# Rotation and Spin and Position Operators in Relativistic Gravity and Quantum Electrodynamics

## Abstract

**:**

## 1. Introduction

## 2. Special Relativity

## 3. Quantum Electrodynamics (QED)

^{k}is calculated to be given by the equation:

^{k}operator. As noted by Equation [8], this result for q

^{k}corresponds to the result given by Pryce [7] in for the case of spin s = 1/2. Pryce investigated how the mass-center of the constituent interacting particles could be defined in relativity and, in particular, he examined various generalizations of the Newtonian definition. He eventually settled on a definition which is the mean of the coordinates of the individual particles weighted with the total energy and the rest mass of the total system. This definition of the mass-center ensures that it is at rest in a frame in which the total momentum is zero but is does depend on the frame in which it is defined. In other words, it is not covariant but it does coincide with the choice of NW. It is written down explicitly in [4]. The equivalence of the Pryce and NW position operators for arbitrary spin has been given by Lorente and Roman [9], who made extensive use of gauge symmetries associated with the inhomogeneous Lorentz group generators.

## 4. Classical Relativistic Systems

_{0}as E and P

_{0}is defined in (5). The total spin is S. To lowest order, P = mv so that in the non-relativistic limit (4) becomes:

_{αβ}and ${S}^{\mu}$ is not unique [18,22] but depends on the choice of the so-called spin supplementary condition, which in turn depends on the coordinate system chosen. Popular choices are ${\mathrm{S}}^{\mathsf{\alpha}\mathsf{\beta}}$ U

_{β}= 0 and ${\mathrm{S}}^{\mathsf{\alpha}\mathsf{\beta}}$ p

_{β}= 0 where U

_{β}and p

_{β}are 4 - velocity and 4 – momentum vectors. However, these choices, because they are covariant, are not suitable for treating an accelerating particle, and so, we introduced a new supplementary condition [3,22]:

