The possibility that the property of mass and the property of space-time curvature are derivable from the free particle plane wave solutions of the Dirac equation is explored. Presenting the Dirac equation and its plane wave solutions, the effects of a small outside force acting on a particle are examined in part 1. By examining the energy-momentum four vector, the dispersion relationship between changes in energy and momentum of the wave solutions is derived. It is shown that the property of mass is equivalent to dispersion of the plane wave solutions.

In part 2 the effects of a small change in wave velocity and its relationship to space-time curvature are examined. It is shown that where there is a gradient in the wave velocity the total energy remains constant therefore no work is done. The dispersion relationship shows that the mass (or rest energy) is no longer invariant. It is a function of the wave velocity and decreases as the particle accelerates in the direction of steepest decent along the velocity gradient.

The Schwarzschild metric is used to show that wave velocity is a function of the polarizability of the Dirac wave solutions. Polarizability is represented by the orientation of the four component column vectors (Spinors) that describe the spin of the wave solutions. It is also shown that the dielectric susceptibility of the environment is a function of the number of particles per unit volume and their polarizability. By consistently using the property of wave dispersion, the properties of mass and space-time curvature are shown to be derivable from the Dirac equation.

The Dirac equation is the main reasons for the success of Quantum Electrodynamics. All of the atoms in our evironment are composed of Fermion particles that obey this wave equation. Therefore the unification of QED and General Relativity has the highest probability of being accomplished here. Using relativistic units hbar = c = 1, the Dirac equation is,

. 1

where the partial derivative with respect to the coordinates is taken of the wave function , where is 0,1,2,3. The quantity m is the rest energy and are the gamma matrices which satisfy the commutation relation,

. 2

The matrix is the metric tensor, and the gamma matrices obey,

. 3

where the prime denotes the adjoint matrix.[1]

The plane wave solutions of equation (1) have both positive and negative energy states designated p and n respectively. They are,

. 4

. 5

Where E is the total energy and p is the relativistic momentum of the free particle obeying the energy momentum relationship;[2]

. 6

or

. 7

The functions and are 4 dimensional column vectors called Spinors. There are 2 possible states for each Spinor. For a particle at rest the unit vector is a positive energy state which has a spin directed either up or down along the z axis, is a negative energy state also with spin directed either up or down along the z-axis. For now only consider a positive energy particle, independent of its spin such that these Spinors have unit magnitude. [3]

and,

Part 1: Wave Dispersion as the property of Mass

By inspection, the wave function can be separated into independent functions of time and space , where . The coordinate is time and are the 3 spacial coordinates i=1,2,3. The vector is the momentum three vector. From equation (4) functions T and R are then,

. 8

. 9

Equation (1) then becomes,

. 10

From the Dirac equation the invariant mass of a free particle is determined by the energy-momentum four vector.

. 11

For the positive energy state solution the magnitude of this Spinor equation is,

. 12

Which is the same as equation (7) for the invariant mass.

. 13

Consider a small constant force acting on the free particle wave solution. Small enough such that the plane wave solutions are still approximately valid. Since the force is constant the magnitude of change in energy must equal the change in momentum such that the invariant mass of equation (13) remains unaffected by the work being done. The change in the energy-momentum four vector must obey the relationship,

. 14

The invariant mass can be described by the Compton wavelength and therefore has an associated wave vector . This term arises naturally in the Heisenberg equations for the of motion of the free particle solutions. It is this term that Dirac called the "Ziterbewegung", or "trembling motion". It is an intrinsic oscillation of the wave solution on the order of magnitude of the Compton wavelength or rest energy of the particle. An electron is a point-like particle. Over distances less than the Compton wavlength, the corresponding momentum spread allows for pair production. It can be thought of as a virtual oscillation between this positive energy electron and a negative energy electron from the "Dirac Sea". [3] It gives the point-like particle an effective physical wavelength extension in space. (See Milonni.) [4]

The Compton wave vector and the momentum wave vector combine to create an effective wave vector magnitude.

