p-adic quantum mechanics

An extension of the formalism of quantum mechanics to the case where the canonical variables are valued in a field ofp-adic numbers is considered. In particular the free particle and the harmonic oscillator are considered. In classicalp-adic mechanics we consider time as ap-adic variable and coordinates and momentum orp-adic or real. For the case ofp-adic coordinates and momentum quantum mechanics with complex amplitudes is constructed. It is shown that the Weyl representation is an adequate formulation in this case. For harmonic oscillator the evolution operator is constructed in an explicit form. For primesp of the form 4l+1 generalized vacuum states are constructed. The spectra of the evolution operator have been investigated. Thep-adic quantum mechanics is also formulated by means of probability measures over the space of generalized functions. This theory obeys an unusual property: the propagator of a massive particle has power decay at infinity, but no exponential one.     

p-adic probability distributions of hidden variables

During the last years large interest was shown in p-adic quantum models (especially, in string theory). As usual, new physical models generate new mathematical methods. In our case a new type of stochastics, p-adic stochastics, was arisen inside p-adic quantum physics. We apply this stochastics to propose a justification of the Einstein-Podolsky-Rosen theory of hidden variables, which was in large contradiction with the Bell type inequality. Our main result is the following: if we consider a p-adic probability distribution of hidden variables, then there are no problems with Bell's inequality.     

Multiple p-adic L-function

An analytic function interpolating the multiple generalized Bernoulli numbers attached to a primitive Dirichlet character X at negative integers in the complex plane is studied. The multiple p-adic L-function is constructed as the p-adic analog of the above function. Finally, the values of the partial derivative of this multiple p-adic L-function at s = 0 are given. Dedicated to the memory of Boris Moiseevich Levitan     

Canp-adic numbers be useful to regularize divergent expectation values of quantum observables?

We show howp-adic analysis can be used in some cases to treat divergent series in quantum mechanics. We consider examples in which the usual theory of the Schrödinger equation would give rise to an infinite expectation value of the energy operator. By usingp-adic analysis, we are able to get a convergent expansion and obtain a finite rational value for the energy. We present also the main ideas to interpret a quantum mechanical state by means ofp-adic statistics.     

p-Adic string compactified on a torus

U(1) ^xD model with the Villain action on ag-loop generalizationF _g of the Bruhat-Tits tree for thep-adic linear groupGL(2, _p ) is considered. All correlation functions and the statistical sum are calculated. We compute also the averages of these correlation functions forN vertices attached to the boundary ofF _g. When the compactification radius tends to infinity the averages provide theg-loopN-point amplitudes of the uncompactifiedp-adic string theory, in particular forg=0 the Freund-Olson amplitudes.     

Harmonic analysis and p-adic strings

A generalization of the Veneziano amplitude is considered which is a convolution of two characters on a field K. Choosing K in an appropriate way, one can obtain the usual Veneziano amplitude, the Virasoro-Shapiro amplitude, the p-adic amplitudes, and the finite Galois field amplitude. The cases when K is an algebra or a group are also discussed. These cases can be of interest in the context of the quantization of spacetime.     

P-adic theta functions and solutions of the KP hierarchy

Based on Schottky uniformization theory of Riemann surfaces, we construct a universal power series for (Riemann) theta function solutions of the KP hierarchy. Specializing this power series to the coordinates associated with Schottky groups overp-adic fields, we show that thep-adic theta functions of Mumford curves give solutions of the KP hierarchy.     

P-adic Feynman and string amplitudes

We derive an explicit representation forp-adic Feynman and Koba-Nielsen amplitudes and we briefly outline the connection between the scalar models ofp-adic quantum field theory and Dyson's hierarchical models.This work was supported in part by the French Government     

Multiloop calculations inp-adic string theory and Bruhat-Tits trees

We treat the openp-adic string world sheet as a coset spaceF=T/, whereT is the Bruhat-Tits tree for thep-adic linear groupGL(2, _p ) and PGL(2, _p ) is some Schottky group. The boundary of this world sheet corresponds to ap-adic Mumford curve of finite genus. The string dynamics is governed by the local gaussian action on the coset spaceF. The tachyon amplitudes expressed in terms ofp-adic -functions are proposed for the Mumford curve of arbitrary genus. We compare them with the corresponding usual archimedean amplitudes. The sum over moduli space of the algebraic curves is conjectured to be expressed in the arithmetic surface terms. We also give the necessary mathematical background including the Mumford approach top-adic algebraic curves. The connection of the problem of closedp-adic strings with the considered topics is discussed.     

