Note on Lorentz Invariants

Basic experimental evidence is sketched: the black body radiation spectrum, the photoeffect, Compton scattering and electron diffraction; the Bohr model of the atom. Quantum mechanics is reviewed using the Copenhagen interpretation: eigenstates, observables, hermitian operators and expectation values are explained. Wave-particle duality, Schrödinger’s equation, and expressions for particle density and current are described. The uncertainty principle, the collapse of the wavefunction, Schrödinger’s cat and the no-cloning theorem are discussed. Dirac delta functions and the usage of wavepackets are explained. An introduction to state vectors in Hilbert space and the bra-ket notation is given. Abstracts of special relativity and Lorentz invariants follow. Minimal electromagnetic coupling and the gauge transformations are explained.

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Lorentz invariants in relativistic fluid dynamics and thermodynamics in non-vacuo

Astrophysics and Space Science ◽

10.1007/bf00661322 ◽

1970 ◽

Vol 9 (2) ◽

pp. 298-303

Author(s):

M. Z. v. Krzywoblocki

Keyword(s):

Fluid Dynamics ◽

Relativistic Fluid Dynamics ◽

Relativistic Fluid ◽

Lorentz Invariants

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On Lorentz invariants in relativistic magnetic reconnection

Physics of Plasmas ◽

10.1063/1.4961431 ◽

2016 ◽

Vol 23 (8) ◽

pp. 082903 ◽

Cited By ~ 6

Author(s):

Shu-Di Yang ◽

Xiao-Gang Wang

Keyword(s):

Magnetic Reconnection ◽

Lorentz Invariants

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The photon element units and their relativistic properties

Physics Essays ◽

10.4006/0836-1398-33.1.38 ◽

2020 ◽

Vol 33 (1) ◽

pp. 38-45

Author(s):

Brian B. K. Min

Keyword(s):

Gravitational Constant ◽

Element Model ◽

Electrical Charge ◽

Speed Of Light ◽

Planck Constant ◽

Natural Basis ◽

Mass Unit ◽

Planck Mass ◽

Time Length ◽

Lorentz Invariants

A set of natural units is determined from the “photon element” model of light, the outcome of an extended Compton analysis. In terms of these units, the speed of light and the electrical and Boltzmann constants are, respectively, on the order of unity, but the Planck constant is ∼1027 or greater and gravitational constant ∼10−59 or greater. This makes the photon element units less convenient than the Planck units. With the mass unit that is only ∼10−43 of the Planck mass, however, the photon element units can correspond better to physical realities than the Planck units. For the spacetime, a photon element forms a set of unit base vectors, a natural basis that is Lorentz covariant. There an analysis shows that (1) of the above five universal constants all are Lorentz invariants except the gravitational constant, and (2) of the five natural units (time, length, mass, electrical charge, and temperature,) only the electrical charge is a Lorentz invariant.

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Theory and applications of parton pseudodistributions

International Journal of Modern Physics A ◽

10.1142/s0217751x20300021 ◽

2020 ◽

Vol 35 (05) ◽

pp. 2030002 ◽

Cited By ~ 4

Author(s):

A. V. Radyushkin

Keyword(s):

Time Distribution ◽

Valence Quark ◽

Basic Feature ◽

Distribution Functions ◽

Ultraviolet Divergence ◽

Parton Distribution Functions ◽

Parton Distributions ◽

Gauge Link ◽

Generalized Parton Distributions ◽

Lorentz Invariants

We review the basic theory of the parton pseudodistributions approach and its applications to lattice extractions of parton distribution functions. The crucial idea of the approach is the realization that the correlator [Formula: see text] of the parton fields is a function [Formula: see text] of Lorentz invariants [Formula: see text], the Ioffe time, and the invariant interval [Formula: see text]. This observation allows to extract the Ioffe-time distribution [Formula: see text] from Euclidean separations [Formula: see text] accessible on the lattice. Another basic feature is the use of the ratio [Formula: see text], that allows to eliminate artificial ultraviolet divergence generated by the gauge link for spacelike intervals. The remaining [Formula: see text]-dependence of the reduced Ioffe-time distribution [Formula: see text] corresponds to perturbative evolution, and can be converted into the scale-dependence of parton distributions [Formula: see text] using matching relations. The [Formula: see text]-dependence of [Formula: see text] governs the [Formula: see text]-dependence of parton densities [Formula: see text]. The perturbative evolution was successfully observed in exploratory quenched lattice calculation. The analysis of its precise data provides a framework for extraction of parton densities using the pseudodistributions approach. It was used in the recently performed calculations of the nucleon and pion valence quark distributions. We also discuss matching conditions for the pion distribution amplitude and generalized parton distributions, the lattice studies of which are now in progress.

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