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 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.     

Non-archimedean strings

A full set of factorized, dual, crossing-symmetric tree-level N-point amplitudes is constructed for non-archimedean closed strings. Momentum components and space-time coordinates are still valued in the field of real numbers, quantum amplitudes in that of complex numbers. It is the world-sheet parameters, which one integrates over, that become p-adic. For the closed string the parameters are valued in quadratic extensions of the fields Q_p of p-adic numbers (p = prime).     

{\hbar}-adic Quantum Vertex Algebras and Their Modules

This is a paper in a series to study vertex algebra-like structures arising from various algebras including quantum affine algebras and Yangians. In this paper, we study notions of (^h/_2p){\hbar}-adic nonlocal vertex algebra and (^h/_2p){\hbar}-adic (weak) quantum vertex algebra, slightly generalizing Etingof-Kazhdan’s notion of quantum vertex operator algebra. For any topologically free \mathbb C[[(^h/_2p)]]{{\mathbb C}\lbrack\lbrack{\hbar}\rbrack\rbrack}-module W, we study (^h/_2p){\hbar}-adically compatible subsets and (^h/_2p){\hbar}-adically S{\mathcal{S}}-local subsets of (End W)[[x, x ⁻¹]]. We prove that any (^h/_2p){\hbar}-adically compatible subset generates an (^h/_2p){\hbar}-adic nonlocal vertex algebra with W as a module and that any (^h/_2p){\hbar}-adically S{\mathcal{S}}-local subset generates an (^h/_2p){\hbar}-adic weak quantum vertex algebra with W as a module. A general construction theorem of (^h/_2p){\hbar}-adic nonlocal vertex algebras and (^h/_2p){\hbar}-adic quantum vertex algebras is obtained. As an application we associate the centrally extended double Yangian of \mathfrak s\mathfrak l₂{{\mathfrak s}{\mathfrak l}_{2}} to (^h/_2p){\hbar}-adic quantum vertex algebras.     

p-adic superstrings

Different possibilities to construct p-adic superstring amplitudes are discussed. To describe one of these possibilities we formulate a p-adic generalization of the conformal technique. p-adic conformal rules allow us to explicitly write down the amplitude for massless particles. They do not have the form of the usual kinematical factors multiplied by the p-adic modification of the usual amplitudes. The relation between open and closed p-adic superstring amplitudes is discussed.     

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     

Airy Functions Over Local Fields

Airy integrals are very classical but in recent years they have been generalized to higher dimensions and these generalizations have proved to be very useful in studying the topology of the moduli spaces of curves. We study a natural generalization of these integrals when the ground field is a non-archimedean local field such as the field of p-adic numbers. We prove that the p-adic Airy integrals are locally constant functions of moderate growth and present evidence that the Airy integrals associated with compact p-adic Lie groups also have these properties.     

Frobenius transformation,mirror map and instanton numbers

We show that one can express Frobenius transformation on middle-dimensional p-adic cohomology of Calabi–Yau threefold in terms of mirror map and instanton numbers. We express the mirror map in terms of Frobenius transformation on p-adic cohomology. We discuss a p-adic interpretation of the conjecture about integrality of Gopakumar–Vafa invariants.     

p-adic analyticity and virasoro algebras for conformal theories in more than two dimensions

The field of p-adic complex numbers has a much richer structure than the field of ordinary complex numbers. This is used in order to extend the powerful tools of two-dimensional conformal field theories to higher dimensions. It is thus proposed that critical systems in more than two dimensions be first studied over the p-adics and then, if possible, recovered by the adelic construction. It is further argued that this higher-dimensional p-adic analyticity may be the key to membrane theories. A natural ansatz for three-brane tree-scattering amplitudes, where p-adic analyticity is instrumental, is given as an explicit example.     

Some <Emphasis Type="Italic">p</Emphasis>-Adic Integrals on ℤ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> Associated with Trigonometric Functions

In this paper, we study some p-adic invariant and fermionic p-adic integrals on ?_p associated with trigonometric functions. By using these p-adic integrals we represent several trigonometric functions as a formal power series involving either Bernoulli or Euler numbers. In addition, we obtain some identities relating various special numbers like zigzag, some ‘trigonometric’, Bernoulli, Euler numbers, and Euler numbers of the second kind.     

