Branko Dragovich

On p-Adic Mathematical Physics

A brief review of some selected topics in p-adic mathematical physics is presented.

THE WAVE FUNCTION OF THE UNIVERSE AND p-ADIC GRAVITY

A new approach to the wave function of the universe is suggested. The key idea is to take into account fluctuating number fields and present the wave function in the form of a Euler product. For this purpose we define a p-adic generalization of both classical and quantum gravitational theory. Elements of p-adic differential geometry are described. The action and gravitation field equations over the p-adic number field are investigated. p-adic analogs of some known solutions to the Einstein equations are presented. It follows that in quantum cosmology one should consider summation only over algebraic manifolds. The correspondence principle with the standard approach is considered.

A p-adic model of DNA sequence and genetic code

Using basic properties of p-adic numbers, we consider a simple new approach to describe main aspects of DNA sequence and the genetic code. In our investigation central role plays an ultrametric p-adic information space whose basic elements are nucleotides, codons and genes. We show that a 5-adicmodel is appropriate for DNA sequence. This 5-adicmodel, combined with 2-adic distance, is also suitable for the genetic code and for amore advanced employment in genomics. We find that genetic code degeneracy is related to the p-adic distance between codons.

Adelic harmonic oscillator

Using the Weyl quantization we formulate one-dimensional adelic quantum mechanics, which unifies and treats ordinary and p-adic quantum mechanics on an equal footing. As an illustration the corresponding harmonic oscillator is considered. It is a simple, exact and instructive adelic model. Eigenstates are Schwartz-Bruhat functions. The Mellin transform of the simplest vacuum state leads to the well-known functional relation for the Riemann zeta function. Some expectation values are calculated. The existence of adelic matter at very high energies is suggested.

p-Adic Modelling of the Genome and the Genetic Code

This paper presents the foundations of p-adic modelling in genomics. Considering nucleotides, codons, DNA and RNA sequences, amino acids and proteins as information systems, we have formulated the corresponding p-adic formalisms for their investigations. Each of these systems has its characteristic prime number used for construction of the related information space. Relevance of this approach is illustrated by some examples. In particular, it is shown that degeneration of the genetic code is a p-adic phenomenon. We have also put a forward a hypothesis on the evolution of the genetic code assuming that primitive code was based on single nucleotides and chronologically first four amino acids. This formalism of p-adic genomic information systems can be implemented in computer programs and applied to various concrete cases.

Adelic model of harmonic oscillator

Adelic quantum mechanics is formulated. The corresponding model of the harmonic oscillator is considered. The adelic harmonic oscillator exhibits many interesting features. One of them is a softening of the uncertainty relation.

p-ADIC AND ADELIC MINISUPERSPACE QUANTUM COSMOLOGY

We consider the formulation and some elaboration of p-adic and adelic quantum cosmology. The adelic generalization of the Hartle–Hawking proposal does not work in models with matter fields. p-adic and adelic minisuperspace quantum cosmology is well defined as an ordinary application of p-adic and adelic quantum mechanics. It is illustrated by a few cosmological models in one, two and three minisuperspace dimensions. As a result of p-adic quantum effects and the adelic approach, these models exhibit some discreteness of the minisuperspace and cosmological constant. In particular, discreteness of the de Sitter space and its cosmological constant is emphasized.

p‐Adic and Adelic Cosmology: p‐Adic Origin of Dark Energy and Dark Matter

A brief review of p‐adic and adelic cosmology is presented. In particular, p‐adic and adelic aspects of gravity, classical cosmology, quantum mechanics, quantum cosmology and the wave function of the universe are considered.p‐Adic worlds made of p‐adic matters, which are different from real world of ordinary matter, are introduced. Real world and p‐adic worlds make the universe as a whole. p‐Adic origin of the dark energy and dark matter are proposed and discussed.

Adelic Path Integrals for Quadratic Lagrangians

Feynmans path integral in adelic quantum mechanics is considered. The propagator for one-dimensional adelic systems with quadratic Lagrangians is analytically evaluated. Obtained exact general formula has the form which is invariant under interchange of the number fields ℝ and ℚp.

p-Adic Path Integrals for Quadratic Actions

The Feynman path integral in p-adic quantum mechanics is considered. The probability amplitude for one-dimensional systems with quadratic actions is calculated in an exact form, which is the same as that in ordinary quantum mechanics.