PHYSICAL REVIEW D 101, 096004 (2020)

Wavelet representation of light-front

W. N. Polyzou Department of Physics and Astronomy, The University of Iowa, Iowa City, Iowa 52242, USA

(Received 28 January 2020; accepted 29 April 2020; published 11 May 2020)

A formally exact discrete multiresolution representation of quantum field theory on a light front is presented. The formulation uses an orthonormal basis of compactly supported wavelets to expand the fields restricted to a light front. The representation has a number of useful properties. First, light-front preserving Poincar´e transformations can be computed by transforming the arguments of the basis functions. The discrete field operators, which are defined by integrating the product of the field and a basis function over the light front, represent localized degrees of freedom on the light-front hyperplane. These discrete fields are irreducible and the is formally trivial. The light-front Hamiltonian and all of the Poincar´e generators are linear combinations of normal ordered products of the discrete field operators with analytically computable constant coefficients. The representation is discrete and has natural resolution and volume truncations like lattice formulations. Because it is formally exact, it is possible to systematically compute corrections for eliminated degrees of freedom.

DOI: 10.1103/PhysRevD.101.096004

I. INTRODUCTION dynamics” that are characterized by having the largest interaction-independent subgroups. A discrete multiresolution representation of quantum Dirac’s “front-form dynamics” is the only form of field theory on a light front is presented. Light-front dynamics with the minimal number, 3, of dynamical formulations of quantum field theory have advantages Poincar´e generators. The interaction-independent subgroup for calculating electroweak current matrix elements in is the seven-parameter subgroup that leaves the hyperplane strongly interacting states in different frames. Lattice truncations have proved to be the most reliable method 0 xþ ¼ x þ nˆ · x ¼ 0 ð1Þ for nonperturbative calculations of strongly interacting states, but and scattering calcula- invariant. The light-front representation of quantum tions are not naturally formulated in the lattice representa- dynamics has several advantages. One is that the kinematic tion. The purpose of this work is to investigate a (interaction-independent) subgroup has a three-parameter representation of quantum field theory that has some of subgroup of Lorentz boosts. The subgroup property means the advantages of both approaches, although this initial that there are no Wigner rotations for light-front boosts. A work is limited to canonical field theory rather than gauge consequence is that the magnetic quantum numbers of the theories. light-front are invariant with respect to these boosts. A In 1939, Wigner [1] showed that the independence of second advantage is that the boosts are independent of quantum observables in different inertial reference frames interactions. This means that boosts can be computed by related by Lorentz transformations and space-time trans- applying the inverse transform to noninteracting basis lations requires the existence of a dynamical unitary states. These properties simplify theoretical treatments of representation of Poincar´e group on the Hilbert space of electroweak probes of strongly interacting systems, where the quantum theory. Because there are many independent the initial and final hadronic states are in different Lorentz paths to the future, consistency of the initial value problem frames. requires that a minimum of three of the infinitesimal In light-front quantum field theory [3–14], there are generators of the Poincar´e group are interaction dependent. additional advantages. These are consequences of the “ In 1949, Dirac [2] introduced three forms of relativistic spectrum of the generator

pþ ¼ p0 þ nˆ · p ≥ 0 ð2Þ Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. of translations in the Further distribution of this work must maintain attribution to ’ the author(s) and the published article s title, journal citation, − 0 − nˆ x and DOI. Funded by SCOAP3. x ¼ x · ð3Þ

2470-0010=2020=101(9)=096004(25) 096004-1 Published by the American Physical Society W. N. POLYZOU PHYS. REV. D 101, 096004 (2020) direction tangent to the light front. The first property is that This work is an extension to the light front of the wavelet free fields restricted to the light front are irreducible. This representations of quantum field theory used in means that both the creation and operators for Refs. [25,28,35]. The notation and development of the a free field can be constructed from the field restricted to the wavelet bases is identical to the development in these light front. It follows that any on the free-field references. The difference is that the algebra generated by Fock space can be expressed as a function of free fields the discrete fields and conjugate generalized momenta in restricted to the light front. The second advantage is that these papers is replaced by the irreducible algebra of fields interactions that commute with the interaction-independent on a light front. The light-front representation is formally subgroup leave the Fock vacuum invariant. This means that exact and has all of the advantages of any other represen- it is possible to express all of the Poincar´e generators as tation of light-front field theories. operators on the free-field Fock space. There are ultraviolet There are several motivations for considering this and infrared (pþ ¼ 0) singularities in the light-front approach. These include the following: Hamiltonian due to local operator products, which could (1) Volume and resolution truncations can be performed impact these properties; however, in an effective theory naturally, the resulting truncated theory is similar to with ultraviolet and infrared cutoffs, the interaction still a lattice truncation [42,43], in the sense that it is a leaves the Fock vacuum invariant and the light-front theory involving a finite number of discrete degrees Hamiltonian can still be represented as a function of the of freedom associated with a given volume and free fields on the light front. resolution. Having an explicit vacuum along with an expression for (2) While the degrees of freedom are discrete, the field the light-front Hamiltonian, operators have a continuous space-time dependence. Kinematic Lorentz transformations can be computed − 0 by transforming the arguments of the basis func- P ¼ P − P · n; ð4Þ tions. While truncations necessarily break kinematic Lorentz invariance, kinematic Lorentz transforma- in terms of the algebra of fields on the light front means that tions can still be approximated by transforming the it is possible to perform nonperturbative calculations by arguments of the basis functions. diagonalizing the light-front Hamiltonian in the light-front (3) Even though some truncations may lead to states Fock space. with below the Fock vacuum energy, the In a given experiment, there is a relevant volume and a error in using the free Fock vacuum as the lowest finite amount of available energy. The available energy state of the truncated theory is due to correc- limits the resolution of the accessible degrees of freedom. tions that arise from the discarded degrees of The number of degrees of freedom with the limiting freedom. resolution that fits in the experimental volume is finite. (4) Since the representation is formally exact and xþ is a It follows that it should be possible to accurately calculate continuous variable, there is a formulation of Haag- experimental observables using only these degrees of Ruelle [44–46] scattering in this representation. freedom. Approximation methods need to be developed in Wavelets can be used to represent fields on the light front the presence of truncations. as linear combinations of discrete field operators with Some of the possible applications of the wavelet repre- different resolutions. This provides a natural representation sentation are discussed in [25] in the context of canonical to make both volume and resolution truncations consistent field theory. There are a number of applications involving with a given reaction. In addition, the representation is free fields that are straightforward and should be instruc- discrete, which is a natural representation for computations. tive. The advantage of free fields is that they can be solved Finally, the basis functions are self-similar, so truncations and used as a testing ground in order to get an initial with different resolutions have a similar form. understanding of the convergence of truncated theories. There are many different types of wavelets that have One such application is understanding the restoration of been discussed in the context of quantum field theory [15– Poincar´e invariance in truncated theories as the resolution is 36]. The common feature is that the different functions have improved. An advantage of the wavelet representation is a common structure related by translations and scale that this can be checked locally, i.e., in a small volume [25]. transformations. This work uses Daubechies’ wavelets Understanding the restoration of Lorentz invariance is [37–41]. These have the property that they are an ortho- important for approximating current matrix elements. normal basis of functions with compact support. The price Another application involving free fields is to test the paid for the compact support is that they have a limited convergence of free-field commutator functions or smoothness. It is also possible to use a wavelet basis of Wightman functions based on truncated fields to the Schwartz functions that are infinitely differentiable, but exact expressions. These can be approximated by iterating these functions do not have compact support. the Heisenberg field equations, which are simple in the

096004-2 WAVELET REPRESENTATION OF LIGHT-FRONT QUANTUM … PHYS. REV. D 101, 096004 (2020) free-field case. This could provide some insight into the light front are defined in Sec. VI. Section VII has a short nature of convergence in interacting theories. In [35], flow discussion on kinematic Poincar´e transformations of equation methods are used to block diagonalize the Hilbert the fields in the light-front representation. In Sec. VIII, space of a truncated free-field theory by resolution, con- the irreducibility of the light-front free-field algebra and the structing an effective Hamiltonian that involves only triviality of the light-front vacuum are used to construct coarse-scale degrees of freedom, but includes the dynamics vectors in the light-front Fock space. Dynamical equations of the eliminated degrees of freedom. This calculation in the light-front wavelet representation are discussed in provided some insight into the complementary roles played Sec. IX. Dynamical computations require expressions for by volume and resolution truncations. the commutator of discrete fields on the light front, which While the elementary calculations discussed above can are computed in Sec. X. In Sec. XI, the coefficients of the provide insight into the nature of approximations, the long- expansion of all ten Poincar´e generators as polynomials of term goal is to use the wavelet representation to perform discrete fields on the light front are computed. Section XII calculations of observables in 3 þ 1-dimensional field discusses truncations and Sec. XIII gives a summary and theories. Calculations in 3 þ 1 dimensions are considerably outlook. more complicated for interacting theories. One computa- tional method is to use the fields to construct a basis by II. NOTATION applying discrete operators to the vacuum and then diag- onalizing the light-front Hamiltonian in that basis. The The light front is a three-dimensional hyperplane that is light-front representation has the advantage that it is not tangent to the . It is defined by the constraint necessary to first solve the vacuum problem. This method xþ ≔ x0 þ nˆ · x ¼ 0: ð5Þ should be useful for modeling composite states that are It is natural to introduce light-front coordinates of the four- spatially localized. This Hamiltonian approach is in the μ same spirit as the basis light-front approach vector x , used in [47]. Variational methods could also be employed 0 x ≔ x nˆ · x; x⊥ ¼ nˆ × ðx × nˆ Þ: ð6Þ for low-lying composite states. Another method that takes advantage of the discrete nature of the wavelet representa- The components tion is to use the light-front Heisenberg equations to − x˜ ≔ ðx ; x⊥Þð7Þ generate an expansion of the field as a linear combination of products of fields restricted to the light front. Correlation are coordinates of points on the light-front hyperplane. This functions can be computed by evaluating products of these will be referred to as a light-front three-vector. In what fields in the light-front vacuum. In this case, while the follows, the light front defined by nˆ ¼ zˆ will be used. algebra is discrete, the number of terms grows with each The contravariant light-front components are iteration. One of the advantages of the wavelet representa- ∓ i x ¼ −x x ⊥ ¼ x ; ð8Þ tion is that interactions involving different modes are i ⊥ self-similar and differ only by multiplicative scaling coef- and the Lorentz-invariant scalar product of two light-front ficients. A detailed study of the scaling properties could vectors is help to formulate efficient approximations to the solution of 1 1 the light-front Heisenberg field equations by eliminating ≔− þ − − − þ x y x · y 2 x y 2 x y þ ⊥ · ⊥ irrelevant degrees of freedom. Another potential use of the 1 wavelet representation would be in quantum computing. In þ − 1 2 ¼ ðx yþ þ x y−Þþx y1 þ x y2: ð9Þ the wavelet representation, the field is replaced by discrete 2 modes that only interact locally. This allows evolution over For computational purposes, it is useful to represent four short time steps to be represented by quantum circuits vectors by 2 × 2 Hermitian matrices. The coordinate matrix involving products of local interactions. is constructed by contracting the four-vector xμ with the This paper consists of 13 sections. The next section Pauli matrices and the identity introduces the notation that will be used in this work, xþ x 1 defines the light-front kinematic subgroup and the Poincar´e μσ ⊥ μ σ X ¼ x μ ¼ − x ¼ 2 Trð μXÞ generators that generate both the kinematic and dynamical x⊥ x 1 2 Poincar´e transformations. Section III discusses the irreduc- x⊥ ¼ x þ ix : ð10Þ ibility of free fields on the light front and properties of kinematically invariant interactions. Section IV discusses In this matrix representation, Poincar´e transformations the structure of Poincar´e generators on the light front using continuously connected to the identity are represented by ’ Noether s theorem. The wavelet basis is constructed from X → X0 ¼ ΛXΛ† þ A Λ ∈ SLð2; CÞ A ¼ A†: the fixed point of a group equation in Sec. V. Wavelet representations of fields restricted to the ð11Þ

