Abstract
We consider the backreaction of a quantum system q on an effectively classical degree of freedom C that is interacting with it. The backreaction equation based on the standard path integral formalism gives the so-called ‘in-out’ backreaction equation, which has several serious pathologies. One can use a different backreaction prescription, referred to as the ‘in-in’ approach, which resolves all the issues of ‘in-out’ backreaction equation. However, this procedure is usually invoked in a rather ad hoc manner. Here we provide a rigorous path integral derivation of the ‘in-in’ approach by exploiting the concept of quantum evolution along complex time contours. It turns out that, this approach can also be used to study both the ‘in-in’ and ‘in-out’ backreaction equations in a unified manner.
Highlights
The probability amplitude A(Q f, t; Qi, ti ) that a system, which was initially, say at ti, in the configuration Qi may be found in the configuration Q f at a later time t is given by [1]A(Q f, t; Qi, ti ) = Q(t f )=Q f D[Q] ei S[Q] h (1)Q(ti )=Qi where D[Q] is an appropriate functional measure
In the classical limit, defined by h → 0, the stationary phase approximation can be invoked to show that the dominant contribution to this integral comes from the configurations that satisfy δS/δ Q = 0
The backreaction of a quantum degree of freedom on an effectively classical system is ubiquitous in physics; it is relevant in the study of black hole evaporation by Hawking radiation and structure formation in the early universe, just to name a few
Summary
The probability amplitude A(Q f , t; Qi , ti ) that a system, which was initially, say at ti , in the configuration Qi may be found in the configuration Q f at a later time t is given by [1]. This corresponds to the SchwingerKeldysh formalism [5,6,7], a path integral based approach adapted to address non-equilibrium quantum systems, which naturally contains a prescription to generate ‘in-in’ expectation values of operators To implement this method, one has to first formulate path integral over a configuration space of the variables qand Cobtained by doubling the degrees of freedom of q and C, respectively, i.e., q ≡ {q+, q−} and C ≡ {C+, C−}. For a given background configuration of C(t), the q system is described by a time dependent harmonic oscillator(TDHO) of mass m(C(t)) and frequency ω(C(t)) This feature of the q system is shared by the Fourier modes of many quantum fields interacting with a classical background [8,9]. Since this is a fairly well-studied subject, we will only quote the results relevant for this work and delegate the details and derivations to the Appendix
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