Quantum mechanics,knot theory,and quantum doubles

In previous papers we have described quantum mechanics as a matrix symplectic geometry and showed the existence of a braiding and Hopf algebra structure behind our lattice quantum phase space. The first aim of this work is to give the defining commutation relations of the quantum Weyl-Schwinger-Heisenberg group associated with our ℜ-matrix solution. The second aim is to describe the knot formalism at work behind the matrix quantum mechanics. In this context, the quantum mechanics of a particle-antiparticle system (pˉp) moving in the quantum phase space is viewed as a quantum double.     

On Julia Set and Chaos in <Emphasis Type="Italic">p</Emphasis>-adic Ising Model on the Cayley Tree

In this paper, we study the chaotic behavior of the p-adic Ising-Potts mapping associated with the p-adic Ising model on the Cayley tree. As an application of this result, we are able to show the existence of periodic (with any period) p-adic quasi Gibbs measures for the model.     

Dynamics of chiral (self-dual) p-forms

An action principle describing the dynamics of a p-form gauge field whose field strength is self-dual is given. The action is local, Lorentz invariant and also invariant under the standard gauge transformation of a p-form. The coupling to gravitation is described. The proposed action permits a consistent passage to quantum mechanics. The path integral is briefly discussed.     

A probabilistic analysis of the difficulties of unifying quantum mechanics with the theory of relativity

A procedure is given for the transformation of quantum mechanical operator equations into stochastic equations. The stochastic equations reveal a simple correlation between quantum mechanics and classical mechanics: Quantum mechanics operates with “optimal estimations,” classical mechanics is the limit of “complete information.” In this connection, Schrödinger's substitution relationsp _x → -i? ?/?x, etc, reveal themselves as exact mathematical transformation formulas. The stochastic version of quantum mechanical equations provides an explanation for the difficulties in correlating quantum mechanics and the theory of relativity: In physics “time” is always thought of as a numerical parameter; but in the present formalism of physics “time” is described by two formally totally different quantities. One of these two “times” is a numerical parameter and the other a random variable. This last concept of time shows all the properties required by the theory of relativity and is therefore to be considered as the relativistic time.     

On the adelic string amplitudes

Some remarkable properties of the adelic string amplitudes for the physical domain of the Mandelstam variables are considered. It is shown that the p-adic four-point functions are always negative. Also, a formula is obtained which expresses the product of moduli of the p-adic amplitudes and the Veneziano amplitude in terms of the zeta functions. This product is absolutely convergent unlike the divergent product of these amplitudes without moduli, recently considered by Freund and Witten. Using the zeta function representation, p-adic interpolation of the Veneziano amplitude is also considered.     

Fractional dimensional Fock space and Haldane's exclusion statistics: q/p case and t−J model

The discussion of fractional dimensional Hilbert spaces in the context of Haldane exclusion statistics is extended from the case of g = 1/p for the statistical parameter to the case of rational g = q/p with q,p coprime positive integers. The corresponding statistical mechanics for a gas of such particles is constructed. This procedure is used to define the statistical mechanics for particles with irrational g. Applications to strongly correlated systems such as the Hubbard and t−J models are discussed.     

Hilbert space representation of an algebra of observables forq-deformed relativistic quantum mechanics

Using a representation of theq-deformed Lorentz algebra as differential operators on quantum Minkowski space, we define an algebra of observables for a q-deformed relativistic quantum mechanics with spin zero. We construct a Hilbert space representation of this algebra in which the square of the massp ² is diagonal.     

Adelic conformal field theory

We stress the use of modular forms in obtaining adelic formulations of field theoretical problems. Supersymmetry then appears in the real section with thep-adic parts as arithmetic completions. We first show how the Casimir effect is naturally interpreted adelically and the coefficient arises from dimensional analysis. We then suggest looking at the zero slope limit of adelic string amplitudes. Finally, we interpret the rationality of the critical exponents for conformal field theories in terms of p-adic analyticity of correlation functions.