Euclidean Maxwell Theory Research Articles

New kernels in quantum gravity

Recent work in the literature has proposed the use of non-local boundary conditions in Euclidean quantum gravity. The present paper studies first a more general form of such a scheme for bosonic gauge theories, by adding to the boundary operator for mixed boundary conditions of local nature a 2 × 2 matrix of pseudo-differential operators with pseudo-homogeneous kernels. The requirement of invariance of such boundary conditions under infinitesimal gauge transformations leads to non-local boundary conditions on ghost fields. In Euclidean quantum gravity, an alternative scheme is proposed, where non-local boundary conditions and the requirement of their complete gauge invariance are sufficient to lead to gauge-field and ghost operators of pseudo-differential nature. The resulting boundary conditions have a Dirichlet and a pseudo-differential sector, and are pure Dirichlet for the ghost. This approach is eventually extended to Euclidean Maxwell theory.

Heat kernel coefficients for Chern-Simons boundary conditions in QED

We consider the four-dimensional Euclidean Maxwell theory with a Chern-Simons term on the boundary. The corresponding gauge-invariant boundary conditions become dependent on tangential derivatives. Taking the 4-sphere as a particular example, we calculate explicitly a number of the first heat kernel coefficients and obtain the general formulae that yield any desired coefficient. A remarkable observation is that the coefficient , which defines the 1-loop counterterm and the conformal anomaly, does not depend on the Chern-Simons coupling constant, while the heat kernel itself becomes singular at a certain (critical) value of the coupling. This could be a reflection of a general property of Chern-Simons theories.

The Faddeev–Popov trick in the presence of boundaries

We formulate criteria of applicability of the Faddeev–Popov trick to gauge theories on manifolds with boundaries. With the example of Euclidean Maxwell theory we demonstrate that the path integral is indeed gauge independent when these criteria are satisfied, and depends on a gauge choice whenever these criteria are violated.

One-loop effective action for Euclidean Maxwell theory on manifolds with a boundary.

This paper studies the one-loop effective action for Euclidean Maxwell theory about flat four-space bounded by one three-sphere, or two concentric three-spheres. The analysis relies on the Faddeev-Popov formalism and $\ensuremath{\zeta}$-function regularization, and the Lorentz gauge-averaging term is used with magnetic boundary conditions. The contributions of transverse, longitudinal, and normal modes of the electromagnetic potential, jointly with ghost modes, are derived in detail. The most difficult part of the analysis consists in the eigenvalue condition given by the determinant of a 2\ifmmode\times\else\texttimes\fi{}2 or a 4\ifmmode\times\else\texttimes\fi{}4 matrix for longitudinal and normal modes. It is shown that the former splits into a sum of Dirichlet and Robin contributions, plus a simpler term. This is the quantum-cosmological case. In the latter case, however, when magnetic boundary conditions are imposed on two bounding three-spheres, the determinant is more involved. Nevertheless, it is evaluated explicitly as well. The whole analysis provides the building block for studying the one-loop effective action in covariant gauges, on manifolds with boundary. The final result differs from the value obtained when only transverse modes are quantized, or when noncovariant gauges are used.

1-loop effective action on the 4-ball

This paper applies -function regularization to evaluate the 1-loop effective action for scalar field theories and Euclidean Maxwell theory in the presence of boundaries. After a comparison of two techniques developed in the recent literature, the vacuum Maxwell theory is studied and the contribution of all perturbative modes to is derived: transverse, longitudinal and normal modes of the electromagnetic potential, jointly with ghost modes. The analysis is performed by imposing magnetic boundary conditions, when the Faddeev - Popov Euclidean action contains the particular gauge-averaging term which leads to a complete decoupling of all perturbative modes. It is shown that there is no cancellation of the contributions to resulting from longitudinal, normal and ghost modes.

Euclidean Maxwell theory in the presence of boundaries. II

-function regularization is applied to complete a recent analysis of the quantized electromagnetic field in the presence of boundaries. The quantum theory is studied by setting to zero on the boundary the magnetic field, the gauge-averaging functional, and hence the Faddeev--Popov ghost field. Electric boundary conditions are also studied. On considering two gauge functionals which involve covariant derivatives of the 4-vector potential, a series of detailed calculations shows that, in the case of flat Euclidean 4-space bounded by two concentric 3-spheres, one-loop quantum amplitudes are gauge independent and their mode-by-mode evaluation agrees with the covariant formulae for such amplitudes and coincides for magnetic or electric boundary conditions. By contrast, if a single 3-sphere boundary is studied, one finds some inconsistencies, i.e. gauge dependence of the amplitudes.

Gauge-averaging functionals for Euclidean Maxwell theory in the presence of boundaries

This paper studies the one-loop expansion of the amplitudes of electromagnetism about flat Euclidean backgrounds bounded by a 3-sphere, recently considered in perturbative quantum cosmology, by using -function regularization. For a specific choice of gauge-averaging functional, the contributions to the full value owed to physical degrees of freedom, decoupled gauge mode, coupled gauge modes, and Faddeev--Popov ghost field are derived in detail, and alternative choices for such a functional are also studied. This analysis enables one to get a better understanding of different quantization techniques for gauge fields and gravitation in the presence of boundaries.