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Proceedings for our meeting ``What comes beyond the Standard Models'', which covered a broad series of subjects., Comment: This is the proceedings for the 25th Workshop in Bled for "What comes beyond the Standard Models'' including also a webinar "meeting'' using Cosmovia on the same subject

Physicists usually believe that physics cannot (and should not) derive the values of $c$ and $\hbar$ but should derive the value of the cosmological constant $\Lambda$. This problem is considered fundamental after the phenomenon of cosmological acceleration (PCA) was discovered in 1998. This phenomenon is usually considered in the framework of General Relativity (GR) and here the main uncertainty is how the background space is treated. If it is flat, PCA is usually treated as a manifestation of dark energy and (as acknowledged in the literature) currently its nature is a mystery. On the other hand, if it is curved then a problem arises why the value of $\Lambda$ is as is. However, in our approach based only on universally recognized results of physics, the solution of the problem does not contain uncertainties because PCA is an inevitable kinematical consequence of quantum theory in semiclassical approximation. Since the de Sitter (dS) algebra is semisimple, it is the most general ten-dimensional Lie algebra because it cannot be obtained by contraction from other ten-dimensional Lie algebras. Let $R$ be the parameter of contraction from the dS algebra to the Poincare one. Then the problem why the quantities $(c,\hbar,R)$ are as are does not arise because they are contraction parameters for transitions from more general Lie algebras to less general ones. In our approach, background space and its geometry (metric and connection) are not used but, in semiclassical approximation, the result for PCA is the same as in GR if $\Lambda=3/R^2$., Comment: 11 pages, no figures. arXiv admin note: substantial text overlap with arXiv:2201.13231

The purpose of this paper is to explain at the simplest possible level why finite mathematics based on a finite ring of characteristic $p$ is more general (fundamental) than standard mathematics. The belief of most mathematicians and physicists that standard mathematics is the most fundamental arose for historical reasons. However, simple {\it mathematical} arguments show that standard mathematics (involving the concept of infinities) is a degenerate case of finite mathematics in the formal limit $p\to\infty$: standard mathematics arises from finite mathematics in the degenerate case when operations modulo a number are discarded. Quantum theory based on a finite ring of characteristic $p$ is more general than standard quantum theory because the latter is a degenerate case of the former in the formal limit $p\to\infty$., Comment: 22 pages, one figure

In standard quantum theory, symmetry is defined in the spirit of Klein's Erlangen Program: the background space has a symmetry group, and the basic operators should commute according to the Lie algebra of that group. We argue that the definition should be the opposite: background space has a direct physical meaning only on classical level while on quantum level symmetry should be defined by a Lie algebra of basic operators. Then the fact that de Sitter symmetry is more general than Poincare one can be proved mathematically. The problem of explaining cosmological acceleration is very difficult but, as follows from our results, there exists a scenario that the phenomenon of cosmological acceleration can be explained proceeding from basic principles of quantum theory. The explanation has nothing to do with existence or nonexistence of dark energy and therefore the cosmological constant problem and the dark energy problem do not arise. We consider finite quantum theory (FQT) where states are elements of a space over a finite ring or field with characteristic $p$ and operators of physical quantities act in this space. We prove that, with the same approach to symmetry, FQT and finite mathematics are more general than standard quantum theory and classical mathematics, respectively: the latter theories are special degenerated cases of the former ones in the formal limit $p\to\infty$., Comment: 32 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:1905.02788, arXiv:1104.4647

Physicists usually understand that physics cannot (and should not) derive that $c\approx 3\cdot 10^8m/s$ and $\hbar \approx 1.054\cdot 10^{-34}kg\cdot m^2/s$. At the same time they usually believe that physics should derive the value of the cosmological constant $\Lambda$ and that the solution of the dark energy problem depends on this value. However, background space in General Relativity (GR) is only a classical notion while on quantum level symmetry is defined by a Lie algebra of basic operators. We prove that the theory based on Poincare Lie algebra is a special degenerate case of the theories based on de Sitter (dS) or anti-de Sitter (AdS) Lie algebras in the formal limit $R\to\infty$ where R is the parameter of contraction from the latter algebras to the former one, and $R$ has nothing to do with the radius of background space. As a consequence, $R$ is necessarily finite, is fundamental to the same extent as $c$ and $\hbar$, and a question why $R$ is as is does not arise. Following our previous publications, we consider a system of two free bodies in dS quantum mechanics and show that in semiclassical approximation the cosmological dS acceleration is necessarily nonzero and is the same as in GR if the radius of dS space equals $R$ and $\Lambda=3/R^2$. This result follows from basic principles of quantum theory. It has nothing to do with existence or nonexistence of dark energy and therefore for explaining cosmological acceleration dark energy is not needed. The result is obtained without using the notion of dS background space (in particular, its metric and connection) but simply as a consequence of quantum mechanics based on the dS Lie algebra. Therefore, $\Lambda$ has a physical meaning only on classical level and the cosmological constant problem and the dark energy problem do not arise., Comment: 16 pages, no figures. A version published in Physics of Particles and Nuclei Letters. arXiv admin note: substantial text overlap with arXiv:1104.4647; text overlap with arXiv:1711.04607

