Nonequilibrium entropy in classical and quantum field theory

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Abstract
This paper proposes a definition of nonequilibrium entropy appropriate for a bosonic classical or quantum field, viewed as a collection of oscillators with equations of motion which satisfy a Liouville theorem (as is guaranteed for a Hamiltonian system). This entropy S is constructed explicitly to provide a measure of correlations and, as such, is conserved absolutely in the absence of couplings between degrees of freedom. This means, e.g., that there can be no entropy generation for a source‐free linear field in flat space, but that S need no longer be conserved in the presence of couplings induced by nonlinearities, material sources, or a nontrivial dynamical background space‐time. Moreover, through the introduction of a ‘‘subdynamics,’’ it is proved that, in the presence of such couplings, the entropy will satisfy an H‐theorem inequality, at least in one particular limit. Specifically, if at some initial time t0 the field is free of any correlations, it then follows rigorously that, at time t0+Δt, the entropy will be increasing: dS/dt>0. Similar arguments demonstrate that this S is the only measure of ‘‘entropy’’ consistent mathematically with the subdynamics. It is argued that this entropy possesses an intrinsic physical meaning, this meaning being especially clear in the context of a quantum theory, where a direct connection exists between entropy generation and particle creation. Reasonable conjectures regarding the more general time dependence of the entropy, which parallel closely the conventional wisdom of particle mechanics, lead to an interpretation of S which corroborates one’s naive intuition as to the behavior of an ‘‘entropy.’’