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arXiv:1605.05165 · Physical review. D/Physical review. D.

Operator description for thermal quantum field theories on an arbitrary path in the real time formalism

Ashok Das, Pushpa Kalauni

Published Jun 22, 2016• 8 citations
arXivHTMLPDF

Abstract

We develop an operator description, much like thermofield dynamics, for quantum field theories on a real time path with an arbitrary parameter $\ensuremath{\sigma}\text{ }(0\ensuremath{\le}\ensuremath{\sigma}\ensuremath{\le}\ensuremath{\beta})$. We point out new features which arise when $\ensuremath{\sigma}\ensuremath{\ne}\frac{\ensuremath{\beta}}{2}$ in that the Hilbert space develops a natural, modified inner product different from the standard Dirac inner product. We construct the Bogoliubov transformation which connects the doubled vacuum state at zero temperature to the thermal vacuum in this case. We obtain the thermal Green's function (propagator) for the real massive Klein-Gordon theory as an expectation value in this thermal vacuum (with a modified inner product). The factorization of the thermal Green's function follows from this analysis. We also discuss, in the main text as well as in two appendices, various other interesting features which arise in such a description.

DOI
10.1103/physrevd.93.125028

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We develop an operator description, much like thermofield dynamics, for quantum field theories on a real time path with an arbitrary parameter \ensuremathσ (0\ensuremath≤\ensuremathσ\ensuremath≤\ensuremathβ)\ensuremath{\sigma}\text{ }(0\ensuremath{\le}\ensuremath{\sigma}\ensuremath{\le}\ensuremath{\beta})\ensuremathσ (0\ensuremath≤\ensuremathσ\ensuremath≤\ensuremathβ). We point out new features which arise when \ensuremathσ\ensuremath≠\ensuremathβ2\ensuremath{\sigma}\ensuremath{\ne}\frac{\ensuremath{\beta}}{2}\ensuremathσ\ensuremath=2\ensuremathβ​ in that the Hilbert space develops a natural, modified inner product different from the standard Dirac inner product. We construct the Bogoliubov transformation which connects the doubled vacuum state at zero temperature to the thermal vacuum in this case. We obtain the thermal Green's function (propagator) for the real massive Klein-Gordon theory as an expectation value in this thermal vacuum (with a modified inner product). The factorization of the thermal Green's function follows from this analysis. We also discuss, in the main text as well as in two appendices, various other interesting features which arise in such a description.
It has been shown earlier [Phys. Rev. D 72, 085006 (2005), Phys. Rev. D 73, 065010 (2006)] that, in the mixed space, there is an unexpected simple relation between any finite temperature graph and its zero temperature counterpart through a multiplicative scalar operator (termed thermal operator) which carries the entire temperature dependence. This holds only in the imaginary time formalism and the closed time path ($\ensuremath{\sigma}=0$) of the real time formalism (as well as for its conjugate $\ensuremath{\sigma}=1$). We study the origin of this operator from the more fundamental Bogoliubov transformation which acts, in the momentum space, on the doubled space of fields in the real time formalisms [Collective Phenomena 2, 55 (1975), Int. J. Mod. Phys. B 10, 1755 (1996), Thermo Field Dynamics and Condensed States (North-Holland, Amsterdam, 1982), Phys. Rev. D 93, 125028 (2016)]. We show how the ($2\ifmmode\times\else\texttimes\fi{}2$) Bogoliubov transformation matrix naturally leads to the scalar thermal operator for $\ensuremath{\sigma}=0$, 1 while it fails for any other value $0<\ensuremath{\sigma}<1$. This analysis also suggests that a generalized scalar thermal operator description, in the mixed space, is possible even for $0<\ensuremath{\sigma}<1$. We also show the existence of a scalar thermal operator relation in the momentum space.
Perturbation theory in thermofield dynamics (a real time finite temperature quantum field theory) is formulated in the interaction representation and a diagrammatic expansion for the expectation value of time ordered products is derived solely on the basis of the thermal state condition and the algebraic properties of the operators. This new formulation has a much wider domain of applicability than those presented previously. The method is applicable even in the case of purely quantum mechanical systems. When certain bosonic operators, which commute with the unperturbed Hamiltonian, contribute to the renormalization of the thermal state vacuum the vacuum diagrams do not cancel in the usual way. It will be shown how these diagrams arise and how they may be computed.
