Nonequilibrium quantum anharmonic oscillator and scalar field: high temperature approximations

Abstract

We treat a relativistic quantum boson gas, described by a scalar quantum field, with quartic self‐interaction (ϕ4) in three spatial dimensions: we review the known equilibrium case and present new proposals off‐equilibrium. For high temperature and large spatial scales, the behaviour of the gas at equilibrium simplifies nonperturbatively (equilibrium dimensional reduction or EDR): its thermodynamics is described by classical statistical mechanics with some quantum field effects. By assumption, the initial state of the gas off‐equilibrium includes interactions and inhomogeneities and is not far from thermal equilibrium. We employ real‐time generating functionals and obtain the free nonequilibrium correlators at non‐zero temperature. The nonequilibrium quantum gas appears to simplify nonperturbatively in the regime of high temperature and large temporal and spatial scales (nonequilibrium dimensional reduction or NEDR), its dynamics being described by classical statistical mechanics with some quantum field effects. We outline the renormalization of the ϕ4 theory, the nonequilibrium statistical mechanics of a quantum anharmonic oscillator and the high temperature simplifications, all of which provide very useful hints for NEDR in the field case. Our main proposals are NEDR and the associated new (renormalized) real‐time nonequilibrium generating functionals for the ϕ4 theory.

Approach to Equilibrium of Statistical Systems: Classical Particles and Quantum Fields Off-Equilibrium

“…Let Λ < +∞. The following new arguments for the field-theoretic case, which generalize the studies in [42] (for the quantum anharmonic oscillator) and [64] (for a quantum particle coupled to a thermal reservoir), will confirm that Equations ( 100) and ( 101) do account for a classical field theory with (unrenormalized) squared mass parameter m 2 r + δm 2 dr . In fact, the structure (−(m 101) does characterize a classical field theory: compare, for instance, with Section 3.…”

“…In fact, the structure (−(m 101) does characterize a classical field theory: compare, for instance, with Section 3. We now outline another supporting argument (see [42]). Let χ ∆ ≡ χ − χ and χ c ≡ 2 −1 (χ + χ ), let t − t 0 be very small, and let us consider…”

“…A quantum anharmonic oscillator of mass m in one spatial dimension (coordinate φ), momentum Π and frequency ω will be treated here, based on [42]. The method in Section 4 could be applied to it, but functional methods will be preferred, as an introduction to those to be applied for the quantum field case in Section 6.…”

“…We shall perform approximations for high temperature (small β eq ) and large distances and long time for Z r [J c , J ∆ , J in ] (non-equilibrium dimensional reduction or NEDR) [42,43]. They will constitute nontrivial generalizations of those for equilibrium in Sections 6.1.2 and 6.1.4.…”

“…The present author has analyzed the classical Liouville and quantum Wigner equations, by using infinite hierarchies in order to implement approaches to equilibrium [33][34][35][36][37][38][39][40][41]. The analysis has been extended, using different techniques (generating functionals), to off-equilibrium evolution in the relativistic quantum φ 4 field theory [42,43]. The above approaches face various complications, which have been treated succesively.…”