PHYSICAL REVIEW D 103, 105017 (2021)
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Relation between instant and light-front formulations of quantum field theory
W. N. Polyzou * Department of Physics and Astronomy, The University of Iowa, Iowa City, Iowa 52242, USA
(Received 16 February 2021; accepted 27 April 2021; published 19 May 2021)
The scattering equivalence of quantum field theories formulated with light-front and instant-form kinematic subgroups is established using nonperturbative methods. The difficulty with field theoretic formulations of Dirac’s forms of dynamics is that the free and interacting unitary representations of the Poincar´e group are defined on inequivalent representations of the Hilbert space, which means that the concept of kinematic transformations must be modified on the Hilbert space of the field theory. This work addresses this problem by assuming the existence of a field theory with the expected properties and constructs equivalent representations with instant and front-form kinematic subgroups. The underlying field theory is not initially associated with an instant form or light-front form of the dynamics. In this construction the existence of a vacuum and one-particle mass eigenstates is assumed and both the light-front and instant-form representations are constructed to share the same vacuum and one-particle states. If there is spontaneous symmetry breaking there will be a 0 mass particle in the mass spectrum (assuming no Higgs mechanism). The free field Fock space plays no role. There is no “quantization” of a classical theory. The property that survives from the perturbative approach is the notion of a kinematic subgroup, which means kinematic Poincar´e transformations can be trivially implemented by acting on suitable basis vectors. This nonperturbative approach avoids dealing with issues that arise in perturbative treatments where is it necessary to have a consistent treatment of renormalization, rotational covariance, and the structure of the light-front vacuum. While addressing these issues in a computational framework is the most important unanswered question for applications, this work may provide some insight into the nature of the expected resolution and identifies the origin of some of differences between the perturbative and nonperturbative approaches.
DOI: 10.1103/PhysRevD.103.105017
I. INTRODUCTION If H ¼ H0 þ V for some interaction V then the operators on the left side of each commutator must also know about This paper discusses the relation between light-front and the interaction. Dirac identified three representations of the instant-form formulations of quantum field theory. The Poincar´e Lie algebra with the minimum number (3–4) of identification of different forms of dynamics is due to Dirac interaction-dependent Poincar´e generators. He called these [1]. In a relativistic quantum theory the invariance of the instant, point and front-forms of the dynamics. In the quantum probabilities in different inertial coordinate sys- instant form the generators of space translations and tems requires that equivalent states in different inertial rotations are free of interactions, in the point form the coordinate systems are related by a unitary ray representa- generators of Lorentz transformations are free of inter- tion of the subgroup of the Poincar´e group continuously actions and in the front form the generators of trans- connected to the identity [2]. The Poincar´e Lie algebra has formations that leave a hyperplane tangent to the light cone 3 independent commutators involving rotationless boost and translation generators that have the Hamiltonian on the invariant are free of interactions. right The equivalence of these different representations of relativistic quantum mechanics was settled by Sokolov and Shatnii [3,4]. In quantum field theory the problem is more ½Ki;Pj ¼iδ H: ð1Þ ij complicated because the free and interacting dynamics are formulated on different inequivalent representations of the *[email protected] Hilbert space [5], so the decomposition of the Hamiltonian into the sum of a free Hamiltonian plus interaction is not Published by the American Physical Society under the terms of defined on the Hilbert space of the field theory. Such a the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to decomposition makes sense in perturbative quantum field the author(s) and the published article’s title, journal citation, theory with cutoffs, so the notion of instant- and light-front and DOI. Funded by SCOAP3. formulations [6–11] of quantum field theory make sense
2470-0010=2021=103(10)=105017(17) 105017-1 Published by the American Physical Society W. N. POLYZOU PHYS. REV. D 103, 105017 (2021) perturbatively. The price paid is that as the cutoffs are of the Haag-Ruelle formulation of scattering is that the removed the theory has infinities that have to be renor- weak limits that are used to define the scattering asymptotic malized. The relation between the different forms of conditions in the LSZ formulation of scattering can be dynamics depends on a consistent treatment of the renorm- replaced by strong limits. alization. The light-front formulation of quantum field A two-Hilbert space representation, which does not theory is of particular interest for applications. The most require the existence of a free dynamics on the Hilbert appealing property of the theory is the apparent triviality of space of the theory, is used to formulate the scattering the light-front vacuum, which reduces the solution of the theory. This avoids any concerns about Haag’s theorem field theory to linear algebra on a Hilbert space, like [5,40], which implies that the interacting and free field nonrelativistic quantum mechanics. In addition to the theories are formulated on inequivalent Hilbert-space computational challenges of implementing this program representations of the field algebra. The second Hilbert in a theory with an infinite number of degrees of freedom, space, which is called the asymptotic Hilbert space, is the there are a number of puzzles that appear in comparing the direct sum of tensor products of single-particle Hilbert two approaches. These include: spaces with physical masses. This representation is natural (i) Is the light-front vacuum the same as the Fock in nonrelativistic many-body scattering. For example, in vacuum? lead-lead scattering the lead nuclei are treated as elemen- (ii) How to understand Pþ ¼ 0 (zero mode) singularities? tary particles in the asymptotic Hilbert space. Products of (iii) How to understand spontaneous symmetry breaking the lead state vectors, jL1; p1; μ1i × jL2; p2; μ2i, define a in a light-front dynamics? mapping from the asymptotic Hilbert space of suitably (iv) How to renormalize the theory consistent with symmetrized square integrable functions, fðp1; μ1; p2; μ2Þ, rotational covariance. to the Hilbert space of the theory. In the field theory case (v) What is the relation between light-front and canoni- the lead wave functions are replaced by products of cal quantization of a quantum field theory? † p μ operators, ALð ; Þ, that create point spectrum lead mass (vi) Are both approaches equivalent? eigenstates out of the vacuum, applied to the vacuum While there is a general consensus that the two formula- † [49,50,51]. In the field theory case the operators A ðp; μÞ tions are equivalent, the answers to the above questions are L are not creation operators. They do not commute or not as clean as desired. These issues have been discussed anticommute with each other and do not satisfy canonical extensively in the literature [6,7,12,13,8–11,14–39]. commutation relations. Inner products of states constructed Most of these discussions are based on perturbation by applying two or more of these operators to the theory; in particular on the assumption that the light-front vacuum can have intermediate states with different particle dynamics can be formulated on the Fock space of a free content. field theory. The continued interest in these questions is In the simplest case the asymptotic Hilbert space is a because a compelling