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Manifestly gauge invariant transition amplitudes and thermal influence functionals in QED and linearized gravity Wilson-Gerow, Jordan

Abstract

Einstein’s theory of General Relativity tells us that gravity is not a force but rather it is the curvature of spacetime itself. Spacetime is a dynamical object evolving and interacting similar to any other system in nature. The equivalence principle requires everything to couple to gravity in the same way. Consequently, as a matter of principle it is impossible to truly isolate a system|it will always be interacting with the dynamical spacetime in which it resides. This may be detrimental for large mass quantum systems since interaction with an environment can decohere a quantum system, rendering it effectively classical. To understand the effect of a ‘spacetime environment’, we compute the Feynman-Vernon influence functional (IF), a useful tool for studying decoherence. We compute the IF for both the electromagnetic and linearized gravitational fields at finite temperature in a manifestly gauge invariant way. Gauge invariance is maintained by using a modification of the Faddeev-Popov technique which results in the integration over all gauge equivalent configurations of the system. As an intermediate step we evaluate the gauge invariant transition amplitude for the gauge fields in the presence of sources. When used as an evolution kernel the transition amplitude projects initial data onto a physical (gauge-invariant) subspace of the Hilbert space and time-evolves the states within that physical subspace. The states in this physical subspace satisfy precisely the same constraint equations which one implements in the constrained quantization method of Dirac. Thus we find that our approach is the path-integral equivalent of Dirac’s. In the gauge invariant computation it is clear that for gauge theories the appropriate separation between system and environment is not a) matter and gauge field, but rather b) matter (dressed with a coherent field) and radiation field. This implies that only the state of the radiation field can be traced out to obtain a reduced description of the matter. We stress the importance of gauge invariance and the implementation of constraints because it resolves the disagreement between results in reported recent literature in which influence functionals were computed in different gauges without consideration of constraints.

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