Cosmic-Time Quantum Mechanics and the Passage-of-Time Problem

Abstract

A new dynamical paradigm merging quantum dynamics with cosmology is discussed. We distinguish between a universe and its background space-time. The universe here is the subset of space-time defined by Ψτ(x)≠0, where Ψτ(x) is a solution of a Schrödinger equation, x is a point in n-dimensional Minkowski space, and τ≥0 is a dimensionless ‘cosmic-time’ evolution parameter. We derive the form of the Schrödinger equation and show that an empty universe is described by a Ψτ(x) that propagates towards the future inside some future-cone V+. The resulting dynamical semigroup is unitary, i.e., ∫V+d4x|Ψτ(x)|2=1 for τ≥0. The initial condition Ψ0(x) is not localized at x=0. Rather, it satisfies the boundary condition Ψ0(x)=0 for x∉V+. For n=1+3 the support of Ψτ(x) is bounded from the past by the ‘gap hyperboloid’ ℓ2τ=c2t2−x2, where ℓ is a fundamental length. Consequently, the points located between the hyperboloid and the light cone c2t2−x2=0 satisfy Ψτ(x)=0, and thus do not belong to the universe. As τ grows, the gap between the support of Ψτ(x) and the light cone increases. The past thus literally disappears. Unitarity of the dynamical semigroup implies that the universe becomes localized in a finite-thickness future-neighbourhood of ℓ2τ=c2t2−x2, simultaneously spreading along the hyperboloid. Effectively, for large τ the subset occupied by the universe resembles a part of the gap hyperboloid itself, but its thickness Δτ is non-zero for finite τ. Finite Δτ implies that the three-dimensional volume of the universe is finite as well. An approximate radius of the universe, rτ, grows with τ due to Δτrτ3=Δ0r03 and Δτ→0. The propagation of Ψτ(x) through space-time matches an intuitive picture of the passage of time. What we regard as the Minkowski-space classical time can be identified with ctτ=∫d4xx0|Ψτ(x)|2, so tτ grows with τ as a consequence of the Ehrenfest theorem, and its present uncertainty can be identified with the Planck time. Assuming that at present values of τ (corresponding to 13–14 billion years) Δτ and rτ are of the order of the Planck length and the Hubble radius, we estimate that the analogous thickness Δ0 of the support of Ψ0(x) is of the order of 1 AU, and r03∼(ctH)3×10−44. The estimates imply that the initial volume of the universe was finite and its uncertainty in time was several minutes. Next, we generalize the formalism in a way that incorporates interactions with matter. We are guided by the correspondence principle with quantum mechanics, which should be asymptotically reconstructed for the present values of τ. We argue that Hamiltonians corresponding to the present values of τ approximately describe quantum mechanics in a conformally Minkowskian space-time. The conformal factor is directly related to |Ψτ(x)|2. As a by-product of the construction, we arrive at a new formulation of conformal invariance of m≠0 fields.