Russian Physics Journal, Vol. 45, No. 2, 2002
PHYSICS OF ELEMENTARY PARTICLES AND FIELD THEORY IDENTITY OF ELEMENTARY PARTICLES AND GAUGE INTERACTIONS UDC 530.12
A. F. Yukin and G. A. Yukin
Problems of gauge invariance and identity of elementary particles are examined. The hypothesis that the gauge field charge is related to the possible increase of the entropy is set up. It is demonstrated that the given hypothesis allows fundamental interactions to be interrelated and numerical relations for elementary particle masses to be derived. Theoretical and practical consequences of the examined hypothesis are discussed. The identity of elementary particles is a fundamental postulate of theoretical physics [1]. The Bose–Einstein statistics for bosons and the Fermi–Dirac statistics for fermions are based on this postulate. In quantum field theory, the identity of particles is assumed in commutation relations for particle creation and annihilation operators. Moreover, as stated in [1], even small deviation from this principle is impermissible within the framework of modern quantum theories, since the identity of particles is included in the mathematical apparatus of the theory. However, the mechanism that ensures the identity of particles is unknown. Within the framework of gauge-invariant field theories, this is the Lagrangian invariance with respect to gauge transformations. Beyond the framework of gauge-invariant interactions, the absolute identity of particles has cast some doubts. Small deviations from this principle under the action of weak external fields and internal nonstationarity were examined in [2] where it was concluded that the number of particles was doubled. Naturally, when a unified gauge-invariant field theory is constructed, the problem of particle identity will be solved completely. However, most of the grand unified theories also call for an increase in the number of particle types. In our opinion, two points are paradoxical. First, the instantaneous correlation of particle properties (according to [1], the electron just created has already been antisymmetrized with all the remaining electrons). Second, to consider gravitational interactions of elementary particles within the framework of a gauge-invariant theory, the theory itself must be considerably complicated. In the present paper, an attempt is made to relate the particle identity principle to the interaction constants. Let us assume that the capability of a gauge field to establish the identity of particles of a specific type is determined by the value of the interaction constant. We designate the dimensionless interaction constant by 1/q2, where q is the field charge (we use the system of units in which Planck’s constant ћ = 1 and the velocity of light c = 1). Let N be the number of identical particles of the given type. If we switch off the interaction, the particles cannot be considered identical. As a result, the entropy of the many-particle system will increase by ln(N!). For sufficiently large N, the entropy of a single particle will increase by 'S = ln(N). Thus, the gauge field decreases the entropy of the many-particle system. From the classic viewpoint, the entropy is the number of degrees of freedom. Moreover, the energy of the system in equilibrium is uniformly distributed over the degrees of freedom. Then q2'S will determine the resultant field action. Considering that the minimum action is equal to ћ, we obtain a relation between the interaction constant q and the number of particles N. To derive the final relation, the pair character of gauge interaction and different particle spin projections onto a given direction must be taken into account. Let us designate by Qthemaximum number of particle pairs with the same spin projections. Then we can write
Qln(N) = 1/q2.
(1)
Kovrov State Engineering Academy. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 16–18, February, 2002. Original article submitted November 5, 2001. 1064-8887/02/4502-0105$27.00 2002 Plenum Publishing Corporation
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For electrons with a spin of 1/2, Q = 3/4. The identity of electrons is determined by electromagnetic interaction with the well-known constant 1/e2 = 137. Then from Eq. (1) we obtain N | 0.2141080. This value of N is close to the estimated number of particles in the Universe. It seems likely that this is not random. A relation between the electron parameters and the parameters of the Universe was pointed out already by Dirac [3]. Using the parameter N, the relation for the fundamental constants can be written as follows: 1/2Gmemp = e2N–1/2.
