Quantum and thermal fluctuations in quantum mechanics and field theories from a new version of semiclassical theory

We develop a new semiclassical approach, which starts with the density matrix given by the Euclidean time path integral with fixed coinciding end points, and proceed by identifying classical (minimal Euclidean action) path, to be referred to as a flucton, which passes through this end point. Fluctuations around a flucton path are included, by standard Feynman diagrams, previously developed for instantons. We calculate the Green function and evaluate the one loop determinant both by direct diagonalization of the fluctuation equation and also via the trick with the Green functions. The two-loop corrections are evaluated by explicit Feynman diagrams, and some curious cancellation of logarithmic and polylog terms is observed. The results are fully consistent with large-distance asymptotics obtained in quantum mechanics. Two classic examples—quartic double-well and sine-Gordon potentials—are discussed in detail, while powerlike potential and quartic anharmonic oscillator are discussed in brief. Unlike other semiclassical methods, like WKB, we do not use the Schrödinger equation, and all the steps generalize to multidimensional or quantum fields cases straightforwardly.

Figure 1

Time dependence of the classical flucton solution $y_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{c}\mathrm{t}}(\tau )$, see (25) (upper plot) and the corresponding potential $(1+W)$, see (37) of the fluctuations (lower plot), both for $x_0=2$, $\lambda =0.1$.

Figure 2

The integrand of (47), $\log (1+p^2)d{\delta }_p/dp$, versus $p$, for $X=4$, see upper plot. The integral (47) vs parameter $X$: points are numerical evaluation, line is defined in the text, see lower plot.

Figure 3

Symbolic one-loop diagram, including variation of the fluctuation potential $\delta V$ and the simplified “single-loop” Green function $G(\tau ,\tau )$.

Figure 4

Diagrams contributing to the two-loop correction $B_1=a+b_1+b_2$. The signs of contributions and symmetry factors are indicated.

Figure 5

The two-loop diagrams $a,b_1,b_2$ and the two-loop correction $B_1\equiv a+b_1+b_2$ as a function of variable $X$ (38).

Figure 6

The probability $P(x_0)$ to find particle at location $x_0$ for $\lambda =0.1$. The thick line is our result (76), thin line is asymptotics derived in Appendix pp1.

Figure 7

The integrand of the sum rule, at $\omega =1$.