scholarly journals Relativistic Inversion, Invariance and Inter-Action

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1117
Author(s):  
Martin B. van der Mark ◽  
John G. Williamson
Keyword(s):  
Maxwell Equations ◽  
Relativistic Quantum ◽  
Rest Mass ◽  
Natural Extension ◽  
Basis Set ◽  
Relativistic Quantum Mechanics ◽  
Dimensional Algebra ◽  
Dirac Algebra ◽  
First Order ◽  
Vector Algebra

A general formula for inversion in a relativistic Clifford–Dirac algebra has been derived. Identifying the base elements of the algebra as those of space and time, the first order differential equations over all quantities proves to encompass the Maxwell equations, leads to a natural extension incorporating rest mass and spin, and allows an integration with relativistic quantum mechanics. Although the algebra is not a division algebra, it parallels reality well: where division is undefined turns out to correspond to physical limits, such as that of the light cone. The divisor corresponds to invariants of dynamical significance, such as the invariant interval, the general invariant quantities in electromagnetism, and the basis set of quantities in the Dirac equation. It is speculated that the apparent 3-dimensionality of nature arises from a beautiful symmetry between the three-vector algebra and each of four sets of three derived spaces in the full 4-dimensional algebra. It is conjectured that elements of inversion may play a role in the interaction of fields and matter.

OCTONIC FIRST-ORDER EQUATIONS OF RELATIVISTIC QUANTUM MECHANICS

2009 ◽  
Vol 24 (22) ◽  
pp. 4157-4167 ◽  
Author(s):  
VICTOR L. MIRONOV ◽  
SERGEY V. MIRONOV
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Maxwell Equations ◽  
Relativistic Quantum ◽  
Order Equation ◽  
Second Order ◽  
Relativistic Quantum Mechanics ◽  
Quantum Fields ◽  
First Order ◽  
Spatial Operators

We demonstrate a generalization of relativistic quantum mechanics using eight-component octonic wave function and octonic spatial operators. It is shown that the second-order equation for octonic wave function describing particles with spin 1/2 can be reformulated in the form of a system of first-order equations for quantum fields, which is analogous to the system of Maxwell equations for the electromagnetic field. It is established that for the special types of wave functions the second-order equation can be reduced to the single first-order equation analogous to the Dirac equation. At the same time it is shown that this first-order equation describes particles, which do not have quantum fields.

scholarly journals Current Developments in the Relativistic Quantum Mechanics of Atoms and Molecules

1986 ◽  
Vol 39 (5) ◽  
pp. 649 ◽  
Author(s):  
IP Grant
Keyword(s):  
Electronic Structure ◽  
Quantum Mechanics ◽  
Atomic Structure ◽  
Quantum Electrodynamics ◽  
Relativistic Quantum ◽  
Finite Basis ◽  
Basis Set ◽  
Structure Theory ◽  
Relativistic Quantum Mechanics ◽  
Atoms And Molecules

Current work in relativistic quantum mechanics by the author and his associates focusses on four topics: atomic structure theory using the GRASP package (Dyall 1986); extension of GRASP to handle electron continuum processes; the relation of quantum electrodynamics and relativistic quantum mechanics of atoms and molecules; and development of methods using finite basis set expansions for studying electronic structure of atoms and molecules. This paper covers only the last three topics, giving emphasis to growing points and outstanding difficulties.

scholarly journals Derivation of the Local Lorentz Gauge Transformation of a Dirac Spinor Field in Quantum Einstein-Cartan Theory

2019 ◽  
Author(s):  
Rainer Kühne
Keyword(s):  
General Relativity ◽  
Quantum Mechanics ◽  
Quantum Electrodynamics ◽  
Relativistic Quantum ◽  
Classical Mechanics ◽  
Relativistic Quantum Mechanics ◽  
Dirac Spinor ◽  
Dirac Algebra ◽  
Gauge Transformations ◽  
Cartan Theory

I examine the groups which underly classical mechanics, non-relativistic quantum mechanics, special relativity, relativistic quantum mechanics, quantum electrodynamics, quantum flavourdynamics, quantum chromodynamics, and general relativity. This examination includes the rotations SO(2) and SO(3), the Pauli algebra, the Lorentz transformations, the Dirac algebra, and the U(1), SU(2), and SU(3) gauge transformations. I argue that general relativity must be generalized to Einstein-Cartan theory, so that Dirac spinors can be described within the framework of gravitation theory.

scholarly journals The passage of fast electrons and the theory of cosmic showers

1937 ◽  
Vol 159 (898) ◽  
pp. 432-458 ◽  
Keyword(s):  
Quantum Mechanics ◽  
Energy Loss ◽  
Large Fraction ◽  
Relativistic Quantum ◽  
Rest Mass ◽  
Cosmic Ray ◽  
Relativistic Quantum Mechanics ◽  
Fast Electrons ◽  
Large Probability ◽  
The Moment