We first give a rigorous mathematical proof that classical mathematics (involving such notions as infinitely small/large, continuity etc.) is a special degenerate case of finite one in the formal limit when the characteristic $p$ of the field or ring in finite mathematics goes to infinity. We consider a finite quantum theory (FQT) based on finite mathematics and prove that standard continuous quantum theory is a special case of FQT in the formal limit $p\to\infty$. The description of states in standard quantum theory contains a big redundancy of elements: the theory is based on real numbers while with any desired accuracy the states can be described by using only integers, i.e. rational and real numbers play only auxiliary role. Therefore, in FQT infinities cannot exist in principle, FQT is based on a more fundamental mathematics than standard quantum theory and the description of states in FQT is much more thrifty than in standard quantum theory. Space and time are purely classical notions and are not present in FQT at all. In the present paper we discuss how classical equations of motions arise as a consequence of the fact that $p$ changes, i.e. $p$ is the evolution parameter. It is shown that there exist scenarios when classical equations of motion for cosmological acceleration and gravity can be formulated exclusively in terms of quantum quantities without using space, time and standard semiclassical approximation., Comment: 48 pages, 1 figure. A version published in Physics of Particles and Nuclei - Springer

The postulate that coordinate and momentum representations are related to each other by the Fourier transform has been accepted from the beginning of quantum theory. As a consequence, coordinate wave functions of photons emitted by stars have cosmic sizes. This results in a paradox because predictions of the theory contradict observations. The reason of the paradox and its resolution are discussed., Comment: 16 pages, no figures

Classical mathematics (involving such notions as infinitely small/large and continuity) is usually treated as fundamental while finite mathematics is treated as inferior which is used only in special applications. We first argue that the situation is the opposite: classical mathematics is only a degenerate special case of finite one and finite mathematics is more pertinent for describing nature than standard one. Then we describe results of a quantum theory based on finite mathematics. Implications for foundation of mathematics are discussed., Comment: 10 pages, 1 figure. A revised version published in Physics of Particles and Nuclei Letters

The postulate that coordinate and momentum representations are related to each other by the Fourier transform has been accepted from the beginning of quantum theory by analogy with classical electrodynamics. As a consequence, an inevitable effect in standard theory is the wave packet spreading (WPS) of the photon coordinate wave function in directions perpendicular to the photon momentum. This leads to several paradoxes. The most striking of them is that coordinate wave functions of photons emitted by stars have cosmic sizes and strong arguments indicate that this contradicts observational data. We argue that the above postulate is based neither on strong theoretical arguments nor on experimental data and propose a new consistent definition of the position operator. Then WPS in directions perpendicular to the particle momentum is absent and the paradoxes are resolved. Different components of the new position operator do not commute with each other and, as a consequence, there is no wave function in coordinate representation. Implications of the results for entanglement, quantum locality and the problem of time in quantum theory are discussed., Comment: 68 pages. A version published in Physics of Particles and Nuclei has been considerably revised in view of our discussions with Anatoly Kamchatnov and his constructive criticism

de Sitter symmetry on quantum level implies that operators describing a given system satisfy commutation relations of the de Sitter algebra. This approach gives a new perspective on fundamental notions of quantum theory. We discuss applications of the approach to the cosmological constant problem, gravity, and particle theory., Comment: 14 pages, no figures. A revised version published in Physical Review D

