DotSimplify[exp] expands and reorders noncommutative
terms in exp. Simplifying relations may be specified by the option
DotSimplifyRelations or by FCCommutator and
FCAntiCommutator definitions. Whether exp is expanded
noncommutatively depends on the option Expanding.
Overview, FCAntiCommutator, FCCommutator, 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
FCCommutator[a, c] = 1
DotSimplify[a . (b - z c) . a]1
a.b.a-z (c.a.a+a)
FCCommutator[a, c] =.
DotSimplify[a . (b - z c) . a]a.b.a-z a.c.a
FCAntiCommutator[b, a] = c
DotSimplify[a . (b - z c) . a]c
-a.a.b-z a.c.a+a.c
FCAntiCommutator[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 . FCCommutator[Q, P] + FCCommutator[Q, P] . Q)/2
rhs = FCCommutator[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
FCCommutator[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;FCCommutator[Q, DOp]
% // DotSimplify // ExpandAll\left[Q,\frac{1}{2} (P.Q+Q.P)\right]
i Q
UnDeclareAllCommutators[]
UnDeclareAllAntiCommutators[]