It is well known that according to relativistic quantum mechanics, electrons and positrons with energy large compared with their rest mass have a very large probability when passing through the field of a nucleus of losing a large fraction of their energy in one process by emitting radiation. Hard quanta have a correspondingly large probability of creating electron pairs. Until recently it was believed that the direct measurements of Anderson and Neddermeyer on the energy loss of fast electrons showed that though this energy loss by radiation existed, it was much smaller for energies greater than about 10 8 e-volts than that theoretically predicted, and it was therefore assumed that the present quantum mechanics began to fail for energies greater than about this value. More recent experiments by Anderson and Neddermeyer (1936) have, however, led them to revise their former conclusions, and their new and more accurate experiments show that up to energies of 300 million e-volts (the highest energies measured in their experiments) and probably higher, the experimentally measured energy loss of fast electrons is in agreement with that predicted theoretically. In fact, one may say that at the moment there are no direct measurements of energy loss by fast electrons which conclusively prove a breakdown of the theory. This is particularly satisfactory, inasmuch as the theoretical reasons for expecting a breakdown of the theoretical formulae at energies greater than about 137 mc 2 , namely the neglect of the classical “radius” of the electron, have been shown by v. Weizsäcker (1934) and Williams (1934) to be unfounded. Under these circumstances, and in view of the experimental evidence mentioned above, it is reasonable as a working hypothesis to assume the theoretical formulae for energy loss and pair creation to be valid for all energies, however high, and to work out the consequences which result from them. It is our aim to deduce results which can be compared directly with cosmic ray experiments and which will then allow one to decide whether or not the theory fails for extremely high energies, and in the latter case, at what point the failure begins.

scholarly journals Unitary quantum lattice simulations for Maxwell equations in vacuum and in dielectric media

2020 ◽  
Vol 86 (5) ◽  
Author(s):  
George Vahala ◽  
Linda Vahala ◽  
Min Soe ◽  
Abhay K. Ram
Keyword(s):  
Wave Packet ◽  
Maxwell Equations ◽  
Quantum Computer ◽  
Relativistic Quantum ◽  
Relativistic Quantum Mechanics ◽  
Spinor Representation ◽  
Quantum Lattice ◽  
Dielectric Media ◽  
Uniform Medium ◽  
Lattice Algorithm

Utilizing the similarity between the spinor representation of the Dirac and the Maxwell equations that has been recognized since the early days of relativistic quantum mechanics, a quantum lattice algorithm (QLA) representation of unitary collision-stream operators of Maxwell's equations is derived for both homogeneous and inhomogeneous media. A second-order accurate 4-spinor scheme is developed and tested successfully for two-dimensional (2-D) propagation of a Gaussian pulse in a uniform medium whereas for normal (1-D) incidence of an electromagnetic Gaussian wave packet onto a dielectric interface requires 8-component spinors because of the coupling between the two electromagnetic polarizations. In particular, the well-known phase change, field amplitudes and profile widths are recovered by the QLA asymptotic profiles without the imposition of electromagnetic boundary conditions at the interface. The QLA simulations yield the time-dependent electromagnetic fields as the wave packet enters and straddles the dielectric boundary. QLA involves unitary interleaved non-commuting collision and streaming operators that can be coded onto a quantum computer: the non-commutation being the very reason why one perturbatively recovers the Maxwell equations.

UPPER AND LOWER LIMITS ON THE EIGENVALUES OF THE SPINLESS SALPETER EQUATION

2005 ◽  
Vol 20 (30) ◽  
pp. 7277-7284 ◽  
Author(s):  
FABIAN BRAU
Keyword(s):  
Differential Equation ◽  
Relativistic Quantum ◽  
Energy Levels ◽  
Relativistic Quantum Mechanics ◽  
Coupling Constant ◽  
Logarithmic Potentials ◽  
First Order ◽  
Cornell Potential ◽  
Meson Spectroscopy ◽  
Lower Limits

In the context of relativistic quantum mechanics, we obtain a nonlinear first order differential equation for the energy as a function of the coupling constant of a central potential. This differential equation is only exact for power law and logarithmic potentials in the massless limit. For other potentials, we discuss under which conditions the differential equation yields rigorous upper and lower limits on the value of energy levels. These results are applied to the Cornell potential used in meson spectroscopy. We also show that the method applies to noncentral potentials.

scholarly journals Derivation of the Local Lorentz Gauge Transformation of a Dirac Spinor Field in Quantum Einstein-Cartan Theory

2018 ◽  
Author(s):  
Rainer Kühne
Keyword(s):  
General Relativity ◽  
Quantum Mechanics ◽  
Quantum Electrodynamics ◽  
Relativistic Quantum ◽  
Classical Mechanics ◽  
Relativistic Quantum Mechanics ◽  
Dirac Spinor ◽  
Dirac Algebra ◽  
Gauge Transformations ◽  