# Presentation: 318164

## Canonical Variables, Metrics, and Parity

- Robert Close, PhD.
- robert.close@classicalmatter.com

**Notes:**
Good day. My name is Robert Close, and today I will be talking about canonical variables, metrics, and parity.

## Outline

- Correcting problems associated with dependent variables and incorrect metrics yields a theory of matter consistent with parity conservation and a soliton wave interpretation of particles.

**Notes:**
Here is a quick outline of the topics I will discuss...

## Dirac Matrices

- Spin (*S**i*) and velocity (*V**i*) are:

- Spin (*** 0 **** 5 **** i*) and velocity (*** 0**** i*) matrices are not independent:

- Dirac Hamiltonian is not in canonical form!

**Notes:**
It is easy to see that the Dirac matrices for spin and velocity are not independent. Therefore the Dirac Hamiltonian constructed from these is not in canonical form.

## Dirac vs. Canonical

- Dirac bispinors have the factorization:
- (amplitude, rotation, 3-D Lorentz boost, velocity-axis rotation)

***i****j****j****i*=0

- The canonical wave factorization is:

- (amplitude, polarization rotation, velocity rotation,
- 1-D Lorentz boost)

## Canonical Variables

- Start with classical 1-D wave equation
- Factor into matrix form (1-D Dirac Eqn.)
- Generalize to 3-D using independent operators for rotation of wave polarization and velocity
- (using common rotation operator yields conventional Dirac form)

**Notes:**
Let me explain quickly how to derive the canonical factorization... 

## 1-D Scalar Waves

- Lorentz invariant  Galilean physical space, Minkowski measurement space
- Solution is sum of forward and backward propagating waves:

**Notes:**
The one-dimensional scalar wave equation is Lorentz invariant, yet the equation is easily derived for numerous physical systems assuming a Galilean space-time. The meaning of Lorentz invariance (and hence Special Relativity) is that the physical space-time is Galilean whereas the space of measurements made with waves is Minkowski space. We will use this interpretation of relativity later when discussing parity.

The solution is a sum of forward and backward propagating waves. Everyone should be familiar with this result, and if not it can be verified by substitution.

## 1-D Scalar Waves

- In 3-D these independent solutions are 180° apart. Hence the wave solutions form a _spin one-half_ system. (Yes, spin one-half has a classical interpretation).

- Forward

- Backward

**Notes:**
In three dimensions these independent solutions are 180° apart. This is the definitive feature of spin 1/2 in physical space. Hence the wave solutions form a spin one-half system. (I was taught that there is no classical interpretation of spin one-half, but obviously that is wrong).

## 1-D Spinors

- The time derivative is separated into 4 components:

-            (→, +)     (→, )      (,+)     (, )

- In spinor form:

**Notes:**
The wave derivatives may be written using one-dimensional spinors. The time derivative is given by this equation, in which each term on the right hand side is positive-definite, F refers to the forward wave, B refers to the backward wave, and + and – subscripts refer to the positive and negative components, respectively. This equation may be written in matrix form as shown here. The subscript v refers to the velocity axis.

## 1-D Scalar Waves

- The spatial derivative is:

**Notes:**
The spatial derivative can be written similarly as shown here, using a different matrix which I call sigma_v beta

## 1-D Scalar Waves

- The forward and backward wave velocities are tabulated in the matrix **v:

**Notes:**
The forward and backward wave velocities are tabulated in the matrix sigma_vas shown here. 

## 1-D Scalar Waves

- Forward and backward waves have arguments (*ct****z*) and (*ct+z*), respectively, so the equation of evolution of the wave function is:

- This 1-D Dirac Equation is equivalent to the second-order homogeneous wave equation.

**Notes:**
Since the different components of the wave function refer to forward and backward waves with arguments (ct-z) and (ct+z), respectively, the equation of evolution of the wave function is this 1-D Dirac equation, which is equivalent to the second-order homogeneous wave equation.

## Corresponding 1-D Wave Equations

- Bispinor Equation                    Scalar Equation

**Notes:**
Bispinor equations with various matrix factors all have equivalent  second-order wave equations as shown here.

