Wigner--Eckart Theorem and the Group SO(2,1)

Details Wigner--Eckart Theorem and the Group SO(2,1) Wigner--Eckart Theorem and the Group SO(2,1)

Apr 2002

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2002aps..apr.x4009z&link_type=abstract

American Physical Society, April Meeting, Jointly Sponsored with the High Energy Astrophysics Division (HEAD) of the American As

Astronomy and Astrophysics

Astrophysics

Scientific paper

We deal here with the use of the Wigner-Eckart theorem to calculate the matrix elements of a hyperbolic vector operator \overrightarrowV belonging to a pseudo-Euclidean space generated by the pseudo-orthogonal group SO(2,1). We particularly focus on calculating the matrix elements of this vector operator within the basis of the hyperbolic angular momentum \overrightarrowT; the components hatT_1, hatT_2, hatT3 of the \overrightarrowT used in this work satisfy an SO(2,1) Lie algebra whose Casimir operator is given by \overrightarrow T^2=hatT_3^2-hatT_1^2- hatT_2^2. We show that the commutation rules between the components of \overrightarrowV and \overrightarrowT can be inferred from the algebra of ordinary angular momentum. We then show that, by analogy to the Wigner-Eckart theorem of ordinary angular momentum, we can calculate the matrix elements of \overrightarrowV within a representation where \overrightarrow T^2 and hatT3 are both diagonal. One of us (NZ) acknowldges the support of Jacksonville State University for this work through a university research grant.

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