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On the Implementation of Constraints through Projection Operators | A. Kempf
; J. R. Klauder
; | Date: |
19 Sep 2000 | Journal: | J.Phys. A34 (2001) 1019-1036 | Subject: | Quantum Physics; Mathematical Physics | quant-ph hep-th math-ph math.MP | 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. | Source: | arXiv, quant-ph/0009072 | Services: | Forum | Review | PDF | Favorites |
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