The Standard Model and General Relativity

Both of these are mathematical systems. Recently I posted some comments on Dr Thomas Love's paper, Towards an Einsteinian Quantum Theory. He has an excellent analysis and literature survey of causal versus abstract quantum field theory controversies. He also puts forward an alternative spacetime of quantum anti-de sitter space, QAds = U(3,2)/U(3,1)xU(1) and finds it allows him to make an analysis of particle physics, and link that to general relativity. However, anti-de Sitter space is not even wrong, as I know from cosmology. So the issue is that I don't really know whether U(3,2)/U(3,1)xU(1) is any better at describing particle physics than the Standard Model, SU(2)xSU(2)xU(1). Certainly there is a lot of experimental evidence for the Standard Model, but on the other hand the electroweak symmetry breaking mechanism requires the Higgs field, and its unification to a superforce at extremely high energies (attained only shortly after the big bang) is supposed to require supersymmetric partners. In addition, the physical reason why there are only 8 gluons when there are 3x3 = 9 colour combinations, remains obscure in the Standard Model. I know that the strong force part, SU(2), works, but the so did Ptolemic epicycles. The electroweak component, SU(2)xU(1), is a lot more rigorous and convincing, although the details Higgs mechanism which it relies upon remains slightly obscure in the mainstream publications.

The chief thing about these theories, the Standard Model and alternatives, is that the underlying physics is symmetry principles. You fill out a matrix as allowed by your principles, and it describes the quarks, leptons, and gauge bosons for the force in question. Love's analysis does allow a lot of important fundamental particles to be covered, but it doesn't predict what their masses or, or predict the strengths of fundamental forces, which is disappointing. I think research should be done into U(3,2)/U(3,1)xU(1). However, from my perspective, it will not help unify quantum field theory and general relativity, as it seems to be just juggling around the abstract mathematical model. It may seem more elegant, but doesn't predict the additional things needed, as far as I can see. So for the sake of argument, I'm going to stick to discussing the Standard Model SU(3)xSU(2)xU(1) until I've finished understanding the implications of U(3,2)/U(3,1)xU(1). I'm not really interested in the abstract mathematical formalism, so much as the physical principles implied, and the difference between the Standard Model and the alternative seems at present to be a squabble between two completely abstract, non-causal, mathematical structures.

Peter Woit in http://arxiv.org/abs/hep-th/0206135 put forward a conjecture: “The quantum field theory of the standard model may be understood purely in terms of the representation theory of the automorphism group of some geometric structure.”

Using Lie spinors and Clifford algebras he comes up with an illustrative model on page 51, which looks as if it will do the job, but then adds the guarded comment: “The above comments are exceedingly speculative and very far from what one needs to construct a consistent theory. They are just meant to indicate how the most basic geometry of spinors and Clifford algebras in low dimensions is rich enough to encompass the standard model and seems to be naturally reflected in the electro-weak symmetry properties of Standard Model particles.”

Woit's work illustrates that string theory is not needed to get the Standard Model to work properly. One thing that caught my eye on p50 is that Woit uses Euclidean 4-dimensional spacetime rather than Minkowski's mess. Minkowski generally confuses measures of say the contraction of real matter (the length of a material ruler, say, which physically contracts when moving due to the radiation pressure of the vacuum which is overcome by the energy taken to accelerate the object, and this energy is stored in electromagnetic field compression of the material), with the idea that spacetime contracts.

The physical mechanism shows that the expansion of the universe drives the compression of matter, by Newton's 3rd law of motion. You always get recoil. I quote Minkowski's only useful statement on my home page: ‘The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.’ – Hermann Minkowski, 1908. Ironically, Minkowski's statement only applies to the Hubble recession, which should be represented as an increase in speeds with definite time past instead of definite distance (distances increase while light comes to use), because velocity/time is a physically important quantity, acceleration. Ironically, Minkowski's statement is applauded for its obscurity, instead of being seen as evidence of the clarity and simplicity it brings to the mechanism for gravity!