(2)
With allowance for Eq. (1), we obtain a relation between the gravitational constant and the electromagnetic field constant. We note that for 1/e2 = 137, the error of Eq. (2) is of the order of 2%. Formulas (1) and (2) are exact for N = 0.20651080 and 1/e2 = 136.97. These exact relations for the fundamental physical constants are hardly random. Therefore, we now examine the physical meaning of Eq. (2). If we restrict ourselves to an electron orbiting in a circle under the action of the proton gravitational field dividing Eq. (2) by the orbit radius we will determine the gravitational interaction energy of the electron. The quantity N1/2 specifies the characteristic fluctuation of the particle number. Then the right side of Eq. (2) determines the characteristic fluctuation of the electromagnetic interaction constant. Thus, Eq. (2) reflects the equality of the gravitational interaction to the fluctuation of the electromagnetic interaction. From the viewpoint of particle identity, this is of crucial importance. The gravitational field does not influence the electromagnetic interaction, because it is compensated by the electromagnetic field fluctuation. Thus, the world is structured so that the gravitational interaction does not influence the electromagnetic interaction of elementary particles. Let us assume that this principle is general, that is, the constant of a less symmetric interaction is determined by the fluctuation of the interaction with a higher degree of symmetry. In accordance with this assumption, the higher the interaction symmetry, the larger its constant. Let us consider a set of elementary charges q1 = 1/3, q2 = 2/3, and q3 = 1 representing relative electric charges of quarks and leptons. From Eq. (1) we obtain N1 = exp (12) | 1.63105, N2 = exp (3) | 20, and N3 = exp (4/3) | 3.8. We note that N1 is numerically equal to the ratio of the charged W-boson mass to the electron mass for me = 0.511 MeV and mW = 83.2 GeV and N2–1/2 = 0.22 | sin2(TW), where TW is the Weinberg angle. Thus, the parameters N1 and N2 completely determine the weak interaction constants. For the electromagnetic interaction constant, we can write the relation e2 | 3N1–1/2 = 1/134.5. Thus, fluctuations in the parameters Ni determine relationships among constants of different interaction types. Therefore, this suggests that the above-indicated fluctuations are significant for the interaction of elementary particles with scalar fields of electroweak theory. This interaction determines particle masses. Therefore, relationships for particle masses will be determined by fluctuations in Ni. Let us introduce a dimensional constant Mdefined as a geometric mean of the electron and W-boson masses: M = 206.2 MeV. Then the electron mass will be me = M/N11/2 = 0.511 MeV, the muon mass mP = M/N31/2 = 105.8 MeV, the lepton mass mW = MN21/2N31/2 = 1.8 GeV, the W-boson mass mW = MN11/2 = 83.2 GeV, the light-quark mass mq = M(N2N3/N1)1/2 = 4.5 MeV, and the nucleon mass mn = MN21/2 = 924 MeV. We note that except quarks and nucleons, all relations are fulfilled to within 1%. Moreover, if we add quark masses to a nucleon mass of 924 MeV, we obtain proton and neutron masses. Obviously, the foregoing does not meet the requirements imposed on a rigorous scientific theory. However, exact relationships for the fundamental constants obtained here suggest that these principles can be used to develop a unified interaction theory based on a single dimensional constant (for example, M). Using three primary charges equal to 1/3, 2/3, and 1, we obtain constants Ni, and with their help calculate electromagnetic and weak interaction constants, lepton and nucleon masses, and finally the gravitational constant. In fact, this has already been done in the zero approximation before the development of a theory; therefore, a more comprehensive analysis of these problems will allow us to increase significantly the accuracy of calculations. However, we believe that in so doing, new and possibly paradoxical effects may arise. In our opinion, new phenomena will be discovered for low and superlow energies rather than for high energies. Indeed, substituting 1/q2 = sin2(TW)/e2 | 30.6, we obtain N4 | 51017. Then meN4–1/2 | 36 cm–1. This energy corresponds to a temperature of ~8 K, that is, to energies of low-temperature physics. We note that the foregoing does not touch the principles of the unified electromagnetic and weak interaction theory and only supplements it by relationships for constants. One of the directions of Grand Unification is the use of theories based on SU(3)SU(2)U(1) symmetries [1] containing three interaction constants. An analysis of these constants yields a characteristic unification energy of ~1015 GeV. However, considering that we succeeded in relating weak and 106
electromagnetic interaction constants to charges of 1, 2/3, and 1/3, we suggest that this value of energy is highly overestimated. It seems likely that the energy of unification is ~mnN11/2 | 370 GeV. We note that masses of nuclei of the heaviest isotopes are close to this value but less than it. In conclusion, we consider the relation for physical constants and the Hubble constant H. Setting H–1 = 1010 years | 0.9461028 cm [4], we obtain 4SGmpN/3H–1 | 1.2. This relation demonstrates that all the parameters of the visible part of the Universe are interrelated. A particular interpretation of this relation depends on the employed cosmological model. We note that the above relation can be interpreted not only within the framework of the dilated Universe. It can be a consequence of the finite radius of action of the gravitational field ~H–1. Then the red shift is caused by the finite radius of action of the electromagnetic field. In this case, the parameter N introduced by us has a local meaning, namely, N1/2 is proportional to the ratio of the electron mass to the photon mass. Then relationships of N with cosmological parameters of the Universe are a consequence of the homogeneity of our part of the Universe.
REFERENCES 1. 2. 3. 4.
L. B. Okun’, Physics of Elementary Particles [in Russian], Nauka, Moscow (1988). Ya. M. Gel’fer, V. L. Lyuvoshits, and M. I. Podgorel’skii, The Gibbs Paradox and the Identity of Particles in Quantum Mechanics [in Russian], Nauka, Moscow (1974). P. A. M. Dirac, Nature, No. 139, 323 (1937). K. Meller, Relativity Theory [Russian translation], Atomizdat, Moscow (1975).
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