PCT Theorem, Wightman Axioms and Conformal Bootstrap
aa r X i v : . [ h e p - t h ] F e b February 4, 2021
PCT Theorem, Wightman Axioms and ConformalBootstrap
Jnanadeva Maharana E-mail [email protected]
Institute of PhysicsandNISERBhubaneswar - 751005, India
Abstract
The axiomatic Wightman formulation for nonderivative conformal field the-ory is adopted to derive conformal bootstrap equation for the four point func-tion. The equivalence between PCT theorem and weak local commutativity ,due to Jost, play a very crucial role in axiomatic field theory. The theorem issuitably adopted for conformal field theory to derive the desired equations inCFT. We demonstrate that the two Wightman functions are analytic continu-ation of each other. Adjunct Professor, NISER, Bhubaneswar he PCT theorem is very profound. The proof of the theorem is based on axiomsof local field theories. It is a fundamental probe for our basic understanding of mi-croscopic physics. Pauli and L¨uder [1, 2, 3] presented the first proof of the theorem.However, at that juncture, possibility of parity violation in weak interaction was notyet proposed by Yang and Lee [4]. Parity nonconservation was subsequently observedexperimentally. Thus, in the proof of the Pauli-L¨uder theorem, the violations of disc-trete symmetries, such as P, C, and T, in field theories, had not been envisaged. Jost[5, 6] proved the PCT theorem from the axioms of local field theories whereas earlierproofs were based on Lagrangian field theories. The fundamental nature of Jost’sproof is that weak local commutativity (WLC) at the Jost points is necessary andsufficient condition for PCT symmetry. This aspect will be elaborated in the sequel.One of the most important consequences of PCT theorem is that masses of particleand antiparticle be equal. The best experimental test comes from the K − ¯ K massdifference [7]. The limit is − . × − GeV < m K − m ¯ K < . × − GeV .Moreover, the violation of PCT invariance of any Wightman function implies theviolation of Lorentz invariance [8]. Therefore, there is so much of premium on thePCT theorem.The purpose of this letter is to derive the conformal bootstrap equation in theWightman’s formulation of axiomatic field theory [6, 9]. The PCT theorem is in-voked to relate two four point Wightman functions. Our motivation is from fol-lowing considerations. Let us envisage two four point Wightman functions: (i) < | φ ( x ) φ ( x ) φ ( x ) φ ( x ) | > and (ii) < | φ ( x ) φ ( x ) φ ( x ) φ ( x ) | > . They arealso boundary values of the analytically continued corresponding analytic functionsof complexified coordinates as will be elaborated shortly. As such, there seems to beno relationship between them at this juncture; however, once PCT theorem is invokedthey get related. As often the case in study of scattering amplitudes, seemingly un-related amplitudes are no longer independent when we invoke a symmetry principle.For example the reactions, πN → πN , are described by several independent ampli-tudes until we invoke the principle of isotpic spin invariance of strong interaction.Then the amplitudes π + p → π + p , π − p → π − p and π − p → π n are no longer indepen-dent. We appeal to PCT theorem, its equivalence to weak local commutativity andinvoke the edge-of-the-wedge theorem to establish relation between the two Wight-man functions, (i) and (ii) defined above. It will be shown that conformal bootstrapequation relating (i) and (ii) would be obtained. This is achieved by employing theconformal partial wave expansion procedure for each of the four point functions. Itis assumed in our derivation that CPW converges for the case under consideration.We feel that it is a novel way to relate two different four point functions and obtainbootstrap equation. In order to derive the aforementioned result, we shall recapitu-late important theorems of axiomatic field theory and recall some of the holomorphicproperties of the Wightman functions [9]. We provide references to original papersand to books for the benefit of readers. The derivation of the conformal bootstrap1quation is presented step by step sequentially starting from the important resultsof