DotSimplify[exp] expands and reorders noncommutative terms in exp. Simplifying relations may be specified by the option DotSimplifyRelations or by Commutator and AntiCommutator definitions. Whether exp is expanded noncommutatively depends on the option Expanding.
Overview, AntiCommutator, Commutator, Calc.
UnDeclareAllCommutators[]
UnDeclareAllAntiCommutators[]GA[\[Mu]] . (2 GS[p] - GS[q]) . GA[\[Nu]]
DotSimplify[%]\bar{\gamma }^{\mu }.\left(2 \bar{\gamma }\cdot \overline{p}-\bar{\gamma }\cdot \overline{q}\right).\bar{\gamma }^{\nu }
2 \bar{\gamma }^{\mu }.\left(\bar{\gamma }\cdot \overline{p}\right).\bar{\gamma }^{\nu }-\bar{\gamma }^{\mu }.\left(\bar{\gamma }\cdot \overline{q}\right).\bar{\gamma }^{\nu }
DeclareNonCommutative[a, b, c]
a . (b - z c) . a
DotSimplify[%]a.(b-c z).a
a.b.a-z a.c.a
Commutator[a, c] = 1
DotSimplify[a . (b - z c) . a]1
a.b.a-z (c.a.a+a)
Commutator[a, c] =.
DotSimplify[a . (b - z c) . a]a.b.a-z a.c.a
AntiCommutator[b, a] = c
DotSimplify[a . (b - z c) . a]c
-a.a.b-z a.c.a+a.c
AntiCommutator[b, a] =.
DotSimplify[a . (b - z c) . a, DotSimplifyRelations -> {a . c -> 1/z}]a.b.a-a
UnDeclareNonCommutative[a, b, c]
DeclareNonCommutative[x]
DotSimplify[x . x . x]x.x.x
DotSimplify[x . x . x, DotPower -> True]
UnDeclareNonCommutative[x]x^3
Check some relations between noncommutative expressions involving two operators Q and P
DeclareNonCommutative[Q, P]lhs = (Q . Commutator[Q, P] + Commutator[Q, P] . Q)/2
rhs = Commutator[Q, Q . P + P . Q]/2
DotSimplify[lhs - rhs]
% // ExpandAll\frac{1}{2} (Q.[Q,P]+[Q,P].Q)
\frac{1}{2} [Q,P.Q+Q.P]
\frac{1}{2} (P.Q.Q-Q.Q.P)+\frac{1}{2} (Q.Q.P-P.Q.Q)
0
Commutator[Q, P] = I;Introduce the dilation operator D from the affine quantization and verify that [Q,D]=i \hbar (cf. arXiv:2108.10713)
DOp = (Q . P + P . Q)/2;Commutator[Q, DOp]
% // DotSimplify // ExpandAll\left[Q,\frac{1}{2} (P.Q+Q.P)\right]
i Q
UnDeclareAllCommutators[]
UnDeclareAllAntiCommutators[]