Relativistic Position Operator for Free Particles

We have revisited the Dirac theory in [Formula: see text] and [Formula: see text] dimensions by using the covariant representation of the parity-extended Poincaré group in their native dimensions. The parity operator plays a crucial role in deriving wave equations in both theories. We studied two position operators, a canonical one and a covariant one that becomes the particle position operator projected onto the particle subspace. In [Formula: see text] dimensions the particle position operator, not the canonical position operator, provides the conserved Lorentz generator. The mass moment defined by the canonical position operator needs an additional unphysical spin-like operator to become the conserved Lorentz generator in [Formula: see text] dimensions. In [Formula: see text] dimensions, the sum of the orbital angular momentum given by the canonical position operator and the spin angular momentum becomes a constant of motion. However, orbital and spin angular momentum do not conserve separately. On the other hand the orbital angular momentum given by the particle position operator and its corresponding spin angular momentum become a constant of motion separately.

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The relativistic position operator at subatomic level

Il Nuovo Cimento A ◽

10.1007/bf02753034 ◽

1967 ◽

Vol 49 (2) ◽

pp. 285-290 ◽

Cited By ~ 18

Author(s):

L. Castell

Keyword(s):

Position Operator ◽

Relativistic Position

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On the Origin of Four Pi in Physics

JOURNAL OF ADVANCES IN PHYSICS ◽

10.24297/jap.v11i8.212 ◽

2016 ◽

pp. 3994-4013

Author(s):

Aaron Hanken

Keyword(s):

Black Hole ◽

Surface Area ◽

Single Particles ◽

Pythagorean Triples ◽

Free Particles ◽

Close Relationship ◽

Dimensional Parameters ◽

Gaussian Surface ◽

Planck Energy ◽

The Universe

We find the highest symmetry between the fields intrinsic to free particles (free particles having only mass, charge and spin), and show these fields symmetries and their close relationship to force and entropy. The Boltzmann Constant is equal to the natural entropy, in that it is The Planck Energy over The Planck Temperature. This completes a needed symmetry in The Bekenstein-Hawking Entropy. Upon substitution of Planck Units into The Schwarzschild Radius, we find that the mass and radius of any black hole define both the gravitational constant and the natural force. We find that the Gaussian Surface area about a particle is equal to the surface area of an equally massed black hole if we define the gravitational field of that particle to be the quotient of The Planck Force and the particles mass. By these simple substitutions we find that gravity is quantized in units of surface entropy. We also find Pythagorean Triples are resting within the dimensional parameters of Special Relativity, and show this to be the dimensional aspects of single particles observing one another, coupled with the intrinsic Hubble nature of the universe.

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The Standard Theory

10.1093/oso/9780198805175.003.0006 ◽

2017 ◽

Author(s):

John Iliopoulos

Keyword(s):

Gauge Theory ◽

Invariant Theory ◽

Standard Theory ◽

Strong Interactions ◽

Gauge Invariant ◽

Electromagnetic Interactions ◽

Protons And Neutrons ◽

Free Particles ◽

Theoretical Predictions ◽

Theory And Experiment

All ingredients of the previous chapters are combined in order to build a gauge invariant theory of the interactions among the elementary particles. We start with a unified model of the weak and the electromagnetic interactions. The gauge symmetry is spontaneously broken through the BEH mechanism and we identify the resulting BEH boson. Then we describe the theory known as quantum chromodynamics (QCD), a gauge theory of the strong interactions. We present the property of confinement which explains why the quarks and the gluons cannot be extracted out of the protons and neutrons to form free particles. The last section contains a comparison of the theoretical predictions based on this theory with the experimental results. The agreement between theory and experiment is spectacular.

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Classical Kinetic Theory

10.1093/oso/9780198797241.003.0003 ◽

2018 ◽

Author(s):

Klaus Morawetz

Keyword(s):

Kinetic Equation ◽

Equation Of State ◽

Mean Field ◽

Ideal Gas ◽

Hydrodynamic Equations ◽

Free Particles ◽

Internal Mechanism ◽

The Mean ◽

Energy Gains ◽

Non Locality

The classical non-ideal gas shows that the two original concepts of the pressure based of the motion and the forces have eventually developed into drift and dissipation contributions. Collisions of realistic particles are nonlocal and non-instant. A collision delay characterizes the effective duration of collisions, and three displacements, describe its effective non-locality. Consequently, the scattering integral of kinetic equation is nonlocal and non-instant. The non-instant and nonlocal corrections to the scattering integral directly result in the virial corrections to the equation of state. The interaction of particles via long-range potential tails is approximated by a mean field which acts as an external field. The effect of the mean field on free particles is covered by the momentum drift. The effect of the mean field on the colliding pairs causes the momentum and the energy gains which enter the scattering integral and lead to an internal mechanism of energy conversion. The entropy production is shown and the nonequilibrium hydrodynamic equations are derived. Two concepts of quasiparticle, the spectral and the variational one, are explored with the help of the virial of forces.

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The stability of a gravitating ring of free particles

Astronomical and Astrophysical Transactions ◽

10.1080/10556799508203275 ◽

1995 ◽

Vol 7 (4) ◽

pp. 279-281

Author(s):

V. A. Antonov ◽

A. A. Vakhidov

Keyword(s):

Free Particles ◽

The Stability

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The application of the Whitehead theory of relativity to non-static, spherically symmetrical systems

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1954.0092 ◽

1954 ◽

Vol 222 (1151) ◽

pp. 509-526 ◽

Cited By ~ 6

Keyword(s):

Theory Of Relativity ◽

General Theory Of Relativity ◽

Current Standard ◽

World Model ◽

Basic Principles ◽

Density Distributions ◽

Free Particles ◽

Static Systems ◽

Solar Field ◽

Good Agreement

In a recent paper Synge (1952) reformulated in current standard notation the basic principles of the theory of relativity of A. N. Whitehead and investigated the gravitational properties of a spherically symmetrical static distribution of matter. He applied this theory to the solar field and verified the result previously indicated by Whitehead (1922) that the theory is in good agreement with the local tests satisfied by Einstein’s general theory of relativity. The present paper extends the theory of Whitehead to non-static systems whose velocity and density distributions possess complete spherical symmetry about a spatial origin in a Galilean frame of reference. In particular, a uniformly expanding homogeneous world-model is constructed which differs significantly both from Milne’s model and that of special relativity. The motion of free particles and photons in this model is investigated and a formula for the law of red-shifts is derived.

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