Wightman formulation. The rigorous results demonstrating intimate relationshipsbetween analyticity, crossing and causality in CFT and their connections with boot-strap will be presented in our forthcoming article [10].The research activities in CFT sprouted following the seminal paper of Mack andSalam [11] and a lot of activties ensued in 1970’s [12, 13, 14, 15]. Migdal introducedthe idea of conformal invariance to derive bootstrap equations for hadronic interac-tions [16]. The conformal bootstrap was proposed by various authors in that period[17, 18, 19]. A rejuvinated activity has emerged in recent years and the conformalbootstrap program has expanded in several ditrections; these have been reviewed re-cently [21, 22, 23]. There has been vigorous research activities in CFT and it hasspread in diverse directions such as the understanding of critical phenomena, super-symmetric conformal field theories in higher dimensions ( D >
AdS/CF T correspondance conjectureof Maldacena [24] which has strongly influenced research in supergravity and stringtheories.The conformal transformation properties of a real scalar field, φ ( x ), are[ P µ , φ ( x )] = i∂ µ φ ( x ) , [ M µν , φ ( x )] = i ( x µ ∂ ν − x ν ∂ µ ) φ ( x ) (1)[ D, φ ( x )] = i ( d + x ν ∂ ν ) φ ( x ) , [ K µ , φ ( x )] = i ( x ∂ µ − x µ x ν ∂ ν − x µ d ) φ ( x ) (2) { P µ , M µν } are the ten generators of the Poincar´e group; D generates dilation and K µ are the four special conformal transformation generators; d is the scale dimension of φ ( x ).Let us discuss some important features of conformal field theories. To begin withwe recall that in the perturbative approach to field theory, crossing property of thescattering amplitude is maintained in each order. All Feynman diagrams, corre-sponding to direct channel and crossed channels, are included in every order. Theanalyticity and unitarity of amplitude are maintained according to stipulated rules inperturbation theoretic computations. In the context of phenomenological S-matrixtheory, which described hadronic collisions, crossing is assumed. The idea of boot-strap evolved from the S-matrix philosophy. The bootstrap equations were used asconsistency conditions in the S-matrix era. The rigorous proof of crossing for scat-tering amplitude was derived in subsequent years [25]. The bootstrap equations wereintroduced in conformal field theory in order to provide a rigorous basis to the phe-nomenological S-matrix notion of bootstrap. The structure of conformal field theoriesis deeply connected with the symmetry principles. In general, a Lagrangian density2s not introduced, nor there is an action principle. Moreover, we do not invoke theconcept of asymptotic fields and interacting fields. As a consequence, the axiomaticfield theoretic techniques to compute scattering amplitude, for example in the LSZformulation [26], turn out to be inadequate. Therefore, many of the rigorous resultsproved for scattering amplitudes from axiomatic field theories do not hold automati-cally in CFT. Important parameters are computed in CFT to test the theories againstparameters of physical systems [23].The correlation functions in CFT are of extreme importance. Furthermore, theanalyticity and crossing properties play a crucial role in the study of CFT. Moreover,there is intimate relationship between causality and analyticity. Therefore, analytic-ity, crossing and causality are three ingredients in the study of CFT. It is natural toadopt Lorenzian metric. Our choice of the metric is g µν = diag (+1 , − , − , −
1) andwe work in four dimensional spacetime, D = 4. We adopt Wighgtman’s axiomatic for-mulation to investigate aforementioned attributes in CFT. The importance of Wight-man function has been emphasized long ago in the intial developmental phase of CFT[12, 13, 14, 15, 19, 20]. The axioms are [9, 39, 6, 40, 41, 42, 43]:(i) There exists a Hilbert space. It is constructed [14] with appropriate definitions forCFT.(ii) The theory is conformally invariant and the vacuum, | > , is unique. It is anni-hilated by all the generators of the conformal group.(iii) Spetrality: The energy and momentum of states are defined such that 0 ≤ p ≤ ∞ and p ≥
