Wigner time-reversal

In quantum mechanics, the operation t — —t (time-reversal) is to be accompanied by complex conjugation (i — —i) so that the Schrodinger equation remains invariant. This operation is called Wigner time-reversal. 

Taking the experimentally measured mass spectrum of hadrons up to 2.5 GeV from the Particle Data Group, Pascalutsa (2003) could show that the hadron level-spacing distribution is remarkably well described by the Wigner surmise for / = 1 (see Fig. 6). This indicates that the fluctuation properties of the hadron spectrum fall into the GOE universality class, and hence hadrons exhibit the quantum chaos phenomenon. One then should be able to describe the statistical properties of hadron spectra using RMT with random Hamiltonians from GOE that are characterized by good time-reversal and rotational symmetry. 

According to the Wigner theorem (see Appendix A), because of time-reversal. symmetry, the functions i ground-state wavefunction does not have a node, we can always make the... 

Wigner, E. (1930). The operation of time reversal in quantum mechanics. Gdttinger Nachrichten, 133-146. See Group Theory and Solid State Physics, edited by P. H. Meijer, Gordon and Breach Science Publishers, 1964, New York, 265-278. 

SU(3) symmetry in hypernuclear physics Radicati, Wigner s supermul-tiplet theory100 Fraunfelder, Parity and Time Reversal in Nuclear Physics Wilkinson, the isobaric analogue symmetry Aage Bohr, the permutation group in light nuclei and J. P. Elliot, the shell model symmetry. 

Here T symbolizes Wigner s time reversal operator. 

To conclude this section we point out that the Wigner statistics is valid only if the system has integer spin and is invariant under an anti-unitary symmetry such as time reversal symmetry. If all the antiunitary symmetries are broken, the nearest neighbour statistics is expected to be... 

As pointed out above, the Wigner surmise and the analysis of statistical distributions require a careful study of which quantum numbers are exact, i.e. which rules a system must obey precisely. A rule which has not been considered so far is time-reversal invariance, which applies when a system evolves backwards in time in precisely the same way as it evolved forward, i.e. when the sign of time can be changed without affecting the basic equations. 

We note that according to considerations by Wigner, the time reversal T is an antiunitary transformation as is the space-time inversion PT. Hence these transformations do not quite fit into the scheme of (80), because they perform an additional complex conjugation of the wave function. We also note that there are several (inequivalent) possibilities to implement these transformations in the Hilbert space of the Dirac equation. 

Note that the action of time reversal on the spin functions precisely corresponds to the C operator and thus is represented hy (Cj). This result may be generalized, in the sense that time reversal can be represented as the product of complex conjugation, denoted as K, and a unitary operator acting on the components of a function space, which we shall denote by the unitary matrix U. We thus write = U/f. When this operator is applied twice, it must return the same state, except possibly for a phase factor, say exp(fx). Following Wigner, we now show that the two cases = 1 are in fact the only possibilities. Hence, the phase factor can be only either H-1 (time-even state) or -1 (time-odd state) [11, Chap. 26]. Taking time reversal twice, we have... 

For the space group Pnam, the compatibility relations are shown in Table IV.5, where symbols follow the notations by Bouckaert, Smolu-chowski, and Wigner (1936). The two symmetry species partitioned by a broken line in a block are degenerate pairs due to the time-reversal symmetry, and accordingly the overall degeneracy of crystal vibrations is twice the degeneracy due to the space symmetry. 

Wigner (1930) has shown that if time is reversible in a quantum-mechanical system, then all wavefunctions can be made real. This theorem enables us to use real wavefunctions whenever possible, which are often more convenient than complex ones. Here we present a simplified proof of Wigner s theorem, with some examples of its applications. 

Fig. 5.9. Comparison of the real-time charge reversal process Ag —> Ags - Ag. (a) Simulation applying the density matrix method in the Wigner representation (Tanions = 300 K, 100fs pulse width), (b) Pump probe experiment (Tanions = 300 K, 100fs pulse width), (c) Simulation applying the density matrix method in the Wigner representation in case of a 5 times shorter probe pulse width (by courtesy of M. Hartmann taken from [136])...

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