Available at: http://www.ictp.trieste.it/~pub-off

IC/99/98

United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

THE SQUARE-ROOT NONLOCAL QUANTUM ELECTRODYNAMICS

Kh. Namsrai Institute of Physics and Technology, Mongolian Academy of Sciences, Ulaanbaatar, 51, Mongolia and The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.

Abstract A finite and gauge invariant nonlocal quantum electrodynamics is constructed within the framework of the square-root operator formalism in which mass of spinor particles has random distributional character with the definite probability measure. It is shown that charge distribution of a charged spinor with random mass (or its wave function is spread-out over space- time) leads to the change of the Coulomb law at short distances and to the nonlocal electromagnetic interaction.

MIRAMARE - TRIESTE August 1999

1

Introduction

Recently, the square-root operator formalism proposed by Weyl (1927) is used in modern particle theory (Smith, 1993) in particular, in applications of the Bethe-Salpeter equation to bound states of quarks (Castorina et al., 1984; Friar and Tomusiak, 1984; Nickisch et al., 1984), in problems of binding in very strong fields ( Hardekopf and Sucher, 1985; Papp, 1985) and of the relativistic string (bosonic)(Kaku, 1988; Fiziev, 1985). In a previous paper (Namsrai, 1998) we have proposed a simple method allowing us to work with the square-root operator and to give its physical interpretation. The purpose of this work is to study local and nonlocal electromagnetic interactions of charged spinors with photons within this scheme. Let us give a basic idea of our method. We have found a solution of the field equation \/m 2 - Uip(x) = 0

(1)

in the form m

(x, A)

(2)

—m

where ip(x, A) is the Dirac spinor field with the random mass A and the distribution 7T

possesses remarkable properties m m

m

I dXp(X) = 1, I dXXp(X) = 0, I dXX2p(X) = ^m 2 , —m

—m

(4)

—m

The Lagrangian corresponding to the equation (1) is given by )

(5)

Instead of (5) we have considered the Lagrangian density

L% = -N { # r , AiX-aMx, A2) + L ^ }

(6)

for the ip(x,\) field. Here notation

m

m

N= I I dX1dX2P(Xl)p(X2),d = iy^ —m —m

(7)

is used. Equations of motion

d\p(\)(d-\)i>(x,\)

= o, I dXp(x)[i

+ A#r, A)) =0

(8)

for the ip(x, A) fields can be obtained from the action A = by using independent variations over the fields ip(y, A) and ip(y, A) by taking the difference 8L°,/8ip(y, A) and S(L°,)T/8ip(y,

A). Here we have used the following obvious relations

and definition O \T

It is easily seen that the propagator of the field ip(x) in (1) is given by (9)

or

{)

D(x) = 2

Vrn - •

p{)

f % J

A+d

= f d\p(X)S(x,X) J

(10)

—m

In the momentum representation expression (10) takes the form m

D{p) = f

d\p(\)S{XS)

(11)

where 1

A

i A* — p* — ie

is the spinor propagator with mass A in momentum space. In expressions (10) and (12) we have used the definitions

p= On the other hand, the propagator of the field ip(x) in the representation (2) can be obtained by using the T-product:

D{x-y)= (d^ — ieA^ip in (6) leads to the interaction Lagrangian

Lm(x) = eN {#r, XjAfrMx, A2)} in our case, where A = j^A^lx)

(17)

and TV is given by (7). With (17) the S*-matrix can be

constructed by the usual rule:

S = Expec.Texp j f d4xLin(x)\

U

(18)

J

where the symbol T is the so-called T-product or T*-operation denned by (14) for the spinor fields. Ex-pec, means to take expectation value over the random variables \ . In the square-root formalism, random variables \ entering into the definition of the spinor propagator with mass Aj are not independent and have strong correlations between them. In other words, functions S(x, Aj) are some stochastic processes over the variable Aj. Expectation values of these processes are defined by the requirement of the gauge invariance of the theory and possess some properties like white noise ones. For example, at least for connected diagrams in the momentum space one assumes m

Expect D(p)\ = f d\p(X)S(p,X), —m

Expec. {-f1D(pi)-f2D(p2)'f3} = \

m

m

> ^ w r \ \ vA [^(Ai - A2)