Operators Kramers permutation

The other auxiliary operator is the Kramers permutation operator, fpq, whose action is given by... 

The Kramers permutation operator must be taken to apply to the Kronecker delta as well as the operators. Thus, for si = S2 = +1 we have... 

Particle interchange symmetry is of course retained. The Kramers permutation operator is effectively a unit operator when applied to a charge density, for... 

However A is not a very suitable operator to use directly to spht the ground doublets. It is Hermitian and invariant under even permutations of Si, S2 and S3. But its behaviour under the Kramers, or time-reversal operator ( ) is rather unsatisfactory in that A = — A. 

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... 

It may be thought that quaternions could be used in the creation and annihilation operators to define a new basis in which all matrices were block-diagonal in the Kramers pairs. However, because of the noncommutative algebra, the step in which the creation operator is permuted over the matrix element to separate the two does not produce the desired result. Therefore, quaternions are useful only at the matrix algebra stage, and not in the formalism. 

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