I would like to expand the expression:
$$(a_x p_x + a_y p_y + a_z p_z + b m)^2$$
where:
$a_x a_y \neq a_y a_z$
Or generally speaking the objects $a_x,a_y,a_z,b$ are non-commutative for multiplication.
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
Asked: 2014-01-09 06:37:57 +0100
Seen: 1,128 times
Last updated: Jan 10 '14
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