Fluctuations in quantum mechanics and field theories from a new version of semiclassical theory. II

Abstract

This is the second paper on semiclassical approach based on the density matrix given by the Euclidean time path integral with fixed coinciding endpoints. The classical path, interpolating between this point and the classical vacuum, called "flucton", plus systematic one- and two-loop corrections, has been calculated in the first paper \cite{Escobar-Ruiz:2016aqv} for double-well potential and now extended for a number of quantum-mechanical problems (anharmonic oscillator, sine-Gordon potential). The method is based on systematic expansion in Feynman diagrams and thus can be extended to QFTs. We show that the loop expansion in QM reminds the leading log-approximations in QFT. In this sequel we present complete set of results obtained using this method in unified way. Alternatively, starting from the Schrödinger equation we derive a {\it generalized} Bloch equation which semiclassical-like, iterative solution generates the loop expansion. We re-derive two loop expansions for all three above potentials and now extend it to three loops, which has not yet been done via Feynman diagrams. All results for both methods are fully consistent with each other. Asymmetric (tilted) double-well potential (non-degenerate minima) is also studied using the second method.

References in corpus (7)

• One-Dimensional Quasi-Exactly Solvable Schrödinger Equations
• Complexified path integrals, exact saddles and supersymmetry
• Three-loop Correction to the Instanton Density. I. The Quartic Double Well Potential
• Double Well Potential: Perturbation Theory, Tunneling, WKB (beyond instantons)
• Three-loop Correction to the Instanton Density. II. The Sine-Gordon potential
• Quantum and thermal fluctuations in quantum mechanics and field theories from a new version of semiclassical theory
• Semiclassical Series at Finite Temperature

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