A Major Revision of Mathematical Physics

Make no mistake this article could have been entitled 'A major revision of physics'. Our mathematical methods are important tools for how we live and how modern civilization can improve on our current standard of living. We can see how technology is helping developing countries like China, India, Indonesia, to mention a few. Perhaps  more than a billion citizens in these countries have been brought out of abject poverty by the application of science and technology. Democracy, politics, and economics may all be important factors but so too is how we calculate within science. Any revision of the computer programming and analytic methods we use within our physical mathematics is likely to benefit this application of science and reduce the errors that can currently occur due to the use of incorrect methods of calculation. This requires a major revision of mathematical physics, including quantum theory, general relativity, even Newton's theory of gravitation, and in fact all field theories.

We discuss a major new revision of mathematical physics. To be precise the change affects the way field theories measure their fields and their distances across all domains. At present field theories such as Newton's law of gravitation, Coulomb's law of electrostatics, and the Biot-Savart law of magnetostatics are measured using Pythagorean distances. Whenever they are coded as computer programs this means distances are taken to be the hypotenuse of a triangle of forces. But at atomic distances the particles do not move along the hypotenuse but along the other two directions of the triangle of atomic forces. While this can be acceptable at the macroscopic and the cosmological domains it is most inaccurate at very small domains. The Pythagorean measurement system was influenced by the history of science and flowed through to both quantum theory and general relativity formulated early in the 20th century. All field theories were influenced by observations made at the macroscopic domain; this is where they were wrong in that macroscopic effects can 'hide' the actual motion.

Looking back we can see why this error in mathematical physics occurred. A look through the history of science shows how mathematics evolved from ancient Greek times, to Newton's time, and over the past few hundred years from macroscopic physics to include microscopic physics, to nanoscopic physics and below.

Looking forward what does it mean? For a start our pharmaceutical industry uses quantum methods to determine the type of drugs they manufacture, the bond lengths, angles, etc.  We need to modify our computerized mathematical methods to reflect this revision to improve upon the process of manufacturing medicinal and therapeutic drugs. The current methods of determining chemical structures may be causing pharmaceutical side-effects. But there are many other areas where such mathematical revisions will bring about a better and clearer picture of the world around us, not to mention within us. Another example may well be microscopy. At present our microscopes hit what some call the diffraction barrier in optical microscopy. If our methods for determining small distance are currently incorrect then we need to do something about it because our medical health for one thing often depends on good medicines and good microscopy. As a final example it may well be possible to calculate more accurately the vast distances across the cosmos enabling us to better observe and eventually better understand our place in the universe and whether there is life somewhere else in the Galaxy.

Perhaps the biggest effect will be to rethink existing quantum theory where probabilistic methods are used to solve the equations instead of the normal closed form solutions obtained from field theories. In other words in these forms of quantum theory there are not enough unknown variables in the equations used to solve them, hence they need probabilistic methods (Arrivederci Uncertainty). When we change our mathematics to suit the presence of the two orthogonal directions instead of the hypotenuse we now find we have enough variables to obtain closed-form solutions.

Historical but Still Relevant Example

Perhaps it is best if we begin by giving the example of how the classical Coulomb's Law of electrostatics and the Biot-Savart Law of magnetostatics are modified by the inclusion of fields that are taken to be measured between centres of rotation rather than directly between charges. In this case both the electric field and magnetic field should have their own unique centre of rotation.

In the SFT formulation for the hydrogen atom the result of this change in metric is that the fields are spinorial, they spin. The fields are

SFT electrostatics eqnSFT version of Coulomb's Law

SFT emagnetostatics eqn     SFT version of Biot-Savart Law

where the form is both relativistic and electromagnetic rather than the electrostatic form of Coulomb’s equation and magnetostatic form of the Biot-Savart equation. The SFT forms are compatible with the classical electrostatic and magnetostatic forms where the orbital and cyclotron frequencies of a particle are assumed to be ωo=0 and ωc=0. In all cases a finite circulating current is required at  ro≠0 and rc≠0.  Both field forms, the classical and the SFT forms, are identical when a time average is taken for the complex exponential term .