resolution of these questions is Fock space with physical particle masses. The mapping obscured by the need to perform a nonperturbative renorm- from the second Hilbert space to the Hilbert space of the alization of the theory. In this work these complications are field theory puts in the internal structure of the physical avoided by assuming the existence of the theory with the particles, including the effects of self-interactions, when the expected properties that are normally assumed in axiomatic particles are asymptotically separated. – formulations of quantum field theory [40 43]. One depar- There is a natural unitary representation of the Poincar´e ’ ture from both of Dirac s Hamiltonian approaches is that in group on the asymptotic Hilbert space, which treats the axiomatic formulations of quantum field theory states are particles as free particles with their observed masses. There constructed by applying fields integrated against Schwartz is sufficient freedom to choose the mappings from the test functions [44] in four space-time variables to the asymptotic Hilbert space to the Hilbert space of the field vacuum. These states can be evolved, but the initial data theory so the unitary representation of the instant or light- is not on a sharp time slice or light front. The existence of front kinematic subgroup of the Poincar´e group on the an asymptotically complete scattering theory is also asymptotic space is mapped to the dynamical unitary assumed, based on the Haag-Ruelle formulation of scatter- representation of the corresponding kinematic subgroup ing theory [45–48]. Haag-Ruelle scattering has the advan- on the field-theory Hilbert space without changing the tage that it can be formulated in terms of wave operators scattering operator. defined by strong limits. The strong limits justify using A theorem due to Ekstein [52] gives necessary and many of the methods that are used in nonrelativistic sufficient conditions for unitary transformations on a scattering theory and they are used in the construction that Hilbert space to preserve the scattering operator. This follows. The difference with the LSZ formulation of result assumes the scattering is formulated in terms of scattering is that the interpolating field operators must asymptotically complete wave operators defined as strong create one-particle states rather than just create states that limits. In the two-Hilbert space formulation Ekstein’s have some overlap with one-particle states. The advantage condition is a constraint on the unitary transformation
105017-2 RELATION BETWEEN INSTANT AND LIGHT-FRONT … PHYS. REV. D 103, 105017 (2021) relating two different Hamiltonians that does not require the absence of mass renormalization suggests the that modifying the Hamiltonian in the asymptotic Hilbert space. simplest types of interactions that change particle number Roughly speaking it means that the unitary operator is a are short-range 2 ↔ 3-particle interactions rather than short-range perturbation of the identity in a manner that 1 ↔ 2 vertex interactions. Ekstein makes precise [see Eq. (34)]. This freedom can be This paper has five sections and two appendices. exploited to construct the most general class of S-matrix Section II summarizes the assumptions used in this work preserving unitary transformations that are invariant under and discusses a two-Hilbert space formulation of Haag- a kinematic subgroup of the dynamical representation of Ruelle scattering that has kinematically invariant injection the Poincar´e group. The proof uses the unitary of the wave operators that map an asymptotic many-particle Hilbert operators which holds provided the vacuum and one-body space to the Hilbert space of the field theory. Haag- channels are included in the asymptotic Hilbert space. Ruelle scattering is the natural generalization of the usual Unitary transformations must leave the one-particle masses formulation of time-dependent scattering theory. The and the mass gap unchanged. In this work these trans- construction of equivalent instant and light-front repre- formations are also constructed to leave the vacuum and sentations of the field theory is given in Sec. III. Section IV one-particle states unchanged. has a brief discussion of spontaneous symmetry breaking. When these unitary transformations are applied to the The construction is compared to the corresponding con- unitary representation of the Poincar´e group on the physical struction in relativistic formulations of quantum mechanics Hilbert space, they result in two new scattering-equivalent of a finite number of degrees of freedom in Sec. V. representations with light-front and instant-form kinematic Section VI has a summary of the results and conclusions. subgroups. These representations have the property that the There are two appendices. The first one contains a injection operators map the kinematic subgroup on the discussion of Ekstein’s treatment of scattering equivalen- asymptotic Hilbert space to the corresponding subgroup of ces [52] that is used in Sec. III. The second appendix the unitary representation of the Poincar´e group on the discusses the construction of Haag-Ruelle injection oper- physical Hilbert space. Both representations are equivalent ators with the kinematic symmetry properties that are used to the original “axiomatic” formulation of the theory in the in Secs. II and III. sense that they are related by S-matrix preserving unitary transformations. One thing that is missing in this construction is the II. GENERAL CONSIDERATIONS existence of a noninteracting representation of the Poincar´e This paper examines the relation between light-front and group on the Hilbert space of the field theory. Haag’s instant-form formulations of quantum field theory from a theorem implies that free fields, even with experimental more abstract perspective that does not assume that the masses, are defined on an inequivalent Hilbert space Hamiltonian can be decomposed as the sum of free and representation of the field algebra, so there is no corre- interacting operators acting on one representation of the sponding local free field theory on the Hilbert space of the Hilbert space. The starting assumptions are typical assump- field theory. In the event that the mapping from the tions about abstract properties of a quantum field theory. asymptotic space to the field theory Hilbert space has These include: dense range, and the unitarized form of this mapping (i) The Hilbert space of the field theory is generated by satisfies Ekstein’s asymptotic condition, Eq. (34), then it applying bounded functions, F, of field operators can be used to map the unitary representation of the smeared over test functions to a unique vacuum Poincar´e group on the asymptotic Hilbert space to a unitary vector, denoted by j0i. The smearing is assumed to representation of the Poincar´e group on the field theory be over Schwartz test functions [44] in four space- Hilbert that plays the role of a free-particle dynamics. time dimensions. It is not assumed that fields Since the two-Hilbert space injection operators are not restricted to a light-front or fixed-time manifold ψ unique and are not local operators, this will not be a local make sense. A dense set of vectors, j i, can be taken field theory. to have the form: The results can be summarized as follows. The con- struction assumes a unitary representation of the Poincar´e XN ϕ ψ 0 i nðfnÞ group on the Hilbert space of the field theory and constructs j i¼Fj i F ¼ cne ð2Þ scattering equivalent representations with light-front and n¼1 instant-form kinematic symmetries. The representations are related by a unitary transformation that satisfies an asymp- where the cn are complex coefficients, the fn are totic condition that preserves the scattering matrix ele- Schwartz functions, ϕn are field operators and N is ments. Because the one-body states are eigenstates of the finite. mass operator, there is no mass renormalization. While (ii) There is a unitary representation of the Poincar´e the form of the interactions is representation dependent, group, UðΛ;aÞ, on this representation of the Hilbert
