Physics – Mathematical Physics
Scientific paper
2008-11-24
Physics
Mathematical Physics
11 pages, 3 figures, 1 table
Scientific paper
We represent vector rotation operators in terms of bras or kets of half-angle exponentials in Clifford (geometric) algebra Cl_{3,0}. We show that SO_3 is a rotation group and we define the dihedral group D_4 as its finite subgroup. We use the Euler-Rodrigues formulas to compute the multiplication table of D_4 and derive its group algebra identities. We take the linear combination of rotation operators in D_4 to represent the four Fermion matrices in Sakurai, which in turn we use to decompose any 2x2 matrix. We show that bra and ket operators generate left- and right-acting matrices, respectively. We also show that the Pauli spin matrices are not vectors but vector rotation operators, except for \sigma_2 which requires a subsequent multiplication by the imaginary number i geometrically interpreted as the unit oriented volume.
Fernandez Carlo B.
McNamara Daniel J.
Sugon Quirino M. Jr.
No associations
LandOfFree
A geometric algebra reformulation of 2x2 matrices: the dihedral group D_4 in bra-ket notation does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with A geometric algebra reformulation of 2x2 matrices: the dihedral group D_4 in bra-ket notation, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and A geometric algebra reformulation of 2x2 matrices: the dihedral group D_4 in bra-ket notation will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-643747