15

. 16

The "sensation" of mass or "inertia" is the reaction force to the applied force as the object is accelerated from p to p+dp. The work required to accelerate a particle is the same as the energy required to increase its frequency, or increase the effective wave vector from k to k+dk, and thereby contract its effective wavelength. Thus inertial mass or resistance to acceleration is the result of wave dispersion.

Historically Lorentz covariance arises because the electromagnetic waves described by Maxwell's equations are Lorentz invariant. The speed of light is a constant in vacuum because the wavelength and frequency of the light obey the dispersion relationship . They are inversely proportional to each other and are observed to be functions of the relative velocity between the source and observer, as illustrated by the relativistic Doppler shift. Similarly the free particle solutions of the Dirac equation obey the same dispersion relationship, equation (16) where is not equal to zero. The change in frequency is still inversely proportional to the change in the effective wavelength. Therefore Lorentz contraction and time dilation as described geometrically by Special Relativity and Lorentz invariance are also described equivalently by quantum mechanical wave dispersion. Changes in the relative frequency and wavelength of the particle account precisely for the observed effects of relativity.

Once again consider equations (8) through (13). Let there be a small constant gradient in the wave velocity such that is a slowly varying function of position only, and the plane wave solutions are still approximately valid. The function is a function of time only, so it is independent of the change in wave velocity which is a function of position only. This means that the energy E of the free particle is constant in time, so no work is being done. Work is defined as the integral of the force tangent to the path, and the change in energy along that path. So there are no "forces" acting on the wave solution.

Only the function is a function of the changing wave velocity. Since the velocity gradient is constant, the magnitude of change in the momentum must be balanced by an equivalent change in the rest energy of the particle, which is a function of c, (mc^2). This is required in order for the energy E to remain constant in equation (8). The rest energy is now to be considered a function of position while the total energy is held constant,

Using equation (14),

. 17

and once again expressing the total energy as the magnitude of an effective wave vector,

. 18

then in order for the total energy E or the effective wave vector to remain constant, the change in the magnitude of the momentum wave vector must be equal and opposite to the change in the Compton wave vector.

. 19

This is applied by using the Schwarzschild metric for the invariant space-time interval ds,

. 20

It describes the invariant interval between events as measured in a spherically symmetric, static space-time curvature. This is the simplest solution of Einstein's equations. The quantity M is the mass of the spherical central body and r is the radial distance from its center. Considering only time and the radial coordinate, while holding the angular coordinates and constant, this metric reduces to,

. 21

One of the experimental predictions of General Relativity is the gravitational red shift. Spectral emissions of light emitted from an atom near the central mass, as observed at a greater distance from the central mass have increased wavelength. This means a lower frequency as compared to light emitted by an identical spectral emission locally at the observers position. The spectral lines are said to be "red shifted". The frequency of a photon as a function of distance is given by the metric to be,

. 22

where r>R. The photon was emitted at the radial coordinate R and is observed at the coordinate r.

For a photon the wavelength and frequency obey the dispersion relationship,

. 23

Due to the experimental evidence in favor of Special Relativity it is believed that the wave velocity is the same constant c=1 everywhere. However the dispersion relationship, equation (19) shows that the wavelength of typical particles that comprise rulers and such devices are a function of the wave velocity, so the length of the ruler depends on where you use it to take measurements. By using objects comprised of particles that obey the dispersion relationship, measurements of the local speed of light will always be constant since an objects length is directly proportional to c.

According to Maxwell's electromagnetic wave equation, the permeability and permittivity of free space are what determine the speed of light.

. 24

They are constants in free space, but in general are derivable from the electromagnetic properties of the local environment, which will be discussed later in this paper. Permeability has units of inductance per unit length and permittivity has units of capacitance per unit length. Equation (23) then gives for the frequency.

