I attempted to create a Clifford Algebra for space-time with the gamma
matrices using the FreeAlgebraQuotient in analogy to the example for
constructing a quarternion algebra from the documentation with the code:
PRGA=FreeAlgebra(QQ,4,'g')
F = PRGA.monoid()
g0,g1,g2,g3 = F.gens()
mons = [ F(1), g0, g1, g2, g3, g0*g1, g0*g2, g0*g3, g1*g2, g2*g3, g3*g1,
g0*g1*g2*g3*g0, g0*g1*g2*g3*g1, g0*g1*g2*g3*g2, g0*g1*g2*g3*g3, g0*g1*g2*g3]
G0=diagonal_matrix([1,1,1,1,-1,-1,-1,-1]) #8x8 gamma matrices
ZR=matrix(2,2,0)
EE=diagonal_matrix([1,1])
II=matrix([[0,1],[-1,0]])
G1=block_matrix([[ZR,ZR,ZR,EE],[ZR,ZR,EE,ZR],[ZR,-EE,ZR,ZR],[-EE,ZR,ZR,ZR]],subdivide=False)
G2=block_matrix([[ZR,ZR,ZR,-II],[ZR,ZR,II,ZR],[ZR,II,ZR,ZR],[-II,ZR,ZR,ZR]],subdivide=False)
G3=block_matrix([[ZR,ZR,EE,ZR],[ZR,ZR,ZR,-EE],[-EE,ZR,ZR,ZR],[ZR,EE,ZR,ZR]],subdivide=False)
mats = [G0,-G1,-G2,-G3]
ST.<g0,g1,g2,g3> = FreeAlgebraQuotient(PRGA, mons, mats)
ST
Free algebra quotient on 4 generators ('g0', 'g1', 'g2', 'g3') and
dimension 16 over Rational Field
everything looked promising here but when I type
g0*g1
Traceback (click to the left of this block for traceback)
...
TypeError: unsupported operand parent(s) for '*': 'Vector space of
dimension 16 over Rational Field' and 'Full MatrixSpace of 8 by 8 dense
matrices over Integer
Ring'
I expected sage to parrot back
g0*g1
Seems like we ought to be able to use a Free Algebra and the code:
PRGA=FreeAlgebra(QQ,4,'g')
MG=matrix(PRGA.gens())
MGG=MG.transpose()*MG
Metric = diagonal_matrix([1,-1,-1,-1])
MGGM=MGG+MGG.transpose()-2*Metric
[MGGM[i,j] for i in range(4) for j in range(4)]
[-2 + 2*g0^2, g0*g1 + g1*g0, g0*g2 + g2*g0, g0*g3 + g3*g0, g0*g1 + g1*g0,
2 + 2*g1^2, g1*g2 + g2*g1, g1*g3 + g3*g1, g0*g2 + g2*g0, g1*g2 + g2*g1, 2 +
2*g2^2,
g2*g3 + g3*g2, g0*g3 + g3*g0, g1*g3 + g3*g1, g2*g3 + g3*g2, 2 + 2*g3^2]
and use this like an ideal. Sage is Clifford averse.
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