This book is addressed to readers who want to have a look at the laws of micro and macro world from a single viewpoint. This is the English translation of our "Theory of Non-Geodesic Motion of Particles'', originally published in Russian in 1999, with some recent amendements.

The background behind the book is as follows. In 1991 we initiated a study to find out what kinds of particles may theoretically inhabit four-dimensional space-time. As the instrument, we equipped ourselves with mathematical apparatus of physical observable values (chronometric invariants) developed by A. L. Zelmanov, a prominent cosmologist.

The study was completed by 1997 to reveal that aside for mass-bearing and massless (light-like) particles, those of third kind may exist. Their trajectories lay beyond regular space-time of General Relativity. For a regular observer the trajectories are of zero four-dimensional length and zero three-dimensional observable length. Besides, along these trajectories interval of observable time is also zero. Mathematically, that means such particles inhabit fully degenerated space-time with non-Riemannian geometry. We called such space "zero-space'' and such particles --- "zero-particles''.

For a regular observer their motion in zero-space is instant, i. e. zero-particles are carriers of long-range action. Through possible interaction with our world's mass-bearing or massless particles zero-particles may instantly transmit signals to any point in our three-dimensional space.

Considering zero-particles in the frames of the wave-particle concept we obtained that for a regular observer they are standing waves and the whole zero-space is a system of standing waves (zero-particles), i. e. a standing-light hologram. This result links with "stop of light'' experiment (Harvard Astronomical Observatory, January 2001).

Using methods of physical observable values we also showed that in basic four-dimensional space-time a mirror world may exist, where coordinate time has reverse flow in respect to regular observer's time.

We published the results in 1997 in two books: Borissova L. B. and Rabounski D. D. The theory of movement of particles in four-dimensional space-time. Lomonossov Workshop, Moscow, 1997; Rabounski D. D. Three forms of existence of matter in four-dimensional space-time: Particles here and beyond the mirror. Lomonossov Workshop, Moscow, 1997.

B. M. Levin, an expert in orthopositronium problem came across these publications. He contacted us immediately and told us about critical situation around anomalies in annihilation of orthopositronium, which had been awaiting theoretical explanation for over a decade.

Rate of annihilation of orthopositronium (the value reciprocal to its life span) is among the references set to verify the basic laws of Quantum Electrodynamics. Hence any anomalies contradict with these reliably proven laws. In 1987 Michigan Group of researchers (Ann Arbor, Michigan, USA) using advanced precision equipment revealed that the measured rate of annihilation of orthopositronium was substantially higher compared to its theoretical value.

That implies that some atoms of orthopositronium annihilate not into three photons as required by laws of conservation, but into lesser number of photons, which breaks those laws. In the same 1987 B. M. Levin discovered what he called "isotope anomaly'' in anomalous annihilation of orthopositronium (Gatchina--St.Petersburg, Russia). Any attempts to explain the anomalies by means of Quantum Electrodynamics over 10 years would fail. This made S. G. Karshenboim, a prominent expert in the field, to resume that all capacities of standard Quantum Electrodynamics to explain the anomalies were exhausted.

In our 1997 publications B. M. Levin saw a means of theoretical explanation of orthopositronium anomalies by methods of General Relativity and suggested a joint research effort in this area.

Solving the problem we obtained that our world and the mirror Universe are separated with a space-time membrane, which is a degenerated space-time (zero-space). We also arrived to physical conditions under which exchanges may occur between our world and the mirror Universe. Thanks to this approach and using methods of General Relativity we developed a geometric concept of virtual interactions: it was mathematically proven that virtual particles are zero-particles that travel in zero-space and carry long-range action. Application of the results to annihilation of orthopositronium showed that two modes of decay are theoretically possible: (a) all three photons are emitted into our Universe; (b) one photon is emitted into our world, while two others go to the mirror Universe and become unavailable for observation.

All the above results stemmed exclusively from application of Zelmanov's mathematical apparatus of physical observable values.

When tackling the problem we had to amend the existing theory with some new techniques. In their famous "The Classical Theory of Fields'', which has long ago become a de-facto standard for a university reference book on General Relativity, L. D. Landau and E. M. Lifshitz give an excellent account of theory of motion of particle in gravitational and electromagnetic fields. But the monograph does not cover motion of spin-particles, which leaves no room for explanations of orthopositronium experiments (as its para and ortho states differ by mutual orientation of electron and positron spins). Besides, L. D. Landau and E. M. Lifschits employed general covariant methods. The technique of physical observable values (chronometric invariants) has not been yet developed by that time by A. L. Zelmanov, which should be also taken into account.