. 25

Where the function L(r) has units of inductance and the function C(r) has units of capacitance. By inserting equation (25) into equation (22) the equivalent solution for the gravitational red shift is,

. 26

The wavelength and frequency are functions of the energy state of the atom emitting the photon. So from this the Schwarzschild metric can be written in terms of the inductance and capacitance per unit length of the environment as seen locally by an atom emitting the light at the coordinate R relative to an identical atom and an observer in the environment at r. Consider these atoms to be "neutral test particles".

. 27

As an analogy consider an electronic circuit where the values for the inductors and capacitors are determined by the current and charge distribution, and the dielectric susceptibility of the materials used in their construction. The Quantum environment has similar properties. The objects that fill our environment are the sources of the Dirac and Maxwell fields. The sources as well as the fields are comprised of polarizable particles, or wave solutions.

Consider just the free particle wave solutions of the Dirac equation, equations (4) and (5). It was stated that the Spinors and represent the possible orientation and rotation of the spin angular momentum. There are four possibilities. A positive energy particle can have spin "up" or "down" along the z-axis. The same is true of a negative energy particle . However in the absence of an electromagnetic field there is nothing in particular that determines the direction of the spin axis .

For instance, the alignment of the spin vector of a positive energy particle along the z-axis is the direction of the magnetic field vector, where the spin axis vector points from South to North in the magnetic field. Iron becomes magnetized when the spin of the outer orbital electrons all line up along the same axis. The number of identical particles whose spin is polarized along the same axis determines the magnetic field strength. Likewise the number of identical particles with the same sense of rotation about the spin axis determines the electric field strength. Positive energy particles obey the right-hand rotation rule whereas negative energy particles obey the left-hand rotation rule. These are the properties of linear and circular polarization of the wave solutions.

However in the absence of an electromagnetic field the orientation of the spin axis is random. This gives the Dirac solutions the property of polarizability necessary to represent the inductance and capacitance of the field. In all wave media the permeability and permittivity determine the speed of light through it. In this case it is the polarizability of the Dirac Spinors in the environment that determine the permeability and permittivity of the environment.

The dielectric susceptibility is a non-linear function of the polarizability of the Spinors. Plotting the strength of the source field vs. flux density, the B-H curve of an iron core inductor and the E-D curve of a dielectric filled capacitor are both prime examples of the non-linear behavior of the materials used in these devices. The permeability and permittivity are functions of the dielectric susceptibility. In general the susceptibility of a dielectric depends on many variables such as the number of particles, the percentage of polarized particles, their temperature and their kinetic energy. Similarly Einstein's field equations are also non-linear, however these similarities are outside the focus of this paper and will not be explored.

The permeability and permittivity of the environment determine the analogous "values of the electronic components" as seen by a free particle wave solution. They are functions of the density of material particles in the environment as is the Schwarzschild metric. In relative units where , the permeability is a function of the magnetic susceptibility of the environment, .

. 28

The susceptibility , is a function of the magnetic moment , and the density of particles per unit volume as a function of position, .

. 29

Likewise permittivity is a function of the electric susceptibility of the environment, .

. 30

The susceptibility , is a function of the electric dipole moment , and also of .

. 31

The wave velocity as seen by a free particle is a function of the dielectric susceptibility of the environment as shown by equation (24), which in turn is a function of the polarizability of the particles it contains. The magnetic moment and the electric dipole moment are identical in that each is proportional to the linear separation between opposite charges. Similarly the Dirac wave solutions have a linear separation between opposite peak amplitudes equal to half the wavelength, . It has previously been stated that mass has units of inverse length and is the magnitude of the energy-momentum four vector. So in a static region where all particles have p=0, the mass density is proportional to the Compton wave vector and the number of particles per unit volume. The total mass is the superposition of all the particle wave solutions of the Dirac equation in the local environment.

Let be the relative number of particles per unit volume of mass , near to the origin of coordinates, at a distance from a test particle under observation, (r >> 0). In relative units L(R)C(R)=1, the relative susceptibility at a distance r is proportional to the central mass.

. 32

This can be expressed as a change in the relative potential between the radius r and the surface of the central mass at R.

