Next: Symmetry in a Up: A discrete model Previous: Properties from discretization

# A unified scalar field

An ambitious goal for this class of models is to unify all the forces and particles in nature using a single scalar field and a simple rule for describing the evolution of that field. The quantum wave function and the electromagnetic field are identical in this model as they are in the Klein Gordon equation  for a single photon and the classical electromagnetic field equation.

All energy is electromagnetic. This requires some way to construct neutral matter from an electromagnetic field. The Klein Gordon equation for a particle with rest mass presents an additional problem.

This is the classical wave equation with a new term involving the rest mass of the particle. How can it be derived from the same rule of evolution that approximates the classical wave equation? This may be possible if there is a high carrier frequency near the highest frequencies that can exist in the discrete model. The Schrödinger wave equation for particles with rest mass would represent the average behavior of the physical wave. It would be the equation for a wave that modulates the high frequency carrier. The carrier itself is not a part of any existing model and would not have significant electromagnetic interactions with ordinary matter because of its high frequency.

Such a model may be able to account for the Klein Gordon equation for a particle with rest mass. A high frequency carrier wave will amplify any truncation effect. Because of this the differential equation that describes the carrier envelope is not necessarily the same as the differential equation that describes the carrier. If the carrier is not detectable by ordinary means then we will only see effects from the envelope of the carrier and not the carrier itself. The minimum time step for the envelope may involve integrating over many carrier cycles. If round off error accumulates during this time in a way that is proportional to the modulation wave amplitude then we will get an equation in the form of the Klein Gordon equation.

The particle mass squared factor in the Klein Gordon equation

can be interpreted as establishing an amplitude scale. The discretized wave equation may describe the full evolution of the carrier and the modulating wave that is a solution of the Klein Gordon equation. However, since no effects (except mass and gravity) of the high frequency carrier are detectable with current technology, we only see the effects of the modulating wave. No matter how localized the particle may be it still must have a surrounding field that falls off in amplitude as . It is this surrounding field that embodies the gravitational field.

If discretization is accomplished by truncating the field values this creates a generalized attractive force. It slows the rate at which a structure diffuses relative to a solution of the corresponding differential equation by a marginal amount. Since the gravitational field is a high frequency electromagnetic field it will alternately act to attract and repel any bit of matter which is also an electromagnetic field. Round off error makes the attraction effect slightly greater and the repulsion slightly less than it is in solutions of the continuous differential equation.

Because everything is electromagnetic in this model special relativity falls out directly. If gravity is a perturbation effect of the electromagnetic force as described it will appear to alter the space time metric and an approximation to general relativity should also be derivable. It is only the metric and not the space time manifold (lattice of discrete points) that is affected by gravity. Thus there is an absolute frame of reference. True singularities will never occur in this class of models. Instead one will expect new structures will appear at the point where the existing theory predicts mass will collapse to a singularity.

Next: Symmetry in a Up: A discrete model Previous: Properties from discretization
Mountain Math Software