Operators Kramers single-replacement

This expression represents a simplification of the original expansion in the primitive Kramers pairs basis. Although there is not a reduction in the size of the one-particle basis, we need only consider half the matrix elements, and there is therefore a 50% reduction in the amount of work. The lack of reduction might be expected because the matrix elements are potentially spin-dependent. The X q operators are called Kramers single-replacement operators, and they define what we will call a Kramers basis. 

There is an even more compact way of defining the Kramers single-replacement operators, Xfq, which also has the advantage of displaying the permutational symmetry. To do this, we introduce two auxiliary operators. One is the bar-reversal operator, Kp, which is the time-reversal operator for a spinor with index p. The effect of this operator is... 

The Kramers single-replacement operators, can now be expressed in terms of these auxiliary operators. 

The reduction of products of second-quantized operators makes use of commutators of the replacement operators. The various commutators of Kramers single-replacement operators are given by application of bar reversal to the basic commutator relation... 

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