## 1-D Scalar Waves

- Average wave velocity is:

- Let:

- Then:

**Notes:**
The average wave velocity is the wave speed multiplied by the ratio between the difference and sum of the forward and backward wave amplitudes. This quantity can be written as a hyperbolic tangent.

## 1-D Scalar Waves

- The Lorentz boost operation is:

- Successive boosts have the same form:

- Hence relativistic wave velocity addition:

**Notes:**
The Lorentz boost operation changes the average wave velocity by increasing either the forward or backward wave amplitude and decreasing the other. Successive Lorentz boosts yield the relativistic velocity addition formula. Note again that even though classical space-time is physically Galilean, waves are Lorentz invariant and subject to all of the laws of special relativity, yielding a Minkowski measurement space.

## 1-D Scalar Waves

- Using Lorentz boosts the 1-D bispinor wave function may be factored as:

- where **F and **B are either one or zero (each
- wave, forward or backward, must be either
- positive or negative, not both).

**Notes:**
The 1-D wave function may be factored using an amplitude, a Lorentz boost which determines the relative amplitudes of forward and backward waves, and delta functions which determine the sign of the forward and backward wave amplitudes, 

## 3-D Scalar Waves

- Extension from 1-D to 3-D scalar waves requires rotation of wave velocity (and gradient). Three orthogonal matrices define unit polar vectors:

- The commutation relations are:

**Notes:**
Extension to three dimensional scalar waves requires rotation of wave velocity and gradient. Three orthogonal matrices are shown here, with 3 replacing the original v. The commutation relations are the same as for the Pauli matrices.

## 3-D Scalar Waves

- The 3-D scalar wave bispinor has the factorization:

- where:

- The gradient is given by:

**Notes:**
The 3-D scalar wave bispinor has this factorization, which starts with an equal superposition of  forward positive and backward negative waves on the z-axis (0), then applies a Lorentz boost along the z-axis, then rotates the velocity by angle zeta. The three dimensional gradient is now given by this formula using the vector of beta matrices.

## 3-D Vector Waves

- Vector polarization uses three orthogonal matrices as unit vectors (axial vectors):

- The commutation relations are:

**Notes:**
Extension to 3-D vector waves uses three polarization matrices, with the original sigma replaced by sigma_3. The commutation relations are again the same as the Pauli Matrices.

## 3-D Vector Waves

- The bispinor wave equation is:

- Terms on the right-hand side account for temporal and spatial rotations of velocity and polarization.

- The 3-D vector bispinor factorization is:

**Notes:**
The 3-D bispinor wave equation is obtained from the 1-D equation by inserting terms corresponding to rotation of velocity and polarization. Including derivatives of these rotations adds additional terms to the 1-D Dirac equation.

## 3-D Vector Waves

- The corresponding second-order wave equation is:

- The additional terms account for temporal (vorticity) and spatial (torsion) rotations of polarization, and divergence of velocity direction (last term not physical).

**Notes:**
The corresponding second-order wave equation is given here. There are three additional terms which represent (1) rotation in time of polarization, (2) rotation in space of polarization, and (3) divergence of velocity direction represented by the parallel component of the curl of the rotation angle. This equation assumes that the 1-D wave equation is valid in all directions. In three dimensions divergence of wave velocity alters the amplitude , so the last term is probably unphysical in most applications.

## 3-D Vector Waves

- The Dirac equation is a factorization of a second-order wave equation.

- Close RA 2002 Torsion Waves in Three Dimensions: Quantum Mechanics with a Twist, *Found. Phys. Lett.*** 15,** 71-83.
- Close R A 2006 Progress Toward a Classical Theory of Matter  Proc. Physical Interpretations of Relativity Theory X, ed M C Duffy (London: British Soc. Philosophy of Science), to be published
- Rowlands P 1998 The physical consequences of a new version of the Dirac equation Causality and Localaity in Modern Physics and Astronomy: Open Questions and Possible Solutions, ed  G Hunter et al (Fundamental Theories of Physics vol. 97 (Dordrecht : Kluwer Academic Publishers) pp 397-402
- Rowlands P and Cullerne J P 2001 The connection between the Han-Nambu quark theory, the Dirac equation and fundamental symmetries Nucl. Phys. A 684, 713-5
- Rowlands P 2005 Removing redundancy in relativistic quantum mechanics Preprint arXiv:physics/0507188
- Rowlands P 2006 Breaking the Dirac code Proc. Physical Interpretations of Relativity Theory X, ed M C Duffy (London: British Soc. Philosophy of Science) to be published

