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KEY WORDS: Dirac equation; electrons; Hamiltonian formalism; spinor factorization; classical field theory; parity

1. INTRODUCTION

In classical electrodynamics the electron is generally regarded as a point-like particle. This view became untenable in 1927 when de Broglie’s hypothesis⁽¹⁾ that matter behaves like waves was confirmed in electron diffraction experiments by Davisson and Germer,⁽²⁾ and independently by Thomson and Reid.⁽³⁾ However, the quantum mechanical equations developed to describe these ‘matter waves’ are first order equations rather than classical second-order wave equations. Although these waves are commonly interpreted as probability waves, the quantum mechanical Dirac equation is also fundamentally a deterministic equation for the evolution of angular momentum density and other physical observables. As such, it should correspond to classical wave theory. Others have reformulated the Dirac theory in terms of deterministic relations between local physical observables.⁽⁴^,⁵⁾ However, these investigators did not construct a corresponding classical wave theory describing evolution of a field variable entirely in terms of its own derivatives.

1.1. Factoring Wave Equations

The Klein-Gordon (or relativistic Schrödinger) operator can be factored into a product of two Dirac operators acting on the wave polarization (or amplitude) a:

(1)

where the gamma factors are related to the Minkowski metric g^mn by:

(2)

The factors g ^m have traditionally been regarded as matrices. However, they can also be interpreted geometrically using multivariate vectors.⁽^5-⁶⁷⁾ The wave polarization a is a classical 3-vector in Galilean space-time. The Minkowski metric of relativity is introduced through the operators and applies to the space of measurements made with waves.

If we define a wave function:

(3)

then the resultant first-order Dirac equation is equivalent to the original Klein-Gordon equation:

(4)

In the above case the two Dirac operators have different sign for the mass term. Rowlands⁽⁸^,9⁾ and Rowlands and Cullerne⁽¹⁰⁾ used a combination of multivariate 4-vectors and quaternions to write the Dirac equation in a nilpotent form in which the two successive Dirac operations are identical. This formulation yields an elegant classification of particle states within the Standard Model.

1.2. Parity Conservation

Historically, parity conservation was a fundamental assumption of physics. Any physical bias toward right- or left-handed processes would be completely arbitrary and therefore unjustifiable. However, Lee and Yang⁽¹¹⁾ proposed that weak interactions may violate parity, and experiments by Wu⁽¹²⁾ demonstrated that beta decay exhibits left-right asymmetry. This asymmetry has been interpreted as implying parity violation, although Lee and Yang mention that their theory could be consistent with parity conservation if protons are not identical (within a rotation) to their mirror images.

If Wu’s experiment could be constructed using antimatter, current theory predicts that such an experiment would behave exactly like a mirror image of the original. This property is called “PC” conservation. Even the decay of neutral kaons, which supposedly exhibits PC violation, is predicted to yield the same experimental results after PC transformation (the supposed PC violation is attributed to a temporal change in parity, not to an asymmetry of the PC transformation itself). The simplest interpretation of the experimental observations is that matter and antimatter are mirror pairs. Therefore a reformulation of the Dirac parity operator is necessary. The logical arguments for the new parity operator are as follows:

First, Lorentz invariance is a property of waves (and measurements made exclusively with waves) propagating in Galilean space-time. Second, the Dirac equation is a wave equation, as evidenced by double application of the time derivative operator. Therefore Dirac waves may be assumed to propagate in a Galilean space-time, and the squared coefficients of temporal and spatial derivatives may have the same sign (in the conventional derivation, it is mistakenly assumed that Dirac waves propagate in a Minkowski space-time requiring squared coefficients of temporal and spatial derivatives to have opposite sign). Third, take the Dirac Hamiltonian operator for a free particle as:

(5)

where s is the matrix operator for spin and has units of frequency. There exists a right-handed set of basis vectors (g⁵, g⁰, ) with commutation relations equivalent to those of Pauli matrices (is the unit imaginary pseudoscalar). These represent directions relative to velocity (cg⁵s) and should all be inverted by the parity operator. The conventional parity operatorinverts only two of these (i.e. ). The correct parity operator is:

(6)

This parity operator inverts all three direction matrices associated with relative velocity (the matrix is inverted because is defined to be a pseudoscalar ). We will see below that this parity operator changes the sign of the electron energy and mass eigenvalues, consistent with an exchange of matter and anti-matter.

