… a great step would be made when we should be able to say of electricity that which we say of light, in saying that it consists of undulations.
Sir George Gabriel Stokes, 1879
What physical properties would a universe have if governed by classical physics with a Galilean space-time? The answer depends on what type of mechanical system makes up this hypothetical universe. Let us consider the simplest possible model: suppose that all of space is (or is filled with) an ideal elastic solid. Disturbances of equilibrium can yield transverse and compressional waves. Historically this model was used to derive the classical properties of light. Variable density results in wave refraction. If anything resembling matter can exist in such a medium, it must take the form of standing or soliton waves whose energy is localized within a limited region.
If matter itself consists of waves, then it is not possible to measure the characteristic wave speed. All distance measurements (even with rulers made of soliton waves) would be equivalent to wave propagation time measurements. The wave speed can be nothing more than a conversion factor relating units of distance and units of time. This is the key feature of Special Relativity: that the speed of light is simply a constant factor. Our standard unit of length, the meter, is defined as the distance light travels in 1/c seconds, where c is called the 'speed of light'. Since classical matter must be described by a wave equation, its description is Lorentz invariant. Measurements by different observers are related by Lorentz transformations. Hence Special Relativity is consistent with a classical universe. For more detail see Chapter2 of The Classical Theory of Matter Waves.
Next, suppose that the soliton waves which compose matter are transverse, rotational waves, just as light consists of transverse waves. Suppose also that these waves alter the medium so as to decrease the wave speed in their vicinity (increased density or decreased elasticity). Waves propagating in a region of decreasing wave speed will be refracted toward the region of slower wave speed. Hence there will be a mutual attraction, or gravity, between matter waves in a classical universe. This interpretation of gravity is similar to General Relativity, which also features a decreased speed of light in the vicinity of matter. For more detail see Chapter4 of The Classical Theory of Matter Waves.
Matter in a classical universe would have other wave characteristics. Spatial localization of a wave increases the magnitude of gradients, which are proportional to wave number and frequency. Hence waves satisfy a classical uncertainty principle which puts a lower limit on the product of spatial localization and wave number localization. The classical wave uncertainty principle is in fact completely equivalent to the quantum mechanical uncertainty principle.
A mathematical description of classical waves must be based on the wave equation. In one dimension, solutions of the wave equation consists of forward and backward waves. These two solutions are separated in space by rotation of 180 degrees. If two independent components are 90 degrees apart they have spin of one. If two independent components are 180 degrees apart they have spin of one-half. Therefore matter waves in a classical universe are described by spin one-half wave functions. Fermions, the fundamental 'particles' in the Standard Model of Particle Physics, are also described by spin one-half wave functions. For more detail see Chapter3 of The Classical Theory of Matter Waves.
Quantization could arise in a classical universe due to wave matching conditions for soliton solutions. Angular periodicity yields quantization of angular parameters. Quantization of wave amplitude can result from nonlinearities.
It is therefore clear that many characteristics of matter are in fact predicted by classical physics. These characteristics include wave-like propagation of matter, Special Relativity, gravity, the uncertainty principle, and spin one-half wave functions. What is not yet clear is whether or not the vacuum can actually be modeled as an ideal elastic solid, or whether some more complicated model is necessary to yield correct quantitative predictions.
Created: 27 February 2006; Last updated: 25 April 2008
Copyright © 2006-2007 Robert A. Close