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 [1] 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 generalized Bloch equation which semiclassicallike, 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.

Baryon preclustering at the freeze-out of heavy-ion collisions and light-nuclei production

“…Systematic application of this method at zero temperature (β → ∞) for a number of quantum mechanical problems has been developed in [18][19][20]. The quantum corrections have been calculated to 3 loops, and shown to be in exact agreement with asymptotic expansion of the ground state wave functions at large distances.…”

Radial anharmonic oscillator: Perturbation theory, new semiclassical expansion, approximating eigenfunctions. II. Quartic and sextic anharmonicity cases

“…Here the expansion of ε in powers of λ is given by (I. 3). Note that all generating functions Z 2k+1 (u), k = 1, 2, ... of odd order λ 2k+1 are absent in expansion (15).…”

Transseries for the ground state density and generalized Bloch equation: Double-well potential case

“…can be constructed self-consistently, without involving non-perturbative, exponentially small terms, c.f. [3], Section III.C.2 . Owing to this property we can separate perturbative and non-perturbative contributions in z!…”

“…The celebrated quartic double-well potential will be taken as the example. Thus, overall, the derivation will be made from two different directions: (i) from quantum mechanics using the the generalized Bloch equation of the type presented in [3] and (ii) from the Euclidian time path integral following a variety of flucton-instanton trajectories.…”

“…In order to proceed let us derive the generalized Bloch equation, c.f. [3], specific for the potential with two degenerate minima. The first step is standard, we begin with the Schrödinger equation for the wave function and go to one on its logarithmic derivative y(x), which eliminates the overall normalization constant from consideration.…”