Abstract
Wavepackets in quantum mechanics spread and the Universe in cosmology expands. We discuss a formalism where the two effects can be unified. The basic assumption is that the Universe is determined by a unitarily evolving wavepacket defined on space-time. Space-time is static but the Universe is dynamic. Spreading analogous to expansion known from observational cosmology is obtained if one regards time evolution as a discrete process with probabilities of jumps determined by a variational principle employing Kolmogorov-Nagumo-R\'enyi averages. The choice of the R\'enyi calculus implies that the form of the Universe involves an implicit fractal structure. The formalism automatically leads to two types of "time" parameters: $\tau$, with dimension of $x^0$, and dimensionless $\varepsilon=\ln \epsilon_\tau$, related to the form of diffeomorphism that defines the dynamics. There is no preferred time foliation, but effectively the dynamics leads to asymptotic concentration of the Universe on spacelike surfaces that propagate in space-time. The analysis is performed explicitly in $1+1$ dimensions, but the unitary evolution operator is brought to a form that makes generalizations to other dimensions and other fields quite natural.
Highlights
Normalizable wavepackets determine regions of space where quantum particles can be found
The issue reduces to finding an appropriate one-parameter group of diffeomorphisms whose pull-back to the level of the wave function implies a Heisenberg picture dynamics of the geodesic position operator qualitatively agreeing with observational cosmology [1]
The problem of combining groups of diffeomorphisms originating from some classical geometric theory with unitary dynamics of the Universe can be formulated in a way which resembles Hilbert-space approaches of Koopman [12] and von Neumann [13, 14], proposed in 1930s in the context of classical mechanics
Summary
Normalizable wavepackets determine regions of space where quantum particles can be found. The issue reduces to finding an appropriate one-parameter group of diffeomorphisms whose pull-back to the level of the wave function implies a Heisenberg picture dynamics of the geodesic position operator qualitatively agreeing with observational cosmology [1] One such model naturally appears if one relates time evolution with an extremal entropy principle of the type discussed in the 1930s by Volterra [2]. Having in mind future generalization in terms of Renyi entropies, let us experiment with a simple exponential map associated with the Volterra process, ητ = eλτ η = (p1/p0)η, ξτ = e−λτ ξ , Jτ = 1 Universe) remains in a superposition of two non-overlapping parallel universes occupying non-overlapping regions of space-time (Figs. 5, 6, 7 and 8)
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