dorsal/arxiv
View SchemaOn the Implementation of Constraints through Projection Operators
| Authors | A. Kempf, J. R. Klauder |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0009072 |
| URL | https://arxiv.org/abs/quant-ph/0009072 |
| DOI | 10.1088/0305-4470/34/5/307 |
| Journal | J.Phys.A34:1019-1036,2001 |
Abstract
Quantum constraints of the type Q \psi = 0 can be straightforwardly implemented in cases where Q is a self-adjoint operator for which zero is an eigenvalue. In that case, the physical Hilbert space is obtained by projecting onto the kernel of Q, i.e. H_phys = ker(Q) = ker(Q*). It is, however, nontrivial to identify and project onto H_phys when zero is not in the point spectrum but instead is in the continuous spectrum of Q, because in this case the kernel of Q is empty. Here, we observe that the topology of the underlying Hilbert space can be harmlessly modified in the direction perpendicular to the constraint surface in such a way that Q becomes non-self-adjoint. This procedure then allows us to conveniently obtain H_phys as the proper Hilbert subspace H_phys = ker(Q*), on which one can project as usual. In the simplest case, the necessary change of topology amounts to passing from an L^2 Hilbert space to a Sobolev space.
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"abstract": "Quantum constraints of the type Q \\psi = 0 can be straightforwardly\nimplemented in cases where Q is a self-adjoint operator for which zero is an\neigenvalue. In that case, the physical Hilbert space is obtained by projecting\nonto the kernel of Q, i.e. H_phys = ker(Q) = ker(Q*). It is, however,\nnontrivial to identify and project onto H_phys when zero is not in the point\nspectrum but instead is in the continuous spectrum of Q, because in this case\nthe kernel of Q is empty.\n Here, we observe that the topology of the underlying Hilbert space can be\nharmlessly modified in the direction perpendicular to the constraint surface in\nsuch a way that Q becomes non-self-adjoint. This procedure then allows us to\nconveniently obtain H_phys as the proper Hilbert subspace H_phys = ker(Q*), on\nwhich one can project as usual. In the simplest case, the necessary change of\ntopology amounts to passing from an L^2 Hilbert space to a Sobolev space.",
"arxiv_id": "quant-ph/0009072",
"authors": [
"A. Kempf",
"J. R. Klauder"
],
"categories": [
"quant-ph",
"hep-th",
"math-ph",
"math.MP"
],
"doi": "10.1088/0305-4470/34/5/307",
"journal_ref": "J.Phys.A34:1019-1036,2001",
"title": "On the Implementation of Constraints through Projection Operators",
"url": "https://arxiv.org/abs/quant-ph/0009072"
},
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