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Register a new userFluctuations in quantum mechanics and field theories from a new version of semiclassical theory. II
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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.
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We develop a new semiclassical approach, which starts with the density matrix given by the Euclidean time path integral with fixed coinciding endpoints, and proceed by identifying classical (minimal Euclidean action) path, to be referred to as {it flucton}, which passes through this endpoint. Fluctuations around 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 power-like potential and quartic anharmonic oscillator are discussed in brief. Unlike other semiclassical methods, like WKB, we do not use the Schr{o}dinger equation, and all the steps generalize to multi-dimensional or quantum fields cases straightforwardly.
We compute the quantum string entropy S_s(m, H) from the microscopic string density of states rho_s (m,H) of mass m in de Sitter space-time. We find for high m, a {bf new} phase transition at the critical string temperature T_s= (1/2 pi k_B)L c^2/alpha, higher than the flat space (Hagedorn) temperature t_s. (L = c/H, the Hubble constant H acts at the transition as producing a smaller string constant alpha and thus, a higher tension). T_s is the precise quantum dual of the semiclassical (QFT Hawking-Gibbons) de Sitter temperature T_sem = hbar c /(2pi k_B L). We find a new formula for the full de Sitter entropy S_sem (H), as a function of the usual Bekenstein-Hawking entropy S_sem^(0)(H). For L << l_{Planck}, ie. for low H << c/l_Planck, S_{sem}^{(0)}(H) is the leading term, but for high H near c/l_Planck, a new phase transition operates and the whole entropy S_sem (H) is drastically different from the Bekenstein-Hawking entropy S_sem^(0)(H). We compute the string quantum emission cross section by a black hole in de Sitter (or asymptotically de Sitter) space-time (bhdS). For T_sem ~ bhdS << T_s, (early evaporation stage), it shows the QFT Hawking emission with temperature T_sem ~ bhdS, (semiclassical regime). For T_sem ~ bhdS near T_{s}, it exhibits a phase transition into a string de Sitter state of size L_s = l_s^2/L}, l_s= sqrt{hbar alpha/c), and string de Sitter temperature T_s. Instead of featuring a single pole singularity in the temperature (Carlitz transition), it features a square root branch point (de Vega-Sanchez transition). New bounds on the black hole radius r_g emerge in the bhdS string regime: it can become r_g = L_s/2, or it can reach a more quantum value, r_g = 0.365 l_s.
We discuss shallow resonances in the nonrelativistic scattering of two particles using an effective field theory (EFT) that includes an auxiliary field with the quantum numbers of the resonance. We construct the manifestly renormalized scattering amplitude up to next-to-leading order in a systematic expansion. For a narrow resonance, the amplitude is perturbative except in the immediate vicinity of the resonance poles. It naturally has a zero in the low-energy region, analogous to the Ramsauer-Townsend effect. For a broad resonance, the leading-order amplitude is nonperturbative almost everywhere in the regime of validity of the EFT. We regain the results of an EFT without the auxiliary field, which is equivalent to the effective-range expansion with large scattering length and effective range. We also consider an additional fine tuning leading to a low-energy amplitude zero even for a broad resonance. We show that in all cases the requirement of renormalizability when the auxiliary field is not a ghost ensures the resonance poles are in the lower half of the complex momentum plane, as expected by other arguments. The systematic character of the EFT expansion is exemplified with a toy model serving as underlying theory.
It is shown that for one-dimensional anharmonic oscillator with potential $V(x)= a x^2+ldots=frac{1}{g^2},hat{V}(gx)$ (and for perturbed Coulomb problem $V(r)=frac{alpha}{r} + ldots = g,tilde{V}(gr)$) the Perturbation Theory in powers of coupling constant $g$ (weak coupling regime) and semiclassical expansion in powers of $hbar^{1/2}$ for energies coincide. %The same is true for strong coupling regime expansion in inverse fractional powers in $g$ of energy. It is related to the fact that the dynamics developed in two spaces: $x (r)$-space and in $gx (gr)$ space, leads to the same energy spectra. The equations which govern dynamics in these two spaces, the Riccati-Bloch equation and the Generalized Bloch(GB) equation, respectively, are presented. It is shown that perturbation theory for logarithmic derivative of wave function in $gx (gr)$ space leads to true semiclassical expansion in powers of $hbar^{1/2}$ and corresponds to flucton calculus for density matrix in path integral formalism in Euclidean (imaginary) time.
Forthcoming exascale digital computers will further advance our knowledge of quantum chromodynamics, but formidable challenges will remain. In particular, Euclidean Monte Carlo methods are not well suited for studying real-time evolution in hadronic collisions, or the properties of hadronic matter at nonzero temperature and chemical potential. Digital computers may never be able to achieve accurate simulations of such phenomena in QCD and other strongly-coupled field theories; quantum computers will do so eventually, though Im not sure when. Progress toward quantum simulation of quantum field theory will require the collaborative efforts of quantumists and field theorists, and though the physics payoff may still be far away, its worthwhile to get started now. Todays research can hasten the arrival of a new era in which quantum simulation fuels rapid progress in fundamental physics.
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