Operators time-reversal

By comparing Eq. (C.6) with Eqs. (C.2) and (C.3), the time-reversal operator can be expressed as a product of an unitary and a complex conjugate operators as follows... 

The above discussion is now generalized to arbitrary spin values. First, we note that twice application of the time-reversal operator leads the system back to its original state v /, that is, T r t = ct t. Thus, we have T = cl. Next, consider the following two relations... 

The transformation U(it) which maps the operator algebra /(x),An x) onto the operator algebra of the time reversed operators is fundamentally different from the unitary mappings previously considered. This can most easily be seen as follows ... 

The Time Reversal Operator.—In this section we show that spatial operators are linear whereas the time reversal operator is antilinear.5 This may be seen by examining the eigenfunctions of the time dependent Schrodinger equation... 

The important difference here between spatial operators and the time reversal operator originates in the way each effects the time displacement of the state functions. We have the following schematic multiplication relations. 

If the time reversal operator is taken to be linear, the left-hand side of Eq. (12-6) becomes... 

A similar argument can be used to show that the spatial operators are linear. It can then be shown that spatial operators are unitary whereas the time reversal operator is anti-unitary. 

The time reversal operator may now be determined.6 The simplest anti-unitary operation is the transition to the complex conjugate. This operator K is clearly antilinear and anti-unitary. 

Furthermore, any antilinear operator can be written as the product of a unitary operator and the operator K. Specifically, we can write the time reversal operator as 6 = UK, and our problem is now that of... 

If the group O contains the time reversal operator itself, we can choose Y0 = E, the identity operator, and Eq. (12-27) reduces to... 

The thirty-two point groups without any form of the time reversal operator form a category of magnetic point groups. The representations of these are obtained by standard means and are presented in Section 12.6. 

The remaining fifty-eight magnetic point groups include the time reversal operator only in combination with rotation and rotation-reflection operators. The representations of these groups may be obtained from Eq. (12-27). 

The effect of time reversal operator T is to reverse the linear momentum (L) and the angular momentum (J), leaving the position operator unchanged. Thus, by definition,... 

Under the action of time reversal operator T the position and momentum commutator [Q,L] = ih becomes... 

According to Eqs.(6) and (7), this becomes — (QL — LQ) = TihT 1, which when compared with the original commutation relation yields T T 1 — i. Therefore the time reversal operator is anti-linear. It can also be shown that the time reversal operator T is anti-unitary. 

Operators that induce transformations in space satisfy eq. (2) and are therefore unitary operators with the property / T = 1. An operator that satisfies eq. (3) is said to be antiunitary. In contrast to spatial symmetry operators, the time-reversal operator is anti-unitary. Let U denote a unitary operator and let T denote an antiunitary operator. 

Therefore, at this level (with spin suppressed) the time-reversal operator is just the complex conjugation operator. it which replaces i by —i. 

For example, A might be the time-reversal operator 0 (Section 13.4)... 

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