dorsal/arxiv
View SchemaQuantum Superposition Principle and Geometry
| Authors | Alejandro Corichi |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0407242 |
| URL | https://arxiv.org/abs/quant-ph/0407242 |
| DOI | 10.1007/s10714-006-0257-6 |
| Journal | Gen. Rel. Grav. 38, 677-687 (2006) |
Abstract
If one takes seriously the postulate of quantum mechanics in which physical states are rays in the standard Hilbert space of the theory, one is naturally lead to a geometric formulation of the theory. Within this formulation of quantum mechanics, the resulting description is very elegant from the geometrical viewpoint, since it allows to cast the main postulates of the theory in terms of two geometric structures, namely a symplectic structure and a Riemannian metric. However, the usual superposition principle of quantum mechanics is not naturally incorporated, since the quantum state space is non-linear. In this note we offer some steps to incorporate the superposition principle within the geometric description. In this respect, we argue that it is necessary to make the distinction between a 'projective superposition principle' and a 'decomposition principle' that extend the standard superposition principle. We illustrate our proposal with two very well known examples, namely the spin 1/2 system and the two slit experiment, where the distinction is clear from the physical perspective. We show that the two principles have also a different mathematical origin within the geometrical formulation of the theory.
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"abstract": "If one takes seriously the postulate of quantum mechanics in which physical\nstates are rays in the standard Hilbert space of the theory, one is naturally\nlead to a geometric formulation of the theory. Within this formulation of\nquantum mechanics, the resulting description is very elegant from the\ngeometrical viewpoint, since it allows to cast the main postulates of the\ntheory in terms of two geometric structures, namely a symplectic structure and\na Riemannian metric. However, the usual superposition principle of quantum\nmechanics is not naturally incorporated, since the quantum state space is\nnon-linear. In this note we offer some steps to incorporate the superposition\nprinciple within the geometric description. In this respect, we argue that it\nis necessary to make the distinction between a \u0027projective superposition\nprinciple\u0027 and a \u0027decomposition principle\u0027 that extend the standard\nsuperposition principle. We illustrate our proposal with two very well known\nexamples, namely the spin 1/2 system and the two slit experiment, where the\ndistinction is clear from the physical perspective. We show that the two\nprinciples have also a different mathematical origin within the geometrical\nformulation of the theory.",
"arxiv_id": "quant-ph/0407242",
"authors": [
"Alejandro Corichi"
],
"categories": [
"quant-ph",
"gr-qc"
],
"doi": "10.1007/s10714-006-0257-6",
"journal_ref": "Gen. Rel. Grav. 38, 677-687 (2006)",
"title": "Quantum Superposition Principle and Geometry",
"url": "https://arxiv.org/abs/quant-ph/0407242"
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