Setting non-commutativity between two variables

Question

Question: I have three functions that I am declaring.

Rx[\[Alpha]_] := Exp[-I \[Alpha] Gx];
Ry[\[Alpha]_] := Exp[-I \[Alpha] Gy];
Rz[\[Alpha]_] := Exp[-I \[Alpha] Gz];

I want to Series expand and simplify the product:
Rx[-v] Ry[-v] Rx[v] Ry[v]

i.e. Series[Rx[-v] Ry[-v] Rx[v] Ry[v], {v, 0, 2}]

The catch is, Gx, Gy and Gz don't commute with one another...

Is there a way to Series expand this product, and then simplify it, given that these three variables alone don't commute? (They commute with everything else...)

Answer 1

One approach would be to define a new multiplication operator (e.g. using CenterDot or any other predefined binary operator without a set definition), and define multiplication rules differentiating between expressions that do and do not contain Gx, Gy, Gz.

For example:

CenterDot[a___, b_?(FreeQ[#, Gx | Gy | Gz] &), c___] :=  b CenterDot[a, c]
CenterDot[a___, b_?(FreeQ[#, Gx | Gy | Gz] &) d_, c___] :=  b CenterDot[a, d, c]
CenterDot[a___, b_ + d_, c___] :=  CenterDot[a, b, c] + CenterDot[a, d, c]
CenterDot[] := 1
CenterDot[a_] := a

Series[
 CenterDot[Exp[I v Gx],Exp[I v Gy],Exp[-I x Gx],Exp[-I v Gy]] /. 
  Exp[A_] :> Normal@Series[Exp[A], {v, 0, 2}]
,
{v, 0, 2}
]

Would produce the result for your example.

Update: Adding the following rule will help in dealing with powers of the same element

CenterDot[a___, (b : (Gx | Gy | Gz))^n_., (b : (Gx | Gy | Gz))^m_.,   c___] :=
    CenterDot[a, b^(n + m), c]