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The subgroup of the Poincar´e group that leaves xþ ¼ 0 while the adjoint and the inverse adjoint of these invariant consists of pairs of matrices ðΛ;AÞ in (11) of the matrices are form 0 1 pffiffiffiffiffiffiffiffiffiffiffiffi 0 0 p⊥=m a b⊥ pþ=m pffiffiffiffiffiffiffiffiffi Λ † B þ C ¼ A ¼ − ; ð12Þ Λ ≔ @ p =m A c 1=a b⊥ b fðp=mÞ pffiffiffiffiffiffiffiffiffiffiffiffi þ − 01= p =m where a, c and b⊥ are complex and b is real. This is a pffiffiffiffiffiffi pffiffiffiffiffiffi ! 2 C þ þ seven-parameter group. The SLð ; Þ matrices with real a v v⊥= v represent light-front preserving boosts. They can be para- ¼ pffiffiffiffiffiffi ; ð15Þ 01þ metrized by the light-front components of the four velocity = v v ¼ p=m, 0 1 0 pffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffiffi p =m þ 1= pþ=m − pffiffiffiffiffiffiffiffiffi⊥ p =m 0 B þ C B C Λ † −1 ≔ p =m Λ ≔ @ pffiffiffiffiffiffiffiffiffiffiffiffi A ðð fÞ Þ ðp=mÞ @ pffiffiffiffiffiffiffiffiffiffiffiffi A fðp=mÞ p⊥=m pffiffiffiffiffiffiffiffiffi 1= pþ=m þ þ 0 p =m p =m ! ffiffiffiffiffiffi ! pffiffiffiffiffiffi pffiffiffiffiffiffi p 1= vþ −v = vþ vþ 0 ⊥ ffiffiffiffiffiffi ¼ pffiffiffiffiffiffi pffiffiffiffiffiffi : ð13Þ ¼ p : ð16Þ þ þ 0 þ v⊥= v 1= v v

These lower triangular matrices form a subgroup. The General Poincar´e transformations are generated by ten inverse light-front boost is given by independent one-parameter subgroups. Seven of the one- 0 pffiffiffiffiffiffiffiffiffiffiffiffi 1 parameter groups leave the light front invariant. The 1= pþ=m 0 −1 B C remaining 3 one-parameter groups map points on the light Λ ðp=mÞ ≔ @ pffiffiffiffiffiffiffiffiffiffiffiffi A f − ppffiffiffiffiffiffiffiffiffi⊥=m pþ=m front to points off of the light front. These are called pþ=m ffiffiffiffiffiffi ! kinematic and dynamical transformations, respectively. The p 2 2 1= vþ 0 kinematic one-parameter groups in the × matrix rep- ¼ pffiffiffiffiffiffi pffiffiffiffiffiffi ; ð14Þ resentation and the corresponding unitary representations − þ þ v⊥= v v of these groups are related by

10 1 10 2 ΛðλÞ¼ UðΛðλÞÞ ¼ eiE λ ΛðλÞ¼ UðΛðλÞÞ ¼ eiE λ; ð17Þ λ 1 iλ 1 λ=2 iλ=2 e 0 3 e 0 3 ΛðλÞ¼ UðΛðλÞÞ ¼ eiK λ ΛðλÞ¼ UðΛðλÞÞ ¼ eiJ λ; ð18Þ 0 e−λ=2 0 e−iλ=2 0 λ 1 0 −iλ 2 AðλÞ¼ UðΛðλÞÞ ¼ eiP λ:AðλÞ¼ UðΛðλÞÞ ¼ eiP λ; ð19Þ λ 0 iλ 0 00 −i þλ AðλÞ¼ UðΛðλÞÞ ¼ e 2P : ð20Þ 0 λ

The corresponding dynamical transformations are 1 λ 1 1 −iλ 2 ΛðλÞ¼ UðΛðλÞÞ ¼ eiF λ ΛðλÞ¼ UðΛðλÞÞ ¼ eiF λ; ð21Þ 01 01 λ 0 −i −λ AðλÞ¼ UðΛðλÞÞ ¼ e 2P : ð22Þ 00

Relations (17)–(20) define the infinitesimal generators

fPþ;P1;P2;E1;E2;K3;J3gð23Þ

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fP−;F1;F2gð24Þ of the dynamical transformations. With these definitions, the light-front Poincar´e generators are related to components of the angular- tensor 0 1 0 −K1 −K2 −K3 B C B K1 0 J3 −J2 C Jμν ¼ B C ð25Þ @ K2 −J3 0 J1 A K3 J2 −J1 0 by

E1 ¼ K1 − J2 E2 ¼ K2 þ J1 F1 ¼ K1 þ J2 F2 ¼ K2 − J1: ð26Þ

The inverse relations are

1 1 1 1 1 1 1 2 2 2 1 2 − 2 2 1 − 1 K ¼ 2 ðE þ F Þ K ¼ 2 ðE þ F Þ J ¼ 2 ðE F Þ J ¼ 2 ðF E Þ: ð27Þ

F1 and F2 could be replaced by J1 and J2 as dynamical with itself generators. 2 μ 2 2 The evolution of a state or operator with initial data on W ¼ W Wμ ¼ M s : ð33Þ the light front is determined by the light-front Schrödinger equation The Pauli-Lubanski vector has components W0 ¼ P · JW¼ HJ þ P × K ð34Þ djψðxþÞi 1 i ¼ P−jψðxþÞi ð28Þ dxþ 2 or expressed in terms of the light-front Poincar´e generators or the light-front Heisenberg equations of motion Wþ ¼ PþJ · zˆ þðP × EÞ · zˆ; ð35Þ

þ 1 dOðx Þ i − þ ˆ − ˆ ˆ ˆ ¼ ½P ;OðxþÞ: ð29Þ W⊥ ¼ ðP z × F − P z × EÞ − ðz · KÞz × P; ð36Þ dxþ 2 2

− − −J zˆ − P F zˆ When P is a self-adjoint operator, the dynamics is well- W ¼ P · ð × Þ · : ð37Þ defined and given by the unitary one-parameter group (22). The Poincar´e Lie algebra has two polynomial invariants. In order to compare the spins of particles in different The mass squared is frames, it is useful to transform both particles to their rest frame using an arbitrary but fixed set of Lorentz trans- 2 − 2 formations parametrized by the four velocity of the particle. M ¼ PþP − P ; ð30Þ ⊥ The light-front spin is the measured in the particle’s or system’s rest frame when the particle or which gives the light-front dispersion relation system are transformed to the rest frame with the inverse light-front preserving boosts (14), M2 þ P2 P− ¼ : ð31Þ Pþ ðE × PÞ · zˆ Wþ s · zˆ ¼ J · zˆ − ¼ ; ð38Þ Pþ Pþ The other invariant is the inner product of the Pauli- þ þ Lubanski vector, s⊥ ¼ðW⊥ − P⊥W =P Þ=M: ð39Þ 1 μ μναβ The components of the light-front spin can also be W ¼ ϵ PνJαβ; ð32Þ 2 expressed directly in terms of Jμν,

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1 i ϵijk Λ−1 j Λ−1 k μν where s ¼ ð lf Þ μðP=MÞð lf Þ νðP=MÞJ ; ð40Þ 2 ∂ 1 2 2 ω p ðp ;p ;p Þ mð Þ where in (40) the P=M in the Lorentz boosts are operators. ¼ ∂ðpþ;p1;p2Þ pþ 1 p2 2 III. FIELDS ⊥ þ m þ þ − p · x ¼ − x þ p x þ p⊥ · x⊥ ð44Þ 2 þ Light-front free fields can be constructed from canonical p free fields by changing variables p → p˜ , where p˜ ≔ ðpþ;p1;p2Þ are the components of the light-front momen- and tum conjugate to x˜. The Fourier representation of a free sffiffiffiffiffiffiffiffiffiffiffiffiffi scalar field of mass m and its conjugate momentum ω p þ mð Þ operator are a˜ðp˜ Þ ≔ a˜ðp ; p⊥Þ¼aðpÞ : ð45Þ pþ Z 1 dp ϕðxÞ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðeip·xaðpÞþe−ip·xa†ðpÞÞ; 2π 3=2 It follows from ð Þ 2ωmðpÞ ð41Þ ½aðpÞ;a†ðp0Þ ¼ δðp − p0Þð46Þ Z rffiffiffiffiffiffiffiffiffiffiffiffiffi i ω ðpÞ πðxÞ¼− dp m ðeip·xaðpÞ − e−ip·xa†ðpÞÞ; and (44) and (45) that ð2πÞ3=2 2 ð42Þ ½aðp˜Þ;a†ðp˜ 0Þ ¼ δðp˜ − p˜ 0Þ: ð47Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 where ωmðpÞ ≔ m þ p is the energy of a The spectral conditions p particle of mass m, is its three-momentum, and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ≔−ω p 0 p x x · p mð Þx þ · . P ¼ H P3 ¼ M2 þ P2 P3 ≥ 0; ð48Þ Changing variables from the three momentum, p, to the light-front components, p˜ ¼ðpþ;p1;p2Þ, of the four M2 þ P2 momentum gives the light-front Fourier representation P− ¼ ≥ 0 ð49Þ of ϕðxÞ, Pþ Z 1 dpþθ pþ imply that it is possible to independently construct both ϕ pffiffiffiffiffiffiffiffiffið Þ p ip·x ˜ p˜ −ip·x ˜ † p˜ ðxÞ¼ 3 2 d ⊥ðe að Þþe a ð ÞÞ; p˜ † p˜ ϕ þ 0 x˜ ð2πÞ = 2pþ að Þ and a ð Þ from the field ðx ¼ ; Þ restricted to the light front. This can be done by computing the partial ð43Þ Fourier transform of the field on the light front,

Z − 1 þ − dx⊥dx þ þ ip x =2−ip⊥·x⊥ þ − ϕðx ¼ 0;p ; p⊥Þ¼ e ϕðx ¼ 0;x ; x⊥Þ : ð50Þ ð2πÞ3=2 2

The creation and annihilation operators can be read off of commutes with ϕðxþ ¼ 0; x˜Þ at all points on the light this expression, front must be a constant multiple of the identity, rffiffiffiffiffiffi pþ ½ϕðxþ ¼ 0; x˜Þ;O¼0 → O ¼ cI: ð53Þ ˜ p˜ θ þ ϕ þ 0 þ p að Þ¼ 2 ðp Þ ðx ¼ ;p ; ⊥Þ; ð51Þ rffiffiffiffiffiffi An important observation is that the only place where the pþ mass of the field appears is in the expression for the ˜ † p˜ θ þ ϕ þ 0 − þ p a ð Þ¼ 2 ðp Þ ðx ¼ ; p ; ⊥Þ: ð52Þ coefficient of xþ. When the field is restricted to the light front, xþ → 0, all information about the mass (and dy- Both operators are constructed directly from the field namics) disappears. restricted to the light front without constructing a gener- This is in contrast to the canonical case because the alized momentum operator. This means that ϕðxÞ restricted canonical transformation that relates free canonical fields to the light front defines an irreducible set of operators. It and their generalized momenta with different follows that any operator O on the Fock space that cannot be realized by a unitary transformation [48].

