Gauge independence in terms of the functional integral

Abstract

Among various approaches in proving gauge independence, models containing an explicit gauge dependence are convenient. The well-known example is the gauge parameter in the covariant gauge fixing which is of course most suitable for the perturbation theory but a negative metric prevents us from imaging a dynamical picture. Noncovariant gauge such as the Coulomb gauge is on the contrary used for many physical situations. Therefore it is desirable to include both cases. More than ten years ago, Steinmann introduced a function (distribution) which can play this role in his attempt on discussing quantum electrodynamics (QED) in terms of the gauge invariant fields solely. The method is, however, broken down in the covariant case: the invariant operators are ill-defined because of 1/p 2 singularity in the Minkowski space. In this paper, we apply his function to the path integral: utilizing the arbitrariness of the function we first restrict it to be able to have a well-defined operator, and then a Hamiltonian with which we can build up the (Euclidean) path integral formula. Although the formula is far from covariant, a full covariant expression is recovered by reviving the components which have been discarded under the construction of the Hamiltonian. There is no pathological defects contrary to the operator formalism. With the aid of the path integral formula, the gauge independence of the free energy as well as the S-matrix is proved. Moreover the reason is clarified why it is so simple and straightforward to argue gauge transformations in the path integral. Discussions on the quark confinement is also presented.

Charges from Dressed Matter: Construction

“…The transformation from the interaction picture to the asymptotic Heisenberg picture is implemented by the usual time ordered product of the Hamiltonian (22). Using the commutator relations (25) and (21), this can be written as the product of two commuting terms:…”

Charges in gauge theories

“…Such pictures have been rediscovered by various authors since Dirac and there have been many attempts to use certain examples of this wide class of dressings over the years [17,18,19,20,21,22,23,24,25,26,27,28,29,30].…”

Path optimization method with use of neural network for the sign problem in field theories

“…In one dimensional integral, we can parameterize the imaginary part by a trial function of the real part, and optimize the trial function by the standard gradient descent method [34]. It is also possible to utilize the neural network [35,36]. Because of this flexibility, similar ideas have been applied to several problems recently: 1+1 dimensional φ 4 theory [35], 0+1 dimensional φ 4 theory [37], 2+1 dimensional finite density Thirring model (their method is referred to as the sign-optimized manifold method (SOMMe)) [16,17], 1+1 dimensional QED [18], and the Polyakov-loop extended Nambu-Jona-Lasinio (PNJL) model [36].…”

“…In this proceedings, we apply the path optimization method with use of the neural network to field theories with the sign problem. After a brief review of the path optimization method, the neural network, and the stochastic gradient method, we discuss 1+1 dimensional φ 4 theory at finite µ [35], 0+1 dimensional QCD at finite µ [38], and the Polyakov-loop extended Nambu-Jona-Lasinio (PNJL) [36].…”