1.3. Contribution of Spin to Hamiltonian

A more serious problem with Dirac theory is that the spin angular momentum is not identifiable in the free particle Hamiltonian (5). The angular momentum operator is conventionally taken as:

(7)

The spin term is justified by the fact that conservation of angular momentum requires commutation with the Hamiltonian. However, the Hamiltonian expression for the angular momentum operator is:

(8)

Clearly the spin angular momentum operator should appear explicitly in the Hamiltonian as the coefficient of an angular velocity. Hestenes⁽⁷⁾ has shown algebraically that the mass can be interpreted as the scalar product of spin S and its associated angular velocity :

(9)

Hence the correct functional form of the Hamiltonian is:

(10)

We will see below that for electron solutions, the spin term is in fact equal to the conventional mass term. It is possible that this Hamiltonian is incomplete, but it is sufficient to explain the dynamics of electrons.

1.4. Classical Interpretation

A classical interpretation of Dirac spinors was presented by Close in describing torsion waves.⁽¹³⁾ In this paper we use a classical approach to factoring a wave equation for angular momentum density. The one-dimensional wave equation is generalized to three dimensions under the assumption that the wave polarization and velocity rotate together. This procedure yields a first-order bispinor wave equation. Wave interference is shown to produce both the Lorenz force and the Pauli Exclusion Principle. Angular correlations between different states are also derived.

2. Spinorial Representation of Waves

Consider a scalar quantity (a) which satisfies a wave equation with wave speed (c) in one spatial dimension (z):

(11)

This equation can be factored:

(12)

The general solution is a superposition of forward (a_F) and backward (a_B) propagating waves:

(13)

This form of the solution to the one-dimensional wave equation can be found in any elementary textbook on waves. We can write the equations for forward and backward waves in matrix form:

(14)

The spatial derivatives are related to the temporal derivatives:

(15)

Let and . We now define a wave function in terms of the time derivatives:

(16)

The wave equation for the forward and backward waves is now:

(17)

We have now reduced the second-order wave equation to a first-order matrix equation.

2.1. Spinors and Bispinors

If we regard the z-axis as one of three orthogonal axes, then the two independent components and differ by a 180 degree rotation. This is the definitive property of independent states in spin one-half systems. Unfortunately, this property is de-emphasized in the physics literature in favor of the more exotic property that complex spinors change sign upon 360 degree rotation. This latter property does not apply to physical observables which are computed from bilinear products of spinors. However, the separation of independent states by 180 degrees does apply to wave velocity, implying that solutions of the wave equation generally form spin one-half systems. Note that unlike positive and negative scalars or vector components (which can also be expressed as bilinear products of spinors), waves with positive and negative velocity are not related by a multiplicative factor of minus one. The forward and backward waves are independent states. The mathematical basis of this property is that wave velocity is a property of the functional arguments and is not simply an amplitude.

The relationship between waves and spinors can be made explicit by further decomposition into positive-definite components () or () representing positive (+) or negative (-) contributions to the wave derivatives:⁽¹³⁾

(18)

and

(19)

From here on the functional arguments will not be written explicitly. Note that the positive-definite components may have discontinuous derivatives where the original signed quantities pass continuously through zero. For example, to make the time derivatives continuous requires matching conditions for:

(20)

Similar relations hold for the backward wave components. Such discontinuities do not affect the validity of the first order equations. However, higher derivatives may be undefined at some points.

Since each component has a unique sign, we can express and in spinorial form with the one-dimensional wave function y_v (the subscript ‘v’ refers to the velocity axis):

(21)

(22)

where the superscript T indicates transposition of the column matrix and the matrix bs tabulates the forward and backward velocities (v):

(23)

This wave function is a one-dimensional bispinor. In one dimension the components of the bispinor may be taken to be real and positive-definite. Extension to three dimensions requires complex components.

Changing the order of terms in the wave function is called a change of ‘representation’. A few important points are:

1. The components of the column matrix wave function are real and positive-definite.

2. Only one forward component and one backward component can be non-zero at any given time and place (for one-dimensional waves).

3. The spatio-temporal variation of each component must be consistent with its location in the column matrix.

Since some of the components must be zero, let d_F and d_B be either zero or one. Then the wave function is:

(24)

Using Lorentz boosts, the wave function can be written as:

(25)

This form has two independent continuous parameters and two binary parameters.

The equation of evolution of the wave components is:

(26)

This is the one-dimensional Dirac equation. This equation can be interpreted as a convective derivative with two opposite velocities represented by the matrix v=cbs.

The relation between one dimensional bispinor equations and scalar wave equations is summarized in Table 1.