My view is that although the metrics of general relativity are at least four dimensional, physically you are dealing with measurement of space in terms of the travel time of the light (think of cosmology in an expanding universe, not of distances over a piece of matter like the earth's surface), and you are dealing with the measurement of matter in terms of distance which can be contracted by radiation pressure in the vacuum when the matter is in motion, or by gravitational contraction.

So you have six distinguishable dimensions, three expanding dimensions of time, and three contractable dimensions of distance in measuring matter. Of course, since distance = velocity multiplied by time, all six dimensions can be expressed as aspects of spacetime. For mathematical purposes, an excellent approximation is that the expansion of the time dimensions are radially symmetrically, so they can all be treated as a single dimension to get the calculations work.

But the underlying reality is six dimensions, 3 contractable dimensions of matter and 3 expanding dimensions of time, because all we can observe about the cosmological expansion is what we see, which is the universe in the past.

Where can we find a 6-dimensional theory to mathematically unify Maxwell's equations and general relativity? Lunsford has such a theory:

D.R. Lunsford has a paper on ‘Gravitation and Electrodynamics over SO(3,3)’ on CERN document server, EXT-2003-090: ‘an approach to field theory is developed in which matter appears by interpreting source-free (homogeneous) fields over a 6-dimensional space of signature (3,3), as interacting (inhomogeneous) fields in spacetime. The extra dimensions are given a physical meaning as ‘coordinatized matter’. The inhomogeneous energy-momentum relations for the interacting fields in spacetime are automatically generated by the simple homogeneous relations in 6-D. We then develop a Weyl geometry over SO(3,3) as base, under which gravity and electromagnetism are essentially unified via an irreducible 6-calibration invariant Lagrange density and corresponding variation principle. The Einstein-Maxwell equations are shown to represent a low-order approximation, and the cosmological constant must vanish in order that this limit exist.’

Lunsford begins with an enlightening overview of attempts to unify electromagnetism and gravitation:

‘The old goal of understanding the long-range forces on a common basis remains a compelling one. The classical attacks on this problem fell into four classes:

‘1. Projective theories (Kaluza, Pauli, Klein)
‘2. Theories with asymmetric metric (Einstein-Mayer)
‘3. Theories with asymmetric connection (Eddington)
‘4. Alternative geometries (Weyl)

‘All these attempts failed. In one way or another, each is reducible and thus any unification achieved is purely formal. The Kaluza theory requires an ad hoc hypothesis about the metric in 5-D, and the unification is non-dynamical. As Pauli showed, any generally covariant theory may be cast in Kaluza’s form. The Einstein-Mayer theory is based on an asymmetric metric, and as with the theories based on asymmetric connection, is essentially algebraically reducible without additional, purely formal hypotheses.

‘Weyl’s theory, however, is based upon the simplest generalization of Riemannian geometry, in which both length and direction are non-transferable. It fails in its original form due to the non-existence of a simple, irreducible calibration invariant Lagrange density in 4-D. One might say that the theory is dynamically reducible. Moreover, the possible scalar densities lead to 4th order equations for the metric, which, even supposing physical solutions could be found, would be differentially reducible. Nevertheless the basic geometric conception is sound, and given a suitable Lagrangian and variational principle, leads almost uniquely to an essential unification of gravitation and electrodynamics with the required source fields and conservation laws.’ Again, the general concepts involved are very interesting: ‘from the current perspective, the Einstein-Maxwell equations are to be regarded as a first-order approximation to the full calibration-invariant system.

‘One striking feature of these equations that distinguishes them from Einstein’s equations is the absent gravitational constant – in fact the ratio of scalars in front of the energy tensor plays that role. This explains the odd role of G in general relativity and its scaling behaviour. The ratio has conformal weight 1 and so G has a natural dimensionfulness that prevents it from being a proper coupling constant – so the theory explains why general relativity, even in the linear approximation and the quantum theory built on it, cannot be regularised.’ [Lunsford goes on to suggest gravity is a residual of the other forces, which is one way to see it.]

