Finiteness of physics and its possible consequences

Lev, Felix M.

doi:10.1063/1.530257

Cited by 22 publications

(80 citation statements)

References 18 publications

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“…However, as shown in Refs. [2,3], in FQT the situation is similar to that in dS theory rather than in AdS theory.…”

Section: Remarks On Fundamental Theoriesmentioning

confidence: 97%

“…As shown in Ref. [2] and in chapters 4 and 8 of Ref. [3] Statement 2: For algebra (2) there exist sets S and representations such that basis vectors in the representation spaces are eigenvectors of the operators from S with eigenvalues belonging to Z.…”

Section: Proof Of the Main Statementmentioning

confidence: 99%

“…As shown in Refs. [2,3,4], FQT sheds a new light on fundamental problems of gravity and particle theory. In the present paper we discuss how FQT can be applied for solving such a fundamental problem of quantum theory as the problem of time.…”

Section: Introductionmentioning

confidence: 99%

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

Lev

2019

Phys. Part. Nuclei

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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.

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“…However, as shown in Refs. [2,3], in FQT the situation is similar to that in dS theory rather than in AdS theory.…”

Section: Remarks On Fundamental Theoriesmentioning

confidence: 97%

Section: Proof Of the Main Statementmentioning

confidence: 99%

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

Lev

2019

Phys. Part. Nuclei

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“…[7,[9][10][11] (where we discussed different properties of dS quantum theory), shows that in the dS case the classical relative acceleration of two free particles is . From the formal point of view, the result is the same as in GR on dS space.…”

Section: Quantum Approach To Cosmological Accelerationmentioning

confidence: 99%

Do We Need Dark Energy to Explain the Cosmological Acceleration?

Lev¹

2012

JMP

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The phenomenon of the cosmological acceleration discovered in 1998 is usually explained as a manifestation of a hypothetical field called dark energy which is believed to contain more than 70% of the energy of the Universe. This explanation is based on the assumption that empty space-time background should be flat and hence a nonzero curvature of the background is a manifestation of a hidden matter. We argue that quantum theory should proceed not from spacetime background but from a symmetry algebra. Then the cosmological acceleration can be easily and naturally explained from first principles of quantum theory without involving empty space-time background, dark energy and other artificial notions. We do not assume that the reader is an expert in the given field and the content of the paper can be understood by a wide audience of physicists.

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“…In Refs. [2,3] we have proposed an approach called GFQT where quantum theory is based on a Galois field with characteristic p. It has been shown that in the formal limit p → ∞ GFQT recovers the predictions of standard continuous theory. Then the fact that standard mathematics describes many experiments with a high accuracy can be explained as a consequence of the fact that in real life the number p is very large.…”

mentioning

confidence: 99%

What Mathematics Is The Most Fundamental?

Lev¹

2015

J Phys Math

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Standard mathematics involves such notions as infinitely small/large, continuity and standard division. However, some of these notions are treated differently in traditional and constructive versions. This mathematics is usually treated as fundamental while finite mathematics is treated as inferior. Standard mathematics has foundational problems (as follows, for example, from Gödel's incompleteness theorems) but people usually believe that this is less important than the fact that it describes many experimental data with high accuracy. We argue that the situation is the opposite: standard mathematics is only a special case of finite one in the formal limit when the characteristic of the ring or field used in finite mathematics goes to infinity. Therefore foundational problems in standard mathematics are not fundamental. MSC: 03AxxKeywords: standard mathematics, finite mathematics, infinity, Galois fields Standard mathematics involves such notions as infinitely small/large, continuity and standard division. However, some of these notions are treated differently in traditional and constructive versions. These notions arose from a belief based on everyday experience that any macroscopic object can be divided into arbitrarily large number of arbitrary small parts. However, the very existence of elementary particles indicates that those notions have only a limited meaning. Indeed, consider the gram-molecule of water having the mass 18 grams. It contains the Avogadro number of molecules 6 · 10 23 . We can divide this gram-molecule by ten, million, billion, but when we begin to divide by numbers greater than the Avogadro one, the division operation loses its meaning and we cannot obtain arbitrarily small parts. This example shows that mathematics involving the set of all rational numbers has only a limited applicability.As a consequence, any description of macroscopic phenomena using continuity and differentiability can be only approximate. Water in the ocean can be described by differential equations of hydrodynamics but this is only an approximation since matter is discrete. The above remarks show that using standard mathematics in quantum physics is at least unnatural.

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Finiteness of physics and its possible consequences

Cited by 22 publications

References 18 publications

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

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

Do We Need Dark Energy to Explain the Cosmological Acceleration?

What Mathematics Is The Most Fundamental?

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