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Some remarks on nonlocal field theory and space-time quantization

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From the point of view that the charge and mass of an electron is of dynamical origin and quantization of charge in units ofe is related to the space-time quantization as developed in an earlier paper, we here show that it is possible to consider
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  International Journal of Theoretical Physics Vol. t 5, No. 5 (1976), pp. 323-331 Some Remarks on Nonlocal Field Theory and Space Time Quantization PRATUL BANDYOPADHYAY and S S R ROY Indian Statistieal Institute Calcutta-35 India Received: 9 June 1975 Abstract From the point of view that the charge and mass of an electron is of dynamical srcin and quantization of charge in units of e is related to the space-time quantization as developed in an earlier paper, we here show that it is possible to consider that the internal space within the elementary domain of the quantized space-time world is not governed by Lorentz invariance. This helps us to develop a consistent theory of non- local fields for extended particles where the infinite mass degeneracy is avoided. More- over, this ensures the convergence of nonlocal field theories and suggests that massless particles like photons and neutrinos, though they may be taken to be of extended structure, will appear only as point particles in the physical world. In this picture, Lorentz invariance appears to be a consequence of the distribution of matter.and energy in the Universe, and this may be taken to be another interpretation of Mach s principle. 1. Introduction In a recent paper Bandyopadhyay, 1973a-hereafter referred to as I) we have argued that the quantization of charge in units of e can be taken to be a consequence of the space-time quantization when charge is considered to be of dynamical origin. In fact, we have shown in I that the charge and mass of an electron as well as of a muon) can be taken to occur as a result of n photon-neutrino weak interactions, when photons and neutrinos are represented as nonlocal fields and n is given by the relation e -~ ng, g being the photon-neutrino weak coupling constant g ~- 10 -1° e) Bandyopadhyay, 1968). In this model electron and muon are depicted as YES) and pus), respectively, where S represents the system of photons interacting weakly at n space-time points with the extended structure of a two-component neutrino. The two other components corresponding to the positive and © 1976 Plenum Publishing Corporation. No part of this publicatio,n may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical photocopying, microfilming, recording, or otherwise, without written permission of the publisher. 323  324 PR TUL B NDYOP DHY Y ND SISIR ROY negative energy states are formed when the form factor associated with the interaction changes its sign, implying that particles and antiparticles are mirror reflections of each other (Bandyopadhyay, 1973b). This procedure helps us to unify weak and electromagnetic interactions and the accompanied violation of symmetry generates the photon as a Goldstone boson (Bandyopadhyay, 1973a). If we take that the charge of a hadron is also due to the presence of a lepton in its structure (Bandyopadhyay, 1975), then the charge spectrum of all hadrons can be interpreted on the basis of this concept of dynamical origin of charge. In this picture, it is possible to show that the quantization of charge in units of e is related to the quantization of space-time, where each quantized space-time domain is determined by the region accommodating the specific n number of weak interactions involving extended structures of n photons and one neutrino, also considered to be of extended structure. However, as we know, it is not yet properly understood how massless particles with extended structures can be described. Moreover, as Yukawa has pointed out, there are difficulties due to mass degeneracy in the simple model of non- local fields (Yukawa, 1965, 1973). In this context it is worthwhile to mention that a model of leptons has been constructed out of n photons weakly interacting with the neutrino in the frame work of nonlocal field theory (Bandyopadhyay, 1973a). This model suggests a preferential direction in the space within the fundamental domain, i.e., the internal space should be such as to violate the Lorentz invariance. Taking this into consideration we shall here show that the nonlocal fields representing the extended particles can be described in a consistent way. 2. Nonlocal Field Theory and the Infinite Degeneracy of Mass States The concept of nonlocal field was first introduced by Yukawa (1950) to incorporate new degrees of freedom that might help us to understand the internal quantum numbers of hadrons. However, the goal could not be successfully achieved; besides, certain inconsistencies like infinite degeneracy of mass states appear in the simple model of nonlocal fields. In fact, in the description of nonlocal fields it is generally considered that the field is a function of two points and it can be represented by a matrix (X~ I ~ [ X~). The principle of reciprocity is understood as a symmetry of natural laws with respect to the commutators [Pu, ~k] and [xg, ~]. Alternatively, ~k can be regarded as a function ~(x , r ) of external coordinates X. = (x .o + x, ,)/2 and internal coordinates rg = xu - xu. For a scalar field, the free field equation can be written as F , r u, ~(X., ru) = 0 (2.1) Assuming that F can. be factorized in terms of a d Alembertian operator for the external coordinates and an operator F(r) depending only on the internal  NONLOCAL FIELD THEORY AND SPACE TIME QUANTIZATION 325 coordinates, we can write + F (r) r u ¢ = 0 (2.2) 3XuaX~ ru, brubru. In this case, q~ can be solved in the product form c~ = U X)X r) (2.3) where U(X) and X r) must satisfy the equations 3X-~X u I~ U X) = t r) - I~)X r) : 0 (2.4) Yukawa (1965, 1973) has considered the case of the harmonic oscillator. In the case of the four-dimensional oscillator model, the operator F is of the form - -- +-- rur (2.5) F= OX~X,, ~ Or~,~ru ~4 Now it is noted that for the solution of the equation ~r) _ la)X r) = 0 (2.6) the eigenvalues/~ are not positive definite. If we assign to each mode of vibration four quantum numbers nl in the