0. We consider a class of theories such that the Fourier transform of thefield, φ ( x ), i.e. ˜ φ ( p ), satisfies spectrality condition stated above [30, 14].(iv) Microcausality: Two local bosonic operators commute when their separation isspacelike i.e. [ O ( x ) , O ( x ′ )] = 0; for ( x − x ′ ) < A ( x ) is any real scalar field) A ( x ) A ( x ) = X f k ( x − x ) C k ( x + x { f k ( x − x ) } are c-number functions whichacquire singularities as ( x − x ) → { C k } are local in ( x + x ). TheOPE, adopted for CFT, is very crucial. We remark that Wilson operator productexpansion [27] was investigated from the Wighgtman axiom perspective by Wilsonand Zimmerman [28]. Subsequently, Otterson and Zimmermann [29] advanced thosetechnques.A nonderivative scalar field φ ( x ), satisfying [ K µ , φ (0)] = 0, K µ being the generatorof special conformal transformation, respects Wightman axioms [30]. The Fouriertransform of φ ( x ), ˜ φ ( p ), satisfies Wightman spectrality condition: p ∈ V + , i.e. p ≥ p ≥
0. Now on, φ ( x ) stands for a nonderivative real conformal field and anyother generic scalar field is denoted as A ( x ). We need to define the the structure ofthe Hilbert space. If we consider OPE of a pair of φ fields the form is φ ( x ) φ ( x ) = X n X χ f χn ( x − x ) C χn ( x + x f χn are c-number coefficients which encode the short distance behaviour. C χn are composite fields belonging to the irreducible representations of SU (2 ,
2) whichis the covering group of the conformal group SO (4 , {C χn } are of nonderivative type [30]. It has been proved by Mack [30]that the expansion is convergent, assuming that it is asymptotic one. Note that weneed infinite number of such fields for the closure of the algebra [14]. The states areconstructed by appealing to state ↔ operator correspondences. Thus, we can identifystate vectors of the underlying Hilbert space, H . In view of preceding remarks, H decomposes into a direct sum of subspaces whose vectors belong to irreduciblerepresentations of SU (2 , H = ⊕H χ (5)where χ collectively stands for all the quantum numbers that characterize an ir-reducible representation such as scale dimension, Lorentz spin etc. Therefore, twonormalised vectors | χ i > ∈ H χ i and | χ j > ∈ H χ j satisfy < χ i | χ j > = δ ij .We recapitulate the essential properties of the n-point Wightman functions, W n ( x , x , ...x n ),defined to be W n ( x , x , ...x n ) = < | φ ( x ) φ ( x ) ...φ ( x n ) | > (6)in order to eventually derive the bootstrap equations. Note that W n ( x , x , ...x n ) arenot ordinary functions but are distributions. They are defined as W n [ f ] = Z d x ..d x n W ( x , x , ...x n ) f ( x , x ...x n ) (7)These are linear functionals; consequently, a complex number is assigned with theintroduction of { f ( x , x ..x n ) } , the Schwarzian-type functions. f ( x , x ..x n ) is in-finitely differentiable function with desired support properties in the spacetime man-ifold. Thus operators of the form φ [ f ] = R d xφ ( x ) f ( x ) are well defined. The n-point Wightman function is a distribution in the light of preceding remarks. Thuswhenever we allude to the properties of W n ( x , x , ...x n ), it is to be kept in mindthat they are distributions; therefore, statements like convergence, limits etc. areto be understood in this context. It follows from translational invariance that W n depends on difference of coordinates: W n ( x , x , ....x n ) = W n ( y , y ....y n − ), where4 j = y j − y j +1 . Furthermore, W n ( { y j } ) are invariant under Lorentz transforma-tions: W n ( y , ...y n − ) = W n (Λ r y , ... Λ r y n − ) where Λ r is a real proper Lorentz trans-formation; det Λ r = 1. That physical momentum states are defined for light-like momenta i.e. p ≥ p ≥
0, implies that the Fourier transforms of W n ( { y j } ), f W ( p , ...p n − ) = 0 , unless { p j } ∈ V + . Define complex valued function W n ( { ξ j } ) , j = 1 , , ..n −
