Fundamental quantal paradox and its resolution

2019

Phys. Part. Nuclei

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 → ∞. 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.

Section: Problem Of Space-time In Quantum Theorymentioning

confidence: 75%

Finite Mathematics, Finite Quantum Theory and a Conjecture on the Nature of Time

2019

Phys. Part. Nuclei

“…Also, Ψ 1 and Ψ 2 = θΨ 1 can be treated as the states having the same coordinates. As discussed e.g., in [14], the position operators can be constructed from the operators M µν . Therefore if for some values of µ and ν, M (1) µν Ψ 1 = λ 2 Ψ 1 then, as follows from Eqs.…”

Section: Dirac Supersingletonmentioning

confidence: 99%

“…Then taking into account Eqs. (4,9,14) and the definition of the basis elements and the coefficients c(j, k), direct calculation shows that, in semiclassical approximation, the operators M ab can be replaced by their numerical values:…”

Section: Dirac Supersingletonmentioning

confidence: 99%

Dirac Singletons As the Only True Elementary Particles

Finite Mathematics as the Foundation of Classical Mathematics and Quantum Theory

2020

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 4µ of the anti-de Sitter algebra are much greater than the eigenvalues of the operators M µν (µ, ν = 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.

Cosmological Acceleration as a Consequence of Quantum de Sitter Symmetry

Phys. Part. Nuclei Lett.

2020

Physicists usually understand that physics cannot (and should not) derive that c ≈ 3·10 8 m/s andh ≈ 1.054·10 −34 kg·m 2 /s. At the same time they usually believe that physics should derive the value of the cosmological constant Λ. We first prove that three parameters defining transitions from more general theories to less general ones are (c,h, R) where R is the parameter defining contraction from the de Sitter (dS) or anti-de Sitter (AdS) algebra to the Poincare algebra. The quantity R is fundamental to the same extent as c andh. In particular, a question why R is as is does not arise, and the answer is simply that R has its value because people want to measure distances in meters. The quantity Λ has a physical meaning only on classical level and equals Λ = ±3/R 2 for dS and AdS spaces, respectively. As a consequence of the fact that quantum dS and AdS symmetries are more general than Poincare symmetry, the cosmological constant problem does not arise, Λ is necessarily not zero and there is no need to involve dark energy for explaining the cosmological acceleration. In the case of those symmetries all physical quantities are dimensionless and no system of units is needed. In particular, the quantities (c,h, s), which are the basic quantities in the modern system of units, are not so fundamental as in relativistic quantum theory, time is not strongly continuous and description of the inflationary stage of the Universe by times (10 −36 s, 10 −32 s) has no physical meaning. Following our previous publications, we consider a system of two free bodies in dS invariant quantum mechanics and show that in semiclassical approximation the dS repulsion is the same as in General Relativity. This result is obtained without using geometry of dS space, metric and connection but simply as a consequence of quantum dS symmetry.