Generalized probabilities taking values in non-Archimedean fields and in topological groups

We develop an analog of probability theory for probabilities taking values in topological groups. We generalize Kolmogorov’s method of axiomatization of probability theory, and the main distinguishing features of frequency probabilities are taken as axioms in the measure-theoretic approach. We also present a survey of non-Kolmogorovian probabilistic models, including models with negative-, complex-, and p-adic-valued probabilities. The last model is discussed in detail. The introduction of probabilities with p-adic values (as well as with more general non-Archimedean values) is one of the main motivations to consider generalized probabilities with values in more general topological groups than the additive group of real numbers. We also discuss applications of non-Kolmogorovian models in physics and cognitive sciences. A part of the paper is devoted to statistical interpretation of probabilities with values in topological groups (in particular, in non-Archimedean fields).     

SU(3) Local Gauge Field Theory as Effective Dynamics of Composite Gluons

The effective dynamics of quarks is described by a nonperturbatively regularized NJL model equation with canonical quantization and probability interpretation. The quantum theory of this model is formulated in functional space and the gluons are considered as relativistic bound states of colored quark-antiquark pairs. Their wave functions are calculated as eigenstates of hardcore equations, and their effective dynamics is derived by weak mapping in functional space. This leads to the phenomenological SU(3) gauge invariant gluon equations in functional formulation, i.e., the local gauge symmetry is a dynamical effect resulting from the dynamics of the quark model.     

P-adic conformal invariance and the Bruhat—Tits tree

It is shown that some Gaussian and non-Gaussian scaling invariant p-adic field theories are invariant under the group of transformations which conserve the p-adic norm of the cross-ratio of any four points. This group can be treated as a p-adic conformal group. It has a continuation on the Bruhat—Tits tree, being an automorphism group of that tree. The models also have tree continuation, in particular the binary correlation function of the tree model is a spherical function.     

Quantum mechanics andp-adic numbers

We study the possibility of representing the proposition lattice associated with a quantum system by a linear vector space with coefficients from ap-adic field. We find inconsistencies if the lattice is assumed, as usual, to be irreducible, complete, orthocomplemented, atomic, and weakly modular.     

First quantization of mass and charge

The proper time is introduced as a parameter into the wave functions of relativistic quantum theory by first quantization of the mass. The classical limit is shown to be given by a recently developed canonical formulation of classical relativistic mechanics. The adjoint spinor is redefined with the help of a sign operator to remove a discrepancy between the classical and quantum actions in the behavior under time inversion. This results in positive energy densities for the Dirac theory. The inclusion of this sign operator into the definition of the probability current then removes negative probabilities from the theory. A five-dimensional formulation with first quantized charge is given.     

A note on p-adic Euler measure on ℤp

In this paper, we give some p-adic approximation of E _n,x for certain n. Finally we will treat p-adic l-function of Kubota-Leopoldt’s type Euler numbers and p-adic measure for Euler numbers.     

On an Essential Spectrum of the Random p-Adic Schrödinger-type Operator in the Anderson Model

The random p-adic Schrödinger-type operators are considered. The p-adic analogue of the Anderson model is defined for these operators and the spectral properties of this model are investigated.     

Field quantization in non-archimedean field theory

A formulation of field quantization of bosonic and fermionic fields defined on the disconnected field of p-acid numbers Q _p is given. We introduce a canonical quantization procedure and exhibit the properties of the nonlocal operators present in the action. Its relevance to string theory on Q _p is also discussed.     

Calculation of generalp-adic Feynman amplitude

The generaln-point masslessp-adic Feynman amplitude with arbitrary parameters of analytic regularization for each line is calculated. This result is presented in the form of a sum over hierarchies of a given graph. The structure of ultraviolet and infrared divergences ofp-adic Feynman amplitudes is characterized and the startriangle uniqueness identity in thep-adic case is derived.Supported by Alexander von Humboldt-Stiftung