canonical commutation and anti-commutation relations: their representations
This is a contributed topic on representations
of canonical commutation and anti-commutation relations.
Consider a Hilbert space
. For a linear operator
O on
, we denote its domain
by
With
Arai's notation, a set
of self-adjoint operators
on
(such as the position
and momentum
operators, for example) is called a representation of the canonical commutation relations (CCR) with
degrees of freedom if there exists a dense subspace
of
such that:
A standard representation of the CCR is the well-known Schrödinger representation
which is given by:
the multiplication operator by the j-th coordinate
, with
, with
being the generalized partial differential operator in
, and with
being the Schwartz space of rapidly decreasing
functions
on
, or
, that is the space of
functions on
with compact support.
One can provide a representation of canonical commutation relations in a
non-Abelian gauge theory
defined on a non-simply connected region in the two-dimensional
Euclidean space. Such representations were shown to provide also a mathematical expression for the non-Abelian, Aharonov-Bohm effect
([6]).
- 1
-
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- 2
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- 3
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Arai A., Analysis on anticommuting self-adjoint operators, Adv. Stud. Pure Math., 1994, v.23, 1-15.
- 4
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Arai A., Scaling limit of anticommuting self-adjoint operators and applications to Dirac operators, Integr. Equat. Oper. Th., 1995, v.21, 139-173.
- 5
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Arai A., Some remarks on scattering theory in supersymmetric quantum mechanics, J. Math. Phys., 1987, V.28, 472-476.
- 6
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Goldin G.A., Menikoff R. and Sharp D.H., Representations of a local current algebra in nonsimply connected space and the Aharonov-Bohm effect, J. Math. Phys., 1981, v.22, 1664-1668.
- 7
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von Neumann J., Die Eindeutigkeit der Schrödingerschen Operatoren,
Math. Ann., 1931, v.104, 570-578.
- 8
-
Pedersen S., Anticommuting self-adjoint operators, J. Funct. Anal., 1990, V.89, 428-443.
- 9
-
Putnam C. R., Commutation Properties of Hilbert Space Operators, Springer, Berlin, 1967.
- 10
-
Reed M. and Simon B., Methods of Modern Mathematical Physics., vol.I, Academic Press, New York, 1972.
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