Light-front computational methods

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,[11][12] 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[13] for a supergravity/super-Yang–Mills duality conjectured by Maldacena.

A systematic approach to discretization of the eigenvalue problem is the DLCQ method originally suggested by Pauli and Brodsky.

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.)

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).

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.

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.

One can instead borrow from the many-body coupled cluster method[27] a construction that computes expectation values from right and left eigenstates.

The concepts of renormalization that appear useful in theories quantized in the front form of dynamics are essentially of two types, as in other areas of theoretical physics.

In other words, one needs to design the Minkowski space-time formulation of a relativistic theory that is not based on any a priori perturbative scheme.

But for mathematical reasons, being forced to use computers for sufficiently precise calculations, one has to work with a finite number of degrees of freedom.

Namely, particles of different momenta are coupled through the dynamics in a nontrivial way, and the calculations aiming at predicting observables yield results that depend on the cutoffs.

Thus, the results take the form However, experiments provide values of observables that characterize natural processes irrespective of the cutoffs in a theory used to explain them.

If the cutoffs do not describe properties of nature and are introduced merely for making a theory computable, one needs to understand how the dependence on

leads to finite, cutoff independent limits for all observables is qualified by the need to use some form of perturbation theory and inclusion of model assumptions concerning bound states.

Thus, not only the possibility that a renormalization group of the first type may exist can be understood, but also the alternative situations are found where the set of required cutoff dependent parameters does not have to be finite.

The first concept allows one to play with a small set of parameters and seek consistency, which is a useful strategy in perturbation theory if one knows from other approaches what to expect.

One can also study sufficiently simplified models for which computers can be used to carry out calculations and see if a procedure suggested by perturbation theory may work beyond it.

The second concept allows one to address the issue of defining a relativistic theory ab initio without limiting the definition to perturbative expansions.

However, to address this issue one needs to overcome certain difficulties that the renormalization group procedures based on the idea of reduction of cutoffs are not capable of easily resolving.

In particular, this difficulty concerns bound states, where interactions must prevent free relative motion of constituents from dominating the scene and a spatially compact systems have to be formed.

As a result, one obtains in the rotated basis an effective Hamiltonian matrix eigenvalue problem in which the dependence on cutoff

The similarity renormalization group procedure, discussed in #Similarity transformations, can be applied to the problem of describing bound states of quarks and gluons using QCD according to the general computational scheme outlined by Wilson et al.[37] and illustrated in a numerically soluble model by Glazek and Wilson.

More recently, similarity has evolved into a computational tool called the renormalization group procedure for effective particles, or RGPEP.

In principle, if one had solved the RGPEP equation for the front form Hamiltonian of QCD exactly, the eigenvalue problem could be written using effective quarks and gluons corresponding to any

, comparable with the size of hadrons, is hoped to take the form of a simple equation that resembles the constituent quark models.

with the kernel corresponding to the same physical content, say, one-boson exchange (which, however, in the both approaches have very different analytical forms) are very close to each other.

[67] Chiral symmetry breaking of quantum chromodynamics is often associated in the instant form with quark and gluon condensates in the QCD vacuum.

In particular, by studying the appearance of degeneracies among the lowest massive states, one can determine the critical coupling strength associated with spontaneous symmetry breaking.