Geometric Deformation of Quantum Mechanics
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909424473341952 |
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| author | Avramidi, Ivan G. Niardi, Roberto |
| author_facet | Avramidi, Ivan G. Niardi, Roberto |
| contents | We develop a novel approach to Quantum Mechanics that we call Curved Quantum Mechanics. We introduce an infinite-dimensional Kähler manifold ${\cal M}$, that we call the state manifold, such that the cotangent space $T_z^*{\cal M}$ is a Hilbert space. In this approach, a state of a quantum system is described by a point in the cotangent bundle $T^*{\cal M}$, that is, by a point $z\in{\cal M}$ in the state manifold and a one-form $ψ\in T^*_z{\cal M}$. The quantum dynamics is described by an infinite-dimensional Hamiltonian system on the state manifold with a magnetic field $H$, which reduces to the Schrödinger equation for zero curvature and reduces to the equations of geodesics for zero magnetic field. The curvature of the state manifold is determined by gravity, that is, by the mass/energy of the system, so that for microscopic systems the manifold is flat and for macroscopic systems it is strongly curved, which prohibits Schrödinger cat type states. We solved the dynamical equations exactly for the complex projective space and the complex hyperbolic space and show that in the case of negative curvature there is a bifurcation at a critical value of the curvature. This means that for small mass all modes are in the quantum regime with the unitary periodic dynamics and for large mass there are classical modes, with not a periodic but rather an exponential time evolution leading to a collapse of the state vector. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_08007 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Geometric Deformation of Quantum Mechanics Avramidi, Ivan G. Niardi, Roberto Quantum Physics High Energy Physics - Theory Mathematical Physics We develop a novel approach to Quantum Mechanics that we call Curved Quantum Mechanics. We introduce an infinite-dimensional Kähler manifold ${\cal M}$, that we call the state manifold, such that the cotangent space $T_z^*{\cal M}$ is a Hilbert space. In this approach, a state of a quantum system is described by a point in the cotangent bundle $T^*{\cal M}$, that is, by a point $z\in{\cal M}$ in the state manifold and a one-form $ψ\in T^*_z{\cal M}$. The quantum dynamics is described by an infinite-dimensional Hamiltonian system on the state manifold with a magnetic field $H$, which reduces to the Schrödinger equation for zero curvature and reduces to the equations of geodesics for zero magnetic field. The curvature of the state manifold is determined by gravity, that is, by the mass/energy of the system, so that for microscopic systems the manifold is flat and for macroscopic systems it is strongly curved, which prohibits Schrödinger cat type states. We solved the dynamical equations exactly for the complex projective space and the complex hyperbolic space and show that in the case of negative curvature there is a bifurcation at a critical value of the curvature. This means that for small mass all modes are in the quantum regime with the unitary periodic dynamics and for large mass there are classical modes, with not a periodic but rather an exponential time evolution leading to a collapse of the state vector. |
| title | Geometric Deformation of Quantum Mechanics |
| topic | Quantum Physics High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2412.08007 |