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

Ψ_{τ} (x) \neq 0

, where

Ψ_{τ} (x)

is a solution of a Schrödinger equation, x is a point in n-dimensional Minkowski space, and

τ \geq 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

Ψ_{τ} (x)

that propagates towards the future inside some future-cone

V_{+}

. The resulting dynamical semigroup is unitary, i.e.,

\int_{V_{+}} d^{4} x {| Ψ_{τ} (x) |}^{2} = 1

for

τ \geq 0

. The initial condition

Ψ_{0} (x)

is not localized at

x = 0

. Rather, it satisfies the boundary condition

Ψ_{0} (x) = 0

for

x \notin V_{+}

. For

n = 1 + 3

the support of

Ψ_{τ} (x)

is bounded from the past by the ‘gap hyperboloid’

ℓ^{2} \sqrt{τ} = 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

Ψ_{τ} (x) = 0

, and thus do not belong to the universe. As

τ

grows, the gap between the support of

Ψ_{τ} (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

ℓ^{2} \sqrt{τ} = c^{2} t^{2} - x^{2}

, 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

Δ_{τ}

is non-zero for finite

τ

. Finite

Δ_{τ}

implies that the three-dimensional volume of the universe is finite as well. An approximate radius of the universe,

r_{τ}

, grows with

τ

due to

Δ_{τ} r_{τ}^{3} = Δ_{0} r_{0}^{3}

and

Δ_{τ} \to 0

. The propagation of

Ψ_{τ} (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

c t_{τ} = \int d^{4} x x^{0} {| Ψ_{τ} (x) |}^{2}

, so

t_{τ}

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)

Δ_{τ}

and

r_{τ}

are of the order of the Planck length and the Hubble radius, we estimate that the analogous thickness

Δ_{0}

of the support of

Ψ_{0} (x)

is of the order of 1 AU, and

r_{0}^{3} \sim {(c t_{H})}^{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

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

| Ψ_{τ} {(x) |}^{2}

. As a by-product of the construction, we arrive at a new formulation of conformal invariance of

m \neq 0

fields.

51 Results Found

A C0-Semigroup of Ulam Unstable Operators

Distributional Chaoticity of C0-Semigroup on a Frechet Space

C-Semigroups and Their Induced Order

The Adjoint of α-Times-Integrated C-Regularized Semigroups

Cuntz Semigroups of Compact-Type Hopf C*-Algebras

Parameter–Elliptic Fourier Multipliers Systems and Generation of Analytic and C∞ Semigroups

A Counterexample Concerning C0-Semigroups of Holomorphic Carathéodory Isometries

C0–Semigroups Approach to the Reliability Model Based on Robot-Safety System

Categorically Closed Unipotent Semigroups

On Some Properties for Cofiniteness of Submonoids and Ideals of an Affine Semigroup

Ulam Stability for the Composition of Operators

Dynamic Analysis of the M/G/1 Stochastic Clearing Queueing Model in a Three-Phase Environment

Self-Similar Inverse Semigroups from Wieler Solenoids

Analysis of a k/n(G) Retrial System with Multiple Working Vacations

Nonlocal Conformable Differential Inclusions Generated by Semigroups of Linear Bounded Operators or by Sectorial Operators with Impulses in Banach Spaces

Subordination Principle for a Class of Fractional Order Differential Equations

On the Evolution Operators of a Class of Linear Time-Delay Systems

On the Non-Hypercyclicity of Normal Operators, Their Exponentials, and Symmetric Operators

Dynamic Analysis of a Standby System with Retrial Strategies and Multiple Working Vacations

A General Class of Differential Hemivariational Inequalities Systems in Reflexive Banach Spaces

Reliability Analysis and Numerical Simulation of Industrial Robot Drive System with Vacation

Polynomial Decay of the Energy of Solutions of the Timoshenko System with Two Boundary Fractional Dissipations

Direct and Inverse Fractional Abstract Cauchy Problems

Groups and Structures of Commutative Semigroups in the Context of Cubic Multi-Polar Structures

Convergence of Implicit Iterative Processes for Semigroups of Nonlinear Operators Acting in Regular Modular Spaces

Categorically Closed Topological Groups

On the Independence Number of Cayley Digraphs of Clifford Semigroups

Idempotent-Aided Factorizations of Regular Elements of a Semigroup

A Class of BCI-Algebra and Quasi-Hyper BCI-Algebra

CDE’ Inequality on Graphs with Unbounded Laplacian

Existence of Hilfer Fractional Stochastic Differential Equations with Nonlocal Conditions and Delay via Almost Sectorial Operators

Relationships among Various Chaos for Linear Semiflows Indexed with Complex Sectors

A Note on the Topological Group c0

Symmetric Properties of (b,c)-Inverses

On a Semilinear Parabolic Problem with Four-Point Boundary Conditions

The Regulator Problem to the Convection–Diffusion Equation

Criterion of the Existence of a Strongly Continuous Resolving Family for a Fractional Differential Equation with the Hilfer Derivative

A Class of Semilinear Parabolic Problems and Analytic Semigroups

A Cotangent Fractional Derivative with the Application

Quasi-Semilattices on Networks

New Discussion on Approximate Controllability for Semilinear Fractional Evolution Systems with Finite Delay Effects in Banach Spaces via Differentiable Resolvent Operators

Constructing C0-Semigroups via Picard Iterations and Generating Functions: An Application to a Black–Scholes Integro-Differential Operator

Strong Stability for a Viscoelastic Transmission Problem Under a Nonlocal Boundary Control

Generalized Hyers-Ulam Stability of Trigonometric Functional Equations

The Covariety of Saturated Numerical Semigroups with Fixed Frobenius Number

Square-Mean S-Asymptotically (ω,c)-Periodic Solutions to Neutral Stochastic Impulsive Equations

Asymptotic Behavior of the Time-Dependent Solution of the M[X]/G/1 Queuing Model with Feedback and Optional Server Vacations Based on a Single Vacation Policy

Viability for Semilinear Differential Equations with Infinite Delay

Left-Symmetric Algebras Arising from Modified DNA Insertion Operations

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

A C₀-Semigroup of Ulam Unstable Operators

Distributional Chaoticity of C₀-Semigroup on a Frechet Space

$C$ -Semigroups and Their Induced Order

Parameter–Elliptic Fourier Multipliers Systems and Generation of Analytic and C^∞ Semigroups

A Counterexample Concerning C₀-Semigroups of Holomorphic Carathéodory Isometries

C₀–Semigroups Approach to the Reliability Model Based on Robot-Safety System

A Note on the Topological Group c₀

Constructing C₀-Semigroups via Picard Iterations and Generating Functions: An Application to a Black–Scholes Integro-Differential Operator

Asymptotic Behavior of the Time-Dependent Solution of the M^[X]/G/1 Queuing Model with Feedback and Optional Server Vacations Based on a Single Vacation Policy