Generalized time-reversal symmetry

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. 

The first term in (13), also called the diagonal term (Berry 1985), originates from periodic orbit pairs (p,p ) related through cyclic permutations of the vertex symbol code. There are typically n orbits of that kind and all these orbits have the same amplitude A and phase L. The corresponding periodic orbit pair contributions is (in general) g n - times degenerate where n is the length of the orbit and g is a symmetry factor (g = 2 for time reversal symmetry). 

The idea of Pollicott-Ruelle resonances relies on this mechanism of spontaneous breaking of the time-reversal symmetry [20, 21]. The Polhcott-Ruelle resonances are generalized eigenvalues sj of LiouviUian operator associated with decaying eigenstates which are singular in the stable phase-space directions but smooth in the unstable ones ... 

In connection to control in dynamics I would like to take here a general point of view in terms of symmetries (see Scheme 1) We would start with control of some symmetries in an initial state and follow their time dependence. This can be used as a test of fundamental symmetries, such as parity, P, time reversal symmetry, T, CP, and CPT, or else we can use the procedure to discover and analyze certain approximate symmetries of the molecular dynamics such as nuclear spin symmetry species [2], or certain structural vibrational, rotational symmetries [3]. 

Describing SR in terms of a susceptibility is particularly advantageous for systems that are in thermal equilibrium, or in quasiequilibrium. In such cases the fluctuation-dissipation relations [9] can be used to express the susceptibility in terms of the spectral density of fluctuations in the absence of the periodic driving. This was used explicitly in the case of noise-protected heterodyning. It is true in general that the analysis of fluctuations is greatly facilitated by the presence of thermal equilibrium when the conditions of detailed balance and of the time reversal symmetry are satisfied [44]. 

Specialized to thermal equilibrium, the velocity distributions for the molecules are the Maxwell-Boltzmann distribution (a special case of the general Boltzmann distribution law). The expression for the rate constant at temperature T, k(T), can be reduced to an integral over the relative speed of the reactants. Also, as a consequence of the time-reversal symmetry of the Schrodinger equation, the ratio of the rate constants for the forward and the reverse reaction is equal to the equilibrium constant (detailed balance). 

As the existence of MChA can be deduced by very general symmetry arguments and the effect does not depend on the presence of a particular polarization, one may wonder if something like MChA can also exist outside optical phenomena, e.g. in electrical conduction or molecular diffusion. Time-reversal symmetry arguments cannot be applied directly to the case of diffusive transport, as diffusion inherently breaks this symmetry. Instead, one has to use the Onsager relation. (For a discussion see, e.g., Refs. 34 and 35.) For any generalized transport coefficient Gy (e.g., the electrical conductivity or molecular diffusion tensor) close to thermodynamic equilibrium, Onsager has shown that one can write... 

In essence, real world-space is not Euclidean and space is generally curved into the time dimension, consistent with the theory of general relativity. The curvature may not be sufficient to become obvious in a local context. However, it is sufficient to break the time-reversal symmetry that seems to characterize the laws of physics. Not only does it cause perpetual time flow with respect to all mass, but actually identifies a fixed direction for this flow. It creates an arrow of time and thereby eliminates an inconsistency in the logic of physics how reversible microscopic laws can underpin an irreversible macroscopic world. General curvature of space breaks the time-reversal symmetry and produces chiral space, manifest in the right-hand... 

The subscript C/ in (4.1.45) stands for unitary since the system Hamiltonian is invariant under general unitary transformations in case all antiunitary symmetries are broken. The effects of time reversal symmetry breaking on the level statistics were recently investigated experimentally by So et al. (1995) and Stoffregen et al. (1995) by measuring the resonance spectrum of quasi-two-dimensional microwave cavities in the presence of time reversal breaking elements such as ferrites and directional couplers. These experiments are of considerable relevance for quantum mechanics since quasi-two-dimensional microwave cavities are generally considered to be excellent models for two-dimensional quantum billiards. 

If we could analytically evaluate the action of the operator in Eq. [144] for any potential energy function or force field, there would be no need for molecular dynamics simulation. Since that is generally not possible, we need to devise a numerical scheme that will solve Eq. [144] to a desired accuracy, while preserving the symmetry of the equations of motion (e.g., time-reversal symmetry). 

Due to time-reversal symmetry the spin-polarization from a paramagnetic solid using circularly polarized radiation is just reversed if the helicity of the radiation is reversed. This feature is of course removed if the solid is magnetically ordered, giving rise to magnetic circular dichroism in valence-band photoemission (Schneider et al. 1991). In a corresponding experiment we have in general the emission direction. 

The formalism can be straightforwardly extended to two-electron operators with the introduction of Kramers double replacement operators [81]. However, the multitude of terms arising in subsequent derivations finally leads to a rather cumbersome formalism. In the author s opinion it is better to derive general formulas and then consider the structure due to time reversal symmetry after the derivation. 

From the above observations we can draw the following conclusions The gradient vector appearing on the right hand side of the first-order response equation (223) generally has positive hermiticity. From (244) and (245) it is then clear that the Hessian and the matrix 5 should be multiplied with Hermi-tian and anti-Hermitian trial vectors, respectively. In the static case (O — O all trial vectors should consequently be restricted to have the Hermitian structure. Furthermore, tried vectors should have the time reversal symmetry of the property gradient since both 4 conserves time reversal symmetry. Then,... 

To illustrate the general applicability of the relaxation equations of Section 11.4 let us study the simple case of a single conserved variable A(q, t) which has the form given by Eq. (11.5.32). The property aj of the jth molecule is presumed to have definite time-reversal symmetry and parity. 

A general statement of this argument is that in an isotropic system flows and forces of different tensorial orders are not coupled. This is known as the Curie principle. Systems that are anisotropic often have some elements of symmetry which reduce the number of nonzero coefficients from the maximum of n2. To prove these relations one must apply the arguments of Chapter 11 involving parity, reflection symmetries, rotational symmetries, and time-reversal symmetries. 

It is well established that the principal results of the generalized kinetic theory, especially the functional form of the slow portion of the memory function, can be obtained also by a direct mode-coupling approach [18, 19, 20]. The basic idea behind the mode-coupling theory is that the fluctuation of a given dynamical variable decays, at intermediate and long times, predominantly into pairs of hydrodynamic modes associated with quasi-conserved dynamical variables. The possible decay channels of a fluctuation are determined by selection rules based, for example, on time-reversal symmetry or on physical considerations. 

---