Narrow Results

The Einstein–Hilbert Lagrangian R is expressed in terms of the chronometrically invariant quantities introduced by Zel'manov for an arbitrary four-dimensional metric g_ij. The chronometrically invariant three-space is the physical space γ_αβ = -g_αβ+e^2ϕ γ_αγ_β, where e^2ϕ = g₀₀ and γ_α = g_0α/g₀₀, and whose determinant is h. The momentum canonically conjugate to γ_αβ is , where and ∂_t≡e^-ϕ∂₀ is the chronometrically invariant derivative with respect to time. The Wheeler–DeWitt equation for the wave function Ψ is derived. For a stationary space-time, such as the Kerr metric, π^αβ vanishes, implying that there is then no dynamics. The most symmetric, chronometrically-invariant space, obtained after setting ϕ = γ_α = 0, is , where δ_αβ is constant and has curvature k. From the Friedmann and Raychaudhuri equations, we find that λ is constant only if k=1 and the source is a perfect fluid of energy-density ρ and pressure p=(γ-1)ρ, with adiabatic index γ=2/3, which is the value for a random ensemble of strings, thus yielding a three-dimensional de Sitter space embedded in four-dimensional space-time. Furthermore, Ψ is only invariant under the time-reversal operator if γ=2/(2n-1), where n is a positive integer, the first two values n=1,2 defining the high-temperature and low-temperature limits ρ ~ T^±2, respectively, of the heterotic superstring theory, which are thus dual to one another in the sense T↔1/2π²α′T.

The time reversal of a completely-positive, non-equilibrium discrete-time quantum Markov evolution is derived via a suitable adjointness relation. Space-time harmonic processes are introduced for the forward and reverse-time transition mechanisms, and their role for relative entropy dynamics is discussed.