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When these fields are restricted to the light front, they Since the fields restricted to the light front are irreduc- become unitarily equivalent [10]. This is because dynami- ible, the canonical commutation relations are replaced by cal information that distinguishes the different representa- the commutator of the fields at different points on the tions is lost as a result of the restriction. light front,

Z −i þ −− − i þ −− − þθ þ 2p ðx y Þ − 2p ðx y Þ þ þ i dp ðp Þ e e ½ϕðx ¼ 0; x˜Þ; ϕðy ¼ 0; y˜Þ ¼ δðx⊥ − y⊥Þ¼ ð54Þ 2π pþ 2i Z þθ þ 1 i dp ðp Þ þ − − i − − − sin p ðx − y Þ δðx⊥ − y⊥Þ¼− ϵðx − y Þδðx⊥ − y⊥Þ: ð55Þ 2π pþ 2 4

Note that while the x− derivative gives

∂ þ þ i − − ½ϕðx ¼ 0; x˜Þ; ϕðy ¼ 0; y˜Þ ¼ − δðx − y Þδðx⊥ − y⊥Þ; ð56Þ ∂x− 2

∂−ϕðxÞ is not the canonical momentum. A more careful analysis shows that the interaction, while Interactions that preserve the light-front kinematic sym- formally leaving the light-front invariant, has singularities metry must commute with the kinematic subgroup. In at pþ ¼ 0, so the formal expressions for the interaction- particular, they must be invariant with respect to trans- dependent generators are not well-defined self-adjoint lations in the x− direction. This means that the interactions operators on the free-field Fock space. This is because must commuteP with Pþ, which is a kinematic operator. the interaction contains products of operator-valued þ þ Since P ¼ i Pi is kinematic, the vacuum of the field distributions which are not defined. Discussions of the theory is invariant with respect to these translations, nontriviality of the light-front vacuum and the associated independent of interactions. This requires that “zero-mode” problem, which is the subject of many papers, can be found in [7,49–55] and the references cited therein. ½Pþ;V¼0 Pþj0i¼0; ð57Þ The expressions for the Poincar´e generators are defined on the free-field Fock space if infrared and ultraviolet which implies cutoffs are introduced, but the cutoffs break the Poincar´e symmetry. The nontrivial problem is how to remove the PþVj0i¼VPþj0i¼0; ð58Þ cutoffs in a manner that recovers the Poincar´e symmetry. While the solution of this last problem is equivalent to where j0i is the free-field Fock vacuum. This means that the unsolved problem of giving a nonperturbative definition Vj0i is an eigenstate of Pþ with eigenvalue 0. Inserting a of the theory, cutoff theories should lead to good approx- complete set of intermediate states between V† and V in imations for observables on scales where the cutoffs are not 0 † 0 þ h jV Vj i, the absolutely continuous spectrum of pi expected to be important. cannot contribute to the sum over intermediate states pþ 0 because i ¼ is a set of measure 0. This means that IV. FORMAL LIGHT-FRONT FIELD DYNAMICS Vj0i¼j0ih0jVj0ið59Þ The Lagrangian density for a scalar field theory is 1 1 or interactions that preserve the kinematic symmetry leave μν 2 2 LðϕðxÞÞ ¼ − η ∂μϕðxÞ∂νϕðxÞ − m ϕðxÞ − VðϕðxÞÞ; the free-field Fock vacuum unchanged. 2 2 The observation that the interaction leaves the vacuum ð60Þ invariant implies that it is an operator on the free-field Fock space. The irreducibility of the light-front Fock algebra where ημν is the metric tensor with signature ð−; þ; þ; þÞ. means that the interaction can be expressed in terms of The action is fields in this algebra. The Poincar´e generators, defined by Z integrating the þ components of the Noether currents that A½V;ϕ¼ d4xLðϕðxÞÞ: ð61Þ come from Poincar´e invariance of the action over the light V front, are also linear in this interaction. This means that it should be possible to solve for the relativistic dynamics of Variations of the field that leave the action stationary satisfy the field on the light-front Fock space. the field equation,

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∂2ϕðxÞ ∂VðϕÞ where for the Lagrangian density (66) − ∇2ϕðxÞþm2ϕðxÞþ ¼ 0: ð62Þ ∂ðx0Þ2 ∂ϕðxÞ Tμν ¼ ημνLðϕðxÞÞ þ ∂μϕðxÞ∂νϕðxÞ; ð72Þ Changing to light-front variables, the partial derivatives become Mμαβ ¼ Tμαxβ − Tμβxα: ð73Þ

− ∂ ∂xþ ∂ ∂x ∂ ∂ ∂ Integrating the þ component of the conserved current over ∂0 ≔ ¼ þ ¼ þ ¼ ∂ þ∂−; ∂x0 ∂x0 ∂xþ ∂x0 ∂x− ∂xþ ∂x− þ the light front, assuming that the fields vanish on the ð63Þ boundary of the light front, gives the light-front conserved (independent of xþ) charges, ∂ ∂xþ ∂ ∂x− ∂ ∂ ∂ ∂3 ≔ ¼ þ ¼ − ¼ ∂ −∂−: d d ∂x3 ∂x3 ∂xþ ∂x3 ∂x− ∂xþ ∂x− þ Pμ ¼ 0 Jαβ ¼ 0; ð74Þ dxþ dxþ ð64Þ where Squaring and subtracting give Z − Z − dx⊥dx dx⊥dx μ ≔ þμ 0μ 3μ ∂2 ∂2 ∂ ∂ P 2 T ¼ 2 ðT þ T Þð75Þ − ¼ 4 : ð65Þ ∂ðx0Þ2 ∂ðx3Þ2 ∂xþ ∂x− and

It follows that the Lagrangian density (60) and the field Z − dx⊥dx equation in light-front variables have the forms αβ ≔ þαβ J 2 M Z 1 − dx⊥dx LðϕðxÞÞ ¼ 2∂−ϕðxÞ∂ ϕðxÞ − ∇⊥ϕðxÞ · ∇⊥ϕðxÞ 0α 3α β − 0β 3β α þ 2 ¼ 2 ððT þ T Þx ðT þ T Þx Þ: ð76Þ 1 − m2ϕðxÞ2 − VðϕðxÞÞ ð66Þ 2 These are the conserved four-momentum and angular- momentum tensors. They are independent of xþ and thus and can be expressed in terms of fields and derivatives of fields restricted to the light front. ∂ ϕ 2 2 Vð Þ In order to construct the Poincar´e generators, the first 4∂ ∂−ϕðxÞ − ∇⊥ϕðxÞþm ϕðxÞþ ¼ 0: ð67Þ þ ∂ϕðxÞ step is to express the þ component of the energy-momen- tum tensor and angular-momentum tensors in terms of Invariance of the action under infinitesimal changes in the fields on the light front, fields and coordinates þþ T ¼ 4∂−ϕðxÞ∂−ϕðxÞ; ð77Þ ϕðxÞ → ϕ0ðx0Þ¼ϕðxÞþδϕðxÞ xμ → x0μ þ δxμðxÞ; þi −2∂ ϕ ∂ ϕ ð68Þ T ¼ − ðxÞ i ðxÞ; ð78Þ

þ− 2 2 along with the field equation, leads to the conserved T ¼ ∇⊥ϕðxÞ · ∇⊥ϕðxÞþm ϕ ðxÞþ2VðϕðxÞÞ; ð79Þ Noether currents, þþ− − M ¼ 4∂−ϕðxÞ∂−ϕðxÞx − ð∇⊥ϕðxÞ · ∇⊥ϕðxÞ μ ∂μJ ðxÞ¼0; ð69Þ þ m2ϕ2ðxÞþ2VðϕðxÞÞxþ; ð80Þ where the Noether current is þþi i þ M ¼ 4∂−ϕðxÞ∂−ϕðxÞx þ 2∂−ϕðxÞ∂iϕðxÞx ; ð81Þ μ μν ∂LðϕÞ ν J ðxÞ¼Lη δxν þ ðδϕðxÞ − ∂νδx Þ: ð70Þ þ−i ∇ ϕ ∇ ϕ 2ϕ2 ∂ð∂μϕÞ M ¼ð ⊥ ðxÞ · ⊥ ðxÞþm ðxÞ i − þ 2VðϕðxÞÞx þ 2∂−ϕðxÞ∂iϕðxÞx ; ð82Þ The Noether currents associated with translational and Lorentz invariance of the action are the energy-momentum, þij j i M ¼ −2∂−ϕðxÞ∂iϕðxÞx þ 2∂−ϕðxÞ∂jϕðxÞx : ð83Þ Tμν, and angular-momentum, Mμαβ, tensors, The Poincar´e generators are constructed by integrating ∂ μν 0 ∂ μαβ 0 μT ¼ μM ¼ ; ð71Þ these operators over the light front,

096004-8 WAVELET REPRESENTATION OF LIGHT-FRONT QUANTUM … PHYS. REV. D 101, 096004 (2020)

Z − 2 dx d x⊥ þ 4 ∂ ϕ ∂ ϕ P ¼ 2 − ðxÞ − ðxÞ; ð84Þ

Z − 2 dx d x⊥ i −2 ∂ ϕ ∂ ϕ P ¼ 2 − ðxÞ i ðxÞ; ð85Þ

Z − 2 − dx d x⊥ ∇ ϕ ∇ ϕ 2ϕ2 2 ϕ P ¼ 2 ð ⊥ ðxÞ · ⊥ ðxÞþm ðxÞþ Vð ðxÞÞ; ð86Þ

Z − 2 dx d x⊥ þ− 4∂ ϕ ∂ ϕ − − ∇ ϕ ∇ ϕ 2ϕ2 2 ϕ þ J ¼ 2 ð − ðxÞ − ðxÞx ð ⊥ ðxÞ · ⊥ ðxÞþm ðxÞþ Vð ðxÞÞx Þ; ð87Þ

Z − 2 dx d x⊥ þi 4∂ ϕ ∂ ϕ i 2∂ ϕ ∂ ϕ þ J ¼ 2 ð − ðxÞ − ðxÞx þ − ðxÞ i ðxÞx Þ; ð88Þ

Z − 2 dx d x⊥ −i ∇ ϕ ∇ ϕ 2ϕ2 2 ϕ i 2∂ ϕ ∂ ϕ − J ¼ 2 ðð ⊥ ðxÞ · ⊥ ðxÞþm ðxÞþ Vð ðxÞÞx þ − ðxÞ i ðxÞx Þ; ð89Þ

Z − 2 dx d x⊥ ij −2∂ ϕ ∂ ϕ j 2∂ ϕ ∂ ϕ i J ¼ 2 ð − ðxÞ i ðxÞx þ − ðxÞ j ðxÞx Þ: ð90Þ

For free fields, these operators can be expressed in terms of the light-front creation and annihilation operators (51) and (52) using the identities Z Z − 2 þ þ 2 dx d x⊥ θðp Þdp d p⊥ ∶ϕðxÞϕðxÞ ≔ a˜ †ðp˜ Þa˜ðp˜ Þ; ð91Þ 2 pþ

Z − 2 Z dx d x⊥ 1 ∶∂ ϕ ∂ ϕ ≔ θ þ þ 2 ˜ † ˜ þ ˜ ˜ 2 − ðxÞ − ðxÞ 4 ðp Þdp d p⊥a ðpÞp aðpÞ; ð92Þ