We argue that the main reason of crisis in quantum theory is that nature, which is fundamentally discrete and even finite, is described by classical mathematics involving the notions of infinitely small, continuity etc. Moreover, since classical mathematics has its own foundational problems which cannot be resolved (as follows, in particular, from G\"{o}del's incompleteness theorems), the ultimate physical theory cannot be based on that mathematics. In the first part of the work we discuss inconsistencies in standard quantum theory and reformulate the theory such that it can be naturally generalized to a formulation based on finite mathematics. It is shown that: a) as a consequence of inconsistent definition of standard position operator, predictions of the theory contradict the data on observations of stars; b) the cosmological acceleration and gravity can be treated simply as {\it kinematical} manifestations of quantum de Sitter symmetry, {\it i.e. the cosmological constant problem does not exist, and for describing those phenomena the notions of dark energy, space-time background and gravitational interaction are not needed}. In the second part we prove that classical mathematics is a special degenerate case of finite mathematics in the formal limit when the characteristic $p$ of the field or ring in the latter goes to infinity. {\bf This implies that mathematics describing nature at the most fundamental level involves only a finite number of numbers while the notions of limit and infinitely small/large and the notions constructed from them (e.g. continuity, derivative and integral) are needed only in calculations describing nature approximately}. In a quantum theory based on finite mathematics, the de Sitter gravitational constant depends on $p$ and disappears in the formal limit $p\to\infty$, i.e. gravity is a consequence of finiteness of nature., Comment: 293 pages, 2 figures. Section 6.3 added

We consider a quantum theory based on a Galois field. In this approach infinities cannot exist, the cosmological constant problem does not arise, and one irreducible representation (IR) of the symmetry algebra splits into independent IRs describing a particle an its antiparticle only in the approximation when de Sitter energies are much less than the characteristic of the field. As a consequence, the very notions of particles and antiparticles are only approximate and such additive quantum numbers as the electric, baryon and lepton charges are conserved only in this approximation. There can be no neutral elementary particles and the spin-statistics theorem can be treated simply as a requirement that standard quantum theory should be based on complex numbers., Comment: 43 pages, Latex, 1 figure

It is shown that in a quantum theory over a Galois field, the famous Dirac's result about antiparticles is generalized such that a particle and its antiparticle are already combined at the level of irreducible representations of the symmetry algebra without assuming the existence of a local covariant equation. We argue that the very existence of antiparticles is a strong indication that nature is described by a finite field rather than by complex numbers., Comment: 9 pages, Latex. Motivation revisited and considerably shortened. arXiv admin note: substantial text overlap with arXiv:hep-th/0209001

We argue that quantum theory should proceed not from a spacetime background but from a Lie algebra, which is treated as a symmetry algebra. Then the fact that the cosmological constant is positive means not that the spacetime background is curved but that the de Sitter (dS) algebra as the symmetry algebra is more relevant than the Poincare or anti de Sitter ones. The physical interpretation of irreducible representations (IRs) of the dS algebra is considerably different from that for the other two algebras. One IR of the dS algebra splits into independent IRs for a particle and its antiparticle only when Poincare approximation works with a high accuracy. Only in this case additive quantum numbers such as electric, baryon and lepton charges are conserved, while at early stages of the Universe they could not be conserved. Another property of IRs of the dS algebra is that only fermions can be elementary and there can be no neutral elementary particles. The cosmological repulsion is a simple kinematical consequence of dS symmetry on quantum level when quasiclassical approximation is valid. Therefore the cosmological constant problem does not exist and there is no need to involve dark energy or other fields for explaining this phenomenon (in agreement with a similar conclusion by Bianchi and Rovelli)., Comment: 44 pages, Latex, no figures. A revised version published in Symmetry, Special Issue: Quantum Symmetry

We first give simple arguments in favor of the "Zero Constants Party", i.e. that quantum theory should not contain fundamental dimensionful constants at all. Then we argue that quantum theory should proceed not from a space-time background but from a Lie algebra, which is treated as a symmetry algebra. With such a formulation of symmetry, the fact that $\Lambda\neq 0$ means not that the space-time background is curved (since the notion of the space-time background is not physical) but that the symmetry algebra is the de Sitter algebra rather than the Poincare one. In particular, there is no need to involve dark energy or other fields for explaining this fact. As a consequence, instead of the cosmological constant problem we have a problem why nowadays Poincare symmetry is so good approximate symmetry. This is rather a problem of cosmology but not fundamental quantum physics., Comment: 12 pages, no figures, Latex. We give additional arguments that the cosmological constant problem (which is often called the dark energy problem) is a purely artificial problem arising as a result of using the notion of space-time background while this notion is not physical.

