Stueckelberg massive electromagnetism in curved spacetime: Hadamard renormalization of the stress-energy tensor and the Casimir effect

Abstract

We discuss Stueckelberg massive electromagnetism on an arbitrary four-dimensional curved spacetime and, in particular, (i) the gauge invariance of the classical theory and its covariant quantization; (ii) the wave equations for the massive spin-1 field ${A}_{\ensuremath{\mu}}$, for the auxiliary Stueckelberg scalar field $\mathrm{\ensuremath{\Phi}}$ and for the ghost fields $C$ and ${C}^{*}$; (iii) Ward identities; (iv) the Hadamard representation of the various Feynman propagators and the covariant Taylor series expansions of the corresponding coefficients. This permits us to construct, for a Hadamard quantum state, the expectation value of the renormalized stress-energy tensor associated with the Stueckelberg theory. We provide two alternative but equivalent expressions for this result. The first one is obtained by removing the contribution of the ``Stueckelberg ghost'' $\mathrm{\ensuremath{\Phi}}$ and only involves state-dependent and geometrical quantities associated with the massive vector field ${A}_{\ensuremath{\mu}}$. The other one involves contributions coming from both the massive vector field and the auxiliary Stueckelberg scalar field, and it has been constructed in such a way that, in the zero-mass limit, the massive vector field contribution reduces smoothly to the result obtained from Maxwell's theory. As an application of our results, we consider the Casimir effect outside a perfectly conducting medium with a plane boundary. We discuss the results obtained using Stueckelberg but also de Broglie-Proca electromagnetism, and we consider the zero-mass limit of the vacuum energy in both theories. We finally compare the de Broglie-Proca and Stueckelberg formalisms and highlight the advantages of the Stueckelberg point of view, even if, in our opinion, the de Broglie-Proca and Stueckelberg approaches of massive electromagnetism are two faces of the same field theory.