научная статья по теме RELATIVISTIC DIFFERENTIAL-DIFFERENCE MOMENTUM OPERATORS AND NONCOMMUTATIVE DIFFERENTIAL CALCULUS Физика

Текст научной статьи на тему «RELATIVISTIC DIFFERENTIAL-DIFFERENCE MOMENTUM OPERATORS AND NONCOMMUTATIVE DIFFERENTIAL CALCULUS»

ЯДЕРНАЯ ФИЗИКА, 2013, том 76, № 9, с. 1240-1246

ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ

RELATIVISTIC DIFFERENTIAL-DIFFERENCE MOMENTUM OPERATORS AND NONCOMMUTATIVE DIFFERENTIAL CALCULUS

©2013 R. M. Mir-Kasimov*

Joint Institute for Nuclear Research, Dubna, Russia; Namik Kemal University, Tekirdag, Turkey Received May 29, 2012

The relativistic kinetic momentum operators are introduced in the framework of the Quantum Mechanics (QM) in the Relativistic Configuration Space (RCS). These operators correspond to the half of the non-Euclidean distance in the Lobachevsky momentum space. In terms of kinetic momentum operators the relativistic kinetic energy is separated as the independent term of the total Hamiltonian. This relativistic kinetic energy term is not distinguishing in form from its nonrelativistic counterpart. The role of the plane wave (wave function of the motion with definite value of momentum and energy) plays the generating function for the matrix elements of the unitary irreps of Lorentz group (generalized Jacobi polynomials). The kinetic momentum operators are the interior derivatives in the framework of the noncommutative differential calculus over the commutative algebra generated by the coordinate functions over the RCS.

DOI: 10.7868/S0044002713080229

1. INTRODUCTION

In the papers of Snyder [1,2] the first attempt to introduce the noncommutative space—time operators or NonCommutative Geometry (NCG) and investigate the immediate physical consequences of it was made. The noncommutative position and time operators can be interpreted (as it was explained by W. Pauli, see the footnote in [1]) as the boost generators of the space of constant curvature.

In Snyder approach the standard quantum-mechanical position and time operators

= ¿Iit^-, a = 0,1,2,3, dpi1'

g^v = diag(1,-1,-1,-1)

(1)

or the generators of translations of the Minkowski momentum space were substituted by noncommuta-tive operators XM, the generators of de Sitter boosts:

д д xß = MAß = Но ( PA—--P,

дР »

[xß,xv ] = -ill MßV,

дР 4

(2)

(3)

PP4 belong to the transitivity surface of the de Sitter group:

2

(4)

PßPß ± P42 = ±

h

lo

E-mail: mirkr@theor.jinr.ru

where l0 , the "fundamental length" (cf. [3—11]), indicates the scale at which the effects of NCG become appreciable (this depends on the physical context),

M are the generators of the isometry group of the momentum space. In the original Snyder paper [2] the Maxwell theory in the framework of his NCG was considered.

This connection between the NCG and the curved momentum space was used by I.E. Tamm, Yu.A. Gol-fand, V.G. Kadyshevsky and others to develop the generalized Quantum Field Theory directly in the curved momentum space [3—8]. The main stimulating idea probably was the possibility to work in terms of the numerical coordinates (coordinates of the continuous de Sitter momentum space) instead of the noncommutative Snyder space—time operators.

The next step in the development of this approach is connected with the understanding of the fundamental fact that the change of the geometry of the momentum space leads to the drastic modification of the dynamical picture. In the axiomatic approach to the Quantum Field Theory (QFT) this change can be understood as the modification of the procedure of extension of the S matrix off the mass shell [6, 11], i.e. to the different dynamical description. In fact, the statement on the Minkowsky geometry of the momentum space off the mass shell is an additional axiom of the standard QFT. It must be emphasized that in the standard QFT, this axiom is accepted without saying. We can think that some background interaction exists, which modifies the geometry of the

momentum space. See in this connection [8] and [ 12]. It was shown [11] that QFT can be generalized to the case of NCG with maintenance of all fundamental physical requirements (axioms).

The translation invariance is fulfilled in the usual sense. This means that there are relative coordinates (properly defined) which are the subject of quantization [6, 8, 12]. In consequence of the change of the geometry of the p space the space—time becomes quantum (noncommutative). We stress that physical meaning of the geometry and topology of the momentum space has not obtained completely clear physical interpretation yet. The geometric symmetries considered in QFT as the covariance groups are connected with the transformations of space—time.