We consider a possibility that gravity is not an interaction but a manifestation of a symmetry based on a Galois field., Comment: 17 pages, Latex, 1 figure. A considerably extended version published in Physics Essays, where 1) a motivation for a quantum theory over a Galois field is given; 2) my hypothesis about gravity is described in greater details; Online access to the journal: http://physicsessays.aip.org

As argued in our previous papers, it would be more natural to modify the standard approach to quantum theory by requiring that i) one unitary irreducible representation (UIR) of the symmetry algebra should describe a particle and its antiparticle simultaneously. This would automatically explain the existence of antiparticles and show that a particle and its antiparticle are different states of the same object. If i) is adopted then among the Poincare, so(2,3) and so(1,4) algebras only the latter is a candidate for constructing elementary particle theory. We extend our analysis in hep-th/0210144 and prove that: 1) UIRs of the so(1,4) algebra can indeed be interpreted in the framework of i) and cannot be interpreted in the framework of the standard approach; 2) as a consequence of a new symmetry (called AB one) between particles and antiparticles for UIRs satisfying i), elementary particles described by UIRs of the so(1,4) algebra can be only fermions; 3) as a consequence of the AB symmetry, the vacuum condition can be consistent only for particles with the half-integer spin (in conventional units) and therefore only such particles can be elementary. In our approach the well known fact that fermions have imaginary parity is a consequence of the AB symmetry., Comment: In view of my new results this paper is of no interest anymore

We consider a version of Shor's quantum factoring algorithm such that the quantum Fourier transform is replaced by an extremely simple one where decomposition coefficients take only the values of $1,i,-1,-i$. In numerous calculations which have been carried out so far, our algorithm has been surprisingly stable and never failed. There are numerical indications that the probability of period finding given by the algorithm is a slowly decreasing function of the number to be factorized and is typically less than in Shor's algorithm. On the other hand, quantum computer (QC), capable of implementing our algorithm, will require a much less amount of resources and will be much less error-sensitive than standard QC. We also propose a modification of Coppersmith' Approximate Fast Fourier Transform. The numerical results show that the probability is signifacantly amplified even in the first post integral approximation. Our algorithm can be very useful at early stages of development of quantum computer., Comment: LaTex, 27 pages, 2 tables

The structure of representations describing systems of free particles in the theory with the invariance group SO(1,4) is investigated. The property of the particles to be free means as usual that the representation describing a many-particle system is the tensor product of the corresponding single-particle representations (i.e. no interaction is introduced). It is shown that the mass operator contains only continuous spectrum in the interval $(-\infty,\infty)$ and such representations are unitarily equivalent to ones describing interactions (gravitational, electromagnetic etc.). This means that there are no bound states in the theory and the Hilbert space of the many-particle system contains a subspace of states with the following property: the action of free representation operators on these states is manifested in the form of different interactions. Possible consequences of the results are discussed., Comment: 35 pages, Latex

The (electromagnetic or weak) current operator responsible for deep inelastic scattering (DIS) should be local and satisfy the well-known commutation relations with the representation operators of the Poincare group. The problem whether these conditions are compatible with the factorization theorem and operator product expansion is investigated in detail. We argue that the current operator contains a nontrivial nonperturbative part which contributes to DIS even in the Bjorken limit. Nevertheless there exists a possibility that many results of the standard theory remain., Comment: 43 pages, LaTeX

Restrictions imposed on the (electromagnetic or weak) current operator by its commutation relations with the representation operators of the Poincare group are considered in detail. We argue that the present theory of deep inelastic scattering based on perturbative QCD does not take into account the dependence of the current operator on the nonperturbative part of the quark-gluon interaction which cannot be neglected even in leading order in $1/Q$, where $Q$ is the magnitude of the momentum transfer., Comment: 27 pages, LaTeX

We consider restrictions imposed on the (electromagnetic or weak) current operator by its commutation relations with the representation operators of the Poincare group and show that the nonperturbative part of the current operator contributes to deep inelastic scattering even in leading order in $1/Q$ where $Q$ is the magnitude of the momentum transfer. Some consequences of this result are discussed., Comment: 14 pages, LaTeX. The revised version does not differ significantly from the original one; only minor inaccuracies have been corrected

We consider restrictions imposed on the electromagnetic and weak current operators by Poincare invariance and show that some assumptions used in deriving the sum rules in deep inelastic scattering (DIS) have no physical ground. In particular there is no ground to neglect the contribution of nonperturbative effects to these operators, even in the Bjorken limit. Therefore the conclusion that the sum rules in DIS unambiguously follow from QCD is not substantiated., Comment: 11 pages, LaTeX

It is shown that in the general case the canonical construction of the current operators in quantum field theory does not render a bona fide vector field since Lorentz invariance is violated by Schwinger terms. We argue that the nonexistence of the canonical current operators for spinor fields follows from a very simple algebraic consideration. As a result, the well-known sum rules in deep inelastic scattering are not substantiated., Comment: 18 pages, LaTeX

