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    Gauge Theory, Maxwell's equations, and the Maxwellian
    By Tony Fleming | June 27th 2012 01:31 AM | 1 comment | Print | E-mail | Track Comments
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    There is an important difference related to the mathematical formulation of Self-Field Theory (SFT) and that is the complete absence of gauge. We know from the various quantum theories that gauge plays a crucial role. To understand this difference we must first look at the basics of gauge theory. And to look at the basics of gauge theory we must first define some terms we have already used in discussing SFT. First let’s define Maxwell’s equations.

    <!--[if gte vml 1]> <![endif]--><!--[if !vml]-->_                    (1a)                                                                             

    <!--[if gte vml 1]> <![endif]--><!--[if !vml]--><!--[endif]--><!--[if gte mso 9]> <![endif]-->                      (1b)

    <!--[if gte vml 1]> <![endif]--><!--[if !vml]--><!--[endif]--><!--[if gte mso 9]> <![endif]-->    (1c)

    <!--[if gte vml 1]> <![endif]--><!--[if !vml]--><!--[endif]--><!--[if gte mso 9]> <![endif]-->    (1d)

    If we just look at the equations above (1a-d), Maxwell’s equations (the inhomogeneous Maxwell's equations), then we have a basis for examining gauge in various systems. The reason being, as they stand, these equations cannot be solved uniquely. Since all equations involve differentials (without the actual variable) there is no definitive form of the equations. In other words, there is a family of solutions, all isomorphs of each other; we do not have a unique solution. We can add any constant we like to them and this will be part of the overall family. Now this is the basis of gauge theory with its symmetries. In terms of SFT, two arbitrary spinors could be a solution as long as they obey the Maxwell equations in (1a-d) above; the solutions 'float' about constants of integration. There is a freedom to choose the constants anyway we like.

    So what is a Maxwellian

    In addition to the inhomogeneous Maxwell equations where 

    becomes (1a) and becomes (1d)

    we use the Lorentz equation that describes the field forces acting on the particles is written as

    <!--[if gte vml 1]> <![endif]--><!--[if !vml]--> (1e)

    where the constitutive equations in free space are
    <!--[if gte vml 1]> <![endif]--><!--[if !vml]--><!--[endif]--><!--[if gte mso 9]> <![endif]-->                      (1f)

    <!--[if gte vml 1]> <![endif]--><!--[if !vml]--><!--[endif]--><!--[if gte mso 9]> <![endif]-->.                      (1g)

    the relationship between the speed of light <!--[if !supportFootnotes]-->[i]<!--[endif]--> and the ratio of the fields

    <!--[endif]--><!--[if gte mso 9]> <![endif]-->                      (1h)

    and finally the atomic energy density per volume is

    <!--[if gte vml 1]> <![endif]--><!--[if !vml]-->(1i)

    which depends upon the E- and H-fields in the atomic region. Equations (1a–i) are termed the Maxwellian, or sometimes the EM field equations.<!--[if !supportFootnotes]-->[ii]<!--[endif]--> In these equations, v is the particle velocity, m is its mass. It is assumed that the volume of integration vn over which the charge density is evaluated, and the area the charge circulates normal to its motion Sn, are calculated during successive periods over which the internal motions of the atom take place. 

    In the Maxwellian, (1a-i), where we use the inhomogeneous forms for (1a) and (1d) there is no gauge because now the the fields are completely defined. We have a unique system of equations. There is no family of isomorphs because we have tied the bispinorial solution down; it is no longer a floating family of similar shaped solutions, but a single unique solution. 

    This is a crucial difference between quantum theory and SFT, a complete absence of gauge, basically due to the use of the Lorentz equation within SFT along with the other defining equations (1e-i). That said, gauge theory has been a very important tool when searching for links between the forces found within nature, for instance when it is used within the Standard Model of particle physics. The Maxwellian has an analog within quantum theory and that is the Lagrangian.

