PlanetPhysics/Canonical Commutation and Anti Commutation Representations

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

Canonical Commutation Relations:
Consider a Hilbert space $$\mathcal{H}$$. For a linear operator {\mathbf O} on $$\mathcal{H}$$, we denote its domain by {\mathbf $$D(O).$$} 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: $$[Q_j,P_k] = i\hbar \delta_{jk},$$ $$[Q_j,Q_k] = 0, \,  [P_j,P_k] = 0, \, j, k = 1,...,d,$$  on $$\mathcal{D}$$, where $$\hbar$$ is the Planck constant $$h$$ divided by $$2 \pi$$.

A standard representation of the CCR is the well-known Schr\"odinger representation $$\left\{Q^S_j,P_j^S \right\}^d_j=1 $$ which is given by: $$\mathcal{H} = L^2(\mathbb{R}^d), \, Q^S_j= x_j, $$

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 .