Many-Particle States and Time Reversal

A many-particle (Fock space) single-determinant state in Kramers-unrestricted form can be written 

To represent the state in Kramers-restricted form we may partition the creation operators into barred and unbarred sets, and anticommute the barred operators to the right. 

The two products of creation operators are in nonrelativistic theory termed a and p strings. To avoid confusion with these operators, but also to retain the analogy between Kramers pairs and spin-orbital pairs, we will term these A and B strings  

The subscripts I and J on the strings are indices of the particular set of Kramers pairs from which the creation operators that make up the string are taken. The spinors that make up the Kramers pairs can, in this notation, be labeled A and B spinors. The many-particle state can now be written 

The behavior of these strings under time reversal is given by 

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