canonical commutation and anti-commutation relations: their representations

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

0.1 Representations of Canonical Commutation Relations (CCR)

0.1.1 Canonical Commutation Relations:

Consider a Hilbert space ℋ. For a linear operator O on ℋ, we denote its domain by D(O). With Arai’s notation, a set {Qj, Pj} _j=1^d of self-adjoint operators on 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 𝒟 of ℋ such that:

(i) 𝒟⊂⋂ _j,k=1^d[D(Q_jP_k) ⋂ D(P_kQ_j) ⋂ D(Q_jQ_k) ⋂ D(P_jP_k)], and
(ii) Q_j and P_j satisfy the CCR relations:

$[Qj, Pk] = iℏδjk,$

$[Qj, Qk] = 0, [Pj,Pk ] = 0, j,k = 1,...,d,$
on 𝒟, where ℏ is the Planck constant h divided by 2π.

A standard representation of the CCR is the well-known Schrödinger representation {QS ,P S}
j j _j^d = 1 which is given by:

2 d S ℋ = L (ℝ ), Q j = xj,

the multiplication operator by the j-th coordinate x_j , with P_j^S = (−1)iℏD_j , with D_j being the generalized partial differential operator in x_j , and with J𝒟 = 𝒮(ℝ^d) being the Schwartz space of rapidly decreasing C_∞ functions on ℝ^d, or 𝒟 = C₀^∞(ℝ^d), that is the space of C^∞ functions on ℝ^d with compact support.

0.1.2 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]).

0.2 Canonical Anticommutation Relations (CAR)

References

[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.

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