arXiv subjects

Felix M. Lev

Publications and source records attributed to Felix M. Lev.

At least 19 recordsLinked to original sources

arXiv2024

Main problems in constructing quantum theory based on finite mathematics

As shown in our publications, quantum theory based on a finite ring of characteristic $p$ (FQT) is more general than standard quantum theory (SQT) because the latter is a degenerate case of the former in the formal limit $p\to\infty$. One of the main differences between SQT and FQT is the following. In SQT, elementary objects are described by irreducible representations (IRs) of a symmetry algebra in which energies are either only positive or only negative and there are no IRs where there are states with different signs of energy. In the first case, objects are called particles, and in the second - antiparticles. As a consequence, in SQT it is possible to introduce conserved quantum numbers (electric charge, baryon number, etc.) so that particles and antiparticles differ in the signs of these numbers. However, in FQT, all IRs necessarily contain states with both signs of energy. The symmetry in FQT is higher than the symmetry in SQT because one IR in FQT splits into two IRs in SQT with positive and negative energies at $p\to\infty$. Consequently, most fundamental quantum theory will not contain the concepts of particle-antiparticle and additive quantum numbers. These concepts are only good approximations at present since at this stage of the universe the value $p$ is very large but it was not so large at earlier stages. The above properties of IRs in SQT and FQT have been discussed in our publications with detailed technical proofs. The purpose of this paper is to consider models where these properties can be derived in a much simpler way.

physics.gen-ph↗

arXiv2024

Why Poincare symmetry is a good approximate symmetry in particle theory

As shown in the famous Dyson's paper "Missed Opportunities", even from purely mathematical considerations (without any physics) it follows that Poincare quantum symmetry is a special degenerate case of de Sitter quantum symmetries. Then the question arises why in particle physics Poincare symmetry works with a very high accuracy. The usual answer to this question is that a theory in de Sitter space becomes a theory in Minkowski space in the formal limit when the radius of de Sitter space tends to infinity. However, de Sitter and Minkowski spaces are purely classical concepts, and in quantum theory the answer to this question must be given only in terms of quantum concepts. At the quantum level, Poincare symmetry is a good approximate symmetry if the eigenvalues of the representation operators $M_{4\mu}$ of the anti-de Sitter algebra are much greater than the eigenvalues of the operators $M_{\mu\nu}$ ($\mu,\nu=0,1,2,3$). We show that explicit solutions with such properties exist within the framework of the approach proposed by Flato and Fronsdal where particles that are considered elementary in the standard theory are bound states of two Dirac singletons.

physics.gen-ph↗

arXiv2020

Cosmological Acceleration as a Consequence of Quantum de Sitter Symmetry

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 $Λ$ 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 $Λ=3/R^2$. 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, $Λ$ has a physical meaning only on classical level and the cosmological constant problem and the dark energy problem do not arise.

physics.gen-ph↗

arXiv2020

Finite Mathematics, Finite Quantum Theory and Applications to Gravity and Particle Theory

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 Gödel'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.

physics.gen-ph↗

arXiv2020

Symmetries in Foundation of Quantum Theory and Mathematics

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 $p\to\infty$.

physics.gen-ph↗

arXiv2017

Why Finite Mathematics Is The Most Fundamental and Ultimate Quantum Theory Will Be Based on Finite Mathematics

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.

physics.gen-ph↗

arXiv2015

A New Look at the Position Operator in Quantum Theory

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.

physics.gen-ph↗

arXiv2014

Existence of Antiparticles as an Indication of Finiteness of Nature

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.

physics.gen-ph↗

arXiv2012

de Sitter Symmetry and Quantum Theory

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.

hep-th↗

arXiv2010

Positive Cosmological Constant and Quantum Theory

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).

physics.gen-ph↗

arXiv2010

Introduction to a Quantum Theory over a Galois Field

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.

physics.gen-ph↗

arXiv2010

Is Gravity an Interaction?

We consider a possibility that gravity is not an interaction but a manifestation of a symmetry based on a Galois field.

physics.gen-ph↗

arXiv2010

Does The Cosmological Constant Problem Exist?

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 $Λ\neq 0$ 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.

physics.gen-ph↗

arXiv2002

Reduced Spin-Statistics Theorem

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.

hep-th↗

arXiv1998

On the problem of interactions in quantum theory

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.

quant-ph↗

arXiv1998

Aspects of the theory of deep inelastic scattering

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.

hep-ph↗

arXiv1996

Effect of Binding in Deep Inelastic Scattering Revisited

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 $ξ$ even in the Bjorken limit. As a result, the data on DIS alone do not make it possible to determine the $ξ$ 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.

hep-ph↗

arXiv1996

Role of Nonperturbative Effects in Deep Inelastic Scattering Revisited

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.

hep-ph↗