As it was stressed above, the change of the geometry of the momentum space can be understood as the effect of some background interaction. In the nonrelativistic Quantum Mechanics (QM) the bright illustration of the connection between the interaction potential and the geometry of momentum space is delivered by V.A. Fock's theory of the higher symmetry of the hydrogen atom [8, 12]. The Fock's explanation of the higher symmetry of the hydrogen atom is connected with the isometry group of the momentum space of constant curvature.

Instead of noncommutative Snyder space another quantum mechanical representation of space—time can be defined. In the Bogolyubov axiomatics of QFT the extension of S matrix off the mass shell is defined by the axioms of Locality and Causality, which can not be formulated in terms of the momentum space. So determining the generalized (quantum) spacetime concept which possesses the "causal" structure, i.e. contains the analogs of the space-like and time-like regions becomes actual. The generalization of the space-time concept to the case of the de Sitter momentum space was elaborated using the Fourier Analysis on the group of isometries of curved momentum space. The standard Fourier transformation which connects the Minkowski space-time with the flat momentum space is the expansion in matrix elements of the unitary representations of the isometry group of the Minkowski momentum space e%p^x>1. The main idea lying in the basis of the concept of the new approach was that the quantum space-time of the generalized theory is dual to the curved momentum space in a sense of the Fourier transformation on its isometry group.

More precisely we determine the maximal set of mutually commuting elements of the universal enveloping algebra of the Lie algebra of the isometry group of the de Sitter momentum space. The points of the common spectrum of this set are called the "points" of the quantum space-time. The "points" of this space-time are commuting. One of the new

coordinates is usually connected with the spectrum of the Casimir operator of the de Sitter Lie algebra. Again, the stimulus for using the new generalized space-time representation was the desire to develop the field theory in terms of the numerical variables (this time the points of the common spectrum of the complete set of commuting operators of the universal enveloping algebra above) instead of using directly the operators of NCG. The kernels of the Fourier transformation connecting de Sitter momentum space and its dual — the new generalized spacetime manifold are the generating functions for the matrix elements of the unitary irreducible representations of the de Sitter group.

The space—time introduced above possesses the necessary causal structure: it contains the generalized invariant "space-like" and "time-like" regions. As a consequence, it becomes possible to generalize QFT axioms to the NCG [11]. The structure of the singular field-theoretic functions in new space—time is entirely reconstructed as compared to the standard QFT, and the corresponding perturbation theory is free of ultraviolet divergences [11, 13].

It is worthwhile to mention about the QM in Relativistic Configuration Space (RCS) and q deformations [13]. For example it was shown that the generalization of the factorization method for the case of the noncommutative relativistic Schrodinger equation leads to q oscillator.

Another manifestation of the ideas described above is connected with 3-dimensional Snyder noncommutative geometry and corresponding 3-dimensional RCS. In this case the curved momentum space is determined by the mass shell of the particle. This approach was first connected with the relativistic 2-body problem on a mass shell [6]. On the one hand this is simple model representing all the main ideas of Snyder theory. On the other hand this is the consistent approach with transparent physical meaning. In this approach the time variable is separated. The mass shell of the relativistic particle is the transitivity surface of the Lorentz group which carries the Lobachevsky geometry.

The present work is devoted to the development of the last approach. The kinetic momentum operators, corresponding to the half of the distance in the Lobachevsky momentum space, based on the noncommutative differential calculus are introduced. Actually it is a version of the QM in the RC S [9]. This work is organized as follows. The short description of QM in the RCS is given in Section 2. Then the kinetic momenta are defined in Section 3. In Section 4 the model with 2-spatial dimensions is introduced and necessary information on the relativistic plane waves for the kinetic momentum case is given, also the Fourier expansion in matrix elements of the

Lorentz group is considered. The noncommutative calculus corresponding to the half-distance operators is developed and essential properties of the QM in the RCS in terms of the kinetic momenta are given in Section 5.

2. RELATIVISTIC CONFIGURATION SPACE

Consider the mass shall of the particle with the mass m

u 2 2 2 2

PuPu = Po - p = m c , P0 > 0, n = 0,1,2.

(5)

X = moi = ih 11 +

p

d

m2c2 dpi

(6)

The explicit character of Snyder's approach to spacetime quantization has a remarkable consequence: we can define the spectrum of a commutative set of operators constructed from x^ and other generators of the de Sitter group.

The important property of the RCS is its continuity. The "price" of this continuity was the necessity to use more general version of the calculus

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