Pérez-Pardo, J. M. (2013) Quadratic forms, unbounded self-adjoint operators and quantum observables. Il nuovo cimento C, 36 (3). pp. 205-214. ISSN 1826-9885
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Abstract
In the context of the geometric formulation of quantum mechanics the observables are characterised by the quadratic forms associated to the selfadjoint operators that describe the corresponding observables in the standard formulation. If the self-adjoint operators are bounded, it can be shown, that their associated quadratic forms are in one-to-one correspondence with the space of real K¨ahlerian functions over the projective Hilbert space defining the system, i.e., over the space of states. However, in the case of unbounded self-adjoint operators such a geometric description is still lacking. The aim of this article is to introduce the main difficulties when dealing with unbounded operators and point out possible generalizations of the geometric notion of observable. In particular, it will be showed how one can work directly with the quadratic forms associated with self-adjoint operators to overcome some of the difficulties. As a motivational example, the case of the Laplace-Beltrami operator is analysed thoroughly.
| Item Type: | Article |
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| Uncontrolled Keywords: | Geometry, differential geometry, and topology ; Geometric mechanics ; Quantum mechanics ; Functional analytical methods |
| Subjects: | 500 Scienze naturali e Matematica > 530 Fisica |
| Depositing User: | Marina Spanti |
| Date Deposited: | 05 May 2020 12:58 |
| Last Modified: | 05 May 2020 12:58 |
| URI: | http://eprints.bice.rm.cnr.it/id/eprint/18278 |
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