The light-front quantization of quantum field theories provides a useful alternative to ordinary equal-time quantization. In particular, it can lead to a relativistic description of bound systems in terms of quantum-mechanical wave functions. The quantization is based on the choice of light-front coordinates, where plays the role of time and the corresponding spatial coordinate is . Here, is the ordinary time, is one Cartesian coordinate, and is the speed of light. The other two Cartesian coordinates, and , are untouched and often called transverse or perpendicular, denoted by symbols of the type . The choice of the frame of reference where the time and -axis are defined can be left unspecified in an exactly soluble relativistic theory, but in practical calculations some choices may be more suitable than others.
The solution of the LFQCD Hamiltonian eigenvalue equation will utilize the available mathematical methods of quantum mechanics and contribute to the development of advanced computing techniques for large quantum systems, including nuclei. For example, in the discretized light-cone quantization method (DLCQ), periodic conditions are introduced such that momenta are discretized and the size of the Fock space is limited without destroying Lorentz invariance. Solving a quantum field theory is then reduced to diagonalizing a large sparse Hermitian matrix. The DLCQ method has been successfully used to obtain the complete spectrum and light-front wave functions in numerous model quantum field theories such as QCD with one or two space dimensions for any number of flavors and quark masses. An extension of this method to supersymmetric theories, SDLCQ, takes advantage of the fact that the light-front Hamiltonian can be factorized as a product of raising and lowering ladder operators. SDLCQ has provided new insights into a number of supersymmetric theories including direct numerical evidence for a supergravity/super-Yang–Mills duality conjectured by Maldacena.
It is convenient to work in a Fock basis where the light-front momenta and are diagonal. The state is given by an expansion
with
is interpreted as the wave function of the contribution from states with particles. The eigenvalue problem is a set of coupled integral equations for these wave functions. Although the notation as presented supports only one particle type, the generalization to more than one is trivial.
Discrete light-cone quantization A systematic approach to discretization of the eigenvalue problem is the DLCQ method originally suggested by Pauli and Brodsky. In essence it is the replacement of integrals by trapezoidal approximations, with equally-spaced intervals in the longitudinal and transverse momenta
corresponding to periodic boundary conditions on the intervals and . The length scales and determine the resolution of the calculation. Because the plus component of momentum is always positive, the limit can be exchanged for a limit in terms of the integer resolution . The combination of momentum components that defines is then independent of . The longitudinal momentum fractions become ratios of integers . Because the are all positive, DLCQ automatically limits the number of particles to be no more than . When a limit on transverse momentum is supplied via a chosen cutoff, a finite matrix problem is obtained; however, the matrix may be too large for present numerical techniques. An explicit truncation in particle number, the light-cone equivalent of the Tamm—Dancoff approximation, can then be made. Large basis sizes require special techniques for matrix diagonalization; the one typically used is the Lanczos algorithm. For the case of one space dimension, one can readily solve for the hadron spectrum of QCD for any quark masses and colors.
Most DLCQ calculations are done without zero modes. However, in principle, any DLCQ basis with periodic boundary conditions may include them as constrained modes, dependent on the other modes with nonzero momentum. The constraint comes from the spatial average of the Euler–Lagrange equation for the field. This constraint equation can be difficult to solve, even for the simplest theories. However, an approximate solution can be found, consistent with the underlying approximations of the DLCQ method itself. This solution generates the effective zero-mode interactions for the light-front Hamiltonian.
Calculations in the massive sector that are done without zero modes will usually yield the correct answer. The neglect of zero modes merely worsens the convergence. One exception is that of cubic scalar theories, where the spectrum extends to minus infinity. A DLCQ calculation without zero modes will require careful extrapolation to detect this infinity, whereas a calculation that includes zero modes yields the correct result immediately. The zero modes are avoided if one uses antiperiodic boundary conditions.
Supersymmetric discrete light-cone quantization The supersymmetric form of DLCQ (SDLCQ) is specifically designed to maintain supersymmetry in the discrete approximation. Ordinary DLCQ violates supersymmetry by terms that do not survive the continuum limit. The SDLCQ construction discretizes the supercharge and defines the Hamiltonian by the superalgebra relation . The range of transverse momentum is limited by a simple cutoff in the momentum value. Effects of zero modes are expected to cancel.
