Time-Reversal Symmetry, 1st ed. 2018 Seven Time-Reversal Operators for Spin Containing Systems Springer Tracts in Modern Physics Series, Vol. 281

This book introduces new developments in the field of Time-Reversal Symmetry presenting, for the first time, the Wigner time-reversal operator in the form of a product of two- or three time-reversal operators of lower symmetry. The action of these operators leads to the sign change of only one or two angular momentum components, not of all of them. It demonstrates that there are six modes of time-reversal symmetry breaking that do not lead to the complete disappearance of the symmetry but to its lowering. The full restoration of the time-reversal symmetry in the six cases mentioned is possible by introducing six types of metaparticles. The book also confirms the presence of six additional time-reversal operators using a group-theoretical method. The problem is only where to seek these metaparticles. 

The book discusses time-reversal symmetry in classical mechanics, classical and relativistic electrodynamics, quantum mechanics and theory of quantized fields, including dynamical reversibility and statistical irreversibility of the time, Wigner?s and Herring?s criteria, Kramers theorem, selection rules due to time-reversal symmetry, Onsager?s relations, Poincaré recurrence theorem, and CPT theorem. It particularly focuses attention on time-reversal symmetry violation. It is proposed a new method of testing the time-reversal symmetry, which is confirmed experimentally by EPR spectroscopy data. 

It shows that the traditional black-white point groups of magnetic symmetry are not applicable to magnetic systems with Kramers degeneration of energy levels and that magnetic groups of four-color symmetry are adequate for them. Further, it addresses the predicted structural distortions in Kramers three-homonuclear magnetic clusters due to time-reversal symmetry that have been identified experimentally. 

Lastly, it proposes a method of synthesis of two-nuclear coordination compounds with predictable magnetic properties, based on the application of the time-reversal transformation that was confirmed experimentally.

1. Time Reversal in Classical and Relativistic Physics

1.1 The Time Conception and Time Translation Invariance

1.2 Cinematically Admissible Transformations and Time Reversal

1.3 Time-Reversal Symmetry in Dynamical Systems

1.4 Painlevé’s Theorem

1.5 Time-Reversal Symmetry in Classical Electrodynamics

1.6 Time-Reversal Symmetry in Relativistic Electrodynamics

1.8 Reversibility of Fluctuations in Closed Systems as Consequence of Onsager Relationships

1.9 Poincaré Recurrence Theorem

2.Time Reversal in Quantum Mechanics and Theory of Quantized Fields

2.1  The Basic Concepts of Quantum Mechanics

2.2   Antilinear and Antiunitary Operators

2.3   Wigner’s Time-Reversal Operator 

2.4   Time-Reversal Operator for High Spin Systems

2.5   Time-Reversal Operator for Symmetry Point Groups

2.6   Wigner’s Criterion of Energy Levels Degeneracy due to Time-Reversal Symmetry

2.7   Herring’s Criterion of Energy Bands Degeneracy due to Time-Reversal Symmetry

2.8   Co-representations of Symmetry Groups

2.10 Nonconventional Time-Reversal Symmetry

2.11 Selection Rules due to Time-Reversal Symmetry

2.12 Time-Reversal and Detailed Balance Principle

2.13 Dynamic Matrix and Time-Reversal Operator

2.14 Time-Reversal Symmetry in Theory of Quantized Fields

2.15 The CPT Theorem

3.1 Magnetic Two-Color Point Symmetry Groups of non-Kramers Systems

3.2 Invariant Spin Arrangement and Admissible Magnetic Point Groups of non-Kramers Systems

3.3 Magnetic Four-Color Point Symmetry Groups of Kramers Systems

4. Anomalous Behavior of Trihomonuclear Kramers Clusters Due to Their Four-Color

    Symmetry

4.1 Structural Asymmetry of Trihomonuclear Kramers Clusters as Consequence of Time-Reversal

      Symmetry

4.2 Trinuclear Chromium(III) and Iron(III) Carboxylate Clusters

4.3 Trinuclear Copper(II) Clusters

4.4 Trinuclear Vanadium(IV) and Cobalt(II) Clusters 

4.5 Concluding Remarks    

5. Time Reversal Symmetry of Quantum Systems with Quasi-Energy Spectrum

5.1 Non-Stationary States of Quantum Systems under Time-Reversal Operator

5.2 Time-Reversal Invariance of Schrödinger Equation for the Green Function

5.3 Quasi-Energy Spectrum and Brillouin Zone in Quasi-Energy Space

5.4 Time-Reversal Symmetry in the Case of Commuting Time Translation and Time-Reversal Operators

5.5 Quasi-Energy Doublets due to non-Commuting Time Translation and Time-Reversal Operators

6. Transformation of Antiferromagnetic Type of Exchange Interactions into Ferromagnetic

    One in Dimer Clusters

6.1 Magnetic Dimer Clusters in Coordination Compounds

      6.1.1 Copper(II) Dimers

      6.1.2 Dimer clusters of Other 3d Elements

      6.1.3 Dimer Clusters of 4f-Elements

6.2 Combined Time-Reversal Transformation 

6.3 Spin Levels Inversion in Cu (II) – Cu (II) Dimers due to Combined Time-Reversal Symmetry

      Symmetry

6.5 Experimental Evidence of Spin Levels Inversion in Dimer Magnetic Clusters due to Combined         

Time-Reversal Symmetry   

7. Is there an Analogy between Jahn-Teller Effect and the Instability of Spin Population in Kramers Clusters with Odd Number of Atoms? 

