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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 NonGeodesic 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 fourdimensional spacetime. 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 massbearing and massless (lightlike) particles, those of third kind may exist. Their trajectories lay beyond regular spacetime of General Relativity. For a regular observer the trajectories are of zero fourdimensional length and zero threedimensional observable length. Besides, along these trajectories interval of observable time is also zero. Mathematically, that means such particles inhabit fully degenerated spacetime with nonRiemannian geometry. We called such space "zerospace'' and such particles  "zeroparticles''. For a regular observer their motion in zerospace is instant, i. e. zeroparticles are carriers of longrange action. Through possible interaction with our world's massbearing or massless particles zeroparticles may instantly transmit signals to any point in our threedimensional space. Considering zeroparticles in the frames of the waveparticle concept we obtained that for a regular observer they are standing waves and the whole zerospace is a system of standing waves (zeroparticles), i. e. a standinglight 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 fourdimensional spacetime 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 fourdimensional spacetime. Lomonossov Workshop, Moscow, 1997; Rabounski D. D. Three forms of existence of matter in fourdimensional spacetime: 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 (GatchinaSt.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 spacetime membrane, which is a degenerated spacetime (zerospace). 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 zeroparticles that travel in zerospace and carry longrange 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 defacto 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 spinparticles, 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 (19131987) and Prof. Kyril Stanyukovich (19161989). 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,
