Abstract
We work in the Heisenberg picture to demonstrate the classical-quantum correspondence (CQC) in which the dynamics of a quantum variable is equivalent to that of a complexified classical variable. The correspondence provides a tool for analyzing quantum backreaction problems which we illustrate by a toy model in which a rolling particle slows down due to quantum radiation. The dynamics found using the CQC is in excellent agreement with that found using the much more laborious full quantum analysis.
Highlights
A large class of physical systems involve classical dynamics that is coupled to quantum degrees of freedom (d.o.f.) that get excited as the classical system evolves
This result can be applied to a mode by mode analysis of a free quantum field to show that the quantum field dynamics can be described in terms of the classical dynamics of a corresponding system with prescribed initial conditions [12]
We have derived the classical-quantum correspondence (CQC) in the Heisenberg picture. This shows that the dynamics of a quantum simple harmonic oscillator with a time-dependent frequency is given by the dynamics of two classical simple harmonic oscillators with the same time-dependent frequency and prescribed initial conditions
Summary
A. Heisenberg equations The Hamiltionian for a simple harmonic oscillator with time-dependent frequency is. Ð1Þ where ω 1⁄4 ωðtÞ is an unspecified function. We define ladder operators in the usual way a 1⁄4 pp−ffiffiiffiffimffiffiffiωffiffi x ; a† 1⁄4 ppþffiffiiffiffimffiffiffiωffiffi x ð2Þ. It is straightforward to check that 1⁄2a; a† 1⁄4 1 even for a time-dependent ω.
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