canonical commutation and anti-commutation relations: their representations

This is a contributed topic on representations of canonical commutation and anti-commutation relations.

Representations of Canonical Commutation Relations (CCR)

Canonical Commutation Relations:

Consider a Hilbert space $ \mathcal{H}$ . For a linear operator O on $ \mathcal{H}$ , we denote its domain by $ D(O).$ With Arai's notation, a set $ \left\{Q_j,P_j\right\} ^d_{j =1}$ of self-adjoint operators on $ \mathcal{H}$ (such as the position and momentum operators, for example) is called a representation of the canonical commutation relations (CCR) with $ d$ degrees of freedom if there exists a dense subspace $ \mathcal{D}$ of $ \mathcal{H}$ such that:

A standard representation of the CCR is the well-known Schrödinger representation $ \left\{Q^S_j,P_j^S \right\}^d_j=1 $ which is given by:

$\displaystyle \mathcal{H} = L^2(\mathbb{R}^d), \, Q^S_j= x_j, $

the multiplication operator by the j-th coordinate $ x_j$ , with $ P^S_j = (-1) i \hbar D_j$ , with $ D_j$ being the generalized partial differential operator in $ x_j$ , and with $ J\mathcal{D} = \mathcal{S}(\mathbb{R}^d)$ being the Schwartz space of rapidly decreasing $ C_{\infty}$ functions on $ \mathbb{R}^d$ , or $ \mathcal{D} = C_0^{\infty}(\mathbb{R}^d)$ , that is the space of $ C^{\infty}$ functions on $ \mathbb{R}^d$ with compact support.

CCR Representations in a Non-Abelian Gauge Theory

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]).

Canonical Anticommutation Relations (CAR)

Bibliography

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Arai A., Characterization of anticommutativity of self-adjoint operators in connection with Clifford algebra and applications, Integr. Equat. Oper. Th., 1993, v.17, 451-463.

2
Arai A., Commutation properties of anticommuting self-adjoint operators, spin representation and Dirac operators, Integr. Equat. Oper. Th., 1993, v.16, 38-63.

3
Arai A., Analysis on anticommuting self-adjoint operators, Adv. Stud. Pure Math., 1994, v.23, 1-15.

4
Arai A., Scaling limit of anticommuting self-adjoint operators and applications to Dirac operators, Integr. Equat. Oper. Th., 1995, v.21, 139-173.

5
Arai A., Some remarks on scattering theory in supersymmetric quantum mechanics, J. Math. Phys., 1987, V.28, 472-476.

6
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
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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