So what difference does this make to us in the real world? Perhaps the effects are subtle and not really a major revision, perhaps calling it a tiny revision is more apt?

Well no, we need to be careful about what we are seeing at the microscopic level and below. Take the following example where an experimental asymmetry in Maxwell’s electrodynamics and a theoretical asymmetry in Maxwell's equations were put down to something wrong with Maxwell's well-known equations

In fact the 'error' appears not to be in Maxwell's equation but the application to so-called stationary, macroscopic effects. The truth is these 'stationary' effects are not stationary but are moving; the electrostatic and magnetostatic effects are not what they seem. We can only know this by carefully examining the spinorial equations given above and not the macroscopic versions known as Coulomb's Law of electrostatics and the Biot-Savart Law of magnetostatics. This applies too to any presumed macroscopic version of Maxwell's equations; if we look at Einstein's 1905 paper on relativity where he mentions 'Maxwell's electrodynamics' we see he was looking at a macroscopic version of the equations, not the complete picture. We need to redesign the experimental setup and look for these unmeasured types of circular currents at nanoscopic levels.

Historically it was at this precise point in scientific history, after Einstein discussed the asymmetry in Maxwell's equations, that all of science began looking for an alternative way to calculate the various problems besetting science. This was the motivation and origin of quantum theory. If we repeat the asymmetry experiment again we may find Maxwell's equations to be correct afterall. What appears to have been incorrect was thinking EM to be a  macroscopic effect rather than an atomic effect discussed above.  

As we noted in all cases a finite circulating current is required at  ro≠0 and rc≠0. Rather than one moving and one stationary particle a true EM effect equires two moving  particles . In having the magnetic field stationary the second equation (the Biot Savart-like equation) becomes singular and degenerate.

It seems either relativity may be wrong where particles are at rest, or perhaps more physically plausible, in the real world there are no particles at rest; we exist in an expanding universe after the Big-Bang.

The Measurement of Distances in Self Field Theory

There are some revolutionary aspects of Self-Field Theory (SFT); the most important by far is its ‘taxicab’ metric space as compared to the Pythagorean metric of relativity (ds^2 = g_{\mu\nu}dx^\mu dx^\nu.\,). But what is a metric and how is it used within a mathematical formulation? A metric is a way to measure distances, whether a distance in the real observable world, or the distance to be used within a mathematical formulation where the two  may be totally different to each other. For instance it is known that in electromagnetics at the atomic level the electric and magnetic forces act so that a charge (e.g. the proton or the electron) moves in two directions at right angles to each other simultaneously. This is not what appears to happen in the macroscopic world where conglomerates of atoms move in ensemble motions. Thus while a planet appears to move along a geodesic, a minimum distance through space, at the atomic level each electron moves in two orthogonal directions simultaneously; it never moves along the Pythagorean distance.

What is Space-Time in GR?

Before we look at the metric space of SFT we first look at how space- time is structured in (general) relativity. In general relativity the metric is a relationship between the space coordinates and that of time. In other words the metric gives a measure of distance in space-time. It is normally a four dimensional space-time being composed of three spatial coordinates and the single time coordinate.

The following extract taken from “On the History of Unified Field Theories” by Hubert F. M. Goenner, University of Göttingen, (Living Rev. Relativity, 7, (2004), 2 http://www.livingreviews.org/lrr-2004-) shows the mathematical elaboration required by GR where time is chosen to be a separated (or isolated) fourth dimension within space-time.

Geometry in GR

The space of physical events will be described by a real, smooth manifold of dimension D coordinatised by local coordinates Xi, Yi and provided with smooth vector fields X, Y,, . . . with components Xi, Yi, . . . and linear forms in the local coordinate system, as well as further geometrical objects such as tensors, spinors, connections. At each point, D linearly independent vectors (linear forms) form a linear space, the tangent space (cotangent space) of. We will assume that the manifold is space- and time-orientable. On it, two independent fundamental structural objects will now be introduced.