Z − 2 Z dx d x⊥ 1 ∶∂ ϕ ∂ ϕ ≔− θ þ þ 2 ˜ † ˜ i ˜ ˜ 2 − ðxÞ i ðxÞ 2 ðp Þdp d p⊥a ðpÞp aðpÞ; ð93Þ Z Z − 2 þ þ 2 dx d x⊥ θðp Þdp d p⊥ ∶∂ ϕðxÞ∂ ϕðxÞ ≔ a˜ †ðp˜ ÞðpiÞ2a˜ðp˜ Þ: ð94Þ 2 i i pþ

Using (91)–(94) in (84)–(90) gives the following expressions for the Poincar´e generators for a free field in terms of the light- front creation and annihilation operators: Z þ þ 2 þ † þ P ¼ dp d p⊥θðp Þa˜ ðp˜ Þp a˜ðp˜ Þ; ð95Þ Z i þ 2 þ † i P ¼ dp d p⊥θðp Þa˜ ðp˜ Þp a˜ðp˜ Þ; ð96Þ Z p2 2 − þ 2 þ † ⊥ þ m P ¼ dp d p⊥θðp Þa˜ ðp˜ Þ a˜ðp˜ Þ; ð97Þ pþ Z ∂ p2 2 þ− þ 2 þ † þ þ ⊥ þ m J ¼ dp d p⊥θðp Þa˜ ðp˜ Þ p −2i − x a˜ðp˜ Þ; ð98Þ ∂pþ pþ Z ∂ þi þ 2 þ † þ i þ J ¼ dp d p⊥θðp Þa˜ ðp˜ Þ p i − p x a˜ðp˜ Þ; ð99Þ ∂pi

096004-9 W. N. POLYZOU PHYS. REV. D 101, 096004 (2020) Z p2 2 ∂ ∂ −i þ 2 þ † ⊥ þ m i J ¼ dp d p⊥θðp Þa˜ ðp˜ Þ i − 2p −i a˜ðp˜ Þ; ð100Þ pþ ∂pi ∂pþ Z ∂ ∂ ij þ 2 þ † j i J ¼ dp d p⊥θðp Þa˜ ðp˜ Þ p −i − p −i a˜ðp˜ Þ: ð101Þ ∂pi ∂pj

Since these are independent of xþ, the expressions with an explicit xþ dependence can be evaluated at xþ ¼ 0. These expressions lead to the following identifications:

Jþ− ¼ −2K3 Jþ1 ¼ K1 − J2 ¼ E1 Jþ2 ¼ K2 þ J1 ¼ E2; ð102Þ

J−1 ¼ K1 þ J2 ¼ F1 J−2 ¼ K2 − J1 ¼ F2: ð103Þ

where V. WAVELET BASIS In this section, the multiresolution basis that is used to pffiffiffi ≔ 2 2 ≔ − 1 represent the irreducible algebra of fields on the light front DfðxÞ fð xÞ and TfðxÞ fðx Þð105Þ is introduced. Wavelets provide a natural means for exactly decomposing a field into independent discrete degrees of are unitary scale transformations and translations. The fixed freedom labeled by volume and resolution. In this repre- point, sðxÞ, is a linear combination of a weighted sum of sentation, there are natural truncations that eliminate translates of itself on a smaller scale by a factor of 2. The degrees of freedom associated with volumes and resolu- weights hl are constant coefficients chosen, so sðxÞ satisfies tions that are expected to be unimportant in modeling a given reaction. Z m n While there are many different types of wavelets, this T sðxÞT sðxÞ¼δmn and application uses Daubechies [37,38] L ¼ 3 wavelets. These X k k n are used to generate an orthonormal basis of functions with x ¼ cnT sðxÞk

096004-10 WAVELET REPRESENTATION OF LIGHT-FRONT QUANTUM … PHYS. REV. D 101, 096004 (2020) Z Sk ⊂ Skþ1: ð114Þ sðxÞdx ¼ 1: ð107Þ This means that the lower resolution subspaces are sub- Because sðxÞ is fractal valued, it cannot be represented in spaces of the higher resolution subspaces. The orthogonal terms of elementary functions; however, it can be exactly complement of Sk in Skþ1 is called Wk, calculated at all dyadic rationals using the renormalization group equation (104). It can also be approximated by Skþ1 ¼ Sk ⊕ Wk: ð115Þ iterating the renormalization group equation starting with a seed function satisfying (107). The evaluation of sðxÞ is not k kþ1 k Since W ⊂ S , orthonormal basis functions wnðxÞ in necessary because most of the integrals that are needed in k kþ1 W are also linear combinations of the sn ðxÞ. These field theory applications can be evaluated exactly using the functions are defined by renormalization group equation. The integrals can be expressed in terms of solutions of finite linear systems k k n wnðxÞ¼D T wðxÞ; ð116Þ of equations involving the numerical weights hl in Table I. The next step in constructing the wavelet basis is to where wðxÞ is the “mother wavelet” defined by construct subspaces of L2ðRÞ with different resolutions defined by 2XL−1 ≔ l  X X  wðxÞ glDT sðxÞ; ð117Þ k k n 2 l¼0 S ≔ fðxÞjfðxÞ¼ cnD T sðxÞ jcnj < ∞ : n n and the coefficients gl are related to the weight coefficients ð108Þ hl by

The resolution is determined by the width of the support of l gl ¼ð−Þ h2L−1−l 0 ≤ l ≤ 2L − 1: ð118Þ these functions, which for L ¼ 3,is5 × 2−k. The functions The orthonormal basis functions wkðxÞ for Wk are called k k n n snðxÞ ≔ D T ðxÞsðxÞ; ð109Þ k wavelets. Since wnðxÞ are finite linear combinations of the skþ1ðxÞ, they have the same number of derivatives as sðxÞ. for fixed k, are orthonormal, have compact support on n wkðxÞ also has the same support as skðxÞ. Finally, it follows 2−k 2−k 5 n n ½ n; ðn þ Þ, satisfy from (106) that Z Z k −k=2 snðxÞdx ¼ 2 ð110Þ m k x wnðxÞ¼00≤ m

Sk 2 d d which indicates that the resolution of can be increased to TD ¼ DT and D ¼ 2D : ð112Þ − − dx dx 2 k n by including additional basis functions in the sub- spaces fWkþn−1; …; Wkg. This can be continued to arbi- k n Applying D T to the renormalization group equation, trarily fine resolutions to get all of L2ðRÞ, using (112), gives 2 R Sk ⊕∞ Wkþn ⊕∞ Wn L ð Þ¼ n¼0 ¼ n¼−∞ : ð121Þ 2XL−1 2XL−1 sk x h Dkþ1T2nþls x h skþ1 x ; 113 nð Þ¼ l ð Þ¼ l 2nþlð Þ ð Þ Since all of the subspaces are orthogonal, an orthonormal l¼0 l¼0 basis for L2ðRÞ consists of Sk which expresses every basis element of as a finite linear ∞ ∞ 1 k ∪ m combination of basis elements of Skþ or fsnðxÞgn¼−∞ fwn ðxÞgn¼−∞;m¼k ð122Þ

096004-11 W. N. POLYZOU PHYS. REV. D 101, 096004 (2020) for any fixed starting resolution 2−k or VI. WAVELET REPRESENTATION OF QUANTUM FIELDS wk x ∞ : f nð Þgk;n¼−∞ ð123Þ In what follows, the basis (122) is used to expand quantum fields restricted to a light front. It is useful to The basis (123) includes functions of arbitrarily large think of the starting scale 2−k in (122) as the resolution that support, while the basis (122) consists of functions with is relevant to experimental measurements. The higher 2−l 2 − 1 ≥ support in intervals of width ð L Þ for l k. resolution degrees of freedom are used to represent shorter 3 The basis (122) is used with L ¼ Daubechies wavelets distance degrees of freedom that couple to experimental- [37,38]. Locally finite linear combinations of the L ¼ 3 k scale degrees of freedom. scaling functions, snðxÞ, can be used to pointwise represent The basis (122) can be used to get a formally exact l polynomials of degree 2. The wavelets, wnðxÞ, are orthogo- representation of the field operators of the form nal to these polynomials. The L ¼ 3 basis functions have one continuous derivative.

X Z ˜ þ þ − 1 2 þ 2 − − 1 2 ˜ þ ϕðx;x Þ ≔ ϕlmnðx Þξlðx Þξmðx Þξnðx Þ where ϕlmnðx Þ¼ d x⊥dx ξlðx Þξmðx Þξnðx Þϕðx;x Þ; ð124Þ

where ξl are the basis functions, functions. This follows from the kinematic covariance of the field ξ ∈ k ∞ ∪ m ∞ lðxÞ fsnðxÞgn¼−∞ fwn ðxÞgn¼−∞;m¼k: ð125Þ † ˜ UðΛ;aÞϕðx˜;xþ ¼ 0ÞU ðΛ;aÞ¼ϕððΛ x˜ þa˜Þ;xþ ¼ 0Þ In what follows, the shorthand notation is used, ð128Þ X X X X − 1 2 ξnðx˜Þ ≔ ξ ðx Þξ ðx Þξ ðx Þ ¼ : n− n1 n2 for ðΛ;aÞ in the light-front kinematic subgroup. Using the n n− n1 n2 discrete representation of the field on both sides of this ð126Þ equation gives the identity X With this notation, (124) has the form þ † UðΛ;aÞ ϕnðx ¼ 0Þξnðx˜ÞU ðΛ;aÞ X n þ þ X ϕðx˜;x Þ ≔ ϕnðx Þξnðx˜Þ; ð127Þ þ ˜ ˜ ˜ n ¼ ϕnðx ¼ 0ÞξnðΛ x þaÞ: ð129Þ n which gives a discrete representation of the field as a linear combination of discrete operators with different resolutions This shows that kinematic transformations can be com- on the light front. puted exactly by transforming the arguments of the Each discrete field operator, ϕnð0Þ, is associated with a expansion functions. degree of freedom that is localized in a given volume on the The transformation property of the discrete field oper- light-front hyperplane. In addition, there are an infinite ators restricted to a light front follows from the orthonor- number of these degrees of freedom that are localized in mality of the basis functions (129), any open set on the light front. X þ † þ ˜ While the fields are operator valued distributions, that UðΛ;aÞϕnðx ¼ 0ÞU ðΛ;aÞ¼ ϕmðx ¼ 0ÞUmnðΛ;a˜Þ; does not preclude the existence of operators constructed by m smearing with functions that have only one derivative. Note ð130Þ that the support condition implies that the Fourier transform of the basis functions are entire. where the matrix Z ˜ 2 − ˜ VII. KINEMATIC POINCARÉ UmnðΛ; a˜Þ ≔ d x⊥dx ξmðΛ x˜ þa˜Þξnðx˜Þð131Þ TRANSFORMATIONS OF FIELDS IN THE WAVELET REPRESENTATION is a discrete representation of the light-front kinematic Since this representation is formally exact, kinematic subgroup. Poincar´e transformations on the algebra of fields restricted This identity implies that in the wavelet representation to the light front can be computed by acting on the basis kinematic Lorentz transformations on the fields can be

