Fluctuations in quantum mechanics and field theories from a new version of semiclassical theory. II.
AbstractThis is the second paper on the semiclassical approach based on the density matrix given by the Euclidean time path integral with fixed coinciding end points. The classical path, interpolating between this point and the classical vacuum (called a “flucton”), as well as systematic one- and two-loop corrections were calculated in the first paper [M. A. Escobar-Ruiz, E. Shuryak, and A. V. Turbiner, Phys. Rev. D 93, 105039 (2016)]PRVDAQ2470-001010.1103/PhysRevD.93.105039 for a double-well potential. Here, we extend them for a number of quantum-mechanical problems, such as an anharmonic oscillator and the sine-Gordon potential. The method is based on a systematic expansion in Feynman diagrams and thus can be extended to quantum field theories (QFTs). We show that the loop expansion in quantum mechanics resembles the leading-log approximations in QFT. In this sequel, we present a complete set of results obtained using this method in a unified way. Alternatively, starting from the Schrödinger equation we derive a generalized Bloch equation whose semiclassical-like, iterative solution generates the loop expansion. We rederive the two-loop expansions for all three of the above potentials and extend them to three loops, which has not yet been done via Feynman diagrams. All results for both methods are fully consistent with each other. An asymmetric (tilted) double-well potential (nondegenerate minima) is also studied using the second method.