direction of X, n2 in the direction of Y, n3 in the direction of Z, and another vibrational quantum number no in the time direction, then the eigenvalue of F(r) is proportional to nl + n2 + n3 - no. Obviously, this leads to the degeneracy of the eigen- values. For example, if we take the case tl = 0, then there are an infinite number of different combinations of nl, n2, n3, and no. Thus, if we accept this formalism, there is an infinite number of different types of particles, all of them having the same mass. Yukawa (1965, 1973) has pointed out that there may be two ways to obviate these difficulties: (i) to consider a nonunitary representation of the Lorentz group, and (ii) to introduce the coupling between external and internal motions. Though by adoption of (ii) the difficulties of infinite degeneracy can be removed, the picture becomes far from simple. However, in this context Pals (t953) emphatically remarked that there is no a priori reason at all why the internal space should be governed by the Lorentz group, and if we demand that the internal space is not governed by the Lorentz group, then the difficulties related to the infinite mass degeneracy problem are removed. But the idea that the internal space does not obey this Lorentz symmetry is not favored for two reasons. Firstly, it was Yukawa s intention to correlate the internal quantum numbers of hadrons like isospin, strangeness and baryon number with the degrees of freedom connected with the internal space. Secondly, if we transform Yukawa s nonlocal field theory to the local field theory with nonlocal interaction, the  326 PRATUL BANDYOPADHYAY AND SISIR ROY form factor of the latter theory becomes connected with the variables of the internal space. So, if the internal space variables do not obey Lorentz symmetry, the form factors of the nonlocal interaction theory will have no relativistic invariance. Here we point out that the description of electron (and muon) on the basis of the dynamical srcin of charge and mass and the necessary requirement of space-time quantization can indeed accommodate the idea that the internal space within the quantized domain is not governed by Lorentz symmetry (Bandyopadhyay, 1974). According to this idea of space-time quantization, each quantized domain becomes the seat of an electron (as well as muon) and within this domain no measurement is possible. When the domain is filled up by the n number of photons interacting at different space-time points with the extended structure of a neutrino Ve(V~), a massive and charged particle like an electron (muon) is formed and we get into the world of Lorentz symmetry. For details, let us recapitulate the previous calculations as presented in I. Let us consider the two-component spinor wave function ~(X, r), where X and r are external and internal space-time variables. It is considered that t)(X, r) satisfies the relation ~(x) = f d4rt~(X, r) 2.7) It is further contended that the nonlocal spinor (X, r) obeys the Dirac equation in terms of the variable X (@ O-~ + M) ~(X, r) = O (2.8) The Spinor current is expressed as C (x) = ~ d4r d4S~(X, r)3'ts~(X, s) (2.9) Now assuming that the electromagnetic field quantity Au(Y, t) also satisfies a similar relation Au(Y ) = ~ d4tAu(X, t) (2.10) we take n photon fields at different space-time points in the external space as follows: Au Y- ½el + Au(Y + ½el) + Au(Y- ½e2 + Au(Y + ½e2) +... + Au(y_ 1era + Au(Y + ½em) (2.11) From this, we see that when e ~ 0, the expression just reduces to the single point potential given by nAu(Y) when n = 2m. Thus the interaction Lagrangian  NONLOCAL FIELD THEORY AND SPACE TIME QUANTIZATION 327 for n photon weak interactions with the spinor takes the form m m Li=ig ~ Cv(X)Au(Y- ½ei) + ~ Cu(X)Au(Y +½ei) i l i l = ig ~ ~ d4r d4s d4t[~(X, r)ru (X, S)Av(Y, t) i=1 01 2.12) where Yi = Y- ½ i, Yi = Y + ½ei, and g is the dimensionless weak coupling constant, which is taken to have the value g = 10 -t° e (Bandyopadhyay~ 1968). Now taking m such that e/2m = g, the weak coupling constant, we note that the system of interactions (2.12) in the limit e -> 0 just reduces to the formal dectromagnetic coupling ie f d4r d4s d4t~(X, r)rU~(X, s)Au(Y, t) (2.13) Thus in the limit e + 0, n photon weak interactions can be considered to be equivalent to the proper electromagnetic interaction (2.13), and by this a geometrical description of e in terms ofg is obtained. To show that the coupling constant e obtained in such a manner actually represents the charge, we have shown in I that the equation (2.13) can give rise to a less symmetric solution that generates the electromagnetic inter- action from a system of n photon weak interactions at different space-time points. This violation of symmetry occurs owing to the fact that (1) the interaction involving nonlocat fields such as equation (2.13) is equivalent to an interaction involving local fields with form factor which introduces a cut- off giving mass to the bare spinor and (2) the positive and negative sign of the form factor is found to correspond to the positive and negative energy states and thus a four-component spinor can be formed from a bare two- component spinor. The generation of mass through the interaction violates the summetry corresponding to the invariance under the transformation -~e ic~y~ ~ inherent in the original system of weak interactions involving the bare two-component spinor as given by equation (2.12), and thus, in this scheme, electromagnetic interaction is generated through the spontaneous breakdown of symmetry and the photon appears as a Goldstone boson. It is to be noted here that the number n of the system of weak interactions in equation (2.12) bears a very crucial sense: n must be a unique number, otherwise we could get any amount of charge and mass of a lepton formed in this manner. Since we know that all charges in nature occur in units of e (provided we assume that there are no fractionally charged particles like quarks), n must be specified by the quantity e/g, where g is the photon- neutrino weak coupling constant. Again in Bandyopadhyay (1974) we have argued that this number n can be specified if we assume that space-time in nature is quantized such that the whole space-time continuum is considered as a collection of elementary space-time domains. Each such domain is
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