096004-12 WAVELET REPRESENTATION OF LIGHT-FRONT QUANTUM … PHYS. REV. D 101, 096004 (2020) computed either by transforming the arguments of the basis From (133) and (137), it follows that the inner product functions or by transforming the discrete field operators. above is a linear combination of two-point functions in the þ discrete fields, ϕnðx ¼ 0Þ. ξ VIII. STATES IN THE WAVELET The basis functions mðxÞ have compact support which REPRESENTATION implies that their Fourier transforms are entire functions of the light-front momenta p˜ . This means that they cannot Because the algebra of free fields restricted to the light vanish in a neighborhood of pþ ¼ 0; however, they can front is irreducible and kinematically invariant interactions have isolated zeroes at pþ ¼ 0. For the wavelet basis leave the Fock vacuum unchanged, the Hilbert space for the l functions, wmðxÞ, the vanishing (119) of the first three dynamical model can be generated by applying functions of 3 þ moments of the L ¼ wavelets implies that the discrete field operators, ϕnðx ¼ 0Þ, to the Fock Z vacuum. 1 l þ l − − Smeared light-front fields can be represented in the w˜ ðp Þ þ ¼ w ðx Þdx ¼ 0 m p ¼0 2π1=2 m discrete representation as linear combinations of the dis- Z 1 crete field operators, d l þ − l − − w˜ ðp Þ þ 0 ¼ − x w ðx Þdx ¼ 0; ð138Þ þ m p ¼ 2π1=2 m X Z dp ϕ þ 0 ≔ 2 − x˜ ξ x˜ ϕ þ 0 ðf; x ¼ Þ d x⊥dx fð Þ nð Þ nðx ¼ Þ: Z n 2 1 d l þ − 2 l − − w˜ ðp Þ þ 0 ¼ − ðx Þ w ðx Þdx ¼ 0: ð132Þ d2pþ m p ¼ 2π1=2 m ð139Þ Equation (132) can be expressed as X þ þ Since the Fourier transforms are entire, this means that they ϕðf; x ¼ 0Þ¼ fnϕnðx ¼ 0Þ; ð133Þ l þ þ 3 l þ l þ n have the form w˜ mðp Þ¼ðp Þ fmðp Þ where fmðp Þ is k entire. For the scaling function basis functions, smðxÞ, the where normalization condition (111) gives Z Z 2 − 1 1 fn ≔ d x⊥dx fðx˜Þξnðx˜Þ: ð134Þ k þ k − − −k=2 s˜ ðp Þ þ 0 ¼ s ðx Þdx ¼ 2 ≠ 0: m p ¼ 2π1=2 m 2π1=2 States can be expressed as polynomials in the smeared ð140Þ fields applied to the light-front Fock vacuum X These results imply that ϕ 0 ϕ 0 0 cm1mn ðfm1 ; Þ ðfmn ; Þj i: ð135Þ þ þ h0jϕmðx ¼ 0Þϕnðx ¼ 0Þj0ið141Þ This representation can be reexpressed as a linear combi- nation of products of discrete fields applied to the Fock is singular if both basis functions have scaling functions in vacuum the x− variable, but are finite if at least one of the basis X − functions has a wavelet in the x variable. cm m ϕm ð0Þϕm ð0Þj0i: ð136Þ 1 n 1 n Since the smearing functions, fðp˜Þ, should all vanish at pþ ¼ 0, the discrete representation will involve linear The inner product of two vectors of this form is a linear combinations of wavelets and scaling functions whose combination of n-point functions. For the free-field alge- Fourier transforms all vanish at pþ ¼ 0. In computing bra, the n-point functions are products of two-point these quantities, the linear combinations of scaling func- functions. The two-point functions have the form tions should be summed before performing the integrals. Z þ þ 2 This can alternatively be done by including a cutoff near θðp Þdp d p⊥ 0 ϕ f; 0 ϕ g; 0 0 f˜ −p˜ g˜ p˜ : pþ ¼ 0, doing the integrals, adding the contributions, and h j ð Þ ð Þj i¼ 2 þ ð Þ ð Þ p then letting the cutoff go to zero. ð137Þ IX. DYNAMICS This integral is logarithmically divergent if the Fourier transforms of the smearing functions do not vanish at The dynamical problem involves diagonalizing P− on pþ ¼ 0. Since pþ ¼ 0 corresponds to infinite three- the free-field Fock space or solving the light-front momentum, this requirement is that the smearing functions Schrödinger (28) or Heisenberg equations (29). The two need to vanish for infinite three-momentum. dynamical equations can be put in integral form

096004-13 W. N. POLYZOU PHYS. REV. D 101, 096004 (2020) Z þ þ i x fields, ϕnð0Þ, in the light-front Fock algebra with x - ΨðxþÞj0i¼Ψðxþ ¼ 0Þj0i − ½P−; Ψðxþ0Þj0idxþ0 2 0 dependent coefficients. What is needed to perform this iteration are the initial operators Ψðxþ ¼ 0Þ and Oðxþ ¼ 0Þ ð142Þ expressed as polynomials in the ϕnð0Þ, the expression for − ϕ 0 or P as a polynomial in the nð Þ, and an expression for the commutator, ½ϕmð0Þ; ϕnð0Þ, of the discrete fields on the Z i xþ light front. OðxþÞ¼Oðxþ ¼ 0Þþ dxþ0½P−;Oðxþ0Þ; ð143Þ 2 0 X. THE COMMUTATOR where Ψðxþ ¼ 0Þ and Oðxþ ¼ 0Þ are operators in the light- front Fock algebra. It follows from (55) that the commutator of the discrete The formal iterative solution of these equations has the fields is structure of a linear combination of products of discrete

Z i − − − − − − ½ϕmð0Þ; ϕnð0Þ ¼ − δ δ ξ − ðx Þϵðx − y Þξ − ðy Þdx dy : ð144Þ 4 m1n1 m2n2 m n

Unlike the inner product, the commutator is always finite Applying DkTn to the renormalization group equation − − since both ξm− ðx Þ and ξn− ðy Þ have compact support. and the expression for wðxÞ gives The commutator (144) can be computed exactly using the renormalization group equations. The computation Xl − DkTnsðxÞ¼ h Dkþ1T2nþlsðxÞð145Þ involves three steps. The first step is to express ξm− ðx Þ l − L¼0 and ξn− ðy Þ as linear combinations of scaling functions on a sufficiently fine common scale. The second step is to and change variables, so the commutator is expressed as a linear combination of commutators involving integer translates of Xl − k n kþ1 2nþl the fixed-point solution sðx Þ of the renormalization group D T wðxÞ¼ glD T sðxÞ: ð146Þ equation. The last step is to use the renormalization group L¼0 equation to construct a finite linear system relating the k k − These equations express snðxÞ and wnðxÞ as linear combi- commutators involving integer translates of the sðx Þ. kþ1 nations of the sn ðxÞ,

2XL−1 2nXþ2L−1 2nXþ2L−1 k kþ1 kþ1 kþ1 ≔ snðxÞ¼ hls2nþlðxÞ¼ hm−2nsm ðxÞ¼ Hn;msm ðxÞ where Hn;m hm−2n ð147Þ l¼0 m¼2n m¼2n and

2XL−1 2nXþ2L−1 2nXþ2L−1 k kþ1 kþ1 kþ1 ≔ wnðxÞ¼ gls2nþlðxÞ¼ gm−2nsm ðxÞ¼ Gn;msm ðxÞ where Gn;m gm−2n: ð148Þ l¼0 m¼2n m¼2n

X While the matrices H ; and G ; are formally infinite, for k m kþm n m n m sn ¼ Hnlsl ; ð150Þ each fixed n, these are 0 unless 2n ≤ m ≤ 2L − 1 þ 2n. l Using powers of the matrices, X k m−1 kþm X wn ¼ Hnt Gtlsl ; ð151Þ m ≔ lt Hnl Hnk1 Hk1k2 Hkml; ð149Þ where the sums in (150) and (151) are finite. Using these and Gnl the basis function can be represented as finite linear identities, all of the integrals can be reduced to finite linear combinations of finer resolution scaling functions combinations of integrals involving a pair of scaling

096004-14 WAVELET REPRESENTATION OF LIGHT-FRONT QUANTUM … PHYS. REV. D 101, 096004 (2020) Z k 2k=2 2k − functions, snðxÞ¼ sð x nÞ, on a common fine I½n ≔ sðx− þ nÞϵðx− − y−Þsðy−Þdx−dy−: ð157Þ scale, 2−k. What remains is linear combinations of products of integrals of the form I½n can be expressed as a difference of two integrals, Z Z Z  Z x− ∞ − − − − − − k − − k − − − I½n¼ sðx þ nÞ sðy Þ sðy Þ dx dy ; smðxÞϵðx −y Þsnðy Þdx dy −∞ x− Z ð158Þ ¼ 2k=2sð2kx− −mÞϵðx− −y−Þ2k=2sð2ky− −nÞdx−dy−: while the normalization condition (107) gives ð152Þ Z Z Z  x− ∞ sðx− þnÞ sðy−Þþ sðy−Þ dx−dy− ¼ 1: ð159Þ Changing variables −∞ x−

y−0 ¼ 2ky− − n; x−0 ¼ 2kx− − n; ð153Þ Adding (158) and (159) gives Z Z x− and noting I½n¼2 sðx− þ nÞ sðy−Þdx−dy− − 1: ð160Þ −∞ ϵðx− − y−Þ¼ϵð2kx− − 2ky−Þ; ð154Þ If the support of sðx− þ nÞ is to the right of the support of − − this becomes sðy Þ, the integral is 1 while if the support of sðx þ nÞ is − −1 Z to the left of the support of sðy Þ the integral is . Thus, for the L ¼ 3 basis functions, 2−ksðx0− −mÞϵðx0− −y0−Þsðy0− −nÞdx0−dy0− ¼; ð155Þ 8 1 ≤ −5 Z < n ¼ −4 ≤ ≤ 4 2−ksðx0− þn−mÞϵðx0− −y0−Þsðy0−Þdx0−dy0− ¼2−kI½n−m; I½n¼: I½n n : ð161Þ −1 n ≥ 5 ð156Þ The I½n for n ∈ ½−4; 4 are related by the renormalization where group equations Z I½n¼ sðx− þ nÞϵðx− − y−Þsðy−Þdx−dy− ¼; ð162Þ X Z − − − − − − 2 hlhk sð2x þ 2n − lÞϵðx − y Þsð2y − kÞdx dy ¼; ð163Þ Z 1 X 2 − 2 − ϵ 2 − − 2 − 2 − − 2 −2 − 2 hlhk sð x þ n lÞ ð x y Þsð y kÞ dx dy ¼; ð164Þ Z 1 X − 2 − ϵ − − − − − − − 2 hlhk sðx þ n lÞ ðx y Þsðy kÞdx dy ¼; ð165Þ Z 1 X − 2 − ϵ − − − − − − 2 hlhk sðx þ n l þ kÞ ðx y Þsðy Þdx dy ¼; ð166Þ 1 X 1 X 1 X 2 − 2 hlhkI½ n þ k l¼2 hmþl−2nhlI½m¼4 am−2nI½m; ð167Þ where X5 ≔ 2 − 5 ≤ ≤ 5 an hlhlþn n : ð168Þ l¼0

The numbers an will appear again. The an are rational numbers [56–58]. For L ¼ 3, the nonzero an are 75 25 3 2 − a0 ¼ a1 ¼ a−1 ¼ 64 a3 ¼ a−3 ¼ 128 a5 ¼ a−5 ¼ 128 : ð169Þ

096004-15 W. N. POLYZOU PHYS. REV. D 101, 096004 (2020)

The 9 × 9 matrix Amn ≔ an−2m (−4 ≤ m; n ≤ 4) has the supports that are sufficiently separated, the integrals vanish 1 1 1 1 9 following rational eigenvalues λ ¼ 2; 1; 2 ; 4 ; 8 ; 16 ; 32 ; because the moments of wavelets vanish. This will also be 9 − 64, so it is invertible. true of linear combinations of scaling functions that þ 0 The nontrivial I½n are solutions of the linear system, represent functions that vanish at p ¼ .