Danny Ross Lunsford’s major paper, published in Int. J. Theor. Phys., v 43 (2004), No. 1, pp.161-177, was submitted to arXiv.org but was removed from arXiv.org by censorship apparently since it investigated a 6-dimensional spacetime which again is not exactly worshipping Witten’s 10/11 dimensional M-theory. It is however on the CERN document server at http://doc.cern.ch//archive/electronic/other/ext/ext-2003-090.pdf , and it shows the errors in the historical attempts by Kaluza, Pauli, Klein, Einstein, Mayer, Eddington and Weyl. It proceeds to the correct unification of general relativity and Maxwell’s equations, finding 4-d spacetime inadequate:

‘… We see now that we are in trouble in 4-d. The first three [dimensions] will lead to 4th order differential equations in the metric. Even if these may be differentially reduced to match up with gravitation as we know it, we cannot be satisfied with such a process, and in all likelihood there is a large excess of unphysical solutions at hand. … Only first in six dimensions can we form simple rational invariants that lead to a sensible variational principle. The volume factor now has weight 3, so the possible scalars are weight -3, and we have the possibilities [equations]. In contrast to the situation in 4-d, all of these will lead to second order equations for the g, and all are irreducible - no arbitrary factors will appear in the variation principle. We pick the first one. The others are unsuitable … It is remarkable that without ever introducing electrons, we have recovered the essential elements of electrodynamics, justifying Einstein’s famous statement …’

D.R. Lunsford shows that 6 dimensions in SO(3,3) should replace the Kaluza-Klein 5-dimensional spacetime, unifying GR and electromagnetism.

Now back to Thomas Love's paper. On page 117 he makes the interesting point: 'If the electromagnetic interaction were due to the exchange of virtual photons then the field strength would exhibit statistical fluctuations, then the electron's energy level would exhibit statistical fluctuations and hence the spectral lines would exhibit statistical fluctuations. They do not. Spectral lines are sharp. ... what is the source of the virtual photon? ... how does an electron know when to create virtual photons?'

Clearly the electron in quantum field theory is in equilibrium, exchanging energy all the time for electromagnetic forces to operate continuously. A photon is then an propagating disturbance to the existing equilibrium field. Similarly, water waves are transverse disturbances to the already existing equilbirium water distribution. You don't try to build up a picture of normal continuous water pressure from a lot of discrete water waves, so you should not try to build up a picture of virtual photons from real photons. The virtual photons are the background equilibrium radiation which creates an exchange force without creating oscillation. The oscillatory emission of radiation causes and results in discrete, quantum changes in energy.

Put that another way: the virtual radiation is more like the classical Maxwell radiation which centripetally accelerating (orbital) electrons emit all the time. The reason why they don't spiral into the nucleus as a result of energy loss is that all particles with spin do the same thing. They radiate energy all the time, normally with an equilibrium between emission and reception.

On page 118, Love also usefully comments that 'QED is supposed to be a theory of electromagnetism ... when the magnetic field of a charged particle acts on another charge d particle, the direction of the force can be perpendicular to the line between the particles. Try to explain that in terms of the exchange of virtual particles! The QED picture of interchanging virtual photons reduces to gibberish!'

The answer to that is that magnetism is carried by the spin of the radiation or particles in the vacuum, as Maxwell suggested after several failed mechanical spacetime fabric models. Maxwell failed to grasp that radiation (gauge bosons) was the mechanism for electric force fields, but he did usefully suggest that:

‘The ... action of magnetism on polarised light [discovered by Faraday not Maxwell] leads ... to the conclusion that in a medium ... is something belonging to the mathematical class as an angular velocity ... This ... cannot be that of any portion of the medium of sensible dimensions rotating as a whole. We must therefore conceive the rotation to be that of very small portions of the medium, each rotating on its own axis [spin] ... The displacements of the medium, during the propagation of light, will produce a disturbance of the vortices ... We shall therefore assume that the variation of vortices caused by the displacement of the medium is subject to the same conditions which Helmholtz, in his great memoir on Vortex-motion, has shewn to regulate the variation of the vortices [spin] of a perfect fluid.’ - Maxwell’s 1873 Treatise on Electricity and Magnetism, Articles 822-3

Compare this to the spin foam vacuum, and the fluid GR model:

‘… the source of the gravitational field can be taken to be a perfect fluid…. A fluid is a continuum that ‘flows’... A perfect fluid is defined as one in which all antislipping forces are zero, and the only force between neighboring fluid elements is pressure.’ – Professor Bernard Schutz, General Relativity, Cambridge University Press, 1986, pp. 89-90.

