Causal signal transmission by quantum fields -- V: Generalised Keldysh rotations and electromagnetic response of the Dirac sea

TL;DR

This paper explores generalized Keldysh rotations in real-time quantum field theory, linking phase-space mappings with operator orderings, and demonstrates their application to electromagnetic response in the Dirac sea, emphasizing causality and fluctuation cancellation.

Contribution

It introduces a broad class of generalized Keldysh rotations based on Cahill-Glauber orderings, extending phase-space mappings to interacting fields and applying this framework to relativistic quantum electrodynamics.

Findings

Response transformation extends normal ordering to time-normal ordering.

Time-normal ordering cancels zero-point fluctuations in the Dirac sea.

Other orderings lead to physically inconsistent electromagnetic field descriptions.

Abstract

The connection between real-time quantum field theory (RTQFT) [see, e.g., A.\ Kamenev and A.\ Levchenko, Advances in Physics {58} (2009) 197] and phase-space techniques [E.\ Wolf and L.\ Mandel, {\em Optical Coherence and Quantum Optics} (Cambridge, 1995)] is investigated. The Keldysh rotation that forms the basis of RTQFT is shown to be a phase-space mapping of the quantum system based on the symmetric (Weyl) ordering. Following this observation, we define generalised Keldysh rotations based on the class of operator orderings introduced by Cahill and Glauber [Phys.\ Rev.\ {177} (1969) 1882]. Each rotation is a phase-space mapping, generalising the corresponding ordering from free to interacting fields. In particular, response transformation [L.P.\ and S.S., Ann.\ Phys. (N.Y.) {323} (2008) 1989] extends the normal ordering of free-field operators to the time-normal ordering of Heisenberg\ operators. Structural properties of the response transformation, such as its association with the nonlinear quantum response problem and the related causality properties, hold for all generalised Keldysh rotations. Furthermore, we argue that response transformation is especially suited for RTQFT formulation of spatial, in particular, relativistic, problems, because it extends cancellation of zero-point fluctuations, characteristic of the normal ordering, to interacting fields. As an example, we consider quantised electromagnetic\ field in the Dirac sea. In the time-normally-ordered representation, dynamics of the field looks essentially classical (fields radiated by currents), without any contribution from zero-point fluctuations. For comparison, we calculate zero-point fluctuations of the interacting electromagnetic\ field under orderings other than time-normal. The resulting expression is physically inconsistent: it does not obey the Lorentz condition, nor Maxwell's equations.