Three matrices ,

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Quantum Mechanics · Symmetry, angular momentum and spin

Invariance, Ch Representations of Lie algebras: spin, products and SU(3)

Problem

Three

4\times4

matrices

J_x

J_y

J_z

satisfy

[J_a,J_b] = i\epsilon_{abc}J_c

. In a basis in which

J_z

is diagonal,

J_z = \mathrm{diag}(1, 0, -1, 0)

and the Casimir operator is

J^2 = \mathrm{diag}(2, 2, 2, 0)

. Someone argues: “Four dimensions means

2j+1 = 4

, so

j = \tfrac32

, and

J^2

must be

\tfrac{15}{4}\mathsf1

. These matrices cannot satisfy the algebra.” ((a)) Find the error in this argument. ((b)) Identify the representation from the eigenvalues of

J^2

, and check your answer against the dimension and the eigenvalues of

J_z