Table 1. Corresponding Bispinor and Scalar Wave Equations in One Dimension

Bispinor Equation	Scalar Equation

2.2. Three Dimensional Waves

Extension to three dimensions requires rotation of the wave velocity to arbitrary direction, and, for vector waves, rotation of polarization. For the Dirac wave functions, extension to three dimensions is achieved by requiring that velocity and polarization rotate together as follows.

Let the polarization matrix s be one component (s₃) of a vector of matrices (s₁, s₂, s₃):

(27)

Another vector of matrices can also be defined (b₁, b₂, b₃):

(28)

These matrices represent directions relative to the velocity, with b₃ representing the parallel direction. Three velocity matrices may be defined as (b₃s₁, b₃s₂, b₃s₃). This choice corresponds to the chiral representation except for a different sign convention. Rotations of these b-matrices change the representation, because they determine which matrix is associated with the wave velocity direction.

Starting from the one-dimensional wave factorization above, we allow the Lorentz boost to have arbitrary direction:

(29)

The Lorentz boost determines the wave velocity direction and the ratio of forward and backward waves along that direction.

For a given representation, there is still freedom in the orientation of two of the b-axes, which may be defined by a rotation angle z:

(30)

Next, we allow co-rotation of polarization and velocity (using ):

(31)

Since Lorentz boosts can exchange forward and backward waves, and rotations can exchange positive and negative components, the delta-functions are no longer necessary. A general wave can be obtained by transforming an initial wave consisting of forward and backward positive components:

(32)

There are eight free parameters in this factorization, enough to uniquely determine all complex components of the bispinor. For the original Dirac representation, we rotate the basis vectors (b₁, b₂, b₃) to let the velocity matrices be represented by (b₁s₁, b₁s₂, b₁s₃):

(33)

so the factorization is:

(34)

This factorization is equivalent to that of Hestenes.⁽⁶⁾ The time derivative is:

(35)

Now we assume that the first term can be replaced by the spatial derivative as in the one-dimensional case. This yields:

(36)

This is the Dirac equation if the final term is replaced by . However, the above bispinor equation is more generally valid.

2.3. Classical Interpretation

We obtain a three dimensional wave equation by multiplying and adding the complex conjugate:

(37)

The classical interpretations of these terms (which define derivatives of the variable Q) are:

(38)

Compared with the one-dimensional case, each component of the time derivative has the same form, but the spatial derivatives now consist of a gradient plus a curl as expected for a vector wave. Using the relation the corresponding wave equation is:

(39)

This is a classical wave equation for a vector wave with polarization Q which rotates with angular frequency .

If we assume that the wave represents rotational oscillations in an elastic solid, then we can interpret Q as an angular potential and derive the following dynamical variables:

(40)

where S is the angular momentum density of rotations of the medium, is the (transverse) displacement, r is the inertial density of the medium, q is the linear momentum density of motion of the medium, j is the rotation angle (proportional to torque density), and w is the vorticity.

If the vorticity is computed from the angular potential as above, the wave equation (39) is nonlinear, as is the corresponding bispinor equation (36). Such nonlinear equations typically have quantized solutions, and we propose that these soliton solutions correspond to elementary particles. Several investigators have attempted to explain quantization using nonlinear Dirac equations.⁽¹⁴ ^-15 ¹⁶¹⁷¹⁸⁾ This appears to be the first time that the form of the proposed nonlinearity has been derived from a physical model. However, solution of this soliton equation is beyond the scope of this paper.

3. Electron Waves

3.1. Free Electron Equation

The free-particle wave equation for a stationary state with energy eigenvalue E is:

(41)

The Dirac solutions are obtained when the spin term is equal to the mass term:

(42)

The operator can be factored:

(43)

Letting , the two-component angular solutions of the eigenvalue equations and are well known.⁽¹⁹⁾ These two angular solutions are related by and yield opposite eigenvalues of the parity (spatial inversion) operation.

Denote two wave functions as:

or (44)

Each of these is an eigenfunction of the conventional parity operator, but they are exchanged by the correct parity operator:

(45)

The spin energy constraint (42) is satisfied if , since:

(46)

Therefore mass is related to the angular frequency, with the integer g-factors (1 for L and 2 for S) appearing as relative coefficients relating angular velocity to the components of angular momentum.

Using yields the coupled radial equations:

(47)

yields similar coupled equations with the signs of M and E reversed. Therefore the new parity operator is associated with exchange of matter and anti-matter.

