New Aspects of a Global Integration Theory

Scientific Paper
Title	New Aspects of a Global Integration Theory
Read in full	Link to paper
Author(s)	Marius Borneas
Keywords	Symmetry, Hypothesis, Mathematical models
Published	2000
No. of pages	14

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Abstract

Some new aspects of a global integration theory are presented, mainly by a Lagrangean formalism. The fundamentals are outlined, then electromagnetic and gravitational phenomena are discussed. The Lagrangean with higher derivatives is decomposed, and elementary equations are derived.

Overview

Published in the Buletinul Ştiinţific of the Politehnica University of Timişoara in 2000, this is an instalment in a research programme Marius Borneas had been developing since the mid-1970s: a "global integration theory" in which space, time, fields and matter are all derived from a single primordial entity by successive breakings of one primordial symmetry. Earlier statements appeared in the International Journal of Theoretical Physics (1976), Naturwissenschaften (1983) and Physical Review D (1984); the higher-derivative Lagrangian machinery on which the paper rests goes back to Borneas's 1969 Physical Review paper. The present article restates the scheme "in a new way, by mainly a Lagrangean formalism".

The structure is a single variational principle from which everything is meant to descend. One writes down a shortest-path condition in the primordial entity, projects it onto a curved cross-section, splits the resulting quantities into independent and dependent variables, and arrives at an action principle whose Lagrangean, Borneas says, "comprises the content of the spacetime frame". That Lagrangean is then decomposed piece by piece: one piece yields Maxwell's equations, another yields the vacuum field equation of general relativity, and the remaining, nonlinear pieces are proposed as the description of matter and short-range fields. It is a unified field theory in the classical sense — not new physics of measurement, but an attempt to derive the known field equations from a common root.

The construction

Two hypotheses and a variational principle

Borneas begins by making his assumptions explicit, on the grounds that every theory has them "even if these are hidden". The first is the principle of background nonpreferentiality: basically there is no preference in physical nature, so a background primordial symmetry exists. The second is that the origin of all physical phenomena is a unique primordial entity (PE), in which space, time and fields are globally integrated; "the multitude of nature's countenances is the result of symmetry reforming". Because the PE includes space and time it must have a geometric character, and because of the basic symmetry that geometry is Euclidean. Observable phenomena are then passages from one point of the PE to another over all intermediate elementary paths, and the fundamental mathematical hypothesis is a stationary-path condition

δ∫dσ = 0

which, in Euclidean PE coordinates α_ε, reads δ∫(Σdα_ε²)^1/2 = 0.

Symmetry breaking and the emergence of an action

Observable spacetime need not be Euclidean, so one takes non-Euclidean cross-sections of the PE with intrinsic coordinates β^ρ, on which the induced metric is the ordinary pull-back γ_ρτ = Σ_ε (∂α_ε/∂β^ρ)(∂α_ε/∂β^τ). The increments Δβ^ρ are then tied to actual physical magnitudes b^ς through operators e_ρ, which Borneas describes as "a kind of informations for the forming of the observables". The first symmetry breaking separates these magnitudes into those belonging to independent variables and those belonging to dependent ones; the independent ones supply the integration measure Πdx_j, and what remains is a Lagrangean density, so that the path principle has become an action principle δ∫Πdx_j L = 0.

A further hypothesis introduces a single universal, nonlocal field W as the content of that frame. The dependent quantities are functionals of W, of the independent variables, and of covariant derivatives of W of arbitrarily high order — this is where the higher-derivative character of the theory enters. Borneas argues that the Lagrangean must be invariant under any formal transformation of W, "because this field being universal, it can interact only with itself, and any transformation cannot lead to a gauge field". Variations are of two kinds: intrinsic variations within one cross-section, and variations that pass to another cross-section. Appealing to "the usual experience evidenciating the three-dimensionality of space", the number of dependent quantities is reduced to two, and the Lagrangean splits into an intrinsic part and a mixed part. The intrinsic part splits again into a piece free of connections and a piece containing them.

Electromagnetism from the connection-free piece

Restricting the independent variables to the four spacetime coordinates and using four components of W, Borneas identifies the combination of the coefficient, the operator and the field component as the electromagnetic potential A_μ. Provided the arbitrary parameter introduced in an auxiliary integral is chosen so that the integral takes the same value for every index, the connection-free Lagrangean takes the familiar quadratic form in ∂A_μ/∂x_λ. With the field tensor T_λν = ∂_λA_ν − ∂_νA_λ, whose components are the electric and magnetic fields in the usual imaginary-time convention (x⁴ = ict), the Euler–Lagrange equations together with the Lorentz condition give Maxwell's equations for the free field.

Borneas then adds a fifth independent variable, in the manner of a Kaluza–Klein extension, and uses five components of W. This produces an "extended potential" B_k and an antisymmetric tensor I_jk whose extra components correspond to a new vector field F and a scalar F₀. The field equations that follow are "Maxwell's equations with supplementary terms": the curl of H acquires a term in ∂F, the divergence of E acquires a term in F₀, and F and F₀ obey their own coupled equations.

