One Approach to the Problem of Fundamental Interactions

Scientific Paper
Title	One Approach to the Problem of Fundamental Interactions
Read in full	Link to paper
Author(s)	Lada Vladimirovna Rudikova, Evgenij Vladimirovich Rudikov
Keywords	elementary particles, Maxwell equations, space-time
Published	2009
No. of pages	12

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Abstract

This article considers one approach to the problem of fundamental interactions based in six-dimensional space-time. It is supposed that space-time consists of elementary particles described by the Maxwell Equations. Every elementary particle corresponds to a specific term in the equation and possesses the implied conservation law. The interaction of elementary particles is described and a series of functional actions is constructed. The sixth term in the series of functional actions is the most important as it determines the dimensionality of observable space-time. It is proven that the first term in the series of actions represents Maxwell electrodynamics and the second term represents the Yang-Mills field.

Overview

The Rudikovs, writing from Yanka Kupala State University of Grodno in Belarus, propose that the observed world is a projection of a six-dimensional manifold R(3,3) — three spatial dimensions and three time dimensions. The choice is argued rather than assumed: adding spatial dimensions "would violate the inverse-square law", which is measured, so the extension is made on the time side instead, and each of the three time axes is put into correspondence with one of the fundamental interactions — gravitational, electro-weak and strong.

Everything else follows from a single guiding maxim the authors call the natural principle: "The universe exists such as we observe it, in 'large' and 'small'." From it they take two experimentally established inputs, Coulomb's law and the gravitational constant, and build a structure in which Maxwell's equations in six dimensions collapse to the single tensor statement dF = 0. The matter content is a set of closed strings, and the interactions of two, three, ... strings generate an infinite functional series of actions S = S⁽¹⁾ + S⁽²⁾ + ... whose successive terms are identified with successive sectors of physics. The claim advertised in the abstract is that the first term is Maxwell electrodynamics and the second is the Yang–Mills field, with the sixth term fixing the dimensionality of the world we see.

The argument

Why a preferred frame

The paper's opening physical argument is against the first postulate of special relativity. Relativity's claim that all inertial frames are equivalent, the authors write, "was only true until 1965 – the year of the discovery of cosmic microwave background radiation." An observer at rest with respect to the background sees it as isotropic; an observer moving with velocity V sees the temperature rise ahead and fall behind by the Doppler effect, breaking the isotropy. They conclude that background radiation "is an essential part of observable space (the vacuum)" and that "inertial frames of reference are only those with isotropic electromagnetic background radiation."

Within that privileged frame they assign relative intensities to the interactions: the electrical components of the gravitational and strong interactions stand to the electromagnetic component as 1 : 10⁻⁴⁰ : 10⁻⁸⁰, while the magnetic components of all interactions are equal, at 10⁻⁴⁰ relative to the electric component of electromagnetism.

Maxwell's equations in R(3,3)

In six dimensions the field equation reduces to ∂F_ij/∂x_k + ∂F_jk/∂x_i + ∂F_ki/∂x_j = 0, or dF = 0 with F = ΣF_ijdxⁱ∧dx^j. The 6×6 antisymmetric tensor has room for three diagonal-block charge entries Q¹₁, Q²₂, Q³₃, which the authors identify with gravitational, electrical and hadronic charge and "arrange in accordance with the colour charges of quantum chromodynamics".

Projecting the six-dimensional system onto R³ and onto T³ separately gives two mirror-image sets of Maxwell-like equations, the second with a magnetic charge density μ where the first has an electric charge density ρ. Under the exchange symmetry R³T³ → T³R³ the two swap: "electrical charge becomes magnetic and vice-versa", written e_R³ = μ_T³. In six dimensions, they note, "charge and field are directly equivalent" — substituting the field components for the currents leaves the form of the field equation unchanged. The vacuum case reduces to a two-dimensional projection "analogous to the Dirac equation in the absence of interactions", from which they conclude that "the objects known as strings are the basis of elementary structural units of matter", and that charge is tied to the non-zero topology of the R³ and T³ subspaces.

The constancy of c derived rather than postulated

The six-dimensional wave equation carries two families of transformation matrices, one corresponding to group velocity and one to phase velocity, whose product in the simplest choice of units is v_g·v_p = c². Because that relation is preserved under change of frame in a homogeneous isotropic vacuum, the authors argue, "to explain constant speed of light in a vacuum in six-dimensional space-time there is no need for axiomatic postulation as is proposed in the Special Theory of Relativity."

