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

${\mathsf{\Psi}}_{\mathsf{\tau}}\left(x\right)\ne 0$, where

${\mathsf{\Psi}}_{\mathsf{\tau}}\left(x\right)$ is a solution of a Schrödinger equation,

*x* is a point in

*n*-dimensional Minkowski space, and

$\mathsf{\tau}\ge 0$ 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

${\mathsf{\Psi}}_{\mathsf{\tau}}\left(x\right)$ that propagates towards the future inside some future-cone

${V}_{+}$. The resulting dynamical semigroup is unitary, i.e.,

${\int}_{{V}_{+}}{d}^{4}x{\left|{\mathsf{\Psi}}_{\mathsf{\tau}}\left(x\right)\right|}^{2}=1$ for

$\mathsf{\tau}\ge 0$. The initial condition

${\mathsf{\Psi}}_{0}\left(x\right)$ is not localized at

$x=0$. Rather, it satisfies the boundary condition

${\mathsf{\Psi}}_{0}\left(x\right)=0$ for

$x\notin {V}_{+}$. For

$n=1+3$ the support of

${\mathsf{\Psi}}_{\mathsf{\tau}}\left(x\right)$ is bounded from the past by the ‘gap hyperboloid’

${\ell}^{2}\sqrt{\mathsf{\tau}}={c}^{2}{t}^{2}-{x}^{2}$, where

*ℓ* is a fundamental length. Consequently, the points located between the hyperboloid and the light cone

${c}^{2}{t}^{2}-{x}^{2}=0$ satisfy

${\mathsf{\Psi}}_{\mathsf{\tau}}\left(x\right)=0$, and thus do not belong to the universe. As

$\mathsf{\tau}$ grows, the gap between the support of

${\mathsf{\Psi}}_{\mathsf{\tau}}\left(x\right)$ 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

${\ell}^{2}\sqrt{\mathsf{\tau}}={c}^{2}{t}^{2}-{x}^{2}$, simultaneously spreading along the hyperboloid. Effectively, for large

$\mathsf{\tau}$ the subset occupied by the universe resembles a part of the gap hyperboloid itself, but its thickness

${\mathsf{\Delta}}_{\mathsf{\tau}}$ is non-zero for finite

$\mathsf{\tau}$. Finite

${\mathsf{\Delta}}_{\mathsf{\tau}}$ implies that the three-dimensional volume of the universe is finite as well. An approximate radius of the universe,

${r}_{\mathsf{\tau}}$, grows with

$\mathsf{\tau}$ due to

${\mathsf{\Delta}}_{\mathsf{\tau}}{r}_{\mathsf{\tau}}^{3}={\mathsf{\Delta}}_{0}{r}_{0}^{3}$ and

${\mathsf{\Delta}}_{\mathsf{\tau}}\to 0$. The propagation of

${\mathsf{\Psi}}_{\mathsf{\tau}}\left(x\right)$ 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

$c{t}_{\mathsf{\tau}}=\int {d}^{4}x\phantom{\rule{0.166667em}{0ex}}{x}^{0}{\left|{\mathsf{\Psi}}_{\mathsf{\tau}}\left(x\right)\right|}^{2}$, so

${t}_{\mathsf{\tau}}$ grows with

$\mathsf{\tau}$ as a consequence of the Ehrenfest theorem, and its present uncertainty can be identified with the Planck time. Assuming that at present values of

$\mathsf{\tau}$ (corresponding to 13–14 billion years)

${\mathsf{\Delta}}_{\mathsf{\tau}}$ and

${r}_{\mathsf{\tau}}$ are of the order of the Planck length and the Hubble radius, we estimate that the analogous thickness

${\mathsf{\Delta}}_{0}$ of the support of

${\mathsf{\Psi}}_{0}\left(x\right)$ is of the order of 1 AU, and

${r}_{0}^{3}\sim {\left(c{t}_{H}\right)}^{3}\times {10}^{-44}$. 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

$\mathsf{\tau}$. We argue that Hamiltonians corresponding to the present values of

$\mathsf{\tau}$ approximately describe quantum mechanics in a conformally Minkowskian space-time. The conformal factor is directly related to

$|{\mathsf{\Psi}}_{\mathsf{\tau}}{\left(x\right)|}^{2}$. As a by-product of the construction, we arrive at a new formulation of conformal invariance of

$m\ne 0$ fields.

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