Quantum variational calculus on a lattice
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911265449836544 |
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| author | Majid, Shahn Simão, Francisco |
| author_facet | Majid, Shahn Simão, Francisco |
| contents | We solve the long-standing problem of variational calculus on a noncommutative space or spacetime for a significant class of models with trivial jet bundle. Our approach entails a quantum version of the Anderson variational double complex $Ω(J^\infty)$ and includes Euler-Lagrange equations and a partial Noether's theorem. We show in detail how this works for a free field on a $\Bbb Z^m$ lattice regarded as a discrete noncommutative geometry, obtaining the Klein-Gordon equation for a scalar field, including with a general metric and gauge field background, as the Euler-Lagrange equations of motion for an action. In the case of a flat metric we also obtain an exactly on-shell conserved stress-energy tensor and Noether charges for a scalar field on the lattice and modified energy-momentum relations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_02628 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantum variational calculus on a lattice Majid, Shahn Simão, Francisco High Energy Physics - Theory General Relativity and Quantum Cosmology Quantum Algebra We solve the long-standing problem of variational calculus on a noncommutative space or spacetime for a significant class of models with trivial jet bundle. Our approach entails a quantum version of the Anderson variational double complex $Ω(J^\infty)$ and includes Euler-Lagrange equations and a partial Noether's theorem. We show in detail how this works for a free field on a $\Bbb Z^m$ lattice regarded as a discrete noncommutative geometry, obtaining the Klein-Gordon equation for a scalar field, including with a general metric and gauge field background, as the Euler-Lagrange equations of motion for an action. In the case of a flat metric we also obtain an exactly on-shell conserved stress-energy tensor and Noether charges for a scalar field on the lattice and modified energy-momentum relations. |
| title | Quantum variational calculus on a lattice |
| topic | High Energy Physics - Theory General Relativity and Quantum Cosmology Quantum Algebra |
| url | https://arxiv.org/abs/2508.02628 |