The Commutator and AntiCommutator commands represent $AB-BA$ and $AB+BA$, respectively, where $A$ and $B$ may be non commutative objects. The output of AntiCommutator(A, B) and Commutator(A, B) is:

If $A$ and $B$ commute, Anticommutator returns $2AB$ and Commutator returns $0$.

You can indicate the result for the Commutator or AntiCommutator by setting algebra rules.

The entered function call may return unevaluated.

The %Commutator and %AntiCommutator commands are the inert forms of Commutator and AntiCommutator; that is, they represent the same mathematical operations while displaying the operations unevaluated. To evaluate the operations, use the value or expand commands.

Commutators and AntiCommutators can also be expanded by using the expand or Physics[Expand] command. When the output is expanded, it is normalized and any repeated indices in products of tensors (defined as such by Define) are automatically simplified by the Simplify command.

Activating the inert operation by using value is the same as expanding it by using expand, except when the result of the Commutator is 0 or the result of the AntiCommutator is 2AB. Otherwise, evaluating just replaces the inert % operators by the active ones in the output.

When the Physics package is loaded, two routines for the enhanced display of unevaluated and inert Commutators and AntiCommutators are loaded together, so that Commutators are shown as a "list" of two operands, subscripted with the `-` sign. AntiCommutators are shown also as a list of two operands, subscripted with the `+` sign. In the literature, AntiCommutators are displayed as two operands inside curly brackets { }. This notation is not used in Maple to avoid potential confusion with Maple sets, displayed using curly brackets, where the order of the operators is not fixed.

$\mathrm{with}\left(\mathrm{Physics}\right)\:$

$\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true}\right)$

$\left[\mathrm{mathematicalnotation}=\mathrm{true}\right]$

$\mathrm{e1}≔\mathrm{\%Commutator}\left(Z\left[1\right]\,Z\left[2\right]\right)$

$\mathrm{e1}≔{\left[Z_1,Z_2\right]}_{-}$

$\mathrm{lprint}\left(\right)$

%Commutator(Z[1],Z[2])

$\mathrm{e2}≔\mathrm{\%AntiCommutator}\left(\mathrm{\theta }\left[1\right]\,\mathrm{\theta }\left[2\right]\right)$

$\mathrm{e2}≔{\left[{\mathrm{\theta }}_1,{\mathrm{\theta }}_2\right]}_{+}$

$\mathrm{lprint}\left(\right)$

%AntiCommutator(theta[1],theta[2])

$\mathrm{expand}\left(\mathrm{e1}\right)=\mathrm{value}\left(\mathrm{e1}\right)$

$0=0$

$\mathrm{expand}\left(\mathrm{e2}\right)=\mathrm{value}\left(\mathrm{e2}\right)$

$2{\mathrm{\theta }}_1{\mathrm{\theta }}_2=2{\mathrm{\theta }}_1{\mathrm{\theta }}_2$

$\mathrm{Commutator}\left(Z\left[1\right]\,Z\left[2\right]\right)$

$0$

$\mathrm{AntiCommutator}\left(\mathrm{\theta }\left[1\right]\,\mathrm{\theta }\left[2\right]\right)$

$2{\mathrm{\theta }}_1{\mathrm{\theta }}_2$

$\mathrm{Setup}\left(\mathrm{anticommutativeprefix}=\mathrm{\theta }\,\mathrm{noncommutativeprefix}=Z\right)$

$\left[\mathrm{anticommutativeprefix}=\left\{\mathrm{\theta }\right\}\,\mathrm{noncommutativeprefix}=\left\{Z\right\}\right]$

$\mathrm{e1}$

${\left[Z_1,Z_2\right]}_{-}$

$\mathrm{expand}\left(\right)=\mathrm{value}\left(\right)$

$Z_1{Z}_2-Z_2{Z}_1={\left[Z_1,Z_2\right]}_{-}$

$\mathrm{e2}$

${\left[{\mathrm{\theta }}_1,{\mathrm{\theta }}_2\right]}_{+}$

$\mathrm{expand}\left(\right)=\mathrm{value}\left(\right)$

$0=0$

$\mathrm{e3}≔\mathrm{\theta }\left[1\right]+\mathrm{\theta }\left[2\right]$

$\mathrm{e3}≔{\mathrm{\theta }}_1+{\mathrm{\theta }}_2$

$\mathrm{e4}≔\mathrm{\theta }\left[1\right]-\mathrm{\theta }\left[2\right]$

$\mathrm{e4}≔{\mathrm{\theta }}_1-{\mathrm{\theta }}_2$

$\mathrm{e5}≔\mathrm{\%Commutator}\left(\mathrm{\theta }\left[1\right]\mathrm{e3}\,\mathrm{\theta }\left[2\right]\mathrm{e4}\right)$

