Symbolic simplificaction without commutativity

savecancel

It sounds like you're looking for FreeAlgebra.

sage: R.<ax,ay,az,px,py,pz> = FreeAlgebra(QQ)
sage: R
Free Algebra on 6 generators (ax, ay, az, px, py, pz) over Rational Field
sage: (ax*px + ay*py + az*pz)^2
ax*px*ax*px + ax*px*ay*py + ax*px*az*pz + ay*py*ax*px + ay*py*ay*py + ay*py*az*pz + az*pz*ax*px + az*pz*ay*py + az*pz*az*pz

If FreeAlgebra doesn't provide what you're looking for, note that Maxima also implements noncommutative symbolic algebra, and you can access it from sage with, e.g., the maxima command:

sage: e = maxima('expand((ax . px + ay . py + az . pz)^^2);')
sage: e
(az.pz)^^2+az.pz.ay.py+az.pz.ax.px+(ay.py)^^2+ay.py.az.pz+ay.py.ax.px+(ax.px)^^2+ax.px.az.pz+ax.px.ay.py

Note that this uses Maxima's syntax, which is generally different from Sage's. You can read more about it starting here (SO) or here (Maxima manual).

In your particular example, you are doing the computation on the free algebra on ax,ay,az,b over the polynomial ring on px,py,pz,m.

sage: R.<px,py,pz,m>=QQ[]
sage: F.<ax,ay,az,b>=FreeAlgebra(R)
sage: (ax*px+ay*py+az*pz+m*b)^2
px^2*ax^2 + px*py*ax*ay + px*pz*ax*az + px*m*ax*b + px*py*ay*ax + py^2*ay^2 + py*pz*ay*az + py*m*ay*b + px*pz*az*ax + py*pz*az*ay + pz^2*az^2 + pz*m*az*b + px*m*b*ax + py*m*b*ay + pz*m*b*az + m^2*b^2