PlanetPhysics/Canonical Commutation and Anti Commutation Representations

This is a contributed topic on [[../CategoricalGroupRepresentation/|representations]] of canonical commutation and [[../AntiCommutationRelations/|anti-commutation relations]].

Representations of Canonical Commutation Relations (CCR)

Canonical Commutation Relations:

Consider a [[../NormInducedByInnerProduct/|Hilbert space]] $\mathscr{H}$. For a [[../Commutator/|linear operator]] {\mathbf O} on $\mathscr{H}$, we denote its [[../Bijective/|domain]] by {\mathbf $D(O).$} With Arai's notation, a set ${\{Q_j,P_j\}}_{j=1}^d$ of self-adjoint [[../QuantumOperatorAlgebra4/|operators]] on $\mathscr{H}$ (such as the [[../Position/|position]] and [[../Momentum/|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 $\mathscr{H}$ such that:

• (i) $𝒟\subset \bigcap _{j,k=1}^d[D(Q_j{P}_k)\bigcap D(P_k{Q}_j)\bigcap D(Q_j{Q}_k)\bigcap D(P_j{P}_k)],$ and
• (ii) $Q_j$ and $P_j$ satisfy the CCR [[../Bijective/|relations]]: $[Q_j,P_k]=i\hslash {\delta }_{jk},$ $[Q_j,Q_k]=0,[P_j,P_k]=0,j,k=1,...,d,$ on $𝒟$, where $\hslash$ is the Planck constant $h$ divided by $2\pi$.

A standard representation of the CCR is the well-known Schr\"odinger representation ${\{Q_j^S,P_j^S\}}_j^d=1$ which is given by: $\mathscr{H}=L^2({ℝ}^d),Q_j^S=x_j,$

the multiplication [[../QuantumSpinNetworkFunctor2/|operator]] by the j-th coordinate $x_j$ , with $P_j^S=(-1)i\hslash D_j$ , 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 $C_{\mathrm{\infty }}$ [[../Bijective/|functions]] on ${ℝ}^d$, or $𝒟=C_0^{\mathrm{\infty }}({ℝ}^d)$, that is the space of $C^{\mathrm{\infty }}$ functions on ${ℝ}^d$ 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 [[../CoriolisEffect/|two-dimensional]] Euclidean space. Such representations were shown to provide also a mathematical expression for the [[../AbelianCategory3/|non-Abelian]], Aharonov-Bohm effect ().