3.2. Velocity Rotation and Mass

It is instructive to compute the effect of mass on the wave velocity:

(48)

The mass term represents a radial acceleration of the wave, which is inward provided that the appropriate sign is chosen for M. Hence it is clear that electrons are soliton waves.

3.3. Hamiltonian Formulation

Hamilton’s equations of motion have the form:⁽²⁰⁾

(49)

where y is a field variable and p_y is the conjugate momentum to the field defined by:

(50)

We can fit the bispinor equation to this form by taking the momentum conjugate to the wave function to be:

(51)

and the Hamiltonian is:

(52)

From here on we will remove the factor of one-half and simply discard the imaginary part. The Hamiltonian H will have units of energy if the wave polarization has units of angular momentum.

We can also define a Hamiltonian operator with :

(53)

3.4. Wave Interference and Potentials

Next we investigate the origin of electromagnetic potentials. Certain observables (scalars and vectors) should be additive when two waves are superposed. This implies that when two waves y_A and y_B are superposed, the total wave y_T has the property that:

(54)

for some linear Hermitian operator G. If we simply added the two wave functions, we would have instead:

(55)

The additional terms are clearly not zero in general. However, they can be forced to zero by introducing phase shifts to the wave functions. We will assume that the spin is a linear observable, requiring:

(56)

In particular, this condition holds when either or is an eigenfunction of a particular spin direction, in which case:

(57)

This is the Pauli Exclusion Principle, which results from the assumption of independent waves. In this case a scalar phase shift is sufficient to achieve the cancelation. A scalar phase shift of the wave function represents rotation about the spin axis.⁽⁶⁾ Let s_A be the spin matrix associated with the spin axis:

(58)

and similarly for y_B. We choose the phase shift to cancel this rotation factor:

(59)

where the subscript ‘0’ indicates the unperturbed wave function. Since the purpose of these phase shifts is to cancel interference with the other wave, it is natural to associate the phase shifts as:

(60)

The relative rotation by p could be distributed between the two waves, but we will treat y_A as the ‘test wave’ and y_B as the ‘source wave’. The constant phase shift has no effect on dynamics. Some observables computed from these independent wave functions may differ from those of the free particle wave. For example:

(61)

Hence the effect of wave interference is to change the operator for wave packet from G to :

(62)

Applying this rule to the operators and H yields:

(63)

Substituting the explicit form of the Hamiltonian for the free electron:

(64)

The final term is necessary because the scalar phase shift is interpreted as rotation about the spin axis. The electromagnetic potentials are therefore:

(65)

Note that the curl of A (magnetic field) may be nonzero because f_A is an orientation angle which may be multi-valued. See ref. 21 for a discussion of multi-valued potentials in electromagnetism. Setting , the modified Hamiltonian is the same as in quantum theory:

(66)

Hence electromagnetic potentials result from wave interference under the assumption that different wave packets are independent.

Multiple source waves may be treated sequentially, at least as a first approximation. For a given test wave, make it independent of the first source wave as above. Then take the modified test wave and make it independent of the second source wave. Repetition of this process for all source waves results in the addition of phase shifts or equivalently, the addition of potentials. Matter and anti-matter solutions have opposite signs of phase shift (), indicating opposite direction of rotation about the spin axis. Hence the potentials for matter and anti-matter pairs of soliton waves are equal and opposite. One may also infer that soliton waves with identical long-range (electromagnetic) potentials (e.g. positrons and protons) also have identical bispinor wave functions at large distances from their centers.

Unlike quantum mechanics, it is not necessary classically to treat various wave packets as independent ‘particles’. Instead, it is likely simpler to solve the single equation for the total angular momentum density, then decompose the solution into soliton ‘particles’ for comparison with experiment.

3.5. Lorenz Force

Since cb₁s is a velocity operator and , the conjugate momentum for r is:

(67)

The wave momentum is familiar from quantum mechanics. The momentum of the medium is .

The conjugate momentum for rotation is:

(68)

This angular momentum operator includes both orbital and spin components just as in Dirac theory. The spin component is the angular momentum associated with motion of the medium, while the orbital component is the angular momentum associated with motion of the wave.

The conjugate momentum for time is simply H itself:

(69)

The time derivative of any observable Q is:

(70)

An example of this is the force density. Substituting the linear wave momentum p for Q yields the Lorenz force law:

(71)

where E and B are the usual electric and magnetic fields, respectively. Hence the Lorenz force has a straightforward interpretation in terms of classical wave interference.

