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canonical commutation and anti-commutation relations: their representations (Topic)

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:

  • (i) $\mathcal{D} \subset \bigcap^d_{j,k=1}[D(Q_jP_k) \bigcap D(P_kQ_j)\bigcap D(Q_jQ_k) \bigcap D(P_jP_k)],$ and
  • (ii) $Q_j$ and $P_j$ satisfy the CCR relations:

    $\displaystyle [Q_j,P_k] = i\hbar \delta_{jk},$

    $\displaystyle [Q_j,Q_k] = 0, \, [P_j,P_k] = 0, \, j, k = 1,...,d,$

    on $\mathcal{D}$, where $\hbar$ is the Planck constant $h$ divided by $2 \pi$.

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

1
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.



"canonical commutation and anti-commutation relations: their representations" is owned by bci1.

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See Also: Representations of canonical anti-commutation relations (CAR)

Also defines:  CCR, CAR, D(O), [Q_j, P_k], D(P_j P_k), Schroedinger representation, representation of the canonical commutation relations, Schwartz space of rapidly decreasing $C_{\infty}$ functions
Keywords:  non-Abelian gauge theory, representation of canonical commutation relations in a non-Abelian gauge theory, Aharonov-Bohm effect

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CCR representation theory (Topic) by bci1

Cross-references: non-Abelian, two-dimensional, functions, operator, relations, momentum, position, operators, domain, linear operator, Hilbert space, anti-commutation relations, representations
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This is version 29 of canonical commutation and anti-commutation relations: their representations, born on 2009-02-21, modified 2009-02-21.
Object id is 547, canonical name is CanonicalCommutationAndAntiCommutationRepresentations.
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Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
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