1. These complex variables are defined as ξ µj = y µj − iη µj ; withthe restrictions on the real set ( { y j , η j } ) , such that η µj ∈ V + , and − ∞ < y µi < + ∞ ;defining a forward tube , T n − . The distributions, W n ( { y j } ), are boundary values ofthe analytic functions W n ( y , y ...y n − ) = lim { η j }→ W n ( ξ , ξ , ...ξ n − ) (8)Notice that W n ( { ξ j } ) are invariant under real Lorentz transformations. The points, { ξ j } ∈ T n − , generate a new set of points Λ ξ , Λ ξ , ... Λ ξ n − under arbitrary complexLorentz transformations where Λ ∈ SL + (2 C ), det Λ = 1. Thus the operation ofΛ on points of T n − generates a new set of points which defines the extended tube T ′ n − . Furthermore, the complex valued analytic function, W n ( { ξ } ), is invariant under SL + (2 C ) and possesses a single values continuation to T ′ n − [49]. Note the importantdifference between points lying in T n − and those lying in T ′ n − : the real points { y j } do not belong to the tube T n − whereas T ′ n − contains the real points { y j } . Wheredo these real points lie? Jost [5] proved an important theorem. The Jost pointsare spacetime points in which all convex combinations of successive differences arespacelike. For W n ( ξ , ξ , ....ξ n − ) a Jost point is an ordered set ( x , x ..., x n ). The Josttheorem states [5]:
A real point of { ξ , ξ , ..ξ n − } lies in the extended tube , T ′ n − , if andonly if all real four vectors of the form P n − λ j ξ µj , λ j ≥ , P n − λ j > are spacelike i.e. ( P n − λ j ξ µj ) < , λ j ≥ , P n − λ j >
0. The necessary and sufficient conditionis that all the real points of T ′ n − are spacelike.Recently, there have been considerable activities to investigate analyticity prop-erties in CFT with Lorentzian signature metric [31, 32, 33, 34, 35, 36, 37]. We recallthat there exists a close relationship between analyticity and crossing. Let us considera four point Wightman function: W ( x , x , x , x ) = < | φ ( x ) φ ( x ) φ ( x ) φ ( x ) | > and the three permutated ones. They are equal < | φ ( x ) φ ( x ) φ ( x ) φ ( x ) | > = < | φ ( x ) φ ( x ) φ ( x ) φ ( x ) | > = < | φ ( x ) φ ( x ) φ ( x ) φ ( x ) | > (9)when ( x − x ) < x − x ) < φ ( x ) , φ ( x )] = 0 and [ φ ( x ) , φ ( x )] = 0due to microcausality. It should be borne in mind that the three Wightman func-tions are boundary values of analytic function, as we discuss later, and they coincidein the domain mentioned above. The three Wightman functions W ( x , x , x , x ), W ( x , x , x , x ) and W ( x , x , x , x ) would be analytically continued to their cor-responding tubes and extended tubes. When we look at their Fourier transforms, the5upport properties will be identified. We have proposed a method of analytic com-pletion when a pair of permuted Wightman functions are considered at a time. Inthis light we have investigated crossing and analyticity of three point and four pointfunctions in the Wightman formulation of CFT [38, 10]. In view of above remarkswe should exercise caution while discussing analyticity and hence the crossing. Weproceed to present how the conformal bootstrap equation follow from the above con-siderations.Consider the first pair of correlators, (9), (i.e. where location of φ ( x ) ↔ φ ( x ) areinterchanged). If we adopt the conformal partial wave technique (CPW) [48, 19, 21,22, 23], on each side of the equation we obtain the desired bootstrap equation when( x − x ) <
0. We recall that, as always, one has to identify the domain where OPEconverges in order to implement CPW expansion. Note, however, that it is desirableto prove that the two permuted Wightman functions are boundary values of analyticfunctions which coincide for spacelike separations of real coordinates. Thus the goal isto identify the domain of holomorphies of the two analytic functions. This is preciselyaccomplished in the proof of dispersion relations in axiomatic QFT [39]. The s and uchannel absorptive parts coincide in a spacelike separated region and then one provesthat they are analytic continuation of each other [39]. We shall accomplish the taskof analytic continuation presently. This proof of analyticity, to our knowledge is notcomprehensively investigated for conformal field theories. We appeal to PCT theoremand its equivalence with WLC [5, 50] in order to derive the bootstrap condition in adomain of holomorphy as has been investigated in [10].