3.6. Measurement Correlations

It is widely believed that the correlations between polarization measurements of entangled particles cannot be predicted classically. This belief is based on correlation predictions using an equation of the form described by Bell:⁽²²⁾

(72)

where l_i represent variables which describe the state of the system, a and b are the measured polarization directions for the two entangled particles, A and B are the theoretical outcomes of the measurement (±1), and C(a,b) is the correlation. The key assumption of Bell’s Theorem is that the measurement results, A and B, represent physical variables (computed from the l_i) which propagate from the source of the entangled particles to the detectors. In fact, however, it is the bispinor wave function y which propagates from place to place since the first order Dirac equation is a kind of convection equation. Therefore Bell’s Theorem does not generally apply to classical bispinor waves, just as it does not apply to quantum mechanical waves. Since y has spin one-half, correlation between states related by rotation about an axis perpendicular to the spin is:

(73)

The correlation for angle is .

Assuming that spin measurements are coincident or anti-coincident in proportion to the correlations between the spinor wave functions, the correlation C_s between spin measurements separated by angle is:

(74)

In the case of pair production in EPR-type experiments, the spins of the two electrons are opposite, thereby changing the sign of the correlation. Hence the classical correlations are in agreement with the quantum correlations.

4. Wave Properties of Matter

We have shown that classical wave theory provides a correct description of electron dynamics. Although a probabilistic particle interpretation of the wave function may also be valid, the dynamics is that of classical waves. This result lends support to recent efforts to revive the classical aether (or ether) as a medium of propagation of matter waves. Duffy⁽²³⁾ has surveyed modern aether theory.

The model of vacuum as an ideal elastic solid was quite successful in explaining classical properties of light in the 19^th century.⁽²⁴⁾ Quantum effects are only apparent in interactions with matter, which evidently consists of classical soliton waves. Our classical equation for the angular potential is essentially empirical, although all of the terms have intuitive interpretations in terms of rotations in an elastic solid. At present there appears to be no rigorous description of rotational waves in an ideal elastic medium. Kleinert⁽²⁵⁾ attempted to include rotations in the elastic energy but was compelled to introduce new elastic constants dependent on an arbitrary scale length. Close⁽¹³⁾ showed that torsion waves (with rotation axis parallel to wave velocity) can be described by a Dirac equation, but did not derive a general equation (and also incorrectly related mass and velocity rotation).

Many physical properties of matter can be derived from a wave model of matter. The Uncertainty Principle applies to all classical waves. Lorentz invariance is also a property of waves, and Special Relativity is therefore a consequence of any wave theory of matter.⁽²⁶⁾ Parity conservation, as demonstrated above, apparently requires a Galilean space-time in which matter waves propagate.

A scalar gravitational field and its effect on the space-time metric may be interpreted as a spatially varying light speed. See Whittaker⁽²⁷⁾ for the historical development of this idea which originates with Einstein⁽²⁸^,²⁹⁾ and has also been investigated more recently.⁽³⁰^,³¹⁾ This interpretation is consistent with general relativity, which predicts a variation of light speed proportional to the gravitational potential.⁽³²⁾ In an elastic solid aether, compression or variations in elasticity imply variable wave speed and hence provide a reasonable physical model for non-Newtonian gravitational effects.

5. Conclusions

In this paper we interpret the Dirac equation as a second-order wave equation whose first order spatial and temporal derivatives are represented by a bispinor wave function. The parity transformation and free electron Hamiltonian are corrected. Half-integer spin is attributable to the co-existence of waves traveling in opposite directions along the gradient axis. The wave function in a given representation can be factored into constant matrix, an amplitude, a three-dimensional Lorentz velocity boost, and a rotation operator. Wave interference yields both the Pauli Exclusion Principle and the Lorenz force. Mass is related to angular velocity and associated with radially inward acceleration of the wave, implying a soliton. The integer g-factors represent relative contributions of orbital and spin angular momentum to the angular velocity. The theory is consistent with parity conservation, which is the simplest explanation for the mirror symmetry ordinarily associated with exchange of matter and anti-matter. Correlations between rotated states are the same as for quantum theory. Bell’s Theorem does not apply to classical electron waves because it is the bispinor wave functions, and not the measurement values, which convect from place to place. Gravity may be incorporated into the theory as wave refraction, consistent with general relativity. Hence classical wave theory can correctly describe electron dynamics and offers the possibility of a simple mechanical model of the vacuum.

Acknowledgments

The author is grateful to Damon Merari for his interest and encouragement. Thanks also to Peter Rowlands and Lorenzo Sadun for helpful discussions during the course of this research.

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