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space which is given by covariance. In the spinless The unitary representation of the Poincar´e group has a set case the action of UðΛ;aÞ on states of the form (2) is of 10 infinitesimal generators that are self-adjoint operators X on the Hilbert space of the field theory satisfying the 0 iϕðfnÞ UðΛ;aÞjψi¼ cne j0i Poincar´e Lie algebra. This follows because each one is the c generator of a unitary one-parameter group. The unitary 0 −1 representation of the Poincar´e group can be decomposed fnðxÞ¼fnðΛ ðx − aÞÞ: ð3Þ into a direct integral of irreducible representations. In this (iii) The vacuum, j0i, is Poincar´e invariant and normal- work a typical spectral condition is assumed, where the ized to unity: representations that appear in the direct integral are assumed to be positive-mass positive-energy representa- UðΛ;aÞj0i¼j0ih0j0i¼1: ð4Þ tions, and the identity on the vacuum subspace. The direct integral of irreducible representations of the (iv) The unitary representation of the Poincar´e group acts Poincar´e group results in a direct integral decomposition of like the identity on the vacuum subspace and can be the Hilbert space into irreducible subspaces or slices. The decomposed into a direct integral of positive-mass next step is to choose a basis on each irreducible subspace. positive-energy irreducible representations of the Two choices will be considered in this work. The first basis Poincar´e group [2,53] on the orthogonal comple- consists of simultaneous eigenstates of the mass, spin, ment of the vacuum subspace. The mass Casimir linear momentum, and the projection of the canonical spin operator is assumed to have point-spectrum eigen- on the z axis. These are all functions of the infinitesimal states representing particles of the theory. generators of the unitary representation of the Poincar´e (v) There is a complete set of asymptotically complete group. There will also be Poincar´e invariant degeneracy Haag-Ruelle [45–48] scattering states. parameters, which can be taken as discrete quantum A characteristic element of Dirac’s forms of dynamics is numbers, since any continuous parameters can be replaced the notion of a kinematic subgroup. Kinematic subgroups by a basis of square integrable functions of the continuous leave a manifold in Minkowski space, that classically degeneracy parameters. The other basis on the same intersects every (massive) particle’s world line once, irreducible subspace consists of simultaneous eigenstates invariant. In an instant-form dynamics the invariant mani- of the mass, spin, the three light-front components of the fold is a fixed-time plane, which is preserved under the four momentum, and the projection of the light-front spin group generated by rotations and space translations. In a on the z axis. These are also functions of the infinitesimal light-front dynamics the manifold is a hyperplane that is generators with the same degeneracy parameters. tangent to the light cone. This is invariant under a seven- These bases can be formally expressed as parameter subgroup of the Poincar´e group. These mani- 0 ∪ p μ folds are prominent in Dirac’s work; in this work the focus j i fjðm; s; dnÞ ; cig and ˜ will be on the subgroups of the Poincar´e group that leave j0i ∪ fjðm; s; dnÞp; μFig ð5Þ these manifolds invariant. These subgroups will be referred to as kinematic subgroups. where the light-front components of the four momentum For quantum mechanical systems of a finite number of are defined by degrees of freedom kinematical and dynamical unitary 0 þ representations of the Poincar´e group exist in the same p˜ ¼ðp · xˆ; p · yˆ;p þ p · zˆÞ ≔ ðp⊥;p Þ: ð6Þ representation of the Hilbert space. In the field theoretic case the representation of the Hilbert space depends on the The variables (6) are eigenvalues of the generators of dynamics. While free fields with physical masses have the translations tangent to the light-front hyperplane. The first “ ” same mass spectrum, they act on an inequivalent Hilbert basis in (5) will be referred to as the instant-form basis and “ ” space representation of the field algebra by Haag’s theorem, the second basis will be referred to as the light-front basis. so there is no free dynamics on the Hilbert space of the These basis states are related by a unitary change of interacting field theory. This is one of the distinctions with basis. The change of basis is block diagonal in the direct the perturbative approach, which attempts to define the integral. The different basis vectors are related by interacting theory as a perturbation of a free field theory. Xs The incompatibility of these formulations leads to the ˜ jðm; s; dnÞp; μIi¼ jðm; s; dnÞpðpÞ; μFi infinities that complicate the analysis of the relation μf¼−s between light-front and instant-form treatments of field sffiffiffiffiffiffiffiffiffiffiffiffiffi þ theory. p s −1 × DμFμI ½BF ðp=mÞBIðp=mÞ The first step in the abstract formulation is to define ωmðpÞ what is meant by the kinematic subgroup in the absence of a noninteracting representation of the Poincar´e group. ð7Þ
105017-4 RELATION BETWEEN INSTANT AND LIGHT-FRONT … PHYS. REV. D 103, 105017 (2021) which assumes that both bases have delta function nor- These equations define the dynamical unitary representa- malizations: tion of the Poincar´e group on these two irreducible bases. The connection of these bases with kinematic subgroups p μ 0 0 p0 μ0 hðm; s; dnÞ ; Ijðm ;s;dkÞ ; Ii is that the coefficients of the basis functions on the right- 0 0 hand side of Eqs. (11) and (13) do not depend on the mass ¼ δðp − p Þδss0 δμ μ0 δnkδ½m − m ð8Þ I I eigenvalue m when ðΛ;aÞ is an element of the kinematic subgroup. For the basis (11) the kinematic subgroup is and generated by rotations and spatial translations while for the basis (13) the kinematic subgroup is the subgroup that p˜ μ 0 0 p˜ 0 μ0 þ 0 hðm; s; dnÞ ; Fjðm ;s;dkÞ ; Fi leaves the light front hyperplane x ¼ x þ x · zˆ ¼ 0 þ þ0 0 0 invariant. ¼ δðp − p Þδðp⊥ − p⊥Þδss0 δμ μ0 δnkδ½m − m ð9Þ F F The next step is to discuss the formulation of scattering theory. A two-Hilbert space representation [51,55], will be where δ½m − m0 is either a Dirac or Kronecker delta used for this purpose. The starting point is the assumption function depending on whether m is in the point or pffiffiffiffiffiffiffiffiffi that the mass operator, M ¼ −p2 has one-particle continuous spectrum of the mass operator. BIðp=mÞ and eigenstates. In this nonperturbative context one-particle BFðp=mÞ are SLð2;CÞ matrices representing a rotationless pffiffiffiffiffiffiffiffiffi boost and a light-front preserving boost from (1, 0, 0, 0) to means that the mass operator, M ¼ −p2, has a nonempty −1 2 p=m. The combination BF ðp=mÞBIðp=mÞ is a SUð Þ point spectrum. The discrete mass eigenvalues are assumed representation of a Melosh rotation [54] that changes to be strictly positive. There is no distinction between s the light-front spin to the canonical spin. Dμν½R ¼ elementary and composite particles. “ ” hs; μjUðRÞjs; νi is the 2s þ 1 dimensionalpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi unitary repre- Normalizable one-particle states in the instant form 2 2 basis can be constructed by integrating the irreducible basis sentation of SUð2Þ and ωmðpÞ¼ p þ m . There are similar transformations relating light-front functions with a wave packet: Z bases corresponding to different orientations of the light Xs front (i.e., replacing zˆ by some other unit vector nˆ ). Like (7) jψ Igi ≔ dp jðm; s; dnÞp; μigðp; μÞð14Þ they involve a variable change, the square root of a μ¼−s Jacobian and a momentum-dependent rotation matrix. Λ The action of Uð ;aÞ on each of these irreducible basis where m is a discrete mass eigenvalue and gðp; μÞ is a states is a consequence of the transformation properties of square integrable wave packet. The corresponding con- the Poincar´e generators and the basis choice. For the struction in the “light-front” basis has the form “instant-form” basis Λ Z Z ∞ Xs ψ ≔ 2p þ p˜ μ ˜ p˜ μ UðΛ;aÞj0i¼j0i; ð10Þ j Fgi d ⊥ dp jðm;s;dnÞ ; igð ; Þð15Þ 0 μ¼−s U Λ;a m; s; d p; μ ð Þjð nÞ i where in the light front case p˜ should vanish as pþ → 0 in a Xs manner that results in a square integrable function of p ia·Λp Λ μ0 ¼ e jðm; s; dnÞ p; i under the variable change (6) p˜ → p. μ0 −s sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi¼ These normalizable vectors are linear in the functions p μ ˜ p˜ μ ω Λp gð ; Þ and gð ; Þ. They can be represented by elements of mð Þ s −1 Λ Λ × Dμ0μ½B ð ðp=mÞÞ BIðp=mÞ ð11Þ the field algebra applied to the vacuum. This means that ω ðpÞ I m they can be expressed as Z and for the “light-front” basis, Xs † jψ i¼ A ðp; μÞj0idpgðp; μÞ Ig m;s;dn UðΛ;aÞj0i¼j0i; ð12Þ μ¼−s ¼ A†ðgÞj0ið16Þ ˜ UðΛ;aÞjðm; s; dnÞp; μi Xs and ia·Λp Λ˜ μ0 ¼ e jðm; s; dnÞ p; i Z s μ0¼−s X ˜ † þ 2 sffiffiffiffiffiffiffiffiffiffiffiffiffi jψ i¼ A ðp;˜ μÞj0idp d p⊥g˜ðp˜ ; μÞ Fg m;s;dn Λp þ μ¼−s ð Þ s −1 Λ Λ × Dμ0μ½B ð ðp=mÞÞ BFðp=mÞ : ð13Þ pþ F ¼ A˜ †ðg˜Þj0i: ð17Þ