7.1Kahn’s Instability of an Equilateral Spin Trimer 1/2 x1/2 x1/2 with Respect to a Weak

      Structural Perturbation

   7.2 Mutual Compensation of Distorted-Induced Spin Polarization in a Trimer 1/2 x1/2 x1/2

       due to Time-Reversal Symmetry                                                                                              

  7.3   Mutual Compensation of Distorted-Induced Spin Polarization in a Trimer 5/2 x5/2 x5/2

       due to Time-Reversal Symmetry 

7.4     Distorted-Induced Spin Populations Instability of Trimer Homonuclear Clusters with

       Kramers Degeneracy of Energy Levels Due to Time-Reversal Symmetry Violation

8. Non-Abelian Symmetry Groups Containing the Time-Reversal Operator

      8.1 Non-Abelian Group G₈^(1/2) of 8^th Order Related to a Particle with the Spin 1/2

      8.2 Extention of the Group G₈^(1/2) to non-Abelian Groups of 16^th Order Related to Kramers Systems   

      8.3 Non-Abelian Groups of 8^th Order Related to non-Kramers Systems Systems

      8.5 Pecularities of the Structure of non-Abelian Groups of of 8^th and 16^th Orders 

9. Factorization of the Wigner’s Time-Reversal Operator and Reduction of Time-Reversal Symmetry

9.1  Six New Types of Time-Reversal Symmetry Related to Kramers Systems

9.2 Violation of the Kramers Theorem

9.3 Six New Types of Time-Reversal Symmetry Related to non-Kramers Systems

9.4 Commutation and Anticommutation Relations for Time-Reversal Operators

9.5 Unitarity of Spinor Operators in Two-Boson Representation of Angular Momentum and 

      Time-Reversal Symmetry

9.6 Boson-Antiboson Representation of Angular Momentum and Its Correlation with

      Factorization of the Wigner’s Time-Reversal Operator

9.7 Restoration of the Wigner Time-Reversal Symmetry: Six Types of Metaparticles and Metaworlds

10. Time-Reversal Symmetry Violation

10.1 Time-Reversal Symmetry Violation in Meson Systems

10.2 Time-Reversal Symmetry Violation in Light and Heavy Nuclei

10.3 Time-Reversal Symmetry Violation in Atoms and Molecules

10.4 Time-Reversal Symmetry Violation in Superconductors

10.5 Time-Reversal Symmetry Violation and Transformation of Zeeman Levels in Energy Bands

10.6 Time-Reversal Symmetry Violation in Hyperfine Interaction

10.7 Time-Reversal Symmetry Violation in Optical Nonlinear Stimulated Radiation and Absorption

10.8 Time-Reversal Symmetry Violation and the Enhancement of Quantum Transport

10.9 Time-Reversal Symmetry Violation and the Unidirectionality of the Time

Appendices

Glossary

References

Index 

Prof. Ion I. Geru got the PhD degree in 1967. In 1986, he became professor of theoretical physics, State University of Moldova. In 1996–1997, he was head of the Department of General Physics. In the years 2005–2008, he was director of the National Center for Analytical Methods and Metrology of the Academy of Sciences of Moldova. In 2008-2013, he was head of the Laboratory “Magnetic Resonance and Laser Spectroscopy”, Center of Chemical Physics and Nanocomposites, Institute of Chemistry of the Academy of Sciences of Moldova. Currently, he is Chief Researcher at the same Institute. 

In the years 2004–2007, Prof. Geru was a part-time visiting professor at the Florida State University. Since 2000, he is a Corresponding Member of the Academy of Sciences of Moldova. In 1998-2017, he was member of the AMPERE Committee. Since 1984, he is the Vice-President of the Moldovan Physical Society. He has more than 330 publications and 4 books, including I. Geru, D. Suter, Resonance Effects of Excitons and Electrons: Basics and Applications, Springer-Verlag, 2013.

Introduces new discrete non-Abelian groups of symmetry, allowing non-trivial factorization of the Wigner time-reversal operator

Develops a method for binuclear coordination compounds with predicted magnetic properties

Indicates that traditional black-white point groups of magnetic symmetry cannot be applied to systems with Kramer´s degeneracy of energy levels

Predicts and gives experimental confirmation for structural distortions in three-homo-nuclear magnetic clusters caused by time-reversal symmetry