X4 XI. POINCARÉ GENERATORS AmnI½n¼dm; ð170Þ n¼−4 The other quantity needed to formulate the dynamics is an expression for P− or one of the other dynamical where Poincar´e generators expressed in terms of operators in the irreducible algebra. Since the generators are conserved − dm ¼ a5−2m a−5−2m: ð171Þ Noether charges, they are independent of xþ, so the generators can be expressed in terms of fields on the light The solution of (170) is front. The discrete representations of the generators can be 0 1 constructed by replacing the fields on the light front by the −3.34201389e þ 00;n¼ −4 B C discrete representation (124), (127) of the fields. The B 8.33333333e þ 00;n¼ −3 C B C integrals over the light front become integrals over products B C B −1.79796007e þ 01;n¼ −2 C of basis functions and their derivatives. This section B C discusses the computation of these integrals using renorm- B 1.94444444e þ 01;n¼ −1 C B C alization group methods. B C I½n¼B 0.00000000e − 00;n¼ 0 C: ð172Þ A scalar ϕ4ðxÞ theory is used for the purpose of B C B −1 94444444 01 1 C illustration. In this case, the problem is to express all of B . e þ ;n¼ C B C the generators as linear combinations of products of B 1.79796007e þ 01;n¼ 2 C B C discrete fields. @ A −8.33333333e þ 00;n¼ 3 The construction of the Poincar´e generators from 3.34201389e þ 00;n¼ 4 Noether’s theorem was given in Sec. IV. Using the discrete representation of fields, the light-front Poincar´e generators While (172) is a numerical solution, the exact solution is (84)–(90) have the following forms: rational since both Amn and dn are rational. X þ þ This solution, along with (161), can be used to construct P ¼ ∶ϕmð0Þϕnð0Þ∶Pm;n; ð174Þ the commutator of any of the discrete field operators using mn (145)–(155). The general structure of the commutators is where Z ϕ 0 ϕ 0 ½ mð Þ; nð Þ ¼ Cm;n þ − 2 ˜ ˜ Pm;n ≔ 2 dx d x⊥∂−ξmðxÞ∂−ξnðxÞ; ð175Þ ¼ðscale factorsÞ×ðpowers of H;GÞ×I½n: X 173 i i ð Þ P ¼ ∶ϕmð0Þϕnð0Þ∶Pm;n; ð176Þ mn Note that while this commutator looks very nonlocal, if the scaling functions in (144) are replaced by wavelets with where

Z i − 2 ˜ ˜ Pm;n ≔− dx d x⊥∂−ξnðxÞ∂iξmðxÞ; ð177Þ X X − ∶ϕ 0 ϕ 0 ∶ − ∶ϕ 0 ϕ 0 ϕ 0 ϕ 0 ∶ − P ¼ mð Þ nð Þ Pm;n þ n1 ð Þ n2 ð Þ n3 ð Þ n4 ð Þ Pn1;n2;n3;n4 ; ð178Þ mn n1n2n4n4 where Z 1 1 − ≔ − 2 ∇ ξ x˜ ∇ ξ x˜ 2ξ x˜ ξ x˜ Pm;n dx d x⊥ 2 ⊥ mð Þ · ⊥ nð Þþ2 m mð Þ nð Þ ð179Þ and

096004-16 WAVELET REPRESENTATION OF LIGHT-FRONT QUANTUM … PHYS. REV. D 101, 096004 (2020) Z − ≔ λ − 2 ξ x˜ ξ x˜ ξ x˜ ξ x˜ Pn1;n2;n3;n4 dx d x⊥ n1 ð Þ n2 ð Þ n3 ð Þ n4 ð Þ; ð180Þ X X 3 ∶ϕ 0 ϕ 0 ∶ 3 ∶ϕ 0 ϕ 0 ϕ 0 ϕ 0 ∶ 3 K ¼ mð Þ nð Þ Km;n þ n1 ð Þ n2 ð Þ n3 ð Þ n4 ð Þ Kn1;n2;n3;n4 ; ð181Þ mn n1n2n4n4 where Z 1 1 3 ≔ − 2 2 −∂ ξ x˜ ∂ ξ x˜ − þ∇ ξ x˜ ∇ ξ x˜ − 2 þξ x˜ ξ x˜ Km;n dx d x⊥x − mð Þ − nð Þ 2 x ⊥ mð Þ · ⊥ nð Þ 2 m x mð Þ nð Þ ð182Þ and Z 3 ≔−λ − 2 þξ x˜ ξ x˜ ξ x˜ ξ x˜ Kn1;n2;n3;n4 dx d x⊥x n1 ð Þ n2 ð Þ n3 ð Þ n4 ð Þ: ð183Þ

Setting xþ ¼ 0, this becomes Z 3 → 2 − 2 −∂ ξ x˜ ∂ ξ x˜ ; 3 → 0 Km;n dx d x⊥x − mð Þ − nð Þ Kn1;n2;n3;n4 : ð184Þ

For the remaining generators, X 1 1 E ¼ ∶ϕmð0Þϕnð0Þ∶Em;n; ð185Þ mn where Z Z 1 − 2 1 ˜ ˜ ˜ ˜ þ 1 ˜ ˜ Em;n ≔ dx d x⊥ð2x ∂−ξmðxÞ∂−ξnðxÞþ∂−ξmðxÞ∂1ξnðxÞx Þ → 2 x ∂−ξmðxÞ∂−ξnðxÞ; ð186Þ X 2 2 E ¼ ∶ϕmð0Þϕnð0Þ∶Em;n; ð187Þ mn where Z Z 2 − 2 2 ˜ ˜ ˜ ˜ þ − 2 2 ˜ ˜ Em;n ≔ dx d x⊥ð2x ∂−ξmðxÞ∂−ξnðxÞþ∂−ξmðxÞ∂2ξnðxÞx Þ → 2 dx d x⊥x ∂−ξmðxÞ∂−ξnðxÞ; ð188Þ X X 1 ∶ϕ 0 ϕ 0 ∶ 1 ∶ϕ 0 ϕ 0 ϕ 0 ϕ 0 ∶ 1 F ¼ mð Þ nð Þ Fm;n þ n1 ð Þ n2 ð Þ n3 ð Þ n4 ð Þ Fn1;n2;n3;n4 ; ð189Þ mn n1n2n3n4 where Z 1 1 1 ≔ − 2 1∇ ξ ∇ ξ 1 2ξ x˜ ξ x˜ −∂ ξ x˜ ∂ ξ x˜ Fm;n dx d x⊥ 2 x ⊥ kðxÞ · ⊥ lðxÞþ2 x m kð Þ lð Þþx − kð Þ 1 lð Þ ð190Þ and Z 1 ≔ λ − 2 1ξ x˜ ξ x˜ ξ x˜ ξ x˜ Fn1;n2;n3;n4 dx d x⊥x n1 ð Þ n2 ð Þ n3 ð Þ n4 ð Þ; ð191Þ X X 2 ∶ϕ 0 ϕ 0 ∶ 2 ∶ϕ 0 ϕ 0 ϕ 0 ϕ 0 ∶ 2 F ¼ mð Þ nð Þ Fm;n þ n1 ð Þ n2 ð Þ n3 ð Þ n4 ð Þ Fn1;n2;n3;n4 ; ð192Þ mn n1n2n4n4 where

096004-17 W. N. POLYZOU PHYS. REV. D 101, 096004 (2020) Z 1 1 2 ≔ − 2 2∇ ξ ∇ ξ 2 2ξ x˜ ξ x˜ −∂ ξ x˜ ∂ ξ x˜ Fm;n dx d x⊥ 2 x ⊥ kðxÞ · ⊥ lðxÞþ2 x m kð Þ lð Þþx − kð Þ 2 lð Þ ð193Þ and Z 2 ≔ λ − 2 2ξ x˜ ξ x˜ ξ x˜ ξ x˜ Fn1;n2;n3;n4 dx d x⊥x n1 ð Þ n2 ð Þ n3 ð Þ n4 ð Þ: ð194Þ

All of these operators have the structure of linear combinations of normal products of discrete fields evaluated at xþ ¼ 0 þ i − − 3 3 i i i times constant coefficients, Pn1;n2 ;Pn1;n2 ;Pn1;n2 ;Pn1;n2;n3;n4 ;Kn1;n2 ;Jn1;n2 ;En1;n2 ;Fn1;n2 ;Fn1;n2;n3;n4 , which are integrals involving products of basis functions and their derivatives. The three-dimensional integrals that need to be evaluated to compute these coefficients are products of 3 one-dimensional integrals that have one of the following eight forms: Z Z Z

dxξmðxÞξnðxÞ dx∂xξmðxÞξnðxÞ dx∂xξmðxÞ∂xξnðxÞ; ð195Þ Z Z Z

dxxξmðxÞξnðxÞ dxx∂xξmðxÞξnðxÞ dxx∂xξmðxÞ∂xξnðxÞ; ð196Þ Z Z ξ ξ ξ ξ ξ ξ ξ ξ dx n1 ðxÞ n2 ðxÞ n3 ðxÞ n4 ðxÞ dxx n1 ðxÞ n2 ðxÞ n3 ðxÞ n4 ðxÞ: ð197Þ

In what follows, it is shown how all of these integrals can be After expressing the integrals in terms of scaling l computed using the renormalization group equation (104). functions, snðxÞ, and their derivatives, the one-dimensional The integrals (195)–(197) are products of basis functions integrals (195)–(197) can be expressed in terms of integrals 2−k which may be scaling functions with scale or wavelets involving products of the snðxÞ. A variable change x → − − of scale 2 k l for l ≥ 0. The same methods that were used x0 ¼ 2−lx can be used to express all of the integrals in terms in the computation of the commutator function, (145)– of translates of the original fixed point sðxÞ. The scale (151), can be used to express the integrals (195)–(197) as factors for each type of integral are shown as follows: linear combinations of integrals involving scaling functions on a common scale fine scale, 2−l.

