Symmetries of Two-Particle States

For systems with more than one electron, spinors appear as part of an -particle wave function. It is therefore of interest to describe how the symmetries of the constituent spinors propagate through a many-particle product. Here we will discuss the simplest case, that of a two-particle state. This case illustrates the principles for extension to more particles. 

For two-particle states the wave function must transform according to a boson irrep. The simplest case is that of a two-particle state made up of a spinor and its Kramers partner. In the absence of other degeneracies this corresponds to a closed shell, and we would expect this product to transform as the totally symmetric irrep. 

From the example of C2, we see that indeed E ji 0 -1/2 = A, and also that 1/2 0 Ei/2 = -1/2 0 -1/2 = B. Thus, for the simple groups where the irreps are all singly degenerate, the symmetries of the two-particle states are easily determined from the multiplication table. This is not the case for groups that have doubly degenerate irreps. 

For the general two-particle case, the wave function is found as the direct product of the 4-spinors. As before, we concentrate on the large- and small-component 2-spinors, and get 

Each product of two 2-spinors consists of four components corresponding to the primitive spin basis aa, Pa, PP). 

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