Abstract
In this work we calculate the closed time path generating functional for the electromagnetic (EM) field interacting with inhomogeneous anisotropic matter. For this purpose, we first find a general expression for the electromagnetic field’s influence action from the interaction of the field with a composite environment consisting in the quantum polarization degrees of freedom in each point of space, at arbitrary temperatures, connected to thermal baths. Then we evaluate the generating functional for the gauge field, in the temporal gauge, by implementing the Faddeev–Popov procedure. Finally, through the point-splitting technique, we calculate closed expressions for the energy, the Poynting vector, and the Maxwell tensor in terms of the Hadamard propagator. We show that all the quantities have contributions from the field’s initial conditions and also from the matter degrees of freedom. Throughout the whole work we discuss how the gauge invariance must be treated in the formalism when the EM-field is interacting with inhomogeneous anisotropic matter. We study the electrodynamics in the temporal gauge, obtaining the EM-field’s equation and a residual condition. Finally we analyze the case of the EM-field in bulk material and also discuss several general implications of our results in relation with the Casimir physics in a non-equilibrium scenario.
Highlights
The very well-known Lifshitz formula [1] describes the forces between dielectrics in terms of their macroscopic EM properties, in thermal equilibrium
A detailed analysis will be performed in relation to how the gauge invariance must be treated in the CTP formalism when the EM-field is interacting with inhomogeneous anisotropic matter
Special care has been taken as regards how the gauge invariance must be treated in the CTP formalism when the EM-field is interacting with inhomogeneous anisotropic matter
Summary
The very well-known Lifshitz formula [1] describes the forces between dielectrics in terms of their macroscopic EM properties, in thermal equilibrium. When dealing with a composite system, in which there are noise, fluctuations, and dissipative effects between different parts of the full system (mirrors, vacuum field, and environment), the theory of open quantum systems [5] is the most appropriate framework to clarify the role of these effects in Casimir physics. In this description, dissipation and noise appear in the effective theory of the relevant degrees of freedom (the EM-field) after integration of the matter and other environmental degrees of freedom. The quantization at the steady situation can be performed starting from the macroscopic Maxwell equations, and including noise terms to account for absorption [6]
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