In addition to calculations of spectra, this technique can be used to calculate expectation values. One such quantity, a correlator of the stress energy tensor, has been computed as a test of a Maldacena conjecture. A very efficient Lanczos-based method was developed for this calculation. The most recent results provide direct evidence for the conjecture.
Transverse lattice The transverse lattice method brings together two powerful ideas in quantum field theory: light-front Hamiltonian quantization and lattice gauge theory. Lattice gauge theory is a very popular means of regulating for calculation the gauge theories that describe all visible matter in the universe; in particular, it manifestly demonstrates the linear confinement of QCD that holds quarks and gluons inside the protons and neutrons of the atomic nucleus. In general, to obtain solutions of a quantum field theory, with its continuously infinite degrees of freedom, one must put kinematical cutoffs or other restrictions on the space of quantum states. To remove the errors this introduces, one may then extrapolate these cutoffs, provided a continuum limit exists, and/or renormalize observables to account for degrees of freedom above the cutoff. For the purposes of Hamiltonian quantization, one must have a continuous time direction. In the case of light-front Hamiltonian quantization, in addition to continuous light-front time , it is necessary to keep the direction continuous if one wants to preserve the manifest Lorentz boost invariance in one direction and to include small light-front energies . Therefore, at most one can impose a lattice cutoff on the remaining transverse spatial directions. Such a transverse lattice gauge theory was first suggested by Bardeen and Pearson in 1976.
Most practical calculations performed with transverse lattice gauge theory have utilized one further ingredient: the color-dielectric expansion. A dielectric formulation is one in which the gauge group elements, whose generators are the gluon fields in the case of QCD, are replaced by collective (smeared, blocked, etc.) variables which represent an average over their fluctuations on short distance scales. These dielectric variables are massive, carry color, and form an effective gauge field theory with classical action minimized at zero field, meaning that color flux is expelled from the vacuum at the classical level. This maintains the triviality of the light-front vacuum structure, but arises only for a low momentum cutoff on the effective theory (corresponding to transverse lattice spacings of order 1/2 fm in QCD). As a result, the effective cutoff Hamiltonian is initially poorly constrained. The color-dielectric expansion, together with requirements of Lorentz symmetry restoration, has nevertheless been successfully used to organize the interactions in the Hamiltonian in a way suitable for practical solution. The most accurate spectrum of large- glueballs has been obtained in this way, and as well as pion light-front wave functions in agreement with a range of experimental data.
Basis Light-Front Quantization The basis light-front quantization (BLFQ) approach uses expansions in products of single-particle basis functions to represent the Fock-state wave functions. Typically, the longitudinal () dependence is represented in the DLCQ basis of plane waves, and the transverse dependence is represented by two-dimensional harmonic oscillator functions. The latter are ideal for applications to confining cavities and are consistent with light-front holographic QCD. The use of products of single particle basis functions is also convenient for incorporation of boson and fermion statistics, because the products are readily (anti)symmetrized. By employing two-dimensional basis functions with rotational symmetry about the longitudinal direction (where the harmonic oscillator functions serve as an example), one preserves the total angular momentum projection quantum number which facilitates determination of the total angular momentum of the mass eigenstates. For applications without an external cavity, where transverse momentum is conserved, a Lagrange multiplier method is used to separate the relative transverse motion from the total system's motion.
The first application of BLFQ to QED solved for the electron in a two-dimensional transverse confining cavity and showed how the anomalous magnetic moment behaved as a function of the strength of the cavity. The second application of BLFQ to QED solved for the electron's anomalous magnetic moment in free space and demonstrated agreement with the Schwinger moment in the appropriate limit.
The extension of BLFQ to the time-dependent regime, namely, time-dependent BLFQ (tBLFQ) is straightforward and is currently under active development. The goal of tBLFQ is to solve light-front field theory in real-time (with or without time-dependent background fields). The typical application areas include intense lasers (see Light-front quantization#Intense lasers}) and relativistic heavy-ion collisions.
Light-front coupled-cluster method The light-front coupled cluster (LFCC) method is a particular form of truncation for the infinite coupled system of integral equations for light-front wave functions. The system of equations that comes from the field-theoretic Schrödinger equation also requires regularization, to make the integral operators finite. The traditional Fock-space truncation of the system, where the allowed number of particles is limited, typically disrupts the regularization by removing infinite parts that would otherwise cancel against parts that are retained. Although there are ways to circumvent this, they are not completely satisfactory.