Gravitation from the connection-containing piece

The piece of the Lagrangean containing the connections is treated by adopting a Riemannian metric — "it seems that this is the metric nearest to observations" — with Christoffel symbols as the connections. After some index rearrangement and a symmetry assumption relating the four coefficient blocks, the Lagrangean reduces to a quadratic form in the connections of the type ΓΓ − ΓΓ. Borneas then invokes the standard identity (quoted from Landau and Lifshitz) which relates the variation of ∫(−g)^1/2R dx with respect to the metric to the variation of the same ΓΓ combination. Selecting from among his index blocks the one that becomes identical with that bracket, on the conditions that the coefficient vanishes for unequal index pairs and that its components with the two remaining indices are all equal, and identifying G^ηθ = (−g)^1/2g^ηθ, he obtains

δ_g∫L_Gdx = δ_g∫(−g)^1/2R dx

so that stationarity of his own Lagrangean gives Einstein's vacuum equation R_λμ − ½g_λμR = 0. Since G is built from W, the metric itself is determined by combinations of the components of the universal field. Borneas remarks that the energy–matter tensor "and perhaps other terms" could be deduced from the remaining parts of the total Lagrangean, and that varying the sum of the gravitational and extended-electromagnetic parts yields more general equations of gravitation of the kind treated in his 1983 and 1984 papers.

Matter from the nonlinear remainder

The mixed part of the Lagrangean is stripped of connections (gravity having been dealt with) and, "for natural simplicity (Ockham's razor)", of its most complicated functional term. What is left is a quadratic kinetic piece plus nonlinear terms of third and fourth order in W and its derivatives. Since the intrinsic part gave the long-range macroscopic fields, Borneas proposes that this part "is liable for the description of the elements of matter and microscopic fields". Decomposing it into a sum of sectors, each of the form of a kinetic term, a quadratic mass-like term and a quartic self-interaction, the Euler–Lagrange equations give nonlinear field equations for the component fields. The nonlinear remainder N is acknowledged to be "very complicated", and parts of it are said to serve variously as interaction with other fields, self-interaction, or the definition of characteristic magnitudes.

Assessment

The programme has a real architectural elegance. One variational principle, one universal field, and a chain of symmetry breakings are asked to deliver, in order, the action principle itself, electromagnetism, gravitation and matter — and Borneas is careful to state his hypotheses at the outset rather than smuggle them in. The gravitational step is technically sound as far as it goes: a Lagrangean quadratic in the connections that reproduces the Einstein–Hilbert variation through the Landau–Lifshitz identity is essentially Einstein's own first-order ΓΓ form, and it is correctly deployed. The idea that the metric is not fundamental but is assembled from components of a deeper field is a serious one, shared with several respectable unification attempts, and the higher-derivative formalism is Borneas's own long-standing technical contribution, developed in refereed venues over three decades.

The central weakness is that at every decisive juncture the known result is reached by identification rather than derivation. The electromagnetic potential is not derived; a combination of a coefficient, an operator and a field component is declared to be A_μ because the resulting Lagrangean then looks like the Maxwell one. The metric is not derived; G^ηθ is set equal to (−g)^1/2g^ηθ because that makes the bracket match Landau and Lifshitz. Between these identifications sit a series of enabling stipulations, each introduced exactly where it is needed: that the arbitrary parameter be chosen so an auxiliary integral is index-independent, that a coefficient vanish for unequal index pairs, that certain of its components be all equal, and that two blocks of coefficients be equal and opposite to two others. Nothing in the theory motivates any of them. A framework flexible enough to be fitted to a target equation by such choices has not explained that equation.

The dimensional structure is likewise imported. The primordial entity has an unspecified number of Euclidean dimensions; the reduction of the dependent quantities to two rests on "the usual experience evidenciating the three-dimensionality of space"; four components of W are used for electromagnetism and five when extra fields are wanted. A theory whose selling point is that space and time emerge from a symmetric primordial entity ought to derive the number of dimensions, not adopt it from experience.

Most seriously, the paper produces no number. There is no coupling constant, no mass, no ratio, and no prediction anywhere in fourteen pages. The one place where contact with measurement is plainly available is the extended electromagnetic sector: the supplementary terms modify the source-free Maxwell equations, and modifications of that kind are among the most tightly constrained things in physics — by Coulomb-law null experiments of the Williams–Faller–Hill type, by the Jovian magnetic field measurements that bound a photon mass, and by the frequency-independence of the arrival times of gamma-ray burst photons. Borneas gives no estimate of the size of F and F₀, so none of these tests can be applied, and the extension is neither supported nor refuted. The recovered gravitational equation is the vacuum one; the energy–matter tensor, which is where a unified theory would have to earn its name, is left to "perhaps" emerge from unexamined terms. Nor does the paper address the standard hazard of higher-derivative Lagrangeans, the Ostrogradsky instability that makes their Hamiltonians unbounded below — a difficulty directly relevant to a formalism built on derivatives of arbitrarily high order.

Read as what it is — a compressed progress report in a long private programme, referring at a dozen points to earlier papers for definitions and proofs — the article is coherent and mathematically literate. Read as a case that electromagnetism, gravitation and matter have been unified, it establishes only that a sufficiently general higher-derivative Lagrangean can be arranged to contain them.

See also

• Marius Borneas
• Unified Field Theory
• Maxwell's Equations
• Electromagnetism
• General Relativity
• Gravity