Entropy generation, particle creation, and quantum field theory in a cosmological spacetime: When do number and entropy increase?

This paper reexamines the statistical quantum field theory of a free, minimally coupled, real scalar field {Φ} in a statically bounded, classical Friedmann cosmology, where the time-dependent scale factor {Ω}(t) tends to constant values Ω₁ and Ω₂ for tt₁ and t>t₂. The principal objective is to investigate the intuition that ``entropy'' S correlates with average particle number 〈N〉, so that increases in 〈N〉 induced by parametric amplification manifest a one-to-one connection with increases in S. The definition of particle number Nₖ becomes unambiguous for t>t₂ and tt₁, where the spacetime is static, the spatial modes ±{}k for early and late times being coupled by Bogoliubov coefficients αₖ and βₖ.The textbook entropy associated with some density matrix {ρ} (for a state either mixed or pure) is conserved since {ρ} evolves unitarily, so that one is led instead to consider a new measure SN(t) defined in terms of P({k,Nₖ}), the probability of observing Nₖ quanta in each mode k, which may be viewed as a diagonal component of {ρ} in a number representation. A key observation then is that 〈Nₖ(t₂)〉-〈Nₖ(t₁)〉 is guaranteed generically to be positive only for special initial data which, in a number representation, are characterized by ``random phases'' in the sense that any relative phase for the projection of {ρ}(t₁) into two different number eigenstates is ``random'' or ``unobservable physically,'' and averaged over in a density matrix. More importantly for the notion of entropy, random-phase initial data also guarantee an increase in the spread of P({k,Nₖ}), so that, e.g., the sum of the variances Δ²{N}_{±{}k}$(${t}₂$) exceeds the initial ${{Δ}}²N_±k(t₁). It is this increasing spread in P, rather than the growth in average numbers per se, which suggests that, for initial data manifesting random phases, SN(t₂)>SN(t₁), a result established rigorously in the limits of strong and weak particle creation.