Metric Structure in GR

The first is a prescription for the definition of the distance ds between two infinitesimally close points on , eventually corresponding to temporal and spatial distances in the external world. For ds, we need positivity, symmetry in the two points, and validity of the triangle equation. We know that ds must be homogeneous of degree one in the coordinate differentials dxi connecting the points. This condition is not very restrictive; it still includes Finsler geometry (…). In the following, ds is linked to a non-degenerate bilinear form g(X, Y), called the first fundamental form; the corresponding quadratic form defines a tensor field, the metrical tensor, with D2 components gij such that

 gr-metric

where the neighbouring points are labeled by xi and xi + dxi, respectively. Besides the norm of a vector , the “angle” between directions X, Y can be defined by help of the metric:

angle of metric in gr

Geometry in SFT

In SFT, unlike GR, the time coordinate is not a separate fourth dimension but is a periodic motion, usually a rotation, in a spatial dimension orthogonal to another spatial dimension. This is expressed via the imaginary unit circle as a motion moving with constant angular speed via the complex exponential complex time. Thus in SFT space-time is three dimensional, and does not require the elaboration of general relativity, but the requirements of mathematical entities within 3-D complex space. This is an extension of Argand space to three dimensions.

What must be realised is the connection between the complex plane and the real plane. In electromagnetics there is a magnetic field and force and an electric field and force. At its most primitive level Maxwell's equations (the ones given in John Jackson's book Classical Electromagnetics) can be divided into two groupings of two equations; one group belong to the electric field while the other group belong to the magnetic field. Both fields move within space-time as a rotation (or curl), one field diverges proportionally with the charge, while the other field rotates (or curls) and doesn't diverge, all the while both fields link together  in a coupled motion, the EM 'pas de deux' as per Maxwell's equation.  Geometrically this is correct. Not bad, these equations have lasted a century and a half without need for amendment. So it's not Maxwell's equations that need revision, just the way we measure within the equations!

Metric Structure of SFT

In SFT the metric is not Pythagorean; you cannot physically get atoms to move along a hypotenuse. At the atomic level they just don't behave like that. Instead they move along lines of motion as dictated by the forces upon them. Think of a snap to grid system of atomic motions; Microsoft Word® has a drawing system with such a tool.

When we pick up an object, our sense of touch is not sensitive enough to feel the atoms moving not along the hypotenuse, but that's what they do at the atomic level. In SFT we find that gravity is a  form of dielectromagnetics ( a form of electromagnetics that is differentiated). So when a planet moves along a geodesic it actually is not moving along the hypotenuse of a triangle but as a series of tiny atomic steps that simulate a geodesic. This is called the taxi-cab metric. and is well known to mathematics.

In the case of electric fields this is a spinor between two centres of rotation; in the case of electrostatic fields this folds down to the classical Coulomb equation after we time average the rotations inherent in the EM motions of the two charged particles.

In the case of magnetic fields this is again a spinor between two centres of rotation; in the case of magnetostatic fields this folds down to the classical Biot-Savart equation after we again time average the rotations inherent in the EM motions of the two charged particles.

So in reality there is no such thing as a 'static' field; nevertheless we can use such a static approximation in the macroscopic domain, as long as we realize that at the atomic level the actual motions are complicated by the rotations of individual atomic particles. This is an example of the case where what we measure depends on what we presume the physics to be;...which can be tricky as in the case of the asymmetry in electrodynamics discussed by Einstein and outlined above in reference to SFT.

The importance of mathematics: Einstein then remembered that he had studied Gauss's theory of surfaces as a student and suddenly realised that the foundations of geometry have physical significance. He consulted his friend Grossmann who was able to tell Einstein of the important developments of RiemannRicci (Ricci-Curbastro) and Levi-CivitaEinstein wrote

... in all my life I have not laboured nearly so hard, and I have become imbued with great respect for mathematics, the subtler part of which I had in my simple-mindedness regarded as pure luxury until now.