The PCT theorem in CFT : W n ( x , x ...x n ), under PCT, transforms as W n ( φ ( x ) , φ ( x ) ...φ ( x n )) → W n ( φ ( − x n ) , φ ( − x n − ) , ...φ ( − x )) (10)The PCT invariance of the theory implies < | φ ( x ) φ ( x ) ...φ ( x n ) | > = < | φ ( − x n ) φ ( x n − ) ...φ ( − x ) | > (11)If the PCT theorem holds then for every x , x , ...x n with each y j = x j − x j +1 , a Jostpoint; the WLC condition implies < | φ ( x ) φ ( x ) ....φ ( x n ) | > = < | φ ( x n ) φ ( x n − ) ....φ ( x ) | > (12)is satisfied.We focus only on the four point function. There is a converse statement to Jost’stheorem: if WLC holds in a real neighbourhood of (12), a Jost point, then thePCT condition (11) is valid everywhere. Note that WLC implies validity of PCTsymmetry for the conformal scalar. We go through the following essential steps and6efer to [10] for detailed expositions. The WLC theorem of Jost will be employed for W ( x , x , x , x ) in what follows. As a consequence, of the WLC W ( x , x , x , x ) = W ( x , x , x , x ) (13)The steps: Step 1 . Assume that CPT theorem is valid for the conformal theory. Recall that W ( ξ , ξ , ξ ) is a holomorphic function and (12) holds, for n = 4, in the extendedtube T ′ and the four point function is a boundary value of W ( ξ , ξ , ξ ),lim { η j }→ W ( ξ , ξ , ξ ) = W ( y , y , y ) (14)Moreover, W ( ξ , ξ , ξ ) is invariant under proper complex Lorentz transformations, SL + (2 C ): { ξ i } → Λ { ξ i } , ξ i ∈ T ′ . Choose a Λ such that the four complex vector ξ µi → − ξ µi , i = 1 , ,
3. Consequently, W ( ξ , ξ , ξ ) = W ( − ξ , − ξ , − ξ ) (15) Step 2 . Note that the r.h.s. of (11), for n = 4, is also boundary value of an analyticfunction.lim { η j }→ W ( ξ , ξ , ξ ) = W ( y , y , y ) = < | φ ( − x ) φ ( − x ) φ ( − x ) φ ( − x ) | > (16) Step 3 . Consider the difference of two 4-point functions: W ( ξ , ξ , ξ ) −W ( ξ , ξ , ξ ).This is holomorphic in the domain T ′ . This difference vanishes for Re ξ i , i = 1 , , edge-of-the-wedge theorem [51, 52] and weconclude W ( ξ , ξ , ξ ) = W ( ξ , ξ , ξ ) (17)Consider the converse of this statement. Following Hall and Wightman [49], if (17)holds good in an arbitrary neighbourhood of T ′ it also holds good in the extendedtube. Moreover, if it is also valid for passing into the boundary in the tube T thenwe recover the condition of PCT invariance (11). Thus we conclude PCT invarianceis equivalent to WLC in CFT. If we utilize the equations (15) and (17) then W ( ξ , ξ , ξ ) = W ( − ξ , − ξ , − ξ ) (18) Remark : Suppose we try to pass to the boundary in the above equation for any setof { y i } in (18). We encounter the following problem. We shall not be able to geta relationship between the two function, in the above equation at these arbitraryreal points. The reason is as ξ , ξ , ξ approach real points the real vectors are in V + ; whereas the real vectors of − ξ , − ξ , − ξ would be in V − . Note the importantinference: at the real point of holomorphy, this is the Jost point. Therefore, we havethe equation W ( ξ , ξ , ξ ) = < | φ ( x ) φ ( x ) φ ( x ) φ ( x ) | > = W ( − ξ , − ξ , − ξ ) = < | φ ( x ) φ ( x ) φ ( x ) φ ( x ) | > (19)7his equation has important implication for the bootstrap equation as we shalldemonstrate presently.Let us consider < | φ ( x ) φ ( x ) φ ( x ) φ ( x ) | > . We assume that conformal partialwave expansion is convergent for the case at hand. Then employ the conformal partialwave expansion by introducing a complete set of states, {| Ψ > } , between the productof two pairs of operators: φ ( x ) φ ( x ) and φ ( x ) φ ( x ). These states, | Ψ > , span all thevectors of the Hilbert space, H , i.e. all irreducible representations of the conformalgroup. The resulting equation is a familiar expression [23] W = < | φ ( x ) φ ( x ) φ ( x ) φ ( x ) | > = X | Ψ > < | φ ( x ) φ ( x ) | Ψ >< Ψ | φ ( x ) φ ( x ) | > (20)Now invoke state ↔ operator correspondence and interprete < | φ ( x ) φ ( x ) | Ψ > asa three point function: < | φ ( x ) φ ( x ) ˆΨ | > with the identification | Ψ > = ˆΨ | > ;ˆΨ represents the complete set of operator belonging to irreducible representations ofthe covering group. The second matrix element on the r.h.s of (20) becomes anotherthree point function where < Ψ | = < | ˆ¯Ψ; ˆ¯Ψ being the adjoint of ˆΨ. Thus we express(20) as < | φ ( x ) φ ( x ) φ ( x ) φ ( x ) | > = X ˆΨ X αβ λ αφ φ W αβφ φ ˆΨ ¯ˆΨ φ φ λ βφ φ (21)now λ αφ φ and λ βφ φ can be read off from the above equation. Furthermore, W αβφ φ ˆΨ ¯ˆΨ φ φ is the conformal partial waves (CPW) [48, 21, 22, 23]. The above equation is to beunderstood in the sense that it holds in a domain where the CPW expansion con-verges. The four point Wightman function appearing on the l.h.s. of (21) is boundaryvalue of an analytic function in T ′ .Now focus on the CPW expansion < | 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 (22)Recall the equivalence theorem relating the PCT theorem and WLC [5, 50]. The twoexpressions for Wightman functions (21) and (22) are equal at those Jost points andwhere conformal bootstrap conditions [48, 19, 21, 22, 23] are valid.It follows from our sequence of arguments that these two functions are analyticfunctions in extended tubes. Thus by invoking Jost’s theorem and Dyson’s [50] proofof analyticity it follows that the two equations (21) and (22) are analytic continuationsof each other since (19) holds at Jost points. The bootstrap relation 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 (23)is an equation involving spacetime coordinates satisfying Jost’s condition. It is quiteremarkable that using the power of PCT theorem and WLC together with the analyt-icity properties alluded to above, it is possible to demonstrate that bootstrap relation8olds since the two four point functions are analytic continuation of each other (byedge-of-wedge theorem). Notice that if we permute a given Wightman function toobtain another one then the pair are analytic continuation of each other. We haveproved in the forthcoming longer paper [10] that a pair of four point Wightman func-tions are analytic in the unions of their domains of holomorphy i.e. the union ofthe corresponding extended tubes, T ′ ’s. Consider, as an example, a permuted fourpoint function: < | φ ( x ) φ ( x ) φ ( x ) φ ( x ) | > . It has been shown by us that thesetwo four point functions are analytic functions in their corresponding extended tubes.Moreover, for ( x − x ) < Remarks:
The above bootstrap condition is not obtained specifically for a scalarconformal field theory. If we consider four point Wightman function for nonderiva-tive conformal fields which belong to irreducible representation of conformal groupthen the above proof will go through with appropriate modifications. Now the corre-sponding Wightman function will carry tensor indices as the fields would transformaccording to the representations of SL (2 C ) ⊗ SL (2 C ) and they will carry their con-formal dimensions [14]. Therefore, the four point function will have a tensor structureinherited from the product of four field operators, each field belonging to represen-tation of SL (2 C ) ⊗ SL (2 C ) in a general setting. Thus the W will be decopmosedaccordingly and transform covariantly under representations of SL (2 C ) ⊗ SL (2 C ).The preceding arguments will essentially go through. Consequently, the analytic-ity properties and bootstrap equations will continue to hold. Therefore, we concludethat the two resulting four point functions will be analytic continuation of each other. Conclusions . We have rigorously derived that the conformal bootstrap equationshold, for conformal scalar field φ ( x ), in the extended tube T ′ , for the four point func-tion. The power of the WLC is crucial to prove this bootstrap condition. The proof isbased on Wightman axioms for CFT, φ ( x ). We have argued that conformal bootstrapconditions will also hold for four point functions of conformal fields belonging irre-ducible representations of conformal group so long as they are of nonderivative typefields. The work is in progress to study analyticity and crossing for n-point functionsin conformal field theories in the present optics.Acknowledgments: I am indebted to Ba-Gaga for their love, affections, patience andfor my raison d ′ ˆetre . 9 eferences
1. W. Pauli, in Niels Bohr and the Development of Physics, McGraw-Hill, NewYork (1955) pp 30.2. G. L¨uders, Danske Videnskabernes Selskab, Mat.-fys. Medd. , No 5 (1954).3. G. Grawert, G. L¨uders and H. Rollnik, Fortscr. der Physik, , 291 (1959).4. C. N. Yang and T. D. Lee, Phys. Rev., , 254 (1956).5. R. Jost, Helv. Phys. Acta, , 409 (1957).6. R. F. Streater and A. S. Wightman, PCT, Spin and Statistics, and All That,W. A. Benjamin, Inc. New York Amsterdam, 1964.7. P. A. Zyla et al.(Particle Data Group), Prog. Th. Phys. , 083C01 (2020).8. O. W. Greenberg, Phys. Rev. Lett. , 231602 (2002).9. A. S. Wightman, Phys. Rev. , 860 (1956).10. J. Maharana, Crossing, Causality and Analyticity in Conformal Field Theory,to be submitted for publication.11. G. Mack and A. Salam, Ann. Phys. , 174 ( 1969).12. S. Ferrara, R. Gatto and A. F. Grillo, Springer Tracts in Mod. Phys. , 1(1973).13. E. S. Fradkin and M. Ya Palchik, Phys. Rep. C44 , 249 (1978).14. E. S. Fradkin and M. Ya Palchik, Conformal Field Theory in D-dimensions,Springer Science Business Media, Dordrecht, 1996.15. I. T. Todorov, M. C. Mintechev and V.R. Petkova, Conformal Invariance inQuantum Field Theory, Publications of Scuola Normale Superiore, Birkh¨auserVerlag, 2007.16. A. A. Migdal, Phys. Lett. , 98 (1971); Phys. Lett. , 386 (1971).17. S. Ferrara, A. F. Grillo and R. Gatto, Annals of Phys. 76, 161 (1973).18. S. Ferrara, A. F. Grillo, R. Gatto and G. Parisi, Nuovo. Cim.