Einstein admitted SR was tragic:‘The special theory of relativity … does not extend to non-uniform motion … The laws of physics must be of such a nature that they apply to systems of reference in any kind of motion. Along this road we arrive at an extension of the postulate of relativity… The general laws of nature are to be expressed by equations which hold good for all systems of co-ordinates, that is, are co-variant with respect to any substitutions whatever (generally co-variant). …’ – Albert Einstein, ‘The Foundation of the General Theory of Relativity’, Annalen der Physik, v49, 1916.

‘Recapitulating, we may say that according to the general theory of relativity, space is endowed with physical qualities... According to the general theory of relativity space without ether is unthinkable.’ – Albert Einstein, Leyden University lecture on ‘Ether and Relativity’, 1920. (Einstein, A., Sidelights on Relativity, Dover, New York, 1952, pp. 15-23.)

‘The Michelson-Morley experiment has thus failed to detect our motion through the aether, because the effect looked for – the delay of one of the light waves – is exactly compensated by an automatic contraction of the matter forming the apparatus…. The great stumbing-block for a philosophy which denies absolute space is the experimental detection of absolute rotation.’ – Professor A.S. Eddington (who confirmed Einstein’s general theory of relativity in 1919), Space Time and Gravitation: An Outline of the General Relativity Theory, Cambridge University Press, Cambridge, 1921, pp. 20, 152.

The radiation (gauge bosons) and virtual particles in the vacuum exert pressure on moving objects, compressing them in the direction of motion. As FitzGerald deduced in 1889, it is not a mathematical effect, but a physical one. Mass increase occurs because of the snowplow effect of Higgs boson (mass ahead of you) when you move quickly, since the Higgs bosons you are moving into can't instantly flow out of your path, so there is mass increase. If you were to approach c, the particles in the vacuum ahead of you would be unable to get out of your way, you'd be going so fast, so your mass would tend towards infinity. This is simply a physical effect, not a mathematical mystery. Time dilation occurs because time is measured by motion, and if as the Standard Model suggests, fundamental spinning particles are just trapped energy (mass being due to the external Higgs field), that energy is going at speed c, perhaps as a spinning loop or vibrating string. When you move that at near speed c, the internal vibration and/or spin speed will slow down, because c would be violated otherwise. Since electromagnetic radiation is a transverse wave, the internal motion at speed x is orthagonal to the direction of propagation at speed v, so x^2 + v^2 = c^2 by Pythagoras. Hence the dynamic measure of time (vibration or spin speed) for the particle is x/c = (1 - v^2/c^2)^1/2, which is the time-dilation formula.As Eddington said, light speed is absolute but undetectable in the Michelson-Morley experiment owing to the fact the instrument contracts in the direction of motion, allowing the slower light beam to cross a smaller distance and thus catch up.

Dr Love helpfully quotes Einstein's admissions that the covariance of the general relativity theory violates the idea in special relativity that the velocity of light is constant:

'This was ... the basis of the law of the constancy of the velocity of light. But ... the general theory of relativity cannot retain this law. On the contrary, we arrived at the result according to this latter theory, the velocity of light must always depend on the coordinates when a gravitational field is present.' - Albert Einstein, Relativity, The Special and General Theory, Henry Holt and Co., 1920, p111.

So general relativity conflicts with, and supersedes, special relativity. General relativity says goodbye to the law of the invariant velocity of light which was used in a fiddle, special relativity:

'... the principle of the constancy of the velocity of light in vacuo must be modified, since we easily recognise that the path of a ray of light ... must in general be curvilinear...' - Albert Einstein, The Principle of Relativity, Dover, 1923, p114.