Therefore we faced the necessity to introduce methods of chronometric invariants into the existing theory of motion of particles in gravitational and electromagnetic fields. Separate consideration was given to motion of particles with inner mechanical momentum (spin). We also added a chapter with account of tensor algebra and analysis. This made our book a contemporary supplement to "The Classical Theory of Fields'' to be used as a reference book in university curricula.

In conclusion we would like to express our sincere gratitude to Dr. Abram Zelmanov (1913--1987) and Prof. Kyril Stanyukovich (1916--1989). Many years of acquaintance and hours of friendly conversations with them have planted seeds of fundamental ideas which by now grew up in our minds to be reflected on these pages.

We are grateful to Dr. Kyril Dombrovski whose works greatly influenced our outlooks.

We highly appreciate contribution from our colleague Dr. Boris Levin. With enthusiasm peculiar to him he stimulated our writing of this book.

Special thanks go to our family for permanent support. Many thanks to Grigory Semyonov, a friend of ours, for translating the manuscript into English. We also are grateful to our publisher Domingo Marin Ricoy for his interest to our works.

February 2001,

*L.B.Borissova and D.D.Rabounski*

Chapter | 1 Introduction |

| 1.1. | Geodesic motion of particles |

| 1.2. | Physical observable values |

| 1.3. | Dynamic equations of motion of free particles |

| 1.4. | Introducing concept of nongeodesic motion of particles. Problem statement |

Chapter | 2 Tensor algebra and the analysis |

| 2.1. | Tensors and tensor algebra |

| 2.2. | Scalar product of vectors |

| 2.3. | Vector product of two vectors. Antisymmetric tensors and pseudotensors |

| 2.4. | Introducing absolute differential and derivative to the direction |

| 2.5. | Divergence and rotor |

| 2.6. | Laplace and d'Alembert operators |

| 2.7. | Conclusions |

Chapter | 3 Charged particle in pseudo-Riemannian space |

| 3.1. | Problem statement |

| 3.2. | Observable components of electromagnetic field tensor. Field invariants |

| 3.3. | Chronometrically invariant Maxwell equations. Law of conservation of electric charge. Lorentz condition |

| 3.4. | Four-dimensional d'Alembert equations for electromagnetic potential and their observable components |

| 3.5. | Chronometrically invariant Lorentz force. Energy-impulse tensor of electromagnetic field |

| 3.6. | Equations of motion of charged particle obtained using parallel transfer method |

| 3.7. | Equations of motion, obtained using the least action principle as a partial case of the previous equations |

| 3.8. | Geometric structure of electromagnetic four-dimensional potential |

| 3.9. | Building Minkowski equations as a partial case of the obtained equations of motion |

| 3.10. | Structure of the space with stationary electromagnetic field |

| 3.11. | Motion of charged particle in stationary electric field |

| 3.12. | Motion of charged particle in stationary magnetic field |

| 3.13. | Motion of charged particle in stationary electromagnetic field |

| 3.14. | Conclusions |

Chapter | 4 Particle with spin in pseudo-Riemannian space |

| 4.1. | Problem statement |

| 4.2. | Spin-impulse of a particle in the equations of motion |

| 4.3. | Equations of motion of spin-particle |

| 4.4. | Physical conditions of spin-interaction |

| 4.5. | Motion of elementary spin-particles |

| 4.6. | Spin-particle in electromagnetic field |

| 4.7. | Motion in stationary magnetic field |

| 4.8. | Law of quantization of masses of elementary particles |

| 4.9. | Compton wave length |

| 4.10. | Massless spin-particle |

| 4.11. | Conclusions |

Chapter | 5 Physical vacuum and the mirror Universe |

| 5.1. | Introduction |

| 5.2. | Observable density of vacuum. T-classification of matter |

| 5.3. | Physical properties of vacuum. Cosmology |

| 5.4. | Concept of Inversional Explosion of the Universe |

| 5.5. | Non-Newtonian gravitational forces |

| 5.6. | Gravitational collapse |

| 5.7. | Inflational collapse |

| 5.8. | Concept of the mirror Universe. Conditions of transition through membrane from our world into the mirror Universe |

| 5.9. | Conclusions |

Chapter | 6 Annihilation and the mirror Universe |

| 6.1. | Isotope anomaly and lambda-T anomaly of orthopositronium. Problem statement |

| 6.2. | Zero-space as home space for virtual particles. Geometric interpretation of Feynmann diagrams |

| 6.3. | Building mathematical concept of annihilation. Parapositronium and orthopositronium |

| 6.4. | Annihilation of orthopositronium: 2+1 split of 3-photon annihilation |

| 6.5. | Isotope anomaly of orthopositronium |

| 6.6. | Conclusions |

Appendix | A Notation |

Appendix | B Special expressions |

Bibliography |

Index |