Dirac Geometry of Time and Space Time as the length of a curve in the space of Dirac operators and space as the spectrum of an algebra with Connes' metric
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| Format: | Recurso digital |
| Langue: | anglais |
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2025
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| _version_ | 1866902064007741440 |
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| author | Szyryngo, Robert |
| author_facet | Szyryngo, Robert |
| contents | <p>This work proposes an axiomatic definition of time and space based on the Dirac operator and spectral geometry. Global time is defined as the length of a curve in the space of Dirac operators equipped with the Hilbert–Schmidt metric. An abstract “square of change” functional on self-adjoint Hilbert–Schmidt operators is characterised, and the axioms single out the Hilbert–Schmidt norm as the unique natural measure of change.</p> <p>Space is defined as the spectrum of an algebra of observables, understood as its pure states, endowed with the Connes spectral distance determined by commutators with the Dirac operator. In the commutative case this reproduces the usual Riemannian distance on a manifold.</p> <p>Combining these structures, the paper introduces the notion of Dirac spacetime: a family of spectral triples with a geodesic time and a time-dependent spatial metric. A corresponding Dirac line element is constructed, with an effective Minkowski signature, and is shown to reduce to the standard Friedmann–Robertson–Walker form in a simple cosmological model. The analysis highlights the special role of the Dirac operator as a common source of both time and space, clarifying conditions under which one can have space without time, or time without nontrivial space.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18008424 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Dirac Geometry of Time and Space Time as the length of a curve in the space of Dirac operators and space as the spectrum of an algebra with Connes' metric Szyryngo, Robert time space Dirac operator spectral geometry noncommutative geometry Connes distance spectral triple geodesic time Dirac spacetime FRW cosmology <p>This work proposes an axiomatic definition of time and space based on the Dirac operator and spectral geometry. Global time is defined as the length of a curve in the space of Dirac operators equipped with the Hilbert–Schmidt metric. An abstract “square of change” functional on self-adjoint Hilbert–Schmidt operators is characterised, and the axioms single out the Hilbert–Schmidt norm as the unique natural measure of change.</p> <p>Space is defined as the spectrum of an algebra of observables, understood as its pure states, endowed with the Connes spectral distance determined by commutators with the Dirac operator. In the commutative case this reproduces the usual Riemannian distance on a manifold.</p> <p>Combining these structures, the paper introduces the notion of Dirac spacetime: a family of spectral triples with a geodesic time and a time-dependent spatial metric. A corresponding Dirac line element is constructed, with an effective Minkowski signature, and is shown to reduce to the standard Friedmann–Robertson–Walker form in a simple cosmological model. The analysis highlights the special role of the Dirac operator as a common source of both time and space, clarifying conditions under which one can have space without time, or time without nontrivial space.</p> |
| title | Dirac Geometry of Time and Space Time as the length of a curve in the space of Dirac operators and space as the spectrum of an algebra with Connes' metric |
| topic | time space Dirac operator spectral geometry noncommutative geometry Connes distance spectral triple geodesic time Dirac spacetime FRW cosmology |
| url | https://doi.org/10.5281/zenodo.18008424 |