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Progress of Theoretical Physics

Vacuum Diagrams in Perturbative Thermo Field Dynamics

H. Matsumoto, Hiroomi Umezawa, J. P. Whitehead

Published Jul 1, 1986• 17 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.76.260

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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.
We formulate the thermofield dynamics for time-dependent systems by combining the Liouville-von Neumann equation, its invariant operators, and the basic notions of thermofield dynamics. The new formulation is applied to time-dependent bosons and fermions by using the time-dependent annihilation and creation operators that satisfy the Liouville-von Neumann equation. It is shown that the thermal state is the time- and temperature-dependent vacuum state and a general formula is derived to calculate the thermal expectation value of operators.
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 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.
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.