The Gaussian wavefunctional approach is developed in thermofield dynamics. We construct the thermal vacuum wavefunctional, its creation as well as annihilation operators, and accordingly the thermo-particle excited states. For a ( D + 1)-dimensional scalar field system with an arbitrary potential whose Fourier representation exists in the sense of tempered distributions, we calculate the finite-temperature Gaussian effective potential (FTGEP), one- and two-thermo-particle-state energies. The zero-temperature limit of each of them is just the corresponding result in quantum field theory, and the FTGEP can lead to the same one for some concrete models as calculated by the imaginary-time Green function.
Thermofield Dynamics (TFD), a real time formalism for thermal quantum field theory, is formulated in terms of a path-integral approach following the Weinberg procedure.In order to assure precise rules for the development, we use a representation of Lie-algebras such that the TFD algebraic ingredients are derived, including the tilde conjugation rules.The association with the canonical formalism is obtained, and for the case of bosons, we introduce the Feynman diagrams and derive the n-point functions.
We present a comprehensive review of the most fundamental and practical aspects of thermo-field dynamics (TFD), including some of the most recent developments in the field. To make TFD fully consistent, some suitable changes in the structure of the thermal doublets and the Bogoliubov transformation matrices have been made. A close comparison between TFD and the Schwinger-Keldysh closed time path formalism (SKF) is presented. We find that TFD and SKF are in many ways the same in form; in particular, the two approaches are identical in stationary situations. However, TFD and SKF are quite different in time-dependent nonequilibrium situations. The main source of this difference is that the time evolution of the density matrix itself is ignored in SKF while in TFD it is replaced by a time-dependent Bogoliubov transformation. In this sense TFD is a better candidate for time-dependent quantum field theory. Even in equilibrium situations, TFD has some remarkable advantages over the Matsubara approach and SKF, the most notable being the Feynman diagram recipes, which we will present. We will show that the calculations of two-point functions are simplified, instead of being complicated, by the matrix nature of the formalism. We will present some explicit calculations using TFD, including space-time inhomogeneous situations and the vacuum polarization in equilibrium relativistic QED.
We introduce a method, numerically exact in principle, for computing the momentum- and frequency-resolved single-particle Green's function of a polaron at finite temperature. The method, which we refer to as the finite-temperature Green's function cluster expansion, combines two ingredients: the generalized Green's function cluster expansion, a numerically exact extension of the momentum average family of methods that solves the polaron problem at zero temperature through a hierarchy of equations of motion for restricted phonon cloud configurations; and the thermofield double formalism, which maps the thermal trace onto a pure-state expectation value over a doubled Hilbert space. The resulting equations of motion have the same algebraic structure as those of the multi-boson zero-temperature theory, with the temperature entering through a Bogoliubov-type mixing angle that controls the coupling to a set of fictitious bath bosons. We implement the method in our open-source software package and benchmark it on the one-dimensional Holstein polaron, computing spectral functions, dispersions, lifetimes, and effective masses across coupling regimes and temperatures up to T/Ω∼1T/Ω\sim 1T/Ω∼1. Where finite-temperature density matrix renormalization group results are available, we find quantitative agreement at affordable computational cost. The method recovers momentum-resolved spectra directly in frequency space, with no time evolution or analytic continuation. We also discuss the practical costs of the approach. In particular, since the doubled phonon Hilbert space has a non-trivial configuration structure in which real and fictitious clouds compete, convergence in the corresponding cloud parameters requires care.