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These states will be equal if gðp; μÞ¼hðm; s; dnÞp; μIjgi ˜ † ˜ † UðR ðϕÞ; 0ÞA ðp˜ 1; μ1Þ A ðp˜ ; μ Þj0i ˜ ˜ z m1;s1;dn mN ;sN ;dn N N and g˜ðp; μÞ¼hðm; s; dnÞp; μFjgi are related by the change 1 N of basis (7). ˜ † ϕ p˜ μ ˜ † ϕ p˜ μ 0 † † ¼ Am1;s1;d ðRzð Þ 1; 1Þ Am ;s ;d ðRð Þ N; NÞj i The operators A ðp; μÞ and A˜ ðp;˜ μÞ are func- n1 N N nN m;s;dn m;s;dn tions of the fields that create particles of mass m, spin s, YN iμnϕ momentum p and magnetic quantum number μ out of the × e : ð23Þ 1 vacuum. The construction of these operators starting from n¼ operators that couple the vacuum to the one-particle states If the Fourier transform of these states are integrated of the theory is discussed in Appendix B. † † against wave packets that are localized in regions separated Since A ðp; μÞ and A˜ ðp;˜ μÞ are in the field m;s;dn m;s;dn by large space-like separations and the field theory satisfies algebra (after smearing with test functions), they can be cluster properties (normally a consequence of uniqueness multiplied. of the vacuum) then the resulting vectors look like states of The following quantity N asymptotically separated particles when they are used in inner products. This property is used [48,50] to prove the † p μ † p μ 0 Am1;s1;d ð 1; 1Þ Am ;s ;d ð k; kÞj ið18Þ n1 k k nk existence of the strong limits that define wave operators in Haag-Ruelle scattering. The interpretation of the operators † † can be considered as a mapping from a Hilbert space of A and A˜ is that they asymptotically behave like m;s;dn m;s;dn square integrable suitably symmetrized functions of creation operators. p μ p μ 1; 1 k; k to the Hilbert space of the field theory. A scattering channel α is associated with a finite † ˜ † Note that while A ðgÞj0i¼A ðg˜Þj0i for g and g˜ related by collection of particles asymptotically. It could correspond (7), because both operators create the same single-particle to the state of a target and an incoming projectile or a states out of the vacuum, this identity is not true for collection of particles that are detected in a scattering products of these operators applied to the vacuum. From experiment. The set of all scattering channels of the Eq. (B17) in Appendix B it follows that this mapping has theory is denoted by A. In what follows both the vacuum the following properties and one-particle states are included in the collection of channels, A. P † p μ † p μ 0 Am1;s1;d ð 1; 1Þ Am ;s ;d ð N; NÞj i Channel injection operators are defined as products of n1 N N nN XN the operators (16) or (17), applied to the vacuum, one for p † p μ † p μ 0 each particle in a scattering channel α. In the canonical ¼ nAm1;s1;d ð 1; 1Þ Am ;s ;d ð N; NÞj ið19Þ n1 N N nN n¼1 basis they are Y and Φ p μ p μ ≔ † p μ 0 αð 1; 1 N; NÞ Am ;s ;d ð k; kÞj ið24Þ k k nk k∈α 0 † p μ † p μ 0 UðR; ÞAm1;s1;d ð 1; 1Þ Am ;s ;d ð 1; NÞj i X n1 N N nN and in the light-front basis they are † p ν † p ν 0 0 ¼ Am1;s1;d ðR 1; 1Þ Am ;s ;d ðR N; NÞj ij i Y n1 N N nN Φ˜ p˜ μ p˜ μ ≔ ˜ † p˜ μ 0 αð 1; 1 N; NÞ Am ;s ;d ð k; kÞj i: ð25Þ YN k k nk sn k∈α × Dνnμn ðRÞ: ð20Þ n¼1 The asymptotic channel Hilbert space Hα is the space of p μ p˜ μ For the mapping in the “light-front” basis the correspond- square integrable functions of the variables k, k or k; k ∈ α ing relations are for k . It is interpreted as a space of N particles of mass mk and spin sk. If any of the particles in the channel α are P˜ ˜ † p˜ μ ˜ † p˜ μ 0 identical then the functions representing identical bosons Am1;s1;d ð 1; 1Þ Am ;s ;d ð N; NÞj i n1 N N nN should be symmetrized and those representing identical XN fermions should anti-symmetrized. The order of the prod- p˜ ˜ † p˜ μ ˜ † p˜ μ 0 ¼ nAm1;s1;d ð 1; 1Þ Am ;s ;d ð N; NÞj ið21Þ n1 N N nN uct in (25) does not matter for suitably symmetrized states n¼1 in the asymptotic Hilbert space. The channel injection ˜ † ˜ ˜ † ˜ operators (24) and (25) are interpreted as mappings from UðBf; 0ÞA ðp1; μ1Þ A ðpN; μNÞj0i m1;s1;dn1 mN ;sN ;dnN the k-particle channel subspace of the asymptotic Hilbert ˜ † B˜ μ ˜ † B˜ μ 0 space Hα to the Hilbert space H of the field theory. ¼ Am1;s1;d ð fp1; 1Þ Am ;s ;d ð fpN; NÞj i sn1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N N nN The asymptotic unitary representation of the Poincar´e H H YN B ðp Þþ group on α is defined by treating α as a space of N × f n : ð22Þ mutually noninteracting particles of mass m and spin s pþ k k n¼1 n for k ∈ α. This representation is defined by the tensor
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Ω Φ˜ − iHtΦ˜ −iHαt ψ 0 product of one-particle irreducible representations in terms lim kð α ðH; α;HαÞ e αe Þj ik ¼ : ð31Þ of “instant-form” variables t→ ∞ The two Hilbert space wave operators satisfy the Poincar´e UαðΛ;aÞjp1;μ1; pk; μki X P covariance condition ia·Λð p Þ ≔ e n n jΛp1; ν1 Λp ; ν i k k Λ Ω Φ Ω Φ Λ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Uð ;aÞ α ðH; α;HαÞ¼ α ðH; α;HαÞUαð ;aÞ Yk ω Λp mn ð nÞ s −1 Λ Ω Φ˜ Ω Φ˜ Λ × Dν μ ½B ðΛp =m ÞΛB ðp =m Þ : Uð ;aÞ α ðH; α;HαÞ¼ α ðH; α;HαÞUαð ;aÞð32Þ ω ðp Þ n n c n n c n n n¼1 mn n Λ ð26Þ where, unlike (28), ð ;aÞ are not restricted to the kinematic subgroup. The corresponding expression in terms of “light-front” The purpose of the injection operators in Haag-Ruelle variables is scattering is to define the asymptotic boundary conditions. They are constructed so they behave like creation operators ˜ ˜ in asymptotically separated regions. They replace the free- UαðΛ;aÞjp1; μ1; pk; μki X P particle asymptotic states of ordinary scattering theory by ia·Λð pnÞ ˜ ˜ ≔ e n jΛp1; ν1 Λpk; νki states in the Hilbert spaces of the field theory that only sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi behave like a system of noninteracting particles in the Yk Λ˜ þ ð pnÞ s −1 asymptotic region. The two injection operators defined in × Dν μ ½B ðΛp =m ÞΛB ðp =m Þ : pþ n n f n n f n n Appendix B both have this property. They differ in which n¼1 n variable enforces the one-particle mass-shell delta function ð27Þ when applied the vacuum. This means that as operators
The states in (26) and (27) are basis vectors in Hα, not H. Ω Φ Ω Φ˜ ˜ α ðH; α;HαÞ¼ α ðH; α;HαÞð33Þ Since the Hilbert space vectors in the range of Φα or Φα are not necessarily N-particle states it follows that in general if they are evaluated in the same one-particle basis. This identity is equivalent to the requirement UðΛ;aÞΦα ≠ ΦαUαðΛ;aÞ: − ˜ ˜ 0 iHt Φ − Φ˜ iHαt ψ UðΛ;aÞΦα ≠ ΦαUαðΛ;aÞ: ð28Þ ¼ lim ke ð α αÞe j αik t→ ∞
˜ −iHα 0 0 ¼ lim kðΦα − ΦαÞe t jψ αik ð34Þ This is because (B6) does not hold for P due to the p t→ ∞ integrals in (B9) and (B6) does not hold for P− due to the p− integrals on (B9) (see Appendix B). which is precisely the condition that the injection operators When ΛK and aK are elements of the instant-form or agree asymptotically. light-front kinematic subgroup, it follows from (19)–(23) The scattering operator for scattering from channel α to that channel β is
