Abstract
The two-dimensional σ-model with the de Sitter target space is locally canonic in the north pole diamond of the Penrose diagram in the cosmological gauge. The left and right moving modes on the cylindrical base space are entangled among themselves and interact with the de Sitter metric. Firstly, we show that the untangled oscillators can be obtained from the entangled operators by applying a set of Bogoliubov transformations constrained by the requirement that the partial evolution generator be diagonal. Secondly, we determine the nonequilibrium dynamics of the untangled modes in the nonequilibrium thermofield dynamics formalism. The thermal modes are represented as thermal doublet oscillators that satisfy partial evolution equations of Heisenberg-type. From these we compute the local free one-body propagator of an arbitrary mode between two times. Thirdly, we discuss the field representation of the thermal modes. We show that there is a set of thermal doublet fields that satisfy the equal time canonical commutation relations, are solutions to the σ-model equations of motion, and can be decomposed in terms of thermal doublet oscillators. Finally, we construct a local partial evolution functional of Hamilton-like form for the thermal doublet fields.
Highlights
The two-dimensional nonlinear σ-model with the de Sitter target space describes a nontrivial class of field theories of which quantum dynamics is still largely unknown
We examine the thermalization of the untangled oscillators in the nonequilibrium thermofield dynamics (NETFD) formalism in which the degrees of freedom of the thermal oscillators are most naturally represented by thermal doublets [14,15,16]
We have continued the analysis of the twodimensional σ-model with the de Sitter target space in the cosmological gauge started in [11]
Summary
The two-dimensional nonlinear σ-model with the de Sitter target space describes a nontrivial class of field theories of which quantum dynamics is still largely unknown. We propose a simple solution to the problem of the untangled oscillator representation by constructing a set of linear time-dependent Bogoliubov transformations that diagonalize the local Hamiltonian-like generator of the partial evolution map. We give a second set of Bogoliubov transformations that map the untangled oscillators into two different representations: the time-independent and the time-dependent pseudoparticle representations, respectively, which are needed to define the vacuum in which the physical quantities should be computed and to determine the dynamics of the thermal oscillators. These are used to calculate the propagator of an arbitrary thermal doublet mode between two different values of time. In Appendix, we briefly review the NETFD relations that are used throughout this work
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