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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., Comment: 15 pages, 5 figure files, LaTeX

In order to estimate the effect of dynamical gluons on chiral condensates, the gap equation of the $\mathrm{SU}(2)$ gauged Nambu--Jona-Lasinio model, under a constant background non-Abelian magnetic field, is investigated up to the two-loop order in $2+1$ and $3+1$ dimensions. We set up a general formulation allowing both cases of electric as well as magnetic background fields. We employ the proper time method which gives a gauge-invariant fermionic functional determinant. In $3+1$ dimensions chiral symmetry breaking $(\ensuremath{\chi}\mathrm{SB})$ is enhanced by gluons even in a zero background magnetic field and becomes more striking as the background field grows larger. In $2+1$ dimensions gluons also enhance $\ensuremath{\chi}\mathrm{SB}$ but the dependence on the background field is not simple; dynamical mass is not a monotone function of a background field for a fixed four-Fermi coupling.

The chiral symmetry breaking in the 4-dimensional QED with the chirally invariant four-fermion interaction is discussed by using a novel path integral expression in terms of the field-strength tensor. In the local potential approximation, we find that the chiral symmetry is spontaneously broken for any nonzero gauge and four-fermion couplings on the tree level of an auxiliary field $\sigma$. The present approach allows us to easily include higher orders of the gauge coupling so that the effective potential up to the sixth order is obtained., Comment: 18 pages, 2 figures

Critical dynamics of the Nambu-Jona-Lasinio model, coupled to a constant electromagnetic field in D=2, 3, and 4, is reconsidered from a viewpoint of infrared behavior and vacuum instability. The latter is associated with constant electric fields and cannot be avoidable in the nonperturbative framework obtained through the proper time method. As for magnetic fields, an infrared cut-off is essential to investigate the critical phenomena. The result reconfirms the fact that the critical coupling in D=3 and 4 goes to zero even under an infinitesimal magnetic field. There also shows that a non-vanishing $F_{\mu\nu}\widetilde F^{\mu\nu}$ causes instability. A perturbation with respect to external fields is adopted to investigate critical quantities, but the resultant asymptotic expansion excellently matches with the exact value., Comment: 27 pages, 17 figure files, LaTeX