The Relativistic Energy-Momentum Equation in the Planck Vacuum Theory
##plugins.themes.bootstrap3.article.main##
This paper examines the relativistic energy-momentum equation and its use in the photon-electron Compton scattering calculations. It provides a better understanding of that equation and reveals the reason for particle spin.
Introduction
The theoretical foundation [1]–[5] of the PV theory rests upon the unification of the Einstein, Newton, and Coulomb superforces:
where the ratio is the curvature superforce that appears in the Einstein field equations. G is Newton’s gravitational constant, c is the speed of light, and are the Planck mass and length respectively [[6], p.1234], and is the massless bare (or coupling) charge. The Planck time is [[6],p.1233]. The fine structure constant is given by the ratio , where e is the observed electronic charge magnitude. The ratio to the right of the arrow is the spin coefficient for the Planck particle (PP), the proton, and the electron cores, where is the reduced Planck constant. One of the s in belongs to the PP under consideration and the other to any one of the remaining PPs making up the PV state.
The electron, proton, and PP Dirac cores associated with the PV theory defined above are
respectively. The ± signs in the equations include the antiparticles. Their coupling to the highly energetic PV state is through the spin equations
where it appears that particle spin and (3) exist to separate (in magnitude) the masses on the left from the mass on the right. The spin is generated in the zero-point PV oscillations [7].
All of the preceding equations are fixed in the sense that their structure is determined at the high PV energy level. This level is roughly nineteen (proton cores) to twenty-two (electron cores) orders-of-magnitude more energetic than the processes taking place at the electron or proton levels. The masses and are assumed to be created along with their Compton radii and in (3) within the PV state.
From (1) the gravity bodies in the PV theory are described by the gravitational constants [8]
where it is noted that does not react to photons because it contains no charge .
Conclusions and Comments
The total (charge and mass) electron core coupling force acting on the PV state in (1) is [9]
with the result that the core propagates through free space unattenuated because its coupling force
vanishes at . From Appendix A, is a constant of the motion that completes (6).
The covariant Dirac equation (A1) shows that the Compton radius is a constant of the motion in the spin equations. Therefor it must be possible to create an equation that includes automatically without being seen. The following relativistic energy-momentum equation is just such an equation
where W is the total energy, cP is the total kinetic energy, and P is the relativistic momentum. Since the in (7) includes particle-spin, W and P must also include particle-spin. In that case, the solution to (7) can be expressed as
which includes an antiparticle.
Assume that a quantum of energy with a momentum moves along the positive x-axis in outer space and strikes a free electron core at rest; and then departs at an angle to the axis, with a wavelength increase . Then [10], [11]
where h is the Planck constant and
It is also shown that
where is the original wavelength. The energy-momentum equation is essential to the above Compton scattering calculations.
The previous calculations lead immediately to the proton core by replacing with .
Appendix A Constant of the Motion
The covariant form [[12], p.90] [Appendix B] of the Dirac equation for the electron and positron cores combined can be expressed as:
where dividing through by and recombining leads to
and where the j partial-derivative terms in (A2) are summed over j = 1, 2, 3. The character of equation (A2) implies that the equations here include particle spin.
It is noted from (A3) that is a constant of the motion as far as the wavefunction is concerned.
Appendix B The γ And β Matrices
The 4 × 4 , , and matrices used in the Dirac theory are defined here: where [[12], p.91]
and (j = 1, 2, 3)
and where I is the 2 × 2 unit matrix and
where the are the 2 × 2 Pauli spin matrices
and . The zeros in (B1)–(B3) are 2 × 2 null matrices.
The zeros on the right end of (A2) and (A3) represent 4 × 4 null matrices.
The coordinates are
where .
References
-
Davies P. Superforce: The Search for a Grand Unified Theory of Nature. New York: Simon and Schuster, Inc.; 1984.
Google Scholar
1
-
Daywitt WC. The Planck Vacuum. Prog Phys. 2009;1:20. Available from: www.planckvacuum.com.
