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

Understand

We discuss Stueckelberg massive electromagnetism on an arbitrary four-dimensional curved spacetime (gauge invariance of the classical theory and covariant quantization, wave equations for the massive spin-1 field $A_\mu$, for the auxiliary Stueckelberg scalar field $\Phi$ and for the ghost fields $C$ and $C^\ast$, Ward identities, Hadamard representation of the various Feynman propagators and 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" $\Phi$ and only involves state-dependent and geometrical quantities associated with the massive vector field $A_\mu$.