James G. Gilson , Relativity, Stochastic Schrödinger Quantum Wave Mechanics, Fine Structure Constant

A short account of my research work in mathematical physics and its background motivations

Motivations

I first came across relativity in the pre-war years in the form of Einstein's famous book The Meaning of Relativity. My fascination with the subject started then but was fuelled and greatly enhanced by remarkably clear, detailed and inspiring lectures on the subject in London in the fifties by Wolfgang Rindler. The second edition of his classic book on the subject Relativity, Special, General and Cosmological was published in 2006 under the new title Relativity, Special, General and Cosmological, Second Edition, a book that had its origins in those brilliant lectures of the fifties. The type of problem that I found most interesting concerned the question of how can paradoxical features real or apparent be resolved or at the very least be clarified? My first concern was with relativity and my next concern was with my second major interest quantum mechanics, the latter subject seemingly abounding with quite terrible inconsistencies brought out by the recognition of paradoxical features such as the Einstein Podolsky Rosen difficulty with information transmission. As far as quantum theory was concerned, I suspected that most of the difficulties stemmed from its apparent departure from the firmly established probabilistic subject stochastic theory in a rather drastic way. This departure arises from the linear superposition principle of quantum theory amplitudes, quantities involving the imaginary unit i, complex quantities, rather than the well understood real positive probabilities of classical theory. Stochastic theory had assumed a definitive form as a result of work by M.S.Bartlett and was described in his book The theory of Stochastic Processes. During my time in the mathematics department at University College London, it was suggestions by M.S.Bartlett that set me off on the search for a sound stochastic basis for quantum mechanics again with the hope of resolving paradoxical aspects. It was the pursuit of this objective that eventually led me to a geometrical description of a quantum state which was sufficiently relativistically orientated so that it could be used to predict a numerically exact value of alpha, the fine structure constant. Many researchers had come to believe that quantum theory is driven by a new logic of its own, very different from the logic generally believed to control the behaviour of the earlier studied so called classical dynamical systems. This view is directly a consequence of the interpretational difficulties in the linear superposition of amplitudes. I thought the idea that a new logic was needed was mistaken so that in the paper Stochastic Simulation of the Three Dimensional Quantum Vacuum (gil0.pdf.), I produced a method of generating the Schrödinger equation from purely classical assumptions about the fluid flow of the local state of the polarized vacuum and used a probability argument to obtain the usual quantum linear state superposition principle. This also gave a means of deriving Born's adhoc quantum probability bilinear Ψ*Ψ structure rigorously from probability considerations. Thus the following proposition seemed inescapable. If there are real paradoxes (inconsistencies) in orthodox Schrödinger quantum theory, then the same inconsistencies must be present in classical fluid dynamics from which it could be derived. This throws a new light onto the possible quantum paradox issue. Such aspects are in the domain of interpretations and philosophy and may be regarded as esoteric. However, some very practical and measurably confirmable results have also come out of this new formulation of the quantum theory. These new developments result from the identification of the basic vacuum polarization units on which the vacuum statistics is built. They will be explained in the next section

Vacuum Elements

The Stochastically based Schrödinger equation is essentially a classical probabilistic structure built about or describing the behaviour of assemblies of vacuum polarization oscillators. The energies of the oscillators involved are indicated by a set of subscripted integers (i). These oscillatory elements are composed of pairs of the oppositely signed monopolar elements associated with the same energy (i=j) or different energies (i≠j) interacting and performing a dance on circular orbits which generates a local angular momentum the mathematical form of which can be found from theory. This same angular momentum is shown to be proportional to the usual quantum probability density functional form Ψ*Ψ which is also identified as a true probabilistic expectation value associated with a quadratic probability distribution given by integer products (n_in_j). The complex conjugate factors in this product are then easily individually identified as a true probabilistic expectation values based on the corresponding linear probability distribution (n_i). This last expectation value is the usual quantum linear superposition principle for the complex amplitudes of quantum mechanics. See gil0.pdf..

The Wave Capture Diagram

A kinematical geometrical image for an individual member of such an assembly can be regarded as a phase space path such as in the wave capture diagram on the right, where χ_N =π/N. Essentially, the diagram represents a system involving a point or some physical marker attached to the boundary of a spinning disk. The angular velocity ω of the disc is its most important characteristic. Thus we can define our oscillator units so that they always have an outer radius L₀ terminating where moving points are at the maximal speed for a physical object, L₀ = c/ω. This, of course, is the largest radius that such a rotating physical unit element can have without violating relativity. These units can have inner structure involving other points on the disks as shown in the wave capture diagram. It turns out that this diagram with its conceptual characteristics is a very powerful geometrical tool for the analysis of the couplings between quantum systems and for finding the actual numerical values for the coupling constants. More about this aspect on page 3.