Time-reversal

Very early in the history of quantum theory, H.A. Kramers introduced a time reversal operator [1-3] for V-electron systems. 

Kramers time-reversal operator, shown in equation (1), is a product of Pauli spin matrices 

In other words, Kramers time reversal operator changes the sign of total spin. Also one can verify that Kramers operator K changes the sign of the total momentum, 

On the other hand, the coordinates are left invariant by Kramers operator  

This behavior is what one would expect from the operation of time reversal. Spin, momentum, and angular momentum all are reversed, while the coordinates are unchanged. 

We have previously commented on the Lorentz invariance of the Dirac equation. Considering that this places time and space coordinates on an equal footing, it may seem inconsistent to discuss transformations in spin space and only. We therefore now turn our attention to time transformations. With only one coordinate, there are only two possible transformations translation and reversal. Translation will be treated in connection with a discussion of the Lorentz transformations in the next section. Here, we will consider the symmetry of the Dirac equation under time reversal. 

Before looking at the consequences of time reversal, we need to introduce some operator concepts not normally encountered in introductory quantum chemistry texts, which deal almost exclusively with linear operators (.A) defined by the relation 

However, there is also a class of antilinear operators B) that have the property 

As before, any linear operator transforms under an antiunitary transformation as 

Antiunitary transformations affect commutation relations between operators. For 

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Conservation laws at a microscopic level of molecular interactions play an important role. In particular, energy as a conserved variable plays a central role in statistical mechanics. Another important concept for equilibrium systems is the law of detailed balance. Molecular motion can be viewed as a sequence of collisions, each of which is akin to a reaction. Most often it is the momentum, energy and angrilar momentum of each of the constituents that is changed during a collision if the molecular structure is altered, one has a chemical reaction. The law of detailed balance implies that, in equilibrium, the number of each reaction in the forward direction is the same as that in the reverse direction i.e. each microscopic reaction is in equilibrium. This is a consequence of the time reversal syimnetry of mechanics. 

A proposal based on Onsager s theory was made by Landau and Lifshitz [27] for the fluctuations that should be added to the Navier-Stokes hydrodynamic equations. Fluctuating stress tensor and heat flux temis were postulated in analogy with the Onsager theory. Flowever, since this is a case where the variables are of mixed time reversal character, tlie derivation was not fiilly rigorous. This situation was remedied by tlie derivation by Fox and Ulilenbeck [13, H, 18] based on general stationary Gaussian-Markov processes [12]. The precise fomi of the Landau proposal is confimied by this approach [14]. 

These are both long ranged in the long-wavelength limit > 0 S q) due to broken translational synnnetry and S q) due to broken time reversal synnnetry. S q) vanishes for flucUiations around equilibrium, and S q) is the same for both NESS and equilibrium. The results above are valid only for iy where the two... 

The second-order nonlinear susceptibility tensor ( 3> 2, fOj) introduced earlier will, in general, consist of 27 distinct elements, each displaying its own dependence on the frequencies oip cci2 and = oi 012). There are, however, constraints associated with spatial and time-reversal symmetry that may reduce the complexity of for a given material [32, 33 and Ml- Flere we examine the role of spatial synnnetry. 

Also we must bear in mind that the advancement of the coordinates fidfds two fiinctions (i) accurate calculation of dynamical properties, especially over times as long as typical correlation times x (ii) accurately staying on the constant-energy hypersurface, for much longer times Exact time reversibility is highly desirable (since the original equations... 

Important features of the Verlet algoritlnn are (a) it is exactly time reversible (b) it is loMf order in time, hence pennitting long time steps (c) it is easy to program. 

Finally, and probably most importantly, the relations show that changes (of a nonhivial type) in the phase imply necessarily a change in the occupation number of the state components and vice versa. This means that for time-reversal-invariant situations, there is (at least) one partner state with which the phase-varying state communicates. 

D. Perturbation Theory, Time-Reversal Symmetry, and Conical Intersections... 

A set of functions will be referred to as time-reversal adapted, provided that for each (j> in the set 7(j> is also in the set. 

We are now in a position to explain the results of Table I. As a consequence of the degeneracy of , at a conical intersection there are four degenerate functions tl/f, tb and Ttbf = Ttb = tb j. By using Eq. (Ic), an otherwise arbitrary Flermitian matrix in this four function time-reversal adapted basis has the form... 

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... 

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