Time Reversal and Kramers-Restricted Representation of Operators

Time Reversal and Kramers-Restricted Representation of Operators 

In the previous discussion of the symmetry of the Dirac equation (chapter 6), it was shown that the Dirac equation was symmetric under time reversal, and that the fermion functions occur in Kramers pairs where the two members are related by time reversal. We will have to deal with a variety of operators, and in most cases the methodologies will be developed in the absence of an external magnetic field, or with the magnetic field considered as a perturbation. Consequently, we can make the developments in terms of a basis of Kramers pairs, which are the natural representation of the wave function in a system that is symmetric under time reversal. The development here is primarily the development of a second-quantized formalism. We will use the term Kramers-restricted to cover techniques and methods based on spinors that in some well-defined way appear as Kramers pairs. The analogous nonrelativistic situation is the spin-restricted formalism, which requires that orbitals appear as pairs with the same spatial part but with a and spins respectively. Spin restriction thus appears as a special case of Kramers restriction, because of the time-reversal connection between a and spin functions. 

The first step in the implementation of time-reversal symmetry is to classify the basic operators according to their behavior under time reversal effected by the operator K, = —iSyKo.  

Our basis of Kramers pairs is 1] , We adopt the convention that general spinors are labeled p, q, r, s, occupied spinors are labeled i, j, k, I, unoccupied spinors are labeled a, b, c, d, and active spinors (where necessary) are labeled t, u, v, w. The time-reversed conjugate of a function or operator is denoted by a bar. We place the bar over the index of a function rather than the function, that is, we use 1] rather than rlf ,. However, the two are equivalent, and we place the bar over the function when there is no index. 

To develop relations under time reversal, we use the creation operator a corresponding to lIt , that is 

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