     [i] In SFT, the speed of light is not proscribed from being variable.  Depending on the energy density of the region being studied, and the photon state, c can vary.  

    [ii] Where a nebular current density is used in (2.1d), the factor 4p comes about from an application of Green’s theorem leading to a surface over the volume enclosed by the charge density.  For the case of discrete charges, the factor p represents the area enclosed by the moving charge point.                                                             


    Tony Fleming
    As you see Maxwell does not hold Einstein's postulate in reverence and awe and I suggest any modern scientist should think about Albert's postulate deeply from a Maxwell's equations viewpoint. (For a start he died before Albert's work) For instance the speed of light within the body is NOT the speed of light we know as c since the energy density is different to that of free space. Again in the inflation era just after the Big Bang, there is a superluminal speed of expansion; this is known as the inflationary period. Where does this superluminal speed come from if you hold to Einstein's postulate about the invariance of the speed of light?

    Albert's postulate about the invariance of the speed of light was an assumption, nothing more, nothing less. Yes it fits the postulates of relativity but there's no other cosmological reason based on physics. To my mind it's an observation of the local region around our part of the Universe. The speed of light as shown in the Maxwellian depends on the energy density at any point in space and this includes the regions where the weak and strong nuclear forces hold within the atom.

    Let me put in a footnote I've just been writing tonight about Newton's gravity, quantum theory, GR; it seems appropriate. Read it and see if you can understand it. 
    Newton empirically determined the gravitational force as a coupling between masses, a mutual interaction similar to SFT. He obtained the inverse square form of the gravitational field first via parabolic calculus. Newton reasoned that if a cannon ball was projected with the right velocity, it would travel completely around the Earth, effectively forming an orbit. A particular gravitational fieldwould lead to a period of revolution. He then validated his results using observations of the Moon around the Earth and the planets around the Sun.The gravitational constant was eventually measured directly by Cavendish in1796. The mutual fields between the two masses are equal and opposite.
            Einstein expanded Newton’s concept of gravitation via the use of a relativistic Lagrangian. This resulted in a theoretical form of gravity that could be used in exotic regions of the Cosmos, not just Newton’s ‘gravitostatics’. In GR the Hilbert action <!--[if gte vml 1]> <![endif]--><!--[if gte mso 9]> <![endif]-->yields   Einstein's field equations through the principle of least action. While Einstein’s equations are called ‘field’ equations they are based on wave equations and written in terms of potential components. In Einstein’s formulation the ‘field’, in reality the potential around a single particle isdetermined via the curvature of space-time.
            As with Newton’s law, SFT obtains the mutual effect between two particles via the equations known as the Maxwellian(see Appendix A on the difference between Maxwell’s equations and the moredefined Maxwellian that includes the Lorentz equation). SFT is thus based onfield equations and not potential wave equations. Like Newton’s gravity SFT isa mutual effect. As with GR, the various quantum field theories are based on waveequations that seek to determine the potential around a single particle. Arestatement of GR and quantum theories as mutual effects between particles can be applied via the mathematics of SFT. The result of such a reformulation isthe replacement of the Lagrangians that involve wave equations of order 2 by Maxwellian equations in the electric and magnetic field variables of order 1. Using the Maxwellian field equations compared with the Lagrangian potential equations induces a significant decrease in complexity. Practitioners and students of quantum theory and GR will attest to the degree of difficulty of these computational methods.
    The way SFT is a mutual description of particle - field interaction is another reason why we can talk about a possible 'cosmostasis'. What we are saying is that just like the hydrogen atom which is a model of the dynamic equilibrium between two particles, the electron and the proton, and the fields between them, can we model the Universe in the same way that provides a system that comes to an equilibrium over time? This gives us more room to compute than previously where we considered the solutions as either flat, open, or closed and we spoke of a critical density; now we also have a possible equilibrium state.

    Tony Fleming Biophotonics Research Institute tfleming@unifiedphysics.com