Z l l dxsmðxÞsnðxÞ¼δmn; ð198Þ Z Z l l l 0 dx∂xsmðxÞsnðxÞ¼2 dxs ðxÞsn−mðxÞ; ð199Þ Z Z l l 2l 0 0 dx∂xsmðxÞ∂xsnðxÞ¼2 dxs ðxÞsn−mðxÞ; ð200Þ Z Z l l l l 2l dxsn1 ðxÞsn2 ðxÞsn3 ðxÞsn4 ðxÞ¼ dxsðxÞsn2−n1 ðxÞsn3−n1 ðxÞsn4−n1 ðxÞ; ð201Þ Z Z l l −l dxxsmðxÞsnðxÞ¼2 dxxsðxÞsn−mðxÞþmδm;n ; ð202Þ Z Z l l 0 dxx∂xsmðxÞsnðxÞ¼ dxðx þ mÞs ðxÞsn−mðxÞ; ð203Þ Z Z l l l 0 0 dxx∂xsmðxÞ∂xsnðxÞ¼2 dxðx þ mÞs ðxÞsn−mðxÞ; ð204Þ Z Z l l l l dxxsn1 ðxÞsn2 ðxÞsn3 ðxÞsn4 ðxÞ¼ dxðx þ n1ÞsðxÞsn2−n1 ðxÞsn3−n1 ðxÞsn4−n1 ðxÞ: ð205Þ

096004-18 WAVELET REPRESENTATION OF LIGHT-FRONT QUANTUM … PHYS. REV. D 101, 096004 (2020) Z These identities express all of the integrals involving scale Γ4 ½m½n½k ≔ dxxsðxÞs ðxÞs ðxÞs ðxÞ 2−l scaling functions in terms of related integrals involving x m n k the snðxÞ. The compact support of the functions snðxÞ −4 ≤ m; n; k; m − n; m − k; k − n ≤ 4: ð213Þ means these integrals are identically zero unless the indices and the absolute values of their differences are less than or The renormalization group equation in the form equal to 2L − 2 which is 4 for L ¼ 3. The integrals of the right side of (199)–(205) are the X5 ffiffiffi following integrals: p sðx − nÞ¼ hl 2sð2x − 2n − lÞð214Þ Z l¼0

δmn ¼ dxsmðxÞsnðxÞ m ¼ n; ð206Þ and a variable change x → x0 ¼ 2x lead to the following Z linear equations relating the nonzero values of these ds D1½m ≔ dx ðxÞs ðxÞ − 4 ≤ m ≤ 4; ð207Þ integrals: dx m Z X4 X4 ds dsm D2½m ≔ dx ðxÞ ðxÞ − 4 ≤ m ≤ 4; ð208Þ D1½n¼ am−2nD1½m¼ AnmD1½m; ð215Þ dx dx m¼−4 m¼−4 Z X4 X4 Γ4½m½n½k ≔ dxsðxÞsmðxÞsnðxÞskðxÞ D2½n¼2 am−2nD2½m¼2 AnmD2½m; ð216Þ −4 ≤ m; n; k; m − n; m − k; k − n ≤ 4; ð209Þ m¼−4 m¼−4 Z where am X½m ≔ dxxsðxÞsmðxÞ − 4 ≤ m ≤ 4; ð210Þ X5 Z ≔ 2 − 5 ≤ ≤ 5 ds am hkþmhk m ð217Þ X1½m ≔ dxx ðxÞs ðxÞ − 4 ≤ m ≤ 4; ð211Þ k 0 dx m ¼ Z is the same quantity (168) and (169) that appeared in the ds dsm X2½m ≔ dxx ðxÞ ðxÞ − 4 ≤ m ≤ 4; ð212Þ computation of the commutator function. A similar quan- dx dx tity appears in the homogeneous equations relating the nonzero Γ4½m½n½k’s,

X5 X Γ ≔ 2 Γ 2 − 2 − 2 − ; 0 0 0 Γ 0 0 0 4½m½n½k hlhlm hln hlk 4½ mþlm l½ nþln l½ kþlk l¼ A4ðm;n;k m ;n ;k Þ 4½m ½n ½k ; ð218Þ 0 0 0 l;lmln;lk¼0 m m k where 1 X 0 0 0 0 0 0 X Γ4x½m½n½k ≔ A4ðm; n; k; m ;n;kÞΓ4x½m ½n ½k −; ; 0 0 0 ≔ 2 2 A4ðm; n; k m ;n;kÞ hlhm0−2mþlhn0−2nþlhk0−2kþl: m0n0k0 l ð223Þ ð219Þ X X Γ 0 0 0 hlhm0−2mþlhn0−2nþlhk0−2kþll 4½m ½n ½k : The relations involving X½n;X1½n;X2½n and Γ4 ½m½n½k x m0n0k0 l have inhomogeneous parts, ð224Þ 1 X4 1 X 9 9 ≔ −4 ≤ ≤ 4 X½n¼ AnmX½mþ lhlhl−2n; ð220Þ Since the × matrix Amn an−2m ( m; n ) has 4 2 1 1 1 1 9 9 m¼−4 l eigenvalues λ ¼ 2; 1; 2 ; 4 ; 8 ; 16 ; 32 ; − 64, it follows that D1½n and D2½n are eigenvectors of A with eigenvalues 1 1 X4 X mn and 1, respectively. The normalization is determined by the X1½n¼2 AnmX1½mþ lhlhl−2nþmD1½m; ð221Þ 2 m¼−4 l equations discussed below. Equation (218) similarly im- X4 X plies that Γ4½m½n½k is an eigenvector with eigenvalue 1 of 2 X2½n¼ AnmX2½mþ lhlhl−2nþmD2½m; ð222Þ the matrix A4 defined by the right-hand side of (218). The m¼−4 l normalization of Γ4½m½n½k is also discussed below.

096004-19 W. N. POLYZOU PHYS. REV. D 101, 096004 (2020)

1 4 The matrix ðI − 4 AÞ in (220) is invertible, so (220) is a X well-posed linear system for X½n, while the matrices nD1½n¼−1: ð231Þ 1 n¼−4 ðI − 2 AÞ and (I − A)in(221) and (222) are singular. To solve them, the Moore-Penrose generalized inverse [59] is Multiplying (230) by s0 x and integrating give applied to the inhomogeneous terms to get specific sol- ð Þ utions. These solutions are substituted back in the equations X4 2 to ensure that the inhomogeneous terms are in the range of n D2½n¼−2: ð232Þ 1 −4 ðI − 2 AÞ and (I − A), respectively, although this must be n¼ the case since the solutions can also be expressed as These conditions determine the normalization of the integrals. The general solutions of (221) and (222) can eigenvectors D1½n and D2½n. Note that the moments do include arbitrary amounts of the solution of the homo- not appear in these normalization conditions, although all geneous equations which are eigenstates of Amn with moments of sðxÞ can be computed recursively using the eigenvalues 2 and 1, respectively. The contribution from renormalization group equation and the normalization the homogeneous equation is determined by the normali- condition (107). Using (229) in (211) and integrating by zation conditions below. parts gives The normalization conditions are derived from the property that polynomials with degree less than L can X4 be pointwise represented as locally finite-linear combina- X1½n¼−1: ð233Þ −4 tion of the snðxÞ. These expansions have the form n¼ X Using (230) in (212) and integrating by parts gives 1 ¼ snðxÞ; ð225Þ 4 X X X nX2½n¼−1: ð234Þ x ¼ ðhxiþnÞsnðxÞ¼hxiþ nsnðxÞ; ð226Þ n¼−4 X X 2 2 2 x ¼ ðhxiþnÞ snðxÞ¼hxi þ 2hxi nsnðxÞ These conditions determine the contribution of the solution X of the homogeneous equations in the general solution. 2 þ n snðxÞ; ð227Þ The normalization conditions for Γ4½m½n½k are obtained using the partition of unity property (225), where Z X4 X4 Γ Γ ; Γ δ hxni ≔ sðxÞxndx ð228Þ 4½m½n½k¼ 3½n½k 3½n½k¼ k0; m¼−4 n¼−4 ð235Þ are moments of sðxÞ. Differentiating (226) and (227) gives X 4 4 0 X X 1 ¼ nsnðxÞ; ð229Þ Γ4x½m½n½k¼Γ3x½n½k; Γ3½n½k¼X½k; 1 X m¼−4 n¼−4 2 0 x ¼hxiþ2 n snðxÞ: ð230Þ ð236Þ

Multiplying (229) by sðxÞ and integrating the result give where

Z

Γ3½m½n ≔ dxxsðxÞsmðxÞsnðxÞ − 2L þ 2 ≤ m; n; m − n ≤ 2L − 2; ð237Þ Z

Γ3x½m½n ≔ dxxsðxÞsmðxÞsnðxÞ − 2L þ 2 ≤ m; n; m − n ≤ 2L − 2; ð238Þ and Γ3½m½n is a solution of the eigenvalue problem X 0 0 0 0 Γ3½m½n¼ a3ðm; n; m n ÞΓ3½m ½n ; ð239Þ m0n0 with normalization (235) and

096004-20 WAVELET REPRESENTATION OF LIGHT-FRONT QUANTUM … PHYS. REV. D 101, 096004 (2020) X ; 0 0 Γ4 m n k a3ðm; n m n Þ¼ hlhm0−2mþlhn0−2nþl: ð240Þ condition (235). Finally, x½ ½ ½ be computed by l applying the Moore Penrose generalized inverse of ðI − a4Þ to the inhomogeneous term in (223) and adding Γ 3x½m½n satisfies an amount of the solution of the eigenvalue problem X ð2I − a4ÞX ¼ 0 consistent with the normalization condi- 0 0 0 0 Γ3x½m½n ≔ a3ðm; n; m n ÞΓ3x½m ½n −; ð241Þ tion (236). m0n0 All the these quantities can alternatively computed by a direct quadrature; however, the fractal nature of the basis X 2XL−1 0 0 functions makes the renormalization group method dis- lh h 0−2 h 0−2 Γ3½m ½n ; ð242Þ l m mþl n nþl cussed above preferable. The values of D1½n;D2½n;X½n; m0n0 l X1½n, and X2½n are given below. with the normalization constraint 0 10 1 1 2 −4 − 3 X D1½−4¼2920 D ½ ¼ 560 Γ3 ½m½n¼X½m: ð243Þ B CB 4 C x B 1 −3 − 16 CB D2½−3¼− C n B D ½ ¼ 1095 CB 35 C B 53 CB 2 −2 92 C B D1½−2¼− CB D ½ ¼105 C These finite linear systems can be solved for all of the B 365 CB C B 272 CB 356 C – 1 2 B D1½−1¼ CB D2½−1¼− 105 C integrals (195) (197). The results for D ½n;D ½n;X½n; B 365 CB C X1½n;X2½n for L ¼ 3, which are needed to compute the B 1 0 0 0 CB 2 0 295 C B D ½ ¼ . CB D ½ ¼ 56 C; ð244Þ constant coefficients for the free-field generators, are given B CB C B D1½1¼− 272 CB 2 1 − 356 C below. The vector Γ4 m n k of coefficients for the B 365 CB D ½ ¼ 105 C ½ ½ ½ B CB C dynamical generators has too many components to display. B 1 2 53 CB 2 2 92 C B D ½ ¼365 CB D ½ ¼105 C They can be computed by finding the eigenvector with B 1 3 − 16 CB 4 C 3 3 0 0 @ D ½ ¼ 1095 A@ D2½3¼− A eigenvalue 1 of the 9 × 9 matrix a4½m½n½m ½n with 35 1 4 − 1 3 normalization given by (235). The normalization condition D ½ ¼ 2920 D2½4¼− 560 requires solving for the eigenvector with eigenvalues 1 of 2 2 0 0 the 9 × 9 matrix a3½m½n½m ½n using the normalization

0 10 10 1 0 −4 −3 96222254 − 06 1 −4 1 75026831 − 06 2 −4 −5 08087952 − 04 B X ½ ¼ . e CB X ½ ¼ . e CB X ½ ¼ . e C B CB CB C B X0½−3¼−6.76219313e − 04 CB X1½−3¼−6.81293512e − 04 CB X2½−3¼−8.68468406e − 03 C B CB CB C B CB CB C B X0½−2¼1.92128831e − 02 CB X1½−2¼−3.98947081e − 02 CB X2½−2¼5.47476157e − 01 C B CB CB C B X0 −1 −1 21043257e − 01 CB X1 −1 3 39841948e − 01 CB X2 −1 −3 01673853e 00 C B ½ ¼ . CB ½ ¼ . CB ½ ¼ . þ C B CB CB C B X0½0¼1.02242228e þ 00 CB X1½0¼−5.00000000e − 01 CB X2½0¼6.95730703e þ 00 C: ð245Þ B CB CB C B 0 1 −1 21043257 − 01 CB 1 1 −1 08504743 00 CB 2 1 −6 40481025 00 C B X ½ ¼ . e CB X ½ ¼ . e þ CB X ½ ¼ . e þ C B CB CB C B X0½2¼1.92128831e − 02 CB X1½2¼3.30305667e − 01 CB X2½2¼2.29938859e þ 00 C B CB CB C B CB CB C @ X0½3¼−6.76219313e − 04 A@ X1½3¼−4.31543229e − 02 A@ X2½3¼−3.51494681e − 01 A X0½4¼−3.96222254e − 06 X1½4¼−1.37161328e − 03 X2½4¼−2.19355544e − 02