A19 , 667 (1974).19. A. M. Polyakov, Z. Eksp. Teor. Fiz, , 23 (1974).20. M. L¨uscher and G. Mack, Commun. Math. Phys. , 203 (1975).101. D. Simon-Duffin, TASI Lectures 2015, arXiv: 1602.07982[hep-th].22. J. Penedones, TASI Lecture 2016, arXiv: 1608.04948[hep-th].23. D. Poland, S. Rychkov and A. Vichi, Rev. Mod. Phys. , 051002 (2019).24. J. Maldacena, Adv. Theor. Math. Phys. , 231 (1998).25. J. Bross, H. Epstein and V. Glaser, Nuovo. Cim. , 1265 (1964).26. H. Lehmann, K. Symanzik and W. Zimmermann, Nuovo. Cim. , 201 (1955).27. K. G. Wilson, Phys. Rev. , 1499 (1969).28. K. G. Wilson and W. Zimmermann, Commun. Math. Phys. , 87 (1972).29. P. Otterson and W. Zimmermann, Commun. Math. Phys. , 107 (1972).30. G. Mack, Commun. Math. Phys. , 155 (1977).31. Z. Komargodski and A. Zhiboedov JHEP, 11, 140 (2013).32. T. Hartman, S. Kundu and A. Tajdini, JHEP 07, 066 (2017).33. T. Hartman, S. Jain and S. Kundu, JHEP 05, 099 (2016).34. M. S. Costa, T. Hansen and J. Penedones, JHEP 10, 197 (2017).35. S. Caron-Huot, JHEP 09, 078 (2017).36. T. Baurista and H. Godazgar, JHEP 01, 142 (2020).37. M. Gillioz, Commun. Math. Phys. , 227 (2020).38. J. Maharana, Mod. Phys. Lett. A35 , 2050186, (2020).39. S. S. Schweber, Introduction to Relativistic Quantum Field Theory, Harper andRow, New York, Evaston and London, 1961.40. R. Jost, General Theory of Quantized Fields, American Mathematical Society,Providence, Rhode Island, 1965.41. R. Haag, Local Quantum Physics: Fields, Particles, Algebras, Springer, 1996.42. C. Itzykson and J. -B. Zubber Quantum Field Theory, Dover Publications Mi-neola, New York, 2008.43. 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. 114. T. Yao, J. Math. Phys. , 1731 (1967).45. T. Yao, J. Math. Phys. , 1615 (1968).46. T. Yao, J. Math. Phys. , 315 (1971).47. G. Mack, Commun. Math. Phys. , 1 (1972).48. S. Ferrara, A. F. Grillo and R. Gatto, Lett. Nuovo, Cimento, , 1363 (1971).49. D. Hall and A. S. Wightman, Mat. Fys. Medd. Dan. Vid. Selsk., , 5 (1957).50. F. J. Dyson, Phys. Rev. , 579 (1958).51. H. J. Bremmermann, R. Oehme and J. G. Taylor, Phys. Rev., , 2178 (1958).The proof was for pion-nucleon scattering in the LSZ formalism.52. H. Epstein, J. Math. Phys.,1