† U Λ ;a Φα ΦαUα Λ ;a ∶H → H ≔ Ω Φ Ω Φ ð K KÞ ¼ ð K KÞ Sβ;α α β βþðH; β;HβÞ α−ðH; α;HαÞ: ð35Þ ˜ ˜ UðΛK;aKÞΦα ¼ ΦαUαðΛK;aKÞð29Þ It follows from the identity (34) that Sαβ is independent of because the coefficients of the kinematic transformations the choice of injection operator. It follows from (32) that ˜ are mass independent. The operators Φα and Φα that map Λ Λ the channel α Hilbert space Hα into the field theory Hilbert Uβð ;aÞSβ;α ¼ Sβ;αUαð ;aÞ: ð36Þ space H are Haag-Ruelle [45–47] injection operators. The A only difference with the standard choices of Haag Ruelle This can be extended to all channels, , by defining the ˜ asymptotic Hilbert space as the direct sum of all channel injection operators is that Φα and Φα are designed to H α ∈ A A preserve the kinematic subgroup (29). Hilbert spaces α for , where is defined to include In the two-Hilbert space formulation [49,50] channel the vacuum and one-particle channels as well as the Ω Ω Φ scattering channels: wave operators α ¼ α ðH; α;HαÞ are defined by the strong limits HA ≔ ⨁ Hα: ð37Þ α∈A Ω Φ − iHtΦ −iHαt ψ 0 lim kð α ðH; α;HαÞ e αe Þj ik ¼ ð30Þ t→ ∞ The asymptotic unitary representation of the Poincar´e or group is defined on HA by
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UAðΛ;aÞ¼ ⨁ UαðΛ;aÞ: ð38Þ kinematic subgroups. This will be used in the next section α∈A to construct unitarily equivalent light-front and instant- form representations of the field theory that preserve the Multi-channel injection operators are defined as the sum scattering matrix. of all channel injection operators X X ˜ ˜ III. CONSTRUCTION ΦA ≔ Φα or ΦA ≔ Φα ð39Þ α∈A α∈A The next step is to use the kinematic symmetries of the injection operators to show that the Hamiltonian H in the where Φα∶Hα → H. These can be used to define multi- expressions for the wave operators can be replaced by channel wave operators the light front Hamiltonian, P−. The relations
Ω Φ Ω˜ Φ˜ 1 A ðH; A;HAÞ¼ A ðH; A;HAÞð40Þ − 3 þ − H ¼ P þ P H ¼ 2 ðP þ P Þð46Þ by the strong limits will be used in what follows. In the first case note because 3 3 Ω Φ − iHtΦ −iHAt ψ 0 P ΦA ΦAP it follows that lim kð A ðH; A;HAÞ e Ae j ik ¼ ð41Þ ¼ A t→ ∞ − − 3 − − 3 iHtΦ iHA iðP þP ÞtΦ iðPAþPAÞt or e Ae ¼ e Ae iP−t −iP− t ¼ e ΦAe A : ð47Þ Ω Φ˜ − iHtΦ˜ −iHAt ψ 0 lim kð A ðH; A;HAÞ e Ae j ik ¼ ð42Þ t→ ∞ þΦ˜ Φ˜ þ P Similarly since P A ¼ APA where HA ≔ α∈A HαΠα and Πα is the projection on the − þ − þ iHt ˜ −iHA iðP þP Þt=2 ˜ −iðP þP Þt=2 subspace Hα of HA. The assumed asymptotic complete- e ΦAe ¼ e ΦAe A A ness of the scattering theory means that the wave operators − 2 − − 2 iP t= Φ˜ iPAt= (including the vacuum and one-body channels) are unitary ¼ e Ae : ð48Þ mappings from the asymptotic Hilbert space, HA, to the This means that the wave operators constructed using H are Hilbert, H, space of the quantum field theory. − With this definition (32) becomes identical to the ones using P with both injection operators. Combining these results with (40) gives the following Λ Ω Φ identifications Uð ;aÞ A ðH; A;HAÞ Ω Φ Λ Ω Φ Ω Φ˜ ¼ A ðH; A;HAÞUAð ;aÞð43Þ ðH; A;HAÞ¼ ðH; A;HAÞ − − ˜ ¼ Ω ðP ; ΦA;P Þ which also holds with ΦA replaced by ΦA. A Ω − Φ˜ − The multi-channel scattering operator is ¼ ðP ; A;PAÞ: ð49Þ Φ H Ω† Φ H Ω Φ H SðH; A; AÞ¼ þðH; A; AÞ −ðH; A; AÞð44Þ The final step in the construction is to introduce two unitary operators VF and VI that act on the field theory Hilbert where it was constructed to be the identity on the vacuum space H with the following properties: and one particle states. Poincar´e invariance of the multichannel scattering oper- VFj0i¼VIj0i¼j0ið50Þ ator on HA follows from the intertwining property of the wave operators (43) [49] Λ 0 Λ 0 ½Uð KI ;aKI Þ;VI ¼ ½Uð KF ;aKF Þ;VF ¼ ð51Þ
UA Λ;a S H; ΦA; HA −iHAt ð Þ ð Þ lim kðΦA − VIΦAÞe jψ αik ¼ 0 ð52Þ t→ ∞ Λ Ω† Φ H Ω Φ H ¼ UAð ;aÞ þðH; A; AÞ −ðH; A; AÞ Φ˜ − Φ˜ −iHAt ψ 0 Ω† Φ H Λ Ω Φ H lim kð A VF AÞe j αik ¼ ð53Þ ¼ þðH; A; AÞUð ;aÞ −ðH; A; AÞ t→ ∞ Ω† Φ H Ω Φ H Λ ¼ þðH; A; AÞ −ðH; A; AÞUAð ;aÞ Λ Λ where ð KI ;aKI Þ and ð KF ;aKF Þ are in the instant-form ¼ SðH; ΦA; HAÞUAðΛ;aÞ: ð45Þ and light-front kinematic subgroups respectively. Note that for Eqs. (52) and (53) to hold the asymptotic Hamiltonian At this point all that has been demonstrated is that the same must have a nontrivial absolutely continuous mass spec- scattering theory is obtained by using injection operators trum. While this is true on the scattering channel subspaces, that agree asymptotically but commute with different it is not true for the vacuum or one-particle states, where
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−iHAt e either factors out because the energy depends on the SðHI; ΦA;HαÞ mass and kinematic variables or is 1 in the case of the † Ω H ; ΦA;HA Ω− H ; ΦA;HA vacuum. In these cases (53) becomes ¼ þð I Þ ð I Þ Ω† Φ˜ † † Ω Φ˜ ¼ þðHF; A;HAÞVFVI VIVF −ðHF; A;HAÞ Φ˜ − Φ˜ ψ 0 kð A VF AÞj αik ¼ : ð54Þ Ω† − Φ˜ − Ω − Φ˜ − ¼ þðPF; A;PAÞ −ðPF; A;PAÞ − ˜ − ¼ SðP ; ΦA;P Þ: ð62Þ In the construction of this section the operators VI and VF F A were chosen to leave the vacuum unchanged. While this is not necessary, this choice gives S-matrix equivalent light- This means both representations of the dynamics result in front and instant-form field theories with the same vacuum the same scattering operators on the asymptotic Hilbert vectors. space. Next consider the two unitary representations of the Λ Λ The next step is to define two new unitary representa- Poincar´e group, UIð ;aÞ and UFð ;aÞ defined above. For P tions of the Poincar´e group on the Hilbert space H of the the first one the operators f ;s;szg are mutually commut- quantum field theory by: ing self-adjoint functions of the instant-form kinematic generators of the representation UIðΛ;aÞ. For the second † ˜ Λ ≔ Λ one the operators fP;sz g are mutually commuting self- UFð ;aÞ VFUð ;aÞVF f adjoint functions of the front-form kinematic generators of Λ ≔ Λ † UIð ;aÞ VIUð ;aÞVI ð55Þ the representation UFðΛ;aÞ. It is possible to construct a basis for UIðΛ;aÞ consisting with the corresponding dynamical generators: of the eigenvalues of fP;s;szIg and some additional kinematically invariant commuting observables xI. ≔ † ≔ † Λ HF VFHVF HI VIHVI ð56Þ Similarly it is possible to construct a basis for UFð ;aÞ ˜ consisting of the eigenvalues of fP;szFg and some addi- † † − − − − tional kinematically invariant commuting observables xF. P ≔ VFP V P ≔ VIP V : ð57Þ F F I I It follows that in these bases wave functions have the form It follows from (51) that p μ ψ p˜ μ ψ Λ Λ h ;s; I;xIj i or h ; F;xFj i: ð63Þ UIð KI ;aKI Þ¼Uð KI ;aKI Þ Λ ≔ Λ UFð KF ;aKF Þ Uð KF ;aKF Þð58Þ Matrix elements of the dynamical operators are nontrivial matrices in the “x” variables: Λ where ð KI ;aKI Þ is an instant-form kinematic Poincar´e Λ hp;s;μ ;x jH jp0;s0; μ0 ;x0 i transformation and ð KF ;aKF Þ is a front-form kinematic I I I I I transformation. 0 0 ¼ δðp − p Þδss0 δμ μ0 hxIkHIðP;sÞkx ið64Þ In Appendix A it is shown [52] that as a consequence of I I I (52) and (53) that and Ω Φ Ω Φ ðHI; A;HAÞ¼VI ðH; A;HAÞð59Þ p˜ μ − p˜ 0 μ0 0 h ; F;xFjPFj ; F;xFi 0 − ˜ 0 ¼ δðp˜ − p˜ Þδμ μ0 hx kP ðP; μ Þkx i: ð65Þ and F F F F F F
˜ ˜ The eigenvalue problems for the dynamical operators Ω ðHF; ΦA;HAÞ¼VFΩ ðH; ΦA;HAÞ: ð60Þ have the forms Taken together with (49) gives: X Z P 0 0 p μ ψ hxIkHIð ;sÞkxIidxIh ;s; I;xIj i Ω Φ Ω Φ ðHI; A;HAÞ¼VI ðH; A;HAÞ ¼ EðP;sÞhp;s;μ ;x jψið66Þ Ω Φ˜ I I ¼ VI ðH; A;HAÞ Ω − Φ˜ − ¼ VI ðP ; A;PAÞ and † Ω − Φ˜ − Z ¼ VIVF ðPF; A;PAÞ: ð61Þ X − P˜ μ 0 0 p˜ μ 0 ψ hxFkPFð ; FÞkxFidxFh ; ;xFj i † Since (61) holds for the same V V for both time limits it I F − P˜ μ p˜ μ ψ follows that ¼ P ð ; FÞh ; F;xFj i: ð67Þ
