019).81. N. Arkani-Hamed, D. Bauman,H. Lee and G. L. Pimentel, ArXiv 1811.200024.82. D. Bauman, C. Duaso Puero, A. Joyce, H. Lee and G. L. Pimentel, arXiv1910.14051.83. C. Sleight, JHEP , 090 (2020).84. C. Sleight and M. Taronna, JHEP , 098 (2020).85. S. Albayrak, C. Chowdhury and S. Kharel, arXiv2001.06777.86. M. Gillioz,X. Lu and A.Luty, JHEP , 171 (2017).87. M. Froissart, Dispersion Relations And their Connection with Causality, Aca-demic, New York; Varrena Summer School Lectures (1964).88. W. Pauli, in Niels Bohr and the Development of Physics, McGraw-Hill, NewYork pp30 (1955). 489. G. L¨uders, Danske Videnskabernes Selskab, Mat.-fys. Medd. , No 5 (1954).90. G. Grawert, G. L¨uders and H. Rollnik, Fortscr. der Physik, , 291 (1959).91. C. N. Yang and T. D. Lee, Phys. Rev., , 254 (1956).92. C. S. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes, and R. P. Hudson Phys.Rev. , 1413 (1957).93. F. J. Dyson, Phys. Rev. , 579 (1958).94. D. Ruelle, Helv. Phys. Acta , 135 (1959).95. O. W. Greenberg, Phys. Rev. Lett. , 231602-1 (2002).96. P. A. Zyla et al.(Particle Data Group), Prog. Th. Phys.97. J. Bros, Phys. Rep.134