p-Adic mathematical physics: the first 30 years

Драгович, Бранко; Khrennikov, Andrei; Козырев, С. В.; Волович, И. В.; Zelenov, E. I.

doi:10.1134/s2070046617020017

Cited by 135 publications

(83 citation statements)

References 335 publications

(454 reference statements)

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“…where, for examples: [4,8,22] and references therein), and L nl = − 1 6 m 2 R✷ −2 R (see [10] and references therein), and R is the scalar curvature. Usage of F (✷) = +∞ n=0 f n ✷ n is influenced by existence of analytic expressions with ✷ in string field theory and p-adic string theory (see [26] and references therein). Some modified gravity models with analytic nonlocality, which have been so far considered, are concrete examples of the action (without matter)…”

Section: Introductionmentioning

confidence: 99%

Cosmological solutions of a nonlocal square root gravity

et al. 2019

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In this paper we consider modification of general relativity extending R−2Λ by nonlocal term of the formis an analytic function of the d'Alembert operator ✷. We have found some exact cosmological solutions of the corresponding equations of motion without matter and with Λ = 0. One of these solutions is a(t) = A t 2 3 e Λ 14 t 2 , which imitates properties similar to an interplay of the dark matter and the dark energy. For this solution we calculated some cosmological parameters which are in a good agreement with observations. This nonlocal gravity model has not the Minkowski space solution. We also found several conditions which function F (✷) has to satisfy.

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Section: Introductionmentioning

confidence: 99%

Cosmological solutions of a nonlocal square root gravity

et al. 2019

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“…The p-adic field and string theories have been studied over the time with some periodic fluctuations in their interest (for some reviews, see [1][2][3][4]). Recently a considerable amount of work has been performed on this topic in the context of the AdS/CFT correspondence [5][6][7][8].…”

Section: Introductionmentioning

confidence: 99%

On p-adic string amplitudes in the limit p approaches to one

Bocardo-Gaspar

García‐Compeán

Zúñiga–Galindo

2018

J. High Energ. Phys.

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In this article we discuss the limit p approaches to one of tree-level p-adic open string amplitudes and its connections with the topological zeta functions. There is empirical evidence that p-adic strings are related to the ordinary strings in the p → 1 limit. Previously, we established that p-adic Koba-Nielsen string amplitudes are finite sums of multivariate Igusa's local zeta functions, consequently, they are convergent integrals that admit meromorphic continuations as rational functions. The meromorphic continuation of local zeta functions has been used for several authors to regularize parametric Feynman amplitudes in field and string theories. Denef and Loeser established that the limit p → 1 of a Igusa's local zeta function gives rise to an object called topological zeta function. By using Denef-Loeser's theory of topological zeta functions, we show that limit p → 1 of tree-level p-adic string amplitudes give rise to certain amplitudes, that we have named Denef-Loeser string amplitudes. Gerasimov and Shatashvili showed that in limit p → 1 the well-known non-local effective Lagrangian (reproducing the tree-level p-adic string amplitudes) gives rise to a simple Lagrangian with a logarithmic potential. We show that the Feynman amplitudes of this last Lagrangian are precisely the amplitudes introduced here. Finally, the amplitudes for four and five points are computed explicitly.

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“…In the literature it is also considered a version of quantum theory based not on real but on p-adic numbers (see e.g. the review paper [45] and references therein). Both, the sets of real and p-adic numbers are the completions of the set of rational numbers but with respect to different metrics.…”

Section: Proof Of Statementmentioning

confidence: 99%

Symmetries in Foundation of Quantum Theory and Mathematics

Lev

2020

Symmetry

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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 → ∞.

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p-Adic mathematical physics: the first 30 years

Cited by 135 publications

References 335 publications

Cosmological solutions of a nonlocal square root gravity

Cosmological solutions of a nonlocal square root gravity

On p-adic string amplitudes in the limit p approaches to one

Symmetries in Foundation of Quantum Theory and Mathematics

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