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P 0 − P˜ μ 0 The matrices hxIkHIð ;sÞkxIi or hxFkPFð ; ÞkxFi must representations are related by S-matrix preserving unitary be diagonalized in order to compute dynamical Poincar´e transformations. Both representation have the same vacuum transformations. On the other hand kinematic transforma- and one-particle subspaces. There are no bare particles in tions can be computed by applying the inverse kinematic these representations. transformation to basis vectors: This construction did not assume the existence of a free dynamics on the Hilbert space of the field theory. If the ˜ hp;s;μ;xIjUIðR; aÞjψi range of Φ is all of H and ψ † a p μ ¼h jUI ðR; Þj ;s; ;xIi ˜ ˜ † −1=2 ˜ W ≔ ðΦAΦAÞ ΦA ð75Þ ia·p −1 s ¼ e hR p;s;ν;xIjψiDμνðRÞ: ð68Þ satisfies Similarly in the light-front case for light-front preserving boosts and translations: ˜ −iHAt lim kðW − ΦAÞe jψ αik ¼ 0 ð76Þ t→ ∞ ˜ hp; μ;xFjUFðΛKF;aKFÞjψi † then W is a kinematically invariant unitary mapping from ¼hψjU ðΛ ;a Þjp˜ ; μ;x i H H Λ Λ † F KF KF sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiF A to and U0ð ;aÞ¼WUAð ;aÞW defines a con- “ ” H Λ−1 þ sistent free dynamics on that satisfies a˜ p˜ ˜ −1 ð k pÞ ¼ ei · hΛ p; μ;x jψi ð69Þ k F pþ UIðΛKI;aKIÞ¼U0ðΛKI;aKIÞð77Þ and for rotations about the z axis with a similar relation in the light-front case. If these representation exist, they are not unique since they depend p˜ μ ϕ 0 ψ h ; ;xFjUFðRzð Þ; Þ i on the choice of injection operator. ψ † ϕ p˜ μ ¼h jUFðRzð ÞÞj ; ;xFi ˜ −1 iμϕ IV. SPONTANEOUS SYMMETRY BREAKING ¼hRzðϕÞ p; μ;xFjψis : ð70Þ One of the puzzling aspects of light-front quantum field
Equation (68) shows that UIðΛ;aÞ has an instant-form theory is how spontaneous symmetry breaking is realized. kinematic subgroup while Eqs. (69) and (70) show that The signal for spontaneous symmetry breaking is a con- μ UFðΛ;aÞ has a light-front kinematic subgroup. In addition served local current, j ðxÞ, that arises from the symmetry the two representations are related by a unitary trans- and the presence of a 0 mass Goldstone boson in the formation: spectrum. For a local scalar field theory the following condition [56] Λ † Λ † UFð ;aÞ¼VFVI UIð ;aÞVIVF ð71Þ lim h0j½Q ; ϕðyÞ j0i ≠ 0 ð78Þ →∞ R 0 † 0 0 R j iF ¼ VFVI j iI ¼j iI: ð72Þ implies the existence of a 0 mass particle. Here In this construction it was chosen so that it preserves the Z vacuum. 0 ϕ 0 ≔ 0 xχ x 0 x ϕ 0 If follows from (62) that both representations give the h j½QR; ðyÞ j i h j d Rðj jÞj ð ;tÞ; ðyÞ j ið79Þ same unitary scattering operators on the asymptotic Hilbert space HA. In addition because VI and Vf are kinematically where χRðjxjÞ is a smooth function that is 1 for jxj
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Current conservation implies Z μ h0j dx∂μj ðx;tÞ; ϕðyÞ j0i¼0: ð81Þ
Inserting a complete set of intermediate states gives X Z μ dp dp μ 0 ¼ dx h0j∂μj ðx;tÞjp; ni hp; njϕðyÞj0i − h0jϕðyÞjp; ni hp; nj∂μj ðx;tÞj0i 2p0 2p0 Z n n X dp dp x 0 ∂ μ 0 0 ϕ 0 0 ip·ðx−yÞ − 0 ϕ 0 ∂ μ 0 0 0 ip·ðy−xÞ ¼ d h j μj ð ; Þjpr;ni 0 hpr;nj ð Þj ie h j ð Þjpr;ni 0 hpr;nj μj ð ; Þj ie ð82Þ 2pn 2pn where Poincar´e covariance has been used to remove the cannot be 0, which means that a translationally invari- nontrivial x, y and p dependence from the matrix elements. ant time-independent charge operator cannot couple the pr is the constant rest four momentum for massive states vacuum to the Goldstone boson. and a constant light-like vector for massless states. For a In the light-front representation it is still possible to scalar field theory the vacuum expectation value of the define QIRðtÞ and the analysis above still applies. The current vanishes so the vacuum does not appear as an assumption that the current is a local function of the fields intermediate state. The Lehmann weights that appear in this means that the light-front dynamics must be used to get its matrix element value on the fixed-time surface where the locality of the theory can be utilized to control the volume integral. 2 μ σðmnÞmn ¼h0j∂μj ð0; 0Þjpr;nihpnr;njϕð0Þj0ið83Þ While the nonvanishing of (78) is a sign of spontaneous symmetry breaking, the argument used above does not and work when it is applied to the light-front charge. The problem is that there is no compact region on the light front 2 μ − σ ðmnÞmn ¼h0jϕð0Þjpnr;nihpnr;nj∂μj ð0; 0Þj0ið84Þ where outside of that region (x y) is always spacelike for fixed y and xþ ¼ 0. This is true even if the light-front are functions of the invariant mass of the intermediate charge is computed in the instant-form representation. If the 2 2 μ states. The factor mn ¼ −pn is due to the ∂μj ð0; 0Þ integrals that define the light-front charge exist and there assuming that the current is a local function of the are no contributions from the divergence theorem, the scalar field. situation is similar to the instant-form case—the charge Current conservation requires that this quantity vanishes operator will commute with the four momentum which 2 means that the light-front charge applied to the vacuum is which follows if either σðmnÞ¼0 or mn ¼ 0. Inserting intermediate states in (79) gives a similar result with an eigenstate of the four momentum with 0 eigenvalue, 2 0 0 which by the argument used above means that the charge σðmnÞmn replaced by pnrσðmnÞ where pnr is a nonzero constant. operator cannot couple the vacuum to the Goldstone boson. Since this does not include the m2 factor, it vanishes by V. MISCELLANEOUS REMARKS current conservation unless σðmnÞ contains a δðmnÞ. Thus for (78) to be nonvanishing the sum over intermediate states This section provides a brief discussion comparing the must have a contribution from a 0 mass particle. This is the structure of the light-front and instant-form representations nonperturbative form of Goldstone’s theorem. discussed in this work to the corresponding relation in In [56] Coleman avoids directly computing the charge relativistic quantum mechanics of a finite number of operator. The current is an operator valued distribution, so degrees of freedom. there is no reason to expect that integrating against the The starting point of this work is the assumed existence constant 1 will result in a well-defined operator. There is of a local field theory with the expected properties. another reason for this. If the integrals that define the Normally in an instant or front-form representation of a charge operator converge and there is no contribution to the quantum field theory “solving the field theory” means current on the surface at infinity, then the charge operator constructing a unitary transformation that decomposes the will be translationally invariant and time independent. This Hilbert space into a direct integral of irreducible repre- means that it would commute with the four momentum sentations. This is the relativistic analog of diagonalizing operator. It follows that even if QIj0i is not 0, it is the Hamiltonian. This is also true in relativistic quantum eigenstate of the four momentum, with 0 eigenvalue. theories of a finite number of degrees of freedom, except in While the Goldstone boson has 0 mass, its momentum that case the interacting and noninteracting theories are
105017-11 W. N. POLYZOU PHYS. REV. D 103, 105017 (2021) formulated on the same representation of the Hilbert space. possibility is to define interactions using matrix elements in In the finite number of degree of freedom case different states constructed by applying products of the operators that choices of independent variables in each irreducible sub- create one-particle states out of vacuum to the vacuum. space are related by a unitary change of variables. This Matrix elements of the Hamiltonian in these states have two defines a unitary, S-matrix preserving transformation relat- properties, the first is that they become products of one- ing the two forms of dynamics. It is nontrivial because there particle states as the particles are asymptotically separated. In is a different (mass-dependent) variable change on each this representation, because the one-particle states are point irreducible subspace. The light-front and instant-form bases spectrum eigenstates of the Hamiltonian, the Hamiltonian have the property that matrix elements of the unitary cannot couple them to the continuum. This means that in this representation of the kinematic subgroups in each case representation a “production vertex” cannot couple to a are mass independent. multiparticle scattering state. This is one way of saying that In this work this process is reversed; the direct integral is stable particle cannot decay, however particle production is assumed to be a property of a quantum field theory with all possible since interactions that couple “two-particle