PlanetPhysics/Canonical Commutation and Anti Commutation 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 {\mathbf O} on ${\mathcal {H}}$, we denote its domain by {\mathbf $D(O).$} With Arai's notation, a set $\left\{Q_{j},P_{j}\right\}_{j=1}^{d}$ 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) ${\mathcal {D}}\subset \bigcap _{j,k=1}^{d}[D(Q_{j}P_{k})\bigcap D(P_{k}Q_{j})\bigcap D(Q_{j}Q_{k})\bigcap D(P_{j}P_{k})],$ 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 $\left\{Q_{j}^{S},P_{j}^{S}\right\}_{j}^{d}=1$ which is given by: ${\mathcal {H}}=L^{2}(\mathbb {R} ^{d}),\,Q_{j}^{S}=x_{j},$

the multiplication operator by the j-th coordinate $x_{j}$ , with $P_{j}^{S}=(-1)i\hbar D_{j}$ , 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 ().