Google Scholar
2
-
Daywitt WC. The Trouble with the Equations of Modern Fundamental Physics. Am JModern Phys. Special Issue: Physics without Higgs and without Supersymmetry. 2016;5(1):22.
Google Scholar
3
-
Daywitt WC. Comparing the Planck-vacuum and the Urantia-book Depictions of the Seven-dimensional Spacetime. Eur J Eng Res Sci. December 2020;5(12):11–13.
Google Scholar
4
-
Daywitt WC. Spin in the Dirac and Schrödinger Equations According to the Planck Vacuum Theory. Eur J Appl Phys. September 2023;5(5):8–10.
Google Scholar
5
-
Carroll BW, Ostlie DA. An Introduction to Modern Astrophysics. San Francisco, Boston, New York, Cape Town, Hong Kong, London, Madrid, Mexico City, Montreal, Munich, Paris, Singapore, Sidney, Tokyo, Toronto: Addison-Wesley; 2007.
Google Scholar
6
-
Daywitt WC. Zero-point Oscillations in the Planck Vacuum State and Its Coordinate Uncertainty. Eur J Eng Res Sci. 2021;6(4):13–15.
Google Scholar
7
-
Daywitt WC. Gravity Bodies in the Planck Vacuum Theory. Eur J Eng Res Sci. April 2023;5(2):26–27.
Google Scholar
8
-
Daywitt WC. Nuclear Cohesion and the Spin of the Planck Particle, the Proton, the Electron and theMesotron as Viewed in the Planck Vacuum Theory. Eur J Appl Phys. December 2021;3(6):21–23.
Google Scholar
9
-
Daywitt WC. The compton Effect in the Planck Vacuum Theory. Eur J Appl Phys. November 2022;4(6):20–21.
Google Scholar
10
-
Daywitt WC. Electron Recoil in the Compton Effect as Viewed in the Planck Vacuum Theory. Eur J Appl Phys. April 2023;5(2):28–30.
Google Scholar
11
-
Gingrich DM. Practical Quantum Electrodynamics. Boca Raton, London, New York: CRC, The Taylor & Francis Group; 2006.
Google Scholar
12
Most read articles by the same author(s)
-
William C. Daywitt,
The Heisenberg Uncertainty Principle in the Planck Vacuum Theory , European Journal of Applied Physics: Vol. 4 No. 5 (2022) -
William C. Daywitt,
The Cosmic Microwave Background Radiation as Viewed in the Planck Vacuum Theory , European Journal of Applied Physics: Vol. 4 No. 1 (2022) -
William C. Daywitt,
The Schrödinger Equation and the Time-Ordering Operator T of the Quantum Field Theory as Viewed in the Planck Vacuum Theory , European Journal of Applied Physics: Vol. 5 No. 4 (2023) -
William C. Daywitt,
General Relativity in the Planck Vacuum Theory , European Journal of Applied Physics: Vol. 4 No. 4 (2022) -
William C. Daywitt,
The Source of the Gravitational Constant for the Dirac Cores in the Planck Vacuum Theory , European Journal of Applied Physics: Vol. 4 No. 3 (2022) -
William C. Daywitt,
The Compton Effect in the Planck Vacuum Theory , European Journal of Applied Physics: Vol. 4 No. 6 (2022) -
William C. Daywitt,
Gravity Bodies in the Planck Vacuum Theory , European Journal of Applied Physics: Vol. 5 No. 2 (2023) -
William C. Daywitt,
Reducing the Covariant Dirac Equation for the Electron or Proton Cores to the Gradients of a Single Wavefunction , European Journal of Applied Physics: Vol. 4 No. 2 (2022) -
William C. Daywitt,
Electron Recoil in the Compton Effect as Viewed in the Planck Vacuum Theory , European Journal of Applied Physics: Vol. 5 No. 2 (2023) -
William C. Daywitt,
Spin in the Dirac and Schr¨odinger Equations According to the Planck Vacuum Theory , European Journal of Applied Physics: Vol. 5 No. 5 (2023)