Quantum geometry from commutators: a Heisenberg-picture framework and a toy application to early structure
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866911518294016000 |
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| author | Kamali, Vahid |
| author_facet | Kamali, Vahid |
| contents | We develop a Heisenberg-picture \emph{kinematical} framework in which (i) time is treated as a quantum observable, admitting both a relational POVM construction for semibounded spectra and a fully self-adjoint realization on an enlarged (conjugate-energy) Hilbert space enabled by a gravitational conjugation symmetry $\mathcal{C}_g$, and (ii) the generators of spacetime translations need not commute in curved backgrounds. The central postulate, $[\,\hat{x}_μ,\hat{P}_ν\,]=\mathrm{i}\hbar\,\hat g_{μν}(\hat{x})$, makes the spacetime metric a \emph{metric operator} defined by the symmetrized commutator. Jacobi identities close the algebra and imply an operator form of metric compatibility; in a worked FRW example we obtain $[\,\hat{P}_0,\hat{P}_i\,]=2\mathrm{i}\hbar\,N^2(t)\,H(t)\,\hat{P}_i$, which reduces to $2\mathrm{i}\hbar\,H\,\hat{P}_i$ in cosmic-time gauge $N=1$, exhibiting Hubble--controlled non-commuting ``translations.'' A key structural ingredient is the symmetry $\mathcal{C}_g$: an antiunitary map that flips all translation generators, $\hat P_μ\!\to\!-Θ\hat P_μΘ^{-1}$, while covariantly transforming the metric and Lorentz sectors, leaving the canonical commutators and the $[P,P]$ algebra invariant. We discuss uncertainty relations and show how metric-operator fluctuations can rescale primordial amplitudes; an explicitly labeled \emph{toy} propagation of such a rescaling to high-$z$ halo abundances is given in Appendix~$D$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_14533 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quantum geometry from commutators: a Heisenberg-picture framework and a toy application to early structure Kamali, Vahid General Physics We develop a Heisenberg-picture \emph{kinematical} framework in which (i) time is treated as a quantum observable, admitting both a relational POVM construction for semibounded spectra and a fully self-adjoint realization on an enlarged (conjugate-energy) Hilbert space enabled by a gravitational conjugation symmetry $\mathcal{C}_g$, and (ii) the generators of spacetime translations need not commute in curved backgrounds. The central postulate, $[\,\hat{x}_μ,\hat{P}_ν\,]=\mathrm{i}\hbar\,\hat g_{μν}(\hat{x})$, makes the spacetime metric a \emph{metric operator} defined by the symmetrized commutator. Jacobi identities close the algebra and imply an operator form of metric compatibility; in a worked FRW example we obtain $[\,\hat{P}_0,\hat{P}_i\,]=2\mathrm{i}\hbar\,N^2(t)\,H(t)\,\hat{P}_i$, which reduces to $2\mathrm{i}\hbar\,H\,\hat{P}_i$ in cosmic-time gauge $N=1$, exhibiting Hubble--controlled non-commuting ``translations.'' A key structural ingredient is the symmetry $\mathcal{C}_g$: an antiunitary map that flips all translation generators, $\hat P_μ\!\to\!-Θ\hat P_μΘ^{-1}$, while covariantly transforming the metric and Lorentz sectors, leaving the canonical commutators and the $[P,P]$ algebra invariant. We discuss uncertainty relations and show how metric-operator fluctuations can rescale primordial amplitudes; an explicitly labeled \emph{toy} propagation of such a rescaling to high-$z$ halo abundances is given in Appendix~$D$. |
| title | Quantum geometry from commutators: a Heisenberg-picture framework and a toy application to early structure |
| topic | General Physics |
| url | https://arxiv.org/abs/2603.14533 |