PHYSICAL REVIEW D 101, 096004 (2020) Wavelet representation of light-front quantum field theory 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 vacuum 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 Lorentz transformation 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 spin 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 annihilation 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 operator 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 energy 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 mass 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.