## 5. Discussion

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- O’Connell, R.F. Rotation and Spin in Physics. In General Relativity and John Archibald Wheeler, 1st ed.; Ciufolini, I., Matzner, R., Eds.; Springer: New York, NY, USA, 2010. [Google Scholar]
- Mathisson, M. Neue mechanik materieller systems. Acta Phys. Pol.
**1937**, 6, 163. [Google Scholar] - Barker, B.M.; O’Connell, R.F. Nongeodesic motion in general relativity. Gen. Relat. Gravit.
**1974**, 5, 539. [Google Scholar] [CrossRef] - Foldy, L.; Wouthuysen, S.A. On the Dirac theory of spin 1/2 particles and its non-relativistic limit. Phys. Rev.
**1950**, 78, 29. [Google Scholar] [CrossRef] - Bjorken, J.D.; Drell, S.D. Relativistic Quantum Mechanics; McGraw Hill: New York, NY, USA, 1964. [Google Scholar]
- Schweber, S.S. An Introduction to Relativistic Quantum Field Theory; Harper and Row: New York, NY, USA, 1962. [Google Scholar]
- Pryce, M.H.L. The mass-centre in the restricted theory of relativity and its connexion with the quantum theory of elementary particles. Proc. Roy. Soc.
**1948**, A195, 62. [Google Scholar] - Newton, T.D.; Wigner, E. Localized states for elementary systems Revs. Modern Phys.
**1949**, 21, 400. [Google Scholar] [CrossRef] [Green Version] - Lorente, M.; Roman, P. General expressions for the position and spin operators of relativistic systems. J. Math. Phys.
**1973**, 15, 70. [Google Scholar] [CrossRef] - Breton, R.P.; Kaspi, V.M.; Kramer, M.; McLaughlin, M.A.; Lyutikov, M.; Ransom, S.M.; Stairs, I.H.; Ferdman, R.D.; Camilo, F.; Possenti, A. Relativistic spin precession in the double pulsar. Science
**2008**, 321, 104. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hanson, A.J.; Regge, T. The relativistic spherical top. Ann Phys.
**1974**, 87, 498. [Google Scholar] [CrossRef] - Dirac, P.A.M. Forms of relativistic dynamics. Rev. Mod. Phys.
**1949**, 21, 392. [Google Scholar] [CrossRef] [Green Version] - Hojman, S.; Regge, T. Studies in Mathematical Physiocs, Essays in Honor of Valentine Bargmann; Lieb, E.H., Simon, B., Wightman, A.S., Hojman, S., Eds.; Princeton University Press: Princeton, NJ, USA, 1976. [Google Scholar]
- Ramond, P. Group Theory. A Physicist’s Survey; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- O’Connell, R.F.; Wigner, E.P. On the relation between momentum and velocity for elementary systems. Phys. Lett.
**1977**, 61A, 353. [Google Scholar] [CrossRef] - Møller, C. On the Definition of the Centre of Gravity of an Arbitrary Closed System in the Theory of Relativity. Available online: https://www.stp.dias.ie/Communications/DIAS-STP-Communications-005-Moller.pdf (accessed on 22 January 2020).
- Møller, C. The Theory of Relativity, 2nd ed.; Oxford University Press: Oxford, UK, 1972. [Google Scholar]
- O’Connell, R.F. Electron interaction with the spin angular momentum of the electromagnetic field. J. Phys. A
**2017**, 50, 085306. [Google Scholar] [CrossRef] [Green Version] - Schiff, L.I. Motion of a gyroscope according to Einstein's theory of gravitation. Proc. Natl. Acad. Sci. USA
**1960**, 46, 871. [Google Scholar] [CrossRef] [Green Version] - Barker, B.M.; O’Connell, R.F. Derivation of the equations of motion of a gyroscope from the quantum theory of gravitation. Phys. Rev. D
**1970**, 2, 1428. [Google Scholar] [CrossRef] - Barker, B.M.; Gupta, S.N.; Haracz, R.D. One-graviton exchange interaction of elementary particles. Phys. Rev.
**1966**, 149, 1027. [Google Scholar] [CrossRef] - Barker, B.M.; O’Connell, R.F. General Relativistic Effects in Binary Systems. Available online: https://ui.adsabs.harvard.edu/abs/1978pans.proc..437B/abstract (accessed on 22 January 2020).
- Barker, B.M.; O’Connell, R.F. Gravitational two-body problem with arbitrary masses, spins, and quadrupole moments. Phys. Rev. D
**1975**, 12, 329. [Google Scholar] [CrossRef] - Jackson, J.D. Classical Electrodynamics, 3rd ed.; Wiley: New York, NY, USA, 1998. [Google Scholar]
- Kidder, L.E. Coalescing binary systems of compact objects to (post)5/2-Newtonian order. V. Spin effects. Phys. Rev. D
**1995**, 52, 821. [Google Scholar] [CrossRef] [PubMed] [Green Version] - O’Connell, R.F. Gravito-Magnetism in one-body and two-body systems: Theory and Experiment, Invited Lecture. In “Atom Optics and Space Physics”, Proc. of Course CLXVIII of the International School of Physics “Enrico Fermi”, Varenna, Italy, 2007; Arimondo, E., Ertmer, W., Schleich, W., Eds.; IOS: Amsterdam, The Netherlands, 2009. [Google Scholar]
- Damour, T.; Jaranowski, P.; Schafer, G. Hamiltonian of two spinning compact bodies with next-to-leading order gravitational spin-orbit coupling. Phys. Rev. D
**2008**, 77, 064032. [Google Scholar] [CrossRef] [Green Version] - Griffiths, D.J. Resource letter EM-1: Electromagnetic momentum. Am. J. Phys.
**2012**, 80, 7. [Google Scholar] [CrossRef] - Barnett, S.M. Comment on “Trouble with the Lorentz law of force: Incompatibility with special relativity and momentum conservation”. Phys. Rev. Lett.
**2013**, 110, 089402. [Google Scholar] [CrossRef] [PubMed] [Green Version] - O’Connell, R.F. Equations of Motion in Relativistic Gravity; Springer Fundamental Theories of Physics; Puetzfeld, D., Ed.; Springer: Berlin, Germany, 2015. [Google Scholar]
- Boyer, T.H. Concerning “hidden momentum”. Am. J. Phys.
**2008**, 76, 190. [Google Scholar] [CrossRef] [Green Version]

© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

O’Connell, R.F.
Rotation and Spin and Position Operators in Relativistic Gravity and Quantum Electrodynamics. *Universe* **2020**, *6*, 24.
https://doi.org/10.3390/universe6020024

**AMA Style**

O’Connell RF.
Rotation and Spin and Position Operators in Relativistic Gravity and Quantum Electrodynamics. *Universe*. 2020; 6(2):24.
https://doi.org/10.3390/universe6020024

**Chicago/Turabian Style**

O’Connell, Robert F.
2020. "Rotation and Spin and Position Operators in Relativistic Gravity and Quantum Electrodynamics" *Universe* 6, no. 2: 24.
https://doi.org/10.3390/universe6020024