In this regard, it has similar properties to a lattice XII. TRUNCATIONS truncation. Unlike a lattice truncation, because the theory The value of the wavelet representation is that, while it is is formally exact, it is straightforward to systematically formally exact, it also admits natural volume and resolution include corrections associated with finer resolution or truncations in the light-front hyperplane. Truncations larger volumes. Some other appealing features are that define effective theories that are expected to be good the truncated fields have a continuous space-time depend- approximations to the theory for reactions associated with ence and can be differentiated, so there is no need to use a volume and energy scale corresponding to the volume and finite difference approximations. Finally, it is possible to resolution of the truncations. The simplest truncation take advantage of some of the advantages of the light-front discards degrees of freedom smaller than some limiting quantization. fine resolution, 2−l, as well as degrees of freedom with One problem that is common to lattice truncations of support outside of some volume on the light front. field theory is that truncations break symmetries. In the

096004-21 W. N. POLYZOU PHYS. REV. D 101, 096004 (2020) light-front case, truncations break the kinematic covari- infinite number of discrete degrees of freedom that are ance. One consequence is that transforming the truncated localized on the light front. Each degree of freedom is field covariantly using (128) is not the same as trans- associated with a compact subset of the light front. These forming the truncated field using the matrix (130) and subsets cover the light front, and there are an infinite truncating the result. The difference between these two number of them in every open subset on the light front. This calculations is due to the discarded degrees of freedom, representation has the property that there are a finite which should be small for a suitable truncation. This number of these degrees of freedom associated with any suggests that kinematic Lorentz transformations can be finite volume and any given maximal resolution on the approximated by using (128) with the truncated fields. The light front. vacuum of the formally exact theory is the trivial Fock Each degree of freedom or mode is represented by a field vacuum if the interaction commutes with the kinematic on the light front integrated over a basis function of subgroup. When the kinematic invariance is broken, the compact support on the light front. The discrete fields lowest mass eigenstate of the truncated P− is not neces- associated with a free-field theory are an irreducible set of sarily the Fock vacuum; however, the Fock vacuum should operators on the free-field Fock space. For interacting become the lowest mass state in the infinite-volume, zero- theories with self-adjoint kinematically invariant inter- resolution limit. This suggests that using trivial Fock actions, the spectral condition on Pþ implies that the vacuum might still be a good approximation. interaction cannot change the Fock vacuum. This means The basis discussed in this work is not the only possible dynamical operators like the Poincar´e generators can be basis choice and may not be the best option for treating the expressed as functions of this irreducible algebra of fields transverse degrees of freedom for fields in 3 þ 1 dimen- acting on the free field Fock space. sions. In this work, the transverse degrees of freedom are The Poincar´e generators involve ill-defined products of expanded in products of multiscale basis functions of fields at the same point, so the formal interactions are not Cartesian coordinates, x and y. Truncations of this basis well-defined self-adjoint operators on the Fock space. In break the rotational symmetry about the z axis. An the multiresolution representation, the ultraviolet singular- alternative is to expand the transverse degrees of freedom ities that arise from local operator products necessarily in a basis consisting of products of functions of the polar appear as nonconvergence of infinite sums of well-defined coordinates r and θ where x ¼ r cosðθÞ and y ¼ r sinðθÞ. operator products. There are also infrared divergences that The basis functions in the θ variable can be taken as the appear in products of scaling function modes even after the 1 θ pffiffiffiffi in periodic functions, 2π e . This choice maintains the smearing. In the light-front case, the ultraviolet and infrared rotational symmetry, but does not give a multiresolution singularities are constrained by rotational covariance, so treatment of the angle degree of freedom. A second option any strategy to nonperturbatively renormalize the theory is to use the multiresolution basis in the angle variable on must treat these problems together. ½0; 2π with periodic boundary conditions. In this case, the Computations necessarily involve both volume and truncations will result in a discrete rotational symmetry that resolution cutoffs, which result in a well-defined truncated depends on the resolution. In both cases, the radial degree theory with a finite number of degrees of freedom. As long of freedom can be expanded in a multiresolution basis. The as the interaction in the truncated theory vanishes at þ 0 only difference is that the radial functions have support on p ¼ , the interaction will leave the Fockpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vacuum 2 2 ½0; ∞ rather than ½−∞; ∞. This requires replacing the unchanged. The variable pþ ¼ zˆ · p þ m þ p basis functions that have support at r ¼ 0 by linear approaches zero in the limit that −zˆ · p → þ∞,soitis combinations of these functions that satisfy the boundary an infinite momentum limit, which involves high-resolu- conditions at the origin. The linear combinations in a tion degrees of freedom. Requiring that the interaction subspace of a given resolution can be constructed so they vanish at pþ ¼ 0 is a resolution cutoff. This can be realized are orthonormal on ½0; ∞, resulting in an orthonormal by discarding products of scaling function modes in the basis on that subspace; however, the modified basis interaction. These modes do not contribute to the operator functions near the origin in subspaces of different reso- product when it is integrated over functions with vanishing lution are no longer orthogonal. This results in additional Fourier transforms at pþ ¼ 0. coupling of degrees of freedom on different scales near Dynamical calculations evolve the fields to points off of r ¼ 0. This is because the exact boundary conditions at the light front. This evolution can be performed by iterating r ¼ 0 involve functions of all resolutions. the light-front Heisenberg field equations or by solving the light-front Schrodinger equation. Both cases involve dis- crete mathematics. Iterating the Heisenberg field equations XIII. SUMMARY AND OUTLOOK results in a representation of the field as an expansion in This work introduced a multiresolution representation of normal products of discrete fields on the light front with xþ- quantum field theory on a light front. This is a formally dependent coefficients. Because fields on the light front are exact representation of the field theory in terms of an irreducible, the different discrete field operators cannot all

096004-22 WAVELET REPRESENTATION OF LIGHT-FRONT QUANTUM … PHYS. REV. D 101, 096004 (2020) commute; however, the commutator can be calculated kinds of problems of interest. These include explicitly and analytically. Vacuum expectation values of problems, scattering problems, studies of correlation func- product of fields can be computed by evaluating the tions, and extensions to gauge theories. While the discrete solution of the Heisenberg equations in the Fock vacuum. nature of the multiresolution representation has some There are a number of problems involving free fields that computational advantages, they will not be of significant can be used to try to understand the convergence of help for these complex problems, especially since the computational strategies in truncated theories. Free-field number of modes scale with dimension and number of theories have the advantage that they can be solved exactly, particles. One of the advantages of multiresolution methods so errors can be calculated by comparing exact computa- is that basis functions are self-similar. The result is that the tions to computations based on truncated theories. Among coupling strength of the various modes differs by different the problems of interest is how is Poincar´e invariance powers of 2. A systematic investigation could help to recovered as the resolution is increased in a truncated identify the most dominant modes in a given application. theory. Because the basis is local, this can be tested in a This could be used to get a rough first approximation that finite volume. Methods for performing this test in the can be improved perturbatively. One interesting property of corresponding wavelet representation of canonical field the multiresolution representation of the theory is that it is theory were discussed in [25]. These methods utilize the both discrete and formally exact. In a formally exact theory, locally finite partition of unity property of the scaling Haag-Ruelle scattering theory can be used to express functions in the expression for the generators in terms of the scattering observables as strong limits. Of interest is to integrals over the energy-momentum and angular-momen- use the exact representation to develop an approximation tum tensor densities. In the light-front case, free fields algorithm for computing scattering observables in this provide a laboratory to investigate the accuracy of kin- discrete representation. This is not trivial, since the ematic Lorentz transformations in truncated theories. time limits will not converge if they are computed after Another important problem is how efficiently can the truncation. multiresolution representation of the light-front Bound state calculations could be computed by diago- Hamiltonian be block diagonalized by resolution. This nalizing the mass operator on a subspace, similar to how was studied for the case of the corresponding wavelet this is done using basis light-front quantization [47]. representation of canonical field theory in [35].One Variational methods could prove useful in this regard. conclusion of that work is that both volume and resolution For gauge theories, the exact representation of the field need to be increased simultaneously in order to converge to theory in terms of a countable number of discrete fields a sensible energy spectrum of the Hamiltonian (i.e., so it with different resolutions suggest that a similar construc- approaches a continuous spectrum that is unbounded tion could be performed in using of gauge-invariant degrees above). In addition, it was found that convergence to a of freedom. To understand how this might work, imagine a block diagonal form slowed as energy separation of the set of gauge-invariant Wilson placquets with a given lattice modes decreased. Another calculation that should be done spacing on a light front. The expectation is that on the light is to compare the Wightman functions or commutator front these form an irreducible algebra of operators of a functions of the truncated theories to the exact quantities. given resolution. Decreasing the lattice spacing by a factor The light-front representation has the advantage that of 2 results in a new algebra that is an irreducible set of these can be computed without solving for an approximate operators for the increased resolution. The coarse-scale vacuum. Another interesting question is what is the con- algebra should be a subalgebra of the fine-scale algebra. In tribution of the product of the infrared singular parts of the the same way, that scaling functions on a file scale can be truncated fields to normal ordered products of free fields. expressed as wavelet and scaling functions on a coarse Does the normal ordering remove these contributions? scale, it can be anticipated that there is something like a The next class of problems of interest are 1 þ 1-dimen- wavelet transform that generates the fine-scale algebra in sional solvable field theories. These are interesting because terms of generators for the coarse-scale algebra and the dynamical equations in the multiresolution representa- independent gauge-invariant operators that generate the tion generate more complicated operators in the algebra of degrees of freedom in the fine-scale algebra that are not in fields on the light front. Reference [15] used Daubechies’ the coarse-scale algebra. As in the wavelet case, this could wavelets methods in a canonical representation of the field be repeated on every scale, leading to a countable set of theory to treat the X-Y model and spontaneous symmetry independent operators that can generate placquets on all breaking in the Landau Ginzburg model. scales. This should result in an irreducible set of gauge- The real interest is to apply multiresolution methods to invariant operators on the light front, with a formally trivial realistic theories in 3 þ 1 dimensions. These are computa- vacuum. While this construction is far from trivial, having a tionally far more complex than problems involving free formally exact representation of gauge theories in terms of fields or problem in low dimensions. There are several local gauge-invariant variables is a desirable goal.

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Another class of applications where the wavelet repre- Refs. [29,33]. The advantage in the light-front case is sentation may be useful is in quantum computing. The the trivial nature of the vacuum. fundamental property is that the local nature of the interactions involving different discrete modes means that ACKNOWLEDGMENTS transfer matrices for small time steps can be expressed as The author would like to acknowledge generous support simple quantum circuits. Some comments on using wavelet from the U.S. Department of Energy, Office of Science, discretized fields in quantum computing appear in Nuclear Theory Program, Grant No. DE-SC16457.

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