states” to of the expected properties. The general class of S-matrix “three-particle states”, etc. are possible in this representation. preserving unitary transformations is used to construct A final comment concerns vacuum polarization in this equivalent representations of the dynamics. The freedom representation. The physical vacuum is orthogonal to all to choose different S—matrix preserving unitary trans- bound and scattering eigenstates of the Hamiltonian, but formations was exploited to choose S-matrix preserving operators that couple the vacuum to other states in the unitary transformations that commute with the instant form Hilbert space are possible and are needed to describe local or light-front kinematic subgroups of the theory, which vacuum excitations. resulted in equivalent light-front or instant-form represen- tations of the field theory. VI. ANALYSIS AND CONCLUSIONS The unitarity of these transformations means that they ’ must preserve the spectrum of the mass operator. This This work provides one way to understand Dirac s forms means that point-spectrum eigenvalues of the mass operator of dynamics in a nonperturbative setting. This approach must be the same in all three representations. The unitary assumed the existence of a local quantum field theory with transformation must also leave the vacuum energy or the all of the expected properties. Using this assumption mass gap unchanged. Preserving the S-matrix means that unitarily equivalent representations with the same scatter- acceptable unitary transformations must be short-range ing operators were constructed. Bases that transform perturbations of the identity in the sense that they do not covariantly with respect to light-front and instant-form require a modification of scattering asymptotic conditions. kinematic subgroups where constructed for each of these The unitary transformations that satisfy all of these con- equivalent representations of the dynamics. Kinematic ditions are not unique and are discussed in Appendix A. Poincar´e transformations on these wave functions could The construction of equivalent instant-form and light- be computed without diagonalizing the dynamical oper- front unitary representations of the Poincar´e group in this ators. One aspect of the construction used in this work is paper is essentially identical to the corresponding con- that both representations of the dynamics leave the vacuum struction in the finite number of degree of freedom case and one-particle states unchanged and both representations [3,4,57,58]. In that case, when there is physical particle give the same Poincar´e invariant scattering operator. While production, the particles all have their physical masses and preserving the vacuum states is not required, the construc- there are no vertex interactions that can change masses [59]. tion demonstrates that it is a consistent assumption. In that case the simplest interactions that change particle A key difference with the perturbative approach is the number are 2 ↔ 3 interactions. nonexistence of an underlying free-field dynamics on the Given the spectrum of one-particle states, the mass Hilbert space of the field theory. If the injection operators have full range on the Hilbert space of the field theory and spectrum of the field theory Hamiltonian includes the one- satisfy (76) there is the analog of a free dynamics on the particle masses and continuous eigenvalues starting at the Hilbert of the field theory. If such a representation exists, it two-particle thresholds. This is identical to the mass spec- cannot be a local free field theory with physical masses. trum of a sum of free-field Hamiltonians with physical While the representations discussed in this analysis are ’ particle masses. The conclusion of Haag s theorem [40] is not useful for applications, the results support the con- that a spectrally equivalent free field theories are defined on clusions of most of the quoted references that the two inequivalent representation of the Hilbert space. This why formulations of the theory are equivalent. The vacuum is there are no free multiparticle states in the field theory and is the same up to a unitary transformation and can be taken as the motivation for using the two-Hilbert space representation the same in both representations. While the light-front in order to formulate the scattering asymptotic conditions. vacuum is kinematically invariant, it is also rotationally In the absence of free multi-particle states in the theory, the invariant, which is an additional dynamical constraint on notion of many-particle interactions needs to be defined. One the light-front vacuum that separates it from the free-field
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≔ Ω 0 Φ0 Ω† Φ Fock vacuum. By starting with the assumed solution to the W þðH ; A;HAÞ þðH; A;HAÞ field theory there are no issues with renormalization, Ω 0 Φ0 Ω† Φ rotational covariance or 0 modes. These issues need to ¼ −ðH ; A;HAÞ −ðH; A;HAÞðA5Þ be addressed in applications. If there is spontaneous is a unitary operator on H. The identification of the symmetry breaking, the nonperturbative condition for the scattering operators (A4) means that W is the same for presence of a Goldstone boson in the mass spectrum is the incoming (t → þ∞) or outgoing (t → −∞) multichan- the same in both representations, but it does not couple to nel wave operator. It follows that the vacuum with a space-time translationally invariant charge operator, whether it is on the fixed-time surface 0 0 WΩ ðH; ΦA;HAÞ¼Ω ðH ; Φ ;HAÞ: ðA6Þ or the light-front hyperplane. A In addition, the intertwining property of the wave ACKNOWLEDGMENTS operators [49],
The author would like to thank the referee for some 0 † 0 0 0 WH ¼ Ω ðH; ΦA;HAÞΩ ðH ; Φ ;HAÞH useful suggestions. This work supported by the U.S. A Ω Φ Ω† 0 Φ0 Department of Energy, Office of Science, Grant No. DE- ¼ ðH; A;HAÞHA ðH ; A;HAÞ SC0016457. Ω Φ Ω† 0 Φ0 ¼ H ðH; A;HAÞ ðH ; A;HAÞ¼HW ðA7Þ
means that the two Hamiltonians are related by the unitary APPENDIX A: SCATTERING EQUIVALENCES transformation W. The two-Hilbert space multichannel wave operators are It follows that defined by the strong limits where the ΦA are two-Hilbert 0 0 space injection operators that map the asymptotic Hilbert Ω−ðH ; ΦA;HAÞ¼WΩ−ðH; ΦA;HAÞ space HA to the Hilbert space H of the quantum field † ¼ Ω−ðWHW ;WΦA;HAÞ theory: 0 ¼ Ω−ðH ;WΦA;HAÞ: ðA8Þ Ω Φ − iHtΦ −iHAt ðH; A;HAÞ¼s lim e Ae : ðA1Þ t→ ∞ Taking the difference of the right and left side of Eq. (A8) gives The wave operators are assumed to exist and be Poincar´e 0 invariant in the sense iH t 0 −iHAt 0 ¼ lim ke ðΦA − WΦAÞe jψik: ðA9Þ t→ ∞ † UðΛ;aÞΩ ðH; ΦA;HAÞ 0 The unitarity of eiH t gives Ω† Φ Λ ¼ ðH; A;HAÞUAð ;aÞ: ðA2Þ 0 −iHAt 0 ¼ lim kðΦA − WΦAÞe jψik: ðA10Þ t→ ∞ where UAðΛ;aÞ is the natural representation of the Poincar´e group on HA that has the form of a direct sum That this condition holds for both time limits is important. of tensor products of irreducible representations. The set of For the vacuum and one-particle channels this condition channels A is assumed to include the vacuum channel, and requires that one-particle channels in addition to multiparticle scattering 0 channels. The wave operators are assumed to be asymp- ΦA − WΦA ¼ 0: ðA11Þ totically complete unitary mappings from asymptotic Hilbert space HA to the Hilbert space H of the field theory. Next consider the converse. Assume that W is a The scattering operator is defined by unitary operator satisfying H0 ¼ WHW†, and the scattering operators Φ Ω† Φ Ω Φ SðH; ;HAÞ¼ þðH; A;HAÞ −ðH; A;HAÞ: ðA3Þ Φ Ω† Φ Ω Φ SðH; ;HAÞ¼ þðH; A;HAÞ −ðH; A;HAÞðA12Þ Consider two identical scattering operators based on differ- 0 Φ0 Ω† 0 Φ0 Ω 0 Φ0 ent Hamiltonians SðH ; A;HAÞ¼ þðH ; A;HAÞ −ðH ; A;HAÞðA13Þ
0 0 SðH; ΦA;HAÞ¼SðH ; ΦA;HAÞðA4Þ both exist. If W satisfies (A10) for both time limits then the S matrices are identical It follows from the definitions and unitarity of the wave 0 Φ0 Φ operators that SðH ; A;HAÞ¼SðH; A;HAÞ: ðA14Þ
