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

## TL;DR

This paper extends a semiclassical density matrix approach to quantum mechanics, calculating flucton paths and loop corrections for various potentials, and introduces a generalized Bloch equation to derive higher-order loop expansions.

## Contribution

It develops a unified method for calculating loop corrections in quantum mechanics and extends the semiclassical approach to more complex potentials and higher loops.

## Key findings

- Calculated two- and three-loop corrections for multiple potentials
- Derived a generalized Bloch equation for iterative loop expansion
- Confirmed consistency between diagrammatic and Schrödinger-based methods

## 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 Schr\"{o}dinger 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.