**Notes:**
In 2002 I published the idea that the first-order Dirac equation with spinors represents a factorization of a second-order wave equation. I later learned that Peter Rowlands of Liverpool had previously interpreted the Dirac equation in multivariate vector form as a factorization of the Klein-Gordon equation. He has derived many features of the Standard Model from this factorization.

## Canonical Wave Matrices

- Spin (*S**i*) and velocity (*v**i*) are:

- Spin (***i*) and velocity (***j*) matrices are independent:

- Hamiltonian can have canonical form!

## Electron Waves

- Replace rotation terms with ()*l* times parity operator (*l* is orbital quantum number) :

- Equivalent local operator:

**Notes:**
Now let’s discuss electron waves. The canonical form of the electron wave equation is obtained by using a modified parity operator for the mass term. The parity operator is nonlocal, but an equivalent local operator is shown here. It seems to represent rotation of velocity, with the second term indicating a component of the rotation axis parallel to L=rxgrad. I don’t yet have a good interpretation of the first term (the “one”).

## Electron Waves

- Two spin-up solutions related by parity operation yield opposite signs of *M* and *E* (matter & antimatter) :

**Notes:**
There are two spin-up solutions which are related by the parity operator and yield opposite signs of M and E. These solutions may be interpreted as matter and anti-matter. The two component wave functions ‘phi’ are familiar from conventional Dirac theory. They have opposite parity and are related by the matrix sigma_r.

## Electron Waves

- The classical radial equations are the same as for the Dirac form:

- Coulomb energy levels are same as Dirac.

## Electron Waves

- Mass term represents rotation of wave velocity toward origin of polar coordinates, indicating soliton wave:

**Notes:**
... The overall sign must be negative for a soliton solution.

## Classical Bispinor Potentials

- Perturbation of velocity angles yields:

- Define the classical bispinor potentials:

- Lorenz condition is automatically satisfied:

**Notes:**
Now we will look at the classical interpretation of potentials. Suppose a soliton wave satisfies the free electron equation with mass M.
Perturbation of the velocity angles yields the following terms containing the divergence and time derivative of differential velocity angle delta zeta...

## Classical Bispinor Potentials

- The perturbed equation is:

- Classical bispinor potentials are equivalent to electromagnetic potentials.
- Potentials have a geometric interpretation as variations in wave velocity rotation.

**Notes:**
Substitution of potentials yields this equation, which is similar to the Dirac equation with electromagnetic potentials (neglecting the curl term). However, electromagnetic potentials have no geometrical interpretation in conventional Dirac theory. Just as ExB drift is due to variations in the radius of curvature of electron orbits in a magnetic field, the acceleration of electrons in electromagnetic fields is due to variations in rotation of wave velocity.

##  Hamiltonian Formalism

- Conjugate momenta:

- Hamiltonian with Bispinor Potentials:

**Notes:**
The Hamiltonian can be written like this...The conjugate momentum for rotation looks just like Dirac theory except that two independent rotation matrices are used. The conjugate momentum for translation is identical to the Dirac form.

##  Hamiltonian Formalism

- Classical wave Lorenz force equation:

- This differs from the Dirac form only in the velocity matrix operator.

**Notes:**
The time derivative of the conjugate momentum is the Lorenz force equation. The only difference from Dirac theory is the form of the velocity matrix.

##  Wave Interference

- Observable *G* computed from sum of wave functions:

- Observables should be additive. Eliminate interference terms with phase factor:

**Notes:**
Now consider wave interference. Observables should be additive, but if wave functions are added then interference terms result. These interference terms can be cancelled using a phase factor.