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The proof follows from To isolate the operators that create the one-body states use space-time translations to define the operator valued Ω 0 Φ0 Ω † Φ0 ðH ; A;HAÞ¼ ðWHW ; A;HAÞ distributions † 0 ¼WΩ ðH;W ΦA;HAÞ BðxÞ ≔ e−iP·xBe−iP·x ¼ U†ðI;xÞBUðI;xÞðB2Þ Ω † Φ0 − ΦA Φ ¼W ðH;W ð A W þW AÞ;HAÞ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} μ →0 where P is the four momentum operator. It follows from Ω Φ (B2) that ¼W ðH; AÞ;HAÞ: ðA15Þ ∂BðxÞ It then follows that ¼ −i½Pμ;BðxÞ : ðB3Þ ∂xμ 0 Φ0 Ω† 0 Φ0 Ω 0 Φ0 SðH ; A;HAÞ¼ þðH ; A;HAÞ −ðH ; A;HAÞ The next step is to compute the Fourier transform of the Ω† Φ † Ω Φ ¼ þðH; A;HAÞW W −ðH; A;HAÞ operator density BðxÞ: Φ Z ¼ SðH; A;HAÞ: ðA16Þ d4x Bˆ q ≔ eiq·xB x : ð Þ 2π 2 ð Þ ðB4Þ Note that unitary equivalence is not a sufficient condition ð Þ for S-matrix equivalence. This is not hard to understand in eiq·x Multiplying both sides of (B3) by 2 and integrating over the case of ordinary quantum mechanics, where two ð2πÞ Hamiltonians with short-range repulsive interactions have d4x by parts, assuming that BðxÞ is an operator valued the same spectrum (so they are unitarily equivalent) but distribution, gives generally have different S-matrix elements or phase shifts. The conclusion of this section is that unitary operators W −i½Pμ; Bˆ ðqÞ ¼ −iqμBˆ ðqÞðB5Þ that satisfy the asymptotic condition (A10) can be used to relate Hamiltonians that have the same scattering matrix. or This asymptotic condition means that W does not disturb the asymptotic structure of the states that define the ½Pμ; Bˆ ðqÞ ¼ qμBˆ ðqÞ: ðB6Þ asymptotic condition. The above condition applies to both quantum mechanics It follows from (B6) that Bˆ ðqÞj0i is either 0 or an eigenstate and quantum field theory assuming that they have asymp- of the four momentum with eigenvalue qμ: totically complete scattering operators. The field theory generalization of multiparticle scattering is the Haag-Ruelle PμBˆ ðqÞj0i¼Bˆ ðqÞPμj0i¼qμBˆ ðqÞj0i¼qμBˆ ðqÞj0i: ðB7Þ formulation of scattering which involves the strong limits pffiffiffiffiffiffiffiffiffi used in this Appendix. Let m be a discrete mass eigenvalue of M ¼ −P2 and let s denote the spin of the particle of mass m. Next let hðqÞ be a smooth Lorentz invariant function of compact support APPENDIX B: TWO HILBERT SPACE in q2 that is 1 when q2 ¼ −m2 and q0 > 0, and identically 0 INJECTION OPERATORS on the rest of the spectrum of intermediate states in (B1) This Appendix discusses the construction of injection and define operators from operators in the field algebra. The starting † ≔ ˆ point is to let B be a function of the fields that creates A ðqÞ BðqÞhðqÞ: ðB8Þ a one-body state out of the vacuum. This means that the completeness sum Additional integrations are used to define the operators: X Z h0jB†Bj0i¼ h0jB†jnihnjBj0iðB1Þ A†ðqÞ ≔ A†ðqÞdq0 and n Z ˜ † q˜ ≔ † − 2 has one-body intermediate states (here one-body means A ð Þ A ðqÞdq = : ðB9Þ states with discrete positive-mass eigenvalues, so there is no distinction between elementary and composite one-body Because hðqÞ is localized, these operators are distributions states). For simplicity it is assumed that the one-body in the three momentum or light-front components of the spectrum is nondegenerate, which means that each one- four momentum. When these operators are applied to the particle state has a different mass, and each one-body mass vacuum they create one-particle states of mass m. Since by eigenstate has a given spin. assumption, there is a unique spin s associated with each
105017-14 RELATION BETWEEN INSTANT AND LIGHT-FRONT … PHYS. REV. D 103, 105017 (2021) discrete mass eigenvalue, the particle created out of the Define the operator vacuum also has spin s. 0 − Because of the integrals over q or q it follows that (B6) † A ðp; μÞ must be replaced by m;s Z ≔ 0 † p † 0 s ½P;A†ðqÞ ¼ qA†ðqÞ: ½P˜ ; A˜ †ðq˜Þ ¼ q˜A˜ †ðq˜ÞðB10Þ UðR; ÞA ðR ÞU ðR; ÞDμsðRÞdR ðB13Þ SUð2Þ which when applied to the vacuum becomes where the integral is over the SUð2Þ Haar measure and † † s 2 PA ðqÞj0i¼qA ðqÞj0i DμsðRÞ is the spin-sSUð Þ Wigner function P˜ A˜ †ðq˜Þj0i¼q˜A˜ †ðq˜Þj0i: ðB11Þ s DμsðRÞ¼hs; μjUðRÞjs; si: ðB14Þ This means that A†ðqÞ creates a one-particle state with q momentum , mass m and spin s out of the vacuum. It follows from (B13) that Similarly
˜ ˜ † † 0 † † 0 PA ðq˜Þj0i¼q˜A ðq˜Þj0iðB12Þ UðR ;0ÞAm;sðp;μÞU ðR ;0Þ Z 0 † −1 † 0 s creates a one-particle state with light-front momentum q˜ ¼ UðR R;0ÞA ðR pÞU ðR R;0ÞDμsðRÞdR: ðB15Þ and mass m and spin s out of the vacuum. SUð2Þ The next step is to construct operators that create simultaneous eigenstates of mass and spin and magnetic Changing variables, R00 ¼ R0R using the invariance of the quantum numbers. Haar measure dR ¼ dR00 for fixed R0 gives
Z 00 † 00−1 0 † 00 00 s 0−1 00 ðB15Þ¼ UðR ; 0ÞA ðR R pÞU ðR ; 0ÞdR DμsðR R Þ SUð2Þ Z Xs 00 † 00−1 0 † 00 00 s 0−1 s 00 ¼ UðR ; 0ÞA ðR R pÞU ðR ; 0ÞdR DμνðR ÞDνsðR Þ SUð2Þ ν − Z ¼ s Xs 00 † 00−1 0 † 00 00 s 0−1 00 s 0 ¼ UðR ; 0ÞA ðR R pÞU ðR ; 0ÞdR DνsðR R ÞDνμðR Þ 2 SUð Þ ν¼−s Xs † 0p ν s 0 ¼ Aðm;jÞðR ; ÞDνμðR Þ: ðB16Þ ν¼−s
† When applied to the vacuum (B16) gives ½P;A ðq; μÞ ðm;sZ Þ Xs P 0 † −1q † 0 s † † ¼ dR½ ;UðR; ÞA ðR ÞU ðR; Þ DμsðRÞ 0 p μ 0 p ν 0 s SUð2Þ UðR; ÞAðm;sÞð ; Þj i¼ Am;sðR ; Þj iDνμðRÞðB17Þ Z ν¼−s † −1 † s ¼ dRUðR; 0Þ½RP;A ðR qÞ U ðR; 0ÞDμsðRÞ SUð2Þ which means that either this vanishes or it transforms like a Z p −1 † −1 † s particle of mass m, spin s, momentum and magnetic ¼ RR q dRUðR; 0ÞA ðR qÞU ðR; 0ÞDμsðRÞ quantum number μ. The spin in these states created out of SUð2Þ the vacuum is the canonical spin. This will vanish if there q † q μ ¼ A m;s ð ; Þ: ðB18Þ are no one-particle intermediate states with mass m and ð Þ spin s in (B1). While the notation is purposely suggestive, The normalization can be chosen so the states created out of † p μ 0 Am;sð ; Þ is pnotffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a creation operator. In addition, p is the vacuum have the normalization (9). 2 2 † only equal to p þ m when Am;sðp; μÞ is applied to the The construction above cannot be used in the “light- vacuum. front” case due to the integral over p− in (B9).Howeverin Because rotations do not change the time component, that case it is enough to project out the magnetic quantum − these operators also satisfy number using a rotation RzðϕÞ which leaves p in (B9)
105017-15 W. N. POLYZOU PHYS. REV. D 103, 105017 (2021) unchanged. The spin is identified with the highest nonzero and μ weight, s ¼ max. In this case Eq. (B13) is replaced by ϕ 0 ˜ † q˜ μ † ϕ 0 ˜ † ˜ UðRzð Þ; ÞAðm;sÞð ; ÞU ðRzð Þ; Þ Am;sðp;μÞ Z ˜ † μϕ 2π ϕ ¼ A ðR ðϕÞq˜; μÞei : ðB22Þ ˜ † −1 † −iμϕ d ðm;sÞ z ≔ UðRzðϕÞ;0ÞA ðRzðϕÞ pÞU ðRzðϕÞ;0Þe ; 0 2π ðB19Þ The proof (B20) is essentially the same as the proof of (B18). To show (B22) note that for rotations about the (B17) and (B18) are replaced by z axis
˜ ˜ † ˜ † ½P; A ðq˜; μÞ ¼ q˜A ðq˜; μÞ; ðB20Þ −1 ðm;sÞ ðm;sÞ BFðp=mÞRz ¼ RzBFðRz p=mÞðB23Þ Λ 0 ˜ † q˜ μ † Λ 0 ˜ † Λ˜ μ Uð K; ÞA m;s ð ; ÞU ð K; Þ¼A m;s ð kq; Þ; ðB21Þ ð Þ ð Þ where BFðp=mÞ is a light-front boost. It follows that
UðB ðp=mÞ; 0ÞA˜ † ðq˜; μÞUðB ðp=mÞ; 0Þ† FZ m;s F 2π ˜ † −1 † −iμϕ ¼ UðBFðp=mÞRzðϕÞ; 0ÞA ðRzðϕÞ qÞU ðBFðp=mÞRzðϕÞ; 0Þe Z0 2π −1 ˜ † −1 † −1 −iμϕ ¼ UðRzðϕÞBFðRz ðϕÞp=mÞ; 0ÞA ðRzðϕÞ qÞU ðRzðϕÞBFðRz ðϕÞp=mÞ; 0Þe Z0 2π −1 ˜ † −1 −1 † † † −iμϕ ¼ UðRzðϕÞ; 0ÞUðBFðRz ðϕÞp=mÞ; 0ÞA ðRzðϕÞ qÞUðBFðRz ðϕÞp=mÞ; 0ÞÞ U ðRzðϕÞ; 0Þ e Z0 2π ˜ † −1 −1 † † −iμϕ ¼ UðRzðϕÞ; 0ÞUð; 0ÞA ðBFðRz ðϕÞp=mÞRzðϕÞ qÞU ðRzðϕÞ; 0Þ e Z0 2π ˜ † −1 † † −iμϕ ¼ UðRzðϕÞ; 0ÞUð; 0ÞA ðRzðϕÞ BFðp=mÞqÞU ðRzðϕÞ; 0Þ e 0 ˜ † ˜ ¼ Am;sðBFðp=mÞq; μÞ ðB24Þ which proves (B21). For (B22) note that Z 2π dϕ ϕ0 0 ˜ † q˜ μ † ϕ0 0 ϕ ϕ0 0 ˜ † ϕ −1p † ϕ ϕ0 0 −iμϕ UðRzð Þ; ÞA m;s ð ; ÞU ðRzð Þ; Þ¼ UðRzð þ Þ; ÞA ðRzð Þ ÞU ðRzð þ Þ; Þe : ðB25Þ ð Þ 0 2π
Let ϕ00 ¼ ϕ þ ϕ0 so (B25) becomes Z 2π dϕ00 ϕ00 0 ˜ † ϕ00 − ϕ0 −1p † ϕ00 0 −iμðϕ00−ϕ0Þ UðRzð Þ; ÞA ðRzð Þ ÞU ðRzð Þ; Þe 2π Z0 2π ϕ00 0 ˜ † ϕ00 −1 ϕ0 p † ϕ00 0 −iμðϕ00 iϕ0μÞ ˜ † ϕ0 q˜ μ iμϕ0 UðRzð Þ; ÞA ðRzð Þ Rzð Þ ÞU ðRzð Þ; Þe e ¼ A m;s ðRzð Þ ; Þe : ðB26Þ 0 ð Þ
The normalization can be chosen so these states created out of the vacuum have the normalization (9).
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