Restoring a Missing Meta-Symmetry of Quantum Mechanics

TL;DR

Standard quantum mechanics treats the momentum–energy sector as a static Fourier label living on $\\mathcal{M}_{kE}$; this work introduces a dual-manifold, enlarged Hilbert space $\\mathcal{H}_{total}=\\mathcal{H}_{xt}\\oplus\\mathcal{H}_{kE}$ with independent $E$-evolution generated by $\\hat{\\mathcal{T}}$, establishing a meta-symmetry $(t,\\hat H) \\\leftrightarrow (E,\\hat{\\mathcal{T}})$. The momentum–energy sector gains its own generator and evolution equation $i\\hbar\\partial_E \\phi = \\hat{\\mathcal{T}} \\phi$, with locality and isotropy constraining $\\hat{\\mathcal{T}}$ to $\\alpha \\hat X^2 + V_0$. Key contributions include (i) a concrete dynamical structure on $\\mathcal{M}_{kE}$, (ii) a finite projection-generated dark-energy density $\\rho_\\Lambda=\\int|A|^2\\,dk\\,dE$, and (iii) an exponential boundary mapping near horizons analogous to Hawking radiation, all arising without general relativity. This framework links quantum mechanics with cosmology and black-hole physics, providing a self-consistent two-sector description and a potential route to empirical parameters.

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 $φ(k,E)$ 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.

Figures (1)

• Figure 1: The singularity in the space–time manifold $\mathcal{M}_{xt}$ corresponds to an infinity in the momentum–energy manifold $\mathcal{M}_{kE}$.