##  Wave Interference

*G*=1 yields Pauli Exclusion Principle:

- Evolution of ***B* :

-                   yields EM potentials:

**Notes:**
Letting G=1, the requirement that interference terms cancel yields the Pauli exclusion principle.

Interpreting the phase factor in terms of velocity rotations yields the electromagnetic potentials.

##  Quantization

- Dynamics of classical electron wave is ‘identical’ to Dirac dynamics.
- Deterministic evolution of momentum, angular momentum, and energy.
- Quantization requires additional property that wave function determines measurement probabilities. (Can this be derived classically for soliton waves?)

**Notes:**
Thus far our description of the electron is completely classical...

## Parity Conservation

**Soliton solutions (*******E***,**** *******M***) related by parity operator **
**	****→**** ****matter and anti-matter are mirror pairs.**
- This is the _simplest_ (best) explanation of myriad observations that processes involving anti-matter behave as  mirror-images of processes involving matter.

**Notes:**
Finally, let’s discuss parity. The soliton solutions with opposite signs of E and M are related by the parity operator, implying that matter and anti-matter are  mirror pairs...

## Parity Conservation

**     ****Matter and anti-matter as mirror pairs.**
- Consistent with left-right asymmetry in weak interactions (Wu 1957).
- Parity conservation can _*not*_ be tested experimentally (it is impossible to invert space).
- The QM parity operator can also be modified to change the signs of both *E* and *M*.

**Notes:**
This interpretation of matter and anti-matter is consistent with left-right asymmetry in weak interactions (Wu 1957).
Everyone should realize that parity conservation can not be tested experimentally because it is impossible to invert space. Lee and Yang [1956] actually mention in their paper that left-right asymmetry does not necessarily imply parity violation.  But they assumed the conventional parity operator to be correct.
The QM parity operator can be changed to include exchange of matter and anti-matter, consistent with parity conservation in weak interactions.

## Parity

- These Dirac matrices form right-handed coordinates:
-    (**0, i**0**5, **5 ) = *relative wave velocity*.
- Conventional parity operator **0**(**r**) corresponds to rotation of velocity by 180.
- Correct parity operator inverts all three relative velocity matrices.

**Notes:**
In Dirac theory, the matrices 0, i05, and 5  form a right-handed coordinate system (based on commutation properties). Gamma5 sigma represents the velocity direction, while gamma0 and i_gamma0_gamma5 can be interpreted as relative orthogonal directions perpendicular to velocity.
The conventional parity operator (0Q0) inverts two of these matrices, corresponding to a rotation of relative velocity.
The correct  parity operator inverts all three relative velocity matrices.

## Conventional Parity

- Observable *Q*:

- Minkowski metric:  (** 0)2 = 1, (** *i*)2 = 1
- Parity: *P***(**r**)* = *** 0**(**r**)     Not conserved
- Assumes* P*(*E*)=*E*, *P*(*M*)=*M*

-   Dirac eqn:

- 0 factor: Galilean → Minkowski space-time

- 0 factor : Minkowski → Galilean space-time

**Notes:**
In QM the matrix gamma_0 is not inverted because it is treated as a unit normal of time in Minkowski space. This assumption implies parity violation.
However, notice that the factor gamma_0 is carefully removed when computing observables. The scientific name for this procedure is “covering your tracks”. It is an implicit admission that physical space-time is Galilean, not Minkowski.

## Alternative Parity (Dirac)

-  **  as time coefficient assumes Galilean physical space, Minkowski measurement space (waves), implies parity conservation.

- Observable *Q*:

- Galilean metric:  *I*2 = 1, (** 0** *i*)2 = 1

-   Dirac eqn:

-   Parity:
*  **P*(*M*** 0)*=*** *M *** 0  *P*(*E*)*=*** *E*

**Notes:**
Taking the identity matrix I as the time coefficient corresponds to a Galilean physical space and Minkowski measurement space. Hence classically it is acceptable to invert gamma_0 along with the other components of relative velocity.

