Coherent States Expectation Values as Semiclassical Trajectories

TL;DR

This paper investigates the semiclassical evolution of quantum anharmonic oscillators using coherent states and deformation quantization, revealing agreements and discrepancies with effective action methods.

Contribution

It introduces a novel semiclassical approach based on coherent states and deformation quantization, providing explicit computable trajectories and comparing them to effective action results.

Findings

Agreement up to order λħ with effective action

Disagreement at order λ²ħ indicating differences in semiclassical dynamics

Exact computability of coherent state trajectories in certain theories

Abstract

We study the time evolution of the expectation value of the anharmonic oscillator coordinate in a coherent state as a toy model for understanding the semiclassical solutions in quantum field theory. By using the deformation quantization techniques, we show that the coherent state expectation value can be expanded in powers of $\hbar$ such that the zeroth-order term is a classical solution while the first-order correction is given as a phase-space Laplacian acting on the classical solution. This is then compared to the effective action solution for the one-dimensional $\f^4$ perturbative quantum field theory. We find an agreement up to the order $\l\hbar$, where $\l$ is the coupling constant, while at the order $\l^2 \hbar$ there is a disagreement. Hence the coherent state expectation values define an alternative semiclassical dynamics to that of the effective action. The coherent state semiclassical trajectories are exactly computable and they can coincide with the effective action trajectories in the case of two-dimensional integrable field theories.