Restoring a Missing Meta-Symmetry of Quantum Mechanics

TL;DR

This paper extends quantum mechanics to include a dual Hilbert space, restoring symmetry between space-time and momentum-energy representations, and links this to cosmological phenomena like dark energy and black-hole horizons.

Contribution

It introduces an enlarged Hilbert space with autonomous evolution in both conjugate sectors, revealing a meta-symmetry that connects quantum states to cosmological effects.

Findings

Restores symmetry between space-time and momentum-energy in quantum theory.

Reproduces dark-energy background and black-hole horizon effects.

Establishes a dual-manifold geometry with implications for cosmology.

Abstract

In conventional quantum mechanics, all unitary evolution takes place within the space-time Hilbert space $\mathcal H_{xt}=L^2(\mathcal M_{xt})$, with time as the sole evolution parameter. The momentum-energy representation $\phi(k,E)$ is treated merely as a Fourier re-expression of the same state-kinematically equivalent but dynamically inert. Here we restore the fundamental symmetry between the conjugate pairs $(x,t)$ and $(k,E)$ by extending the quantum theory to an enlarged Hilbert space $\mathcal H_{\text{total}} = \mathcal H_{xt} \oplus \mathcal H_{kE}$, within which the momentum-energy sector $\mathcal H_{kE}=L^2(\mathcal M_{kE})$ carries its own autonomous unitary evolution generated by a self-adjoint operator $\hat{\mathcal T}$. The resulting structure establishes a meta-symmetry: a symmetry between two conjugate dynamical projections of a single global quantum state. It produces a dual-manifold geometry in which each domain is locally complete yet globally open, with divergent limits in one mapping onto extended regions in the other. Remarkably, the dual-manifold symmetry alone reproduces both the uniform dark-energy background and the exponential boundary mapping near black-hole horizons that underlies Hawking radiation. This framework thus opens a quantum-theoretic route to cosmological phenomena that are ordinarily treated within general relativity.