. 33

Note as r approaches R the term 2M/r goes to zero. So according to the metric equation (21) space-time is flat locally, even on the surface of a black hole! As 2M/r approaches 1 the value of R/r approaches zero and you have gravitational collapse into a point-like Black Hole, as observed from the coordinate r. If the Black Hole is considered to be a single particle of mass M, where the radius of the particle R is equal to half the wavelength,

. 34

then as M approaches infinity the value of R asymptotically goes to zero. Once again equation (34) is the dispersion relationship. It shows that the radius of the event horizon of a Black Hole beyond which light emitted will not reach an observer at r, asymptotically goes to zero as observed from the coordinate r. This is consistent with Special Relativity where infinite energy is required to exceed the speed of light. Now infinite energy (zero wavelength) would also be required to create an event horizon with an escape velocity faster than light. The Schwarzschild solution is then seen to be equivalent to gamma in Special Relativity.

. 35

General Relativity is interpreted such that it predicts Black Holes exist with only a finite amount of mass. By using the dispersion relationship the wavelength becomes twice the radius of the event horizon. Since the wavelength collapses to zero as mass goes to infinity, the speed of light is never exceeded.

It was shown that the mass of a particle is the reaction force caused by wave dispersion. The effective wavelength is inversely proportional to the energy of the particle, and directly proportional to the wave velocity. Hence the length of an object is directly proportional to the speed of light. It was also shown that wave velocity as seen by a free particle solution is a function of the density of polarizable particles in the local Quantum environment. From this it can be argued that the revised metric equation (27) is no different than the Schwarzschild metric (21), and the "Dispersion Interpretation" presented here is no different than Minkowski's "Geometrical Interpretation" of Relativity Theory. The theory fits the observed data while giving a vivid account of the quantum mechanical "reasons" which lie at the heart of Relativity Theory. By consistently applying the principle of wave dispersion the intrinsic properties of mass and space-time curvature are shown to be complimentary aspects of the dispersion phenomena. The results lead to an interpretation of the electromagnetic field as the polarization strength of the Quantum environment, and an interpretation of the gravitational field as the polarizability of that environment.

The evidence for gravitational red shift is well known. As it is used here, it is consistent with the field equations of both Maxwell and Einstein. Likewise magnetization in materials such as iron shows us that the polarization of spin axes and the rotation about them determines the strength of the electromagnetic field. As a result of the Dirac equation and the dispersion interpretation, a quantum mechanical basis for a formal theory that unifies Quantum Electrodynamics and General Relativity is shown to be possible in the "weak" linear region of the field.

In the dispersion interpretation presented here gravitation has a fundamental relationship with the random orientation of the spin vectors, and therefore its non-linear relationship to entropy, kinetic energy and polarization needs to be explored further. The Klien-Gordon equation and the electromagnetic field were outside the focus of this paper, however they too may lead to similar results when the polarizability of scalar and Maxwell fields are considered. The possibility of deriving Einstein's field equations, the geodesic equation or the effects of strong fields from the dispersion relationship were also not discussed. Nor was any notion of "second quantization" of the field. However the metric equation for the dispersion intepretation is derived in the Appendix. There is much more that needs to be done to show the dispersion interpretation is equivalent to General Relativity. The most astonishing result of the dispersion interpretation is that the speed of light is shown to be constant locally only, and is a variable function of the environment over global distances. The implications of this may lead to new and exciting technologies.

Thanks to John Anderson for debating the Twin paradox over and over at different levels until I could grasp the subtleties of the experiment. Those arguments helped me to formulate this paper clearly. Also thanks to Jon Kemper and Richard G. Harrison for their revisions and comments.

A special thanks to Steve Carlip for pointing out a major error in my notation on 8/19/98.

P. W. Milonni The Quantum Vacuum; and Introduction to Quantum Electrodynamics Academic Press, 1994. Pages listed below, or go to my Dirac page.

1. 306-307

2. 308-309

3. 310-311

4. 322-323

Appendix A: Derivation of the Refractive Metric Tensor

Copyright 1998 Todd Desiato