## PC Violation?

- Interpretation of *PC* violation (actually *P* violation) is model-dependent.
*PC* operator yields valid results.
- Find a better explanation...

**Notes:**
One phenomenon which is not yet explained is so-called PC violation in neutral Kaon decays (which should really be called P violation). Just keep in mind that the interpretation is model-dependent and remember that it is impossible to perform a direct test by inverting space. Also, even with the current interpretation the PC operator is believed to yield valid physical processes for decay of anti-particles.

Also, the rarity of the supposed violation makes it quite likely that a better explanation can be found. 

## Summary

- Dirac Hamiltonian is not in canonical form.
- Classical wave processes have spin one-half solutions due to 180° separation of independent forward and backward waves.
- Waves propagating in Galilean physical space have Minkowski measurement space.

## Summary

- Mass represents rotation of wave velocity consistent with soliton wave interpretation of particles.
- Electromagnetic potentials are implicit in classical bispinor factorization (unlike the Dirac factorization).
- Wave interference yields EM potentials and Pauli exclusion principle.

## Summary

- Matter and anti-matter are mirror pairs.
- There is no _direct_ experimental evidence of parity violation.
- Apparent parity violation in beta decay resulted from dubious extension of Minkowski metric from measurement space to physical space.

## Bibliography

- de Broglie LV 1924 *Recherches sur la Theorie des Quanta*, PhD Thesis, (Paris: University of Sorbonne).
- Close RA 2002 Torsion Waves in Three Dimensions: Quantum Mechanics with a Twist, *Found. Phys. Lett.*** 15,** 71-83.
- Close R A 2006 Progress Toward a Classical Theory of Matter  Proc. Physical Interpretations of Relativity Theory X, ed M C Duffy (London: British Soc. Philosophy of Science), to be published
- Close RA 2006 *The Classical Theory of Matter Waves*, ( http://www.classicalmatter.com/ ClassicalMatterWaves.html) Chapter 2.
- Gu YQ 1998 Some Properties of the Spinor Soliton, *Advances in Applied Clifford Algebras* **8**(1) 17-29.
- Hestenes D 1967 Real Spinor Fields, *J. Math. Phys.* **8**(4) 798-808.
- Hestenes D 1973 Local observables in the Dirac theory, *J. Math. Phys.* **14**(7) 893-905.
- Hestenes D 1990 The Zitterbewegung Interpretation of Quantum Mechanics, *Found. Phys.* **20**(10) 1213-32.
- Lee TD and Yang CN 1956 Question of Parity Conservation in Weak Interactions, *Phys. Rev.* **104**, 254.
- Ranada AF  1983 Classical Nonlinear Dirac Field Models of Extended Particles *Quantum Theory, Groups, Fields, and Particles* ed A O Barut (Amsterdam: Reidel) pp 271-88.
- Rowlands P 1998 The physical consequences of a new version of the Dirac equation Causality and Localaity in Modern Physics and Astronomy: Open Questions and Possible Solutions, ed  G Hunter et al (Fundamental Theories of Physics vol. 97 (Dordrecht : Kluwer Academic Publishers) pp 397-402
- Rowlands P and Cullerne J P 2001 The connection between the Han-Nambu quark theory, the Dirac equation and fundamental symmetries Nucl. Phys. A 684, 713-5
- Rowlands P 2005 Removing redundancy in relativistic quantum mechanics Preprint arXiv:physics/0507188
- Rowlands P 2006 Breaking the Dirac code Proc. Physical Interpretations of Relativity Theory X, ed M C Duffy (London: British Soc. Philosophy of Science) to be published
- Takabayashi Y 1957 Relativistic hydrodynamics of the Dirac matter, *Suppl. Prog. Theor. Phys.* **4**(1)** **1-80.
- Whittaker E 1951 *A History of the Theories of Aether and Electricity*, (Edinburgh: Thomas Nelson and Sons Ltd.).
- Wu CS, et al. 1957  Experimental test of parity conservation in beta decay,* Phys. Rev.* **105**, 1413.

## More at:

- http://www.classicalmatter.com
- Please contact me at:
- robert.close@classicalmatter.com