to particle theory gives that: a) the electric charge and the baryon and lepton quantum numbers can be only approximately conserved (i.e. the notion of a particle and its antiparticle is only approximate); b) particles which in standard theory are treated as neutral (i.e. coinciding with their antiparticles) cannot be elementary. We consider a possibility that only Dirac singletons can be true elementary particles. Finally we discuss a conjecture that classical time t manifests itself as a consequence of the fact that p changes, i.e.
p and not
t is the true evolution parameter.
The monograph will be based on my results published in:
[1] F. M. Lev, Some Group-theoretical Aspects of SO(1,4)-Invariant Theory. J. Phys., A21, 599–615 (1988).
[2] F. Lev, Representations of the de Sitter Algebra Over a Finite Field and Their Possible Physical Interpretation. Yad. Fiz., 48, 903–912 (1988).
[3] F. Lev, Modular Representations as a Possible Basis of Finite Physics. J. Math. Phys., 30, 1985–1998 (1989).
[4] F. Lev, Finiteness of Physics and its Possible Consequences. J. Math. Phys., 34, 490–527 (1993).
[5] F. Lev, Exact Construction of the Electromagnetic Current Operator in Relativistic Quantum Mechanics. Ann. Phys. 237, 355–419 (1995).
[6] F. M. Lev, The Problem of Interactions in de Sitter Invariant Theories. J. Phys., A32, 1225–1239 (1999).
[7] F. Lev, Massless Elementary Particles in a Quantum Theory over a Galois Field. Theor. Math. Phys., 138, 208–225 (2004). The journal is published by Springer.
[8] F. M. Lev, Could Only Fermions Be Elementary? J. Phys., A37, 3287–3304 (2004).
[9] F. Lev, Why is Quantum Theory Based on Complex Numbers? Finite Fields and Their Applications, 12, 336–356 (2006).
[10] F. M. Lev, Quantum Theory and Galois Fields, International J. Mod. Phys. B20, 1761–1777 (2006).
[11] F. M. Lev, Positive Cosmological Constant and Quantum Theory. Symmetry 2(4), 1401–1436 (2010).
[12] F. M. Lev, Introduction to a Quantum Theory over a Galois Field. Symmetry 2(4), 1810–1845 (2010).
[13] F. M. Lev, Is Gravity an Interaction? Physics Essays, 23, 355–362 (2010).
[14] F. Lev, Do We Need Dark Energy to Explain the Cosmological Acceleration? J. Mod. Phys. 9A, 1185–1189 (2012).
[15] F. Lev, de Sitter Symmetry and Quantum Theory. Phys. Rev. D85, 065003 (2012).
[16] F. M. Lev, A New Look at the Position Operator in Quantum Theory. Physics of Particles and Nuclei, 46, 24–59 (2015). The journal is published by Springer.
[17] F. M. Lev, Why Finite Mathematics Is The Most Fundamental and Ultimate Quantum Theory Will Be Based on Finite Mathematics. Physics of Elementary Particles and Atomic Nuclei Letters, 14, 77–82 (2017). The journal is published by Springer.
[18] F. M. Lev, Fundamental Quantal Paradox and its Resolution. Physics of Elementary Particles and Atomic Nuclei Letters, 14, 444–452 (2017). The journal is published by Springer.
and possibly in other journals.
I graduated from the Moscow Institute for Physics and Technology, got a PhD from the Institute of Theoretical and Experimental Physics in Moscow and a Dr. Sci. degree from the Institute for High Energy Physics (also known as the Serpukhov Accelerator). In Russia there are two doctoral degrees; Dr. Sci. degree is probably an analog of Habilitationsschrift in Germany. In Russia I worked at the Joint Institute for Nuclear Research (Dubna, Moscow region) and now I work at a software company in Los Angeles, USA.
I have many papers published in known journals (Ann. Phys., Few Body Systems, J. Math. Phys., J. Phys. A, Nucl. Phys. C, Phys. Rev. C and D, Phys. Rev. Letters and others). The majority of those papers are done in the framework of more or less mainstream approaches. On the other hand, the proposed monograph will be done in the fully new approach which I am working on for many years. In this approach quantum theory is based on finite mathematics.
I think that the main problems in convincing physicists that ultimate quantum theory will be based on finite mathematics are not scientific but subjective. First of all, the majority of physicists do not have even a very basic knowledge in finite mathematics. This is not a drawback because everybody knows something and does not know something and it is impossible to know everything. However, many physicists have a mentality that only their vision of physics is correct, they do not accept that different approaches should be published and if they do not understand something or something is not in the spirit of their dogmas then this is pathology or exotics which has nothing to do with physics.
Probably this situation has happened in view of several reasons. For example, the successes of QED at the end of the 40th were very impressive and it is of course impressive that the theory gives correct eight digits for the electron and muon magnetic moments and five digits for the Lamb shift. From mathematical point of view QED has several inconsistencies the reasons of which are clear. The above famous results are obtained by subtracting infinities from each other. However, in view of these and other results the mentality of the majority of physicists is that agreement with the data is much more important than mathematical consistency and many of those physicists believe that all fundamental problems of quantum theory can be solved in the framework of QFT or string theory (which has similar mathematical inconsistencies).
The meaning of «quantum» is discrete and historically the name «quantum theory» has arisen because it was realized that some physical quantities have discrete spectrum. The founders of quantum theory were highly educated physicists but they used only standard continuous mathematics, and even now discrete and finite mathematics is not a part of standard mathematical education at physics departments. Several famous physicists (e.g. Schwinger, Wigner, Nambu, Gross and others) discussed a possibility that ultimate quantum theory will be based on finite mathematics. One of the reasons is that in this case