The physical picture offered is that a heated surface emits photons across a Planck distribution with λ₁ > λ₂ > λ₃; for them to arrive together the longer-wavelength photon must have the smaller group velocity and the larger phase velocity, with phase velocity interpreted as "the speed at which photons are born".

Strings, spin, statistics and the series

Strings carry dual energy–momentum parameters, and their rotation — "closed string's rotational charge" — is spin. Each string has a time component and a space component corresponding to electric and magnetic parts, and the reciprocal relation of these to the observer "defines the type of observable object; either fermions or bosons." The Pauli exclusion principle is rederived geometrically: strings rotating identically make identical magnetic fields and repel when coplanar with the observer (fermion case), while for a rotation plane perpendicular to the observer's, like-rotating strings attract and opposite ones repel (boson case). Supersymmetry becomes a change of orientation of the string in space-time relative to the detector.

The action for n interacting strings is expanded as a series, and the terms are assigned: S⁽¹⁾ gives Maxwell electrodynamics and the light particles (electrons, neutrinos, photons, gravitons); S⁽²⁾ the Yang–Mills field, the space-time metric and quantum weak fields; S⁽³⁾ the π-mesons; S⁽⁴⁾ the ρ and ω mesons; S⁽⁵⁾ the η-mesons; and S⁽⁶⁾ the strong interaction and protons. Terms beyond the sixth describe nuclei and atoms, and are argued to be negligible "considering the minuteness of the number 10⁻⁴⁰". The coupling strengths are locked together by α⁽ⁿ⁾ = (α⁽²⁾)ⁿ⁻¹, with α⁽¹⁾ = h/h₀ equal to the Sommerfeld (fine structure) constant — the paper introduces two Planck constants, h "minimal possible for matter" and h₀ "maximum possible for a vacuum" — and α⁽²⁾ ≈ 10⁻⁴⁰ presented as "the effect of gravity upon the strength of electromagnetism for two interacting electrons".

Six-dimensional kinematics and the three generations

Writing the differential of the action for homogeneous vacuum in terms of three group velocities c₁, c₂ = c, c₃ and three phase velocities w₁, w₂, w₃ yields a six-dimensional analogue of the relativistic factor γ. Two of its terms are said to be "very small numbers (such that they need not be calculated)", so that for v < c "the general form of γ coincides with the form for Special Theory of Relativity" — with the difference that here "all particles possess invariant mass with respect to the relativistic calculation."

The closing figures are geometric. A graph of e/m and c²/w against r_g/r and v/c has two marked points: point A, where the stability condition for leptons is met, and point B, where electromagnetic and gravitational interactions become equal. The ratio of the segments OA to OB is said to give the gravitational-to-electric charge ratio for electrons, α⁽²⁾ ≈ 10⁻⁴⁰. At B "geometry with positive curvature crosses to geometry with negative curvature" — a phase crossing at which the chromodynamic charge shifts and baryons take over the electromagnetic role.

The final diagram shows three momentary planes of projection identified with the three primary colour charges, linked to the Weinberg–Salam angle, and offered as the explanation of why there are exactly three generations. In this scheme electrons and neutrinos carry colour; baryons are the colourless state described by S⁽⁶⁾; quarks are unobservable because fractional charge creates colour in light particles and the experimenter cannot be sure of observing the same electron that entered an interaction — from which "a more correct interpretation of quantum mechanics was proposed by H Everett". A striking prediction is stated plainly: "On one electron level there won't be two electrons but six times more than that – 12. However only two electrons are directly observable."

An anti-universe rotates the same diagram in the opposite sense; the mutual expansion of universe and anti-universe is attributed to Ampère's circuital law and the repulsion of anti-parallel charges. The number of baryons in the universe, about 10⁸⁰, is matched to a baryon interaction intensity of 10⁻⁸⁰ relative to electromagnetism, and "for the hypothetical observer, located within the boundaries of our Universe, the Universe will appear as an elementary particle, for example a baryon."