Nuclear deep inelastic scattering is considered in the framework of a model in which the current operator explicitly satisfies Poincare invariance and current conservation. The results considerably differ from the standard ones at small values of the Bjorken variable $x$. In particular, it is impossible to extract the neutron structure functions from the deuteron data at x<0.01 and we predict that the behavior of the deuteron structure functions at low x and large momentum transfer considerably differs from the behavior of the nucleon structure functions at such conditions. We also argue that for heavier nuclei the effect of the final state interaction is important even in the Bjorken limit., Comment: 24 pages, LaTeX

Lorentz invariance of the current operators implies that they satisfy the well-known commutation relations with the representation operators of the Lorentz group. It is shown that if the standard construction of the current operators in quantum field theory is used then the commutation relations are broken by the Schwinger terms., Comment: 7 pages, LaTeX

In the Bjorken limit of the present theory of deep inelastic scattering (DIS) the structure functions (up to anomalous dimensions and perturbative QCD corrections) are described by the parton model. However the current operator in the parton model does not properly commute with the representation operators corresponding to the Lorentz group, space reflection and time reversal. To investigate the violation of these symmetries in the parton model we consider a model in which the current operator explicitly satisfies extended Poincare invariance and current conservation. It is shown that due to binding of quarks in the nucleon the Bjorken variable x no longer can be interpreted as the internal light cone momentum fraction $\xi$ even in the Bjorken limit. As a result, the data on DIS alone do not make it possible to determine the $\xi$ distribution of quarks in the nucleon. We also consider a qualitative explanation of the fact that in the parton model the values given by the sum rules exceed the corresponding experimental quantities while the quark contribution to the nucleon momentum and spin is underestimated., Comment: 41 pages. The revised version is self-contained and additional arguments that perturbative QCD does not apply to DIS are given

The three well-known forms of relativistic dynamics are unitarily equivalent and the problem of constructing the current operators can be solved in any form. However the notion of the impulse approximation is reasonable only in the point form. In particular, the parton model which is the consequence of the impulse approximation in the front form is incompatible with Lorentz invariance, P invariance and T invariance. The results for deep inelastic scattering based on the impulse approximation in the point form give natural qualitative explanation of the fact that the values given by the parton model sum rules exceed the corresponding experimental quantities., Comment: 9 pages, LaTeX

In our recent paper (hep-ph/9501348) we argued that the Bjorken variable $x$ in deep inelastic scattering cannot be interpreted as the light cone momentum fraction $\xi$ even in the Bjorken limit and in zero order of the perturbation theory. The purpose of the present paper is to qualitatively explain this fact using only a few simplest kinematical relations., Comment: 6 pages, LaTeX.

The phenomenon of scaling in deep inelastic lepton-nucleon scattering is usually explained in terms of the Feynman parton model, and the logarithmic corrections to scaling are explained in the framework of perturbative QCD. For testing the validity of the parton model, we consider the deep inelastic electron scattering in a model in which the system electromagnetic current operator explicitly satisfies relativistic invariance and current conservation. Let the struck particle have the fraction $\xi$ of the total momentum in the infinite momentum frame. Then it is shown that, due to binding of particles in the system under consideration, the Bjorken variable $x$ no longer can be interpreted as $\xi$, even in the Bjorken limit and in zero order of the perturbation theory. We argue that, as a result, the data on deep inelastic scattering alone do not make it possible to determine the $\xi$ distribution of quarks in the nucleon., Comment: 30 pages, LaTeX.

The electromagnetic current operator of a composite system must be a relativistic vector operator satisfying current conservation, cluster separability and the condition that interactions between the constituents do not renormalize the total electric charge. Assuming that these interactions are described in the framework of relativistic quantum mechanics of systems with a fixed number of particles we explicitly construct the current operators satisfying the above properties in cases of two and three particles and prove that solutions exist for any number of particles. The consideration is essentially based on the method of packing operators in the point form of relativistic dynamics developed by Sokolov. Using the method developed by Sokolov and Shatny we also construct the current operators in the instant and front forms and prove that the corresponding results are physically equivalent. The paper is self-contained what makes it possible for the reader to learn both, relativistic quantum mechanics of systems with a fixed number of particles and the problem of constructing the electromagnetic current operator for such systems., Comment: (lev@nusun.jinr.dubna.su), 100 pages LaTeX