$\mathrm{e5}≔{\left[{\mathrm{\theta }}_1\left({\mathrm{\theta }}_1+{\mathrm{\theta }}_2\right),{\mathrm{\theta }}_2\left({\mathrm{\theta }}_1-{\mathrm{\theta }}_2\right)\right]}_{-}$

$\mathrm{expand}\left(\mathrm{e5}\right)$

$0$

$\mathrm{value}\left(\mathrm{e5}\right)$

$0$

$\mathrm{Setup}\left(\mathrm{quantumop}=\left\{P\,R\right\}\,\mathrm{algebrarule}=\left\{\mathrm{\%Commutator}\left(R\left[j\right]\,P\left[k\right]\right)=\mathrm{I}\mathrm{KroneckerDelta}\left[j,k\right]\right\}\right)$

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{quantumop}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{quantumoperators}\text{'}$

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{algebrarule}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{algebrarules}\text{'}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left[\mathrm{algebrarules}=\left\{{\left[R_j,P_k\right]}_{-}=\mathrm{I}{\mathrm{\delta }}_{j,k}\right\}\,\mathrm{quantumoperators}=\left\{P\,R\right\}\right]$

$\mathrm{Commutator}\left(R\left[1\right]\,P\left[2\right]\right)$

$0$

$\mathrm{Commutator}\left(R\left[2\right]\,P\left[2\right]\right)$

$\mathrm{I}$

$\mathrm{Commutator}\left(R\left[n\right]\,P\left[m\right]\right)$

$\mathrm{I}{\mathrm{\delta }}_{m,n}$

$\mathrm{Setup}\left(\mathrm{quantumop}=\left\{P\,Q\right\}\,\mathrm{algebrarule}=\left\{\mathrm{\%Commutator}\left(Q\,P\right)=\mathrm{I}\right\}\right)$

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{quantumop}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{quantumoperators}\text{'}$

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{algebrarule}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{algebrarules}\text{'}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left[\mathrm{algebrarules}=\left\{{\left[Q,P\right]}_{-}=\mathrm{I}\,{\left[R_j,P_k\right]}_{-}=\mathrm{I}{\mathrm{\delta }}_{j,k}\right\}\,\mathrm{quantumoperators}=\left\{P\,Q\,R\right\}\right]$

$\mathrm{Commutator}\left(Q\,P\right)$

$\mathrm{I}$

$\mathrm{Commutator}\left(Q\,P^2\right)$

$2\mathrm{I}P$

$\mathrm{Commutator}\left(Q^2\,P^3\right)$

$3\mathrm{I}{\left[Q,P^2\right]}_{+}$

$\mathrm{Simplify}\left(\right)$

$6\mathrm{I}Q{P}^2+6P$

$\mathrm{Commutator}\left(Q\,P^m\right)$

${\left[Q,P^m\right]}_{-}$

$\mathrm{Setup}\left(\mathrm{redo}\,\mathrm{algebra}=\left\{\mathrm{\%Commutator}\left(Q\,P^m\right)=\mathrm{I}m{P}^{m-1}\right\}\right)$

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{algebra}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{algebrarules}\text{'}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left[\mathrm{algebrarules}=\left\{{\left[Q,P^m\right]}_{-}=\mathrm{I}m{P}^{m-1}\right\}\right]$

$\mathrm{Commutator}\left(Q\,P\right)$

$\mathrm{I}$

$\mathrm{Commutator}\left(Q^2\,P^3\right)$

$3\mathrm{I}\left(2\mathrm{I}P+2P^{2}Q\right)$

$\mathrm{Commutator}\left(Q^2\,P^s\right)$

$\mathrm{I}s\left(\mathrm{I}\left(s-1\right)P^{s-2}+2P^{s-1}Q\right)$