Assessment

There is a real idea underneath the formalism, and it is stated more crisply than most papers of this kind manage. The observation that one cannot add spatial dimensions without spoiling the inverse-square law — which is measured, to sub-millimetre separations in torsion-balance experiments — and that the extension must therefore go on the time side, is a genuine argument rather than an assertion. The reduction of Maxwell's equations to dF = 0 in R(3,3), the appearance of a magnetic-charge sector as the mirror of the electric one under R³T³ → T³R³, and the derivation of the invariance of c from the preservation of v_gv_p = c² rather than by postulate are all coherent as formal moves. And the authors are explicit about their preferred frame rather than smuggling one in.

But the paper's numbers do not survive checking, and several of them are load-bearing. The constant α⁽²⁾ ≈ 10⁻⁴⁰, on which the whole coupling hierarchy α⁽ⁿ⁾ = (α⁽²⁾)ⁿ⁻¹ rests, is described as the ratio of gravitational to electric force "for two interacting electrons". That ratio is Gm_e²/ke² = 2.4×10⁻⁴³, not 10⁻⁴⁰. The figure 10⁻⁴⁰ is the electron–proton value, Gm_em_p/ke² = 4.4×10⁻⁴⁰; the paper has taken a mixed-particle ratio and attached an electron–electron label to it, a factor of about 1800. Since the same number is then read off the geometry of Figure 8 as the segment ratio OA:OB, the geometric "derivation" inherits the error.

More serious is the placement of the strong interaction. The paper puts its intensity at 10⁻⁸⁰ relative to electromagnetism, and repeats the figure for baryon interactions. The strong interaction is not weaker than electromagnetism — it is the strongest of the four. Its coupling at hadronic scales is of order unity against the fine structure constant's 1/137, and the direct measurement is in every nuclear binding energy: about 8 MeV per nucleon against a few eV for atomic electron binding, six orders of magnitude the other way. A framework whose whole architecture is a descending series of couplings cannot accommodate an interaction that is stronger than the one it starts from, and the paper does not acknowledge the difficulty. There is also an internal tension: α⁽ⁿ⁾ = (α⁽²⁾)ⁿ⁻¹ puts 10⁻⁸⁰ at n = 3, the π-meson term, while the strong interaction is assigned to S⁽⁶⁾, where the same rule would give 10⁻²⁰⁰.

The experimental support offered at the end does not support the theory. The listed reactions in which a neutrino and a neutron produce three pions violate baryon number — the neutron carries baryon number 1 and pions carry none — and lepton number as well; no such process has been observed, and the neutron's measured behaviour is beta decay, n → p + e⁻ + antineutrino, with a lifetime of about 878 seconds. Citing an unobserved and conservation-violating process as evidence is the weakest passage in the paper.

The twelve-electron claim runs into the most familiar measurement in atomic physics. If each electron level held twelve states with ten unobservable, the shell closures would not fall where they do; the periodic table's structure — the noble gases at Z = 2, 10, 18, 36 — and the observed multiplicities of atomic spectral lines are direct counts of the available states, and they give exactly two per orbital. Saying that ten are hidden requires an account of why the hidden ones never affect a count that is made by energy level, not by direct observation of the electron, and none is given.

Finally, the argument from the microwave background misidentifies what relativity claims. The background does define a frame in which it is isotropic — the dipole anisotropy measuring the Solar System's motion at about 370 km/s is one of the best-determined numbers in cosmology — but so does any physical medium, and the existence of a preferred medium has never been in conflict with the principle that the laws take the same form in every inertial frame. The relevant test is whether local physics is anisotropic in that frame, and modern Michelson–Morley experiments using cryogenic optical resonators bound any such anisotropy at the level of a few parts in 10¹⁷. Nothing in the paper engages with that bound.

The presentation compounds these problems. The functional series is introduced with its terms already labelled — mesons, Yang–Mills, protons — rather than with any calculation showing that the n-th term reproduces the corresponding sector's known spectrum, coupling or cross-section. No mass is computed, no decay rate, no scattering amplitude. The claim in the abstract that "it is proven" that the first term is Maxwell electrodynamics rests on the structural resemblance of equation (33) to an electrodynamic action rather than on any derivation of an electrodynamic result. Read as a programme sketch the paper has some interesting formal ingredients; read as the demonstration it announces itself to be, the case is not made.