Time reversal invariance

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. 

Time-reversal invariant the reverse sequence is obtained by inverting the last two configurations obtained in the forward direction and applying the same, rule, 4>tz-... 

Apart from their pedagogical value, reversible rules may be used to explore possible relationships between discrete dynamical systems and the dynamics of real mechanical systems, for which the microscopic laws are known to be time-reversal invariant. What sets such systems apart from continuous idealizations is their exact reversibility, discreteness assures us that computer simulations run for arbitrarily long times will never suffer from roundoff or truncation errors. As Toffoli points out, ...the results that one obtains have thus the force of theorems [toff84a]. ... 

In the context of CA systems, it turns out that there is a difference between rules that are invertible and rules that are time-reversal invariant. A global CA rule S —> S, mapping a global state ct S to some other global state ct S, is said to be invertible if for all states ct S there exists exactly one predecessor state O S such that (cr) = a. The state transition graphs G for all such rules must therefore consist entirely of cycles. 

As suggested above, time-reversal invariance is a stronger property than simple invertibility and means that the dynamics is not only invertible, but that the time-reversed evolution proceeds according to the same rule as well. We shall call all invertible CA rules that are also time-reversal invariant, reversible rules. ... 

The emission line is centered at the mean energy Eq of the transition (Fig. 2.2). One can immediately see that I E) = 1/2 I Eq) for E = Eq E/2, which renders r the full width of the spectral line at half maximum. F is called the natural width of the nuclear excited state. The emission line is normalized so that the integral is one f l(E)dE = 1. The probability distribution for the corresponding absorption process, the absorption line, has the same shape as the emission line for reasons of time-reversal invariance. 

Kramers degeneracy theorem states that the energy levels of systems with an odd number of electrons remain at least doubly degenerate in the presence of purely electric fields (i.e. no magnetic fields). This is a consequence of the time-reversal invariance of electric fields, and follows from an application of the antiunitary T-operator to the wavefunction of an odd number of electrons [51]. 

After the discovery of the combined charge and space symmetry violation, or CP violation, in the decay of neutral mesons [2], the search for the EDMs of elementary particles has become one of the fundamental problems in physics. A permanent EDM is induced by the super-weak interactions that violate both space inversion symmetry and time reversal invariance [11], Considerable experimental efforts have been invested in probing for atomic EDMs (da) induced by EDMs of the proton, neutron, and electron, and by the P,T-odd interactions between them. The best available limit for the electron EDM, de, was obtained from atomic T1 experiments [12], which established an upper limit of de < 1.6 x 10 27e-cm. The benchmark upper limit on a nuclear EDM is obtained from the atomic EDM experiment on Iyt,Hg [13] as d ig < 2.1 x 10 2 e-cm, from which the best restriction on the proton EDM, dp < 5.4 x 10 24e-cm, was also obtained by Dmitriev and Senkov [14]. The previous upper limit on the proton EDM was estimated from the molecular T1F experiments by Hinds and co-workers [15]. 

As mentioned earlier, heavy polar diatomic molecules, such as BaF, YbF, T1F, and PbO, are the prime experimental probes for the search of the violation of space inversion symmetry (P) and time reversal invariance (T). The experimental detection of these effects has important consequences [37, 38] for the theory of fundamental interactions or for physics beyond the standard model [39, 40]. For instance, a series of experiments on T1F [41] have already been reported, which provide the tightest limit available on the tensor coupling constant Cj, proton electric dipole moment (EDM) dp, and so on. Experiments on the YbF and BaF molecules are also of fundamental significance for the study of symmetry violation in nature, as these experiments have the potential to detect effects due to the electron EDM de. Accurate theoretical calculations are also absolutely necessary to interpret these ongoing (and perhaps forthcoming) experimental outcomes. For example, knowledge of the effective electric field E (characterized by Wd) on the unpaired electron is required to link the experimentally determined P,T-odd frequency shift with the electron s EDM de in the ground (X2X /2) state of YbF and BaF. 

At low stellar temperatures, nuclear reactions occur predominantly in the direction leading to positive values of Q, but at higher temperatures the inverse reactions become increasingly significant. In Eq. (2.15), owing to time-reversal invariance, the matrix elements are the same for both forward and reverse reactions, so that the ratio of the two cross-sections is... 

Very accurate results were obtained for the classically chaotic Sinai billiard by Bohigas, Giannoni, and Schmit (see Fig. 2) which led them to the important conclusion (Bohigas, Giannoni and Schmit, 1984) Spectra of time-reversal invariant systems whose classical analogues are K systems show the same fluctuation properties as predicted by the Gaussian orthogonal ensemble (GOE) of random-matrix theory... 

RMT). K systems are most strongly mixing classical systems with a positive Kolmogorov entropy. The conjecture turned out valid also for less chaotic (ergodic) systems without time-reversal invariance leading to the Gaussian unitary ensemble (GUE). 

One distinguishes between three different types depending on space-time symmetry classified by the Dyson parameter (3d = 1,2,4 (Guhr, Muller-Groeling and Weidenmuller, 1998). The Gaussian orthogonal ensemble (GOE, (3d = 1) holds for time-reversal invariance and rotational symmetry of the Hamiltonian... 

In short, the distributivity of the transformation f/t implies that retains the reducibility of the Liouville equation into a pair of Schrodinger equations. Furthermore, this transformation retains the time-reversal invariance of these equations, since the free-motion equations [Eqs. (15)] are time-reversal invariant. 

This equation is explicitly time-reversal invariant, because it only contains the square of the momentum p = - ihV. By taking the complex conjugate of Eq. (A.l), we have... 

The latter condition is commonly known as microscopic reversibility or local detailed balance. This property is equivalent to time reversal invariance in deterministic (e.g., thermostatted) dynamics. Although it can be relaxed by requiring just global (rather than detailed) balance, it is physically natural to think of equilibrium as a local property. Microscopic reversibility, a common assumption in nonequilibrium statistical mechanics, is the crucial ingredient in the present derivation. 

S. R. Jain When Prof. Rice talks about optimal control schemes, his Lagrange function follows a time-reversed Schrodinger equation. Is it assumed in the variational deduction that the Hamiltonian is time reversal invariant that is, is it always diagonalizable by orthogonal transformations ... 

The form displayed in eq. (2-40) implies that the ratios of the amplitudes for scattering into different exit channels are independent of the entrance channel. This, of course, will only be true if the resonance is long lived, so that memory of the initial state can be lost. Note that Aga is a symmetric function, which is a consequence of time-reversal invariance. Note also that, within the approximations used, the phase shift associated with a given channel is just the elastic scattering phase shift for that channel. Finally, the partial widths are proportional to the probability of decay from channel fi. Equation (2-41) is, then, merely a statement that the total probability of decay from channel is the sum of the probabilities of decay into individual channels. 

Time reversal invariance describes the fact that in reactions between elementary particles, it does not make any difference if the direction of the time coordinate is reversed. Since all reactions are invariant to simultaneous application of mirror inversion, charge conjugation, and time reversal, the combination of all three is called CPT symmetry and is considered to be a very fundamental symmetry of nature. 

This equation is not time-reversal-invariant. The second term on the right-hand side indicates that there will be a dissipation of energy during the propagation of a photon. 

Since T(q, R) and. if both commute with D(q), the T(q, AR)) commute with D(q). This is referred to as the time-reversal invariance of the dynamical matrix. 

In the following we also assume that H is time reversal invariant. Therefore, states and matrix elements can be assumed to be real (Messiah (1979)). Differentiating the orthogonality relation in (4.1.50) with respect to e we obtain... 

The Hamiltonian (5.3.2) of the quantum kicked rotor is time reversal invariant. The question is, what happens to the localization length of the kicked rotor if a time reversal violating interaction is switched on The difficulty here is to add time reversal violating terms to (5.3.2) in such a way that the classical properties of the Hamiltonian are not affected. This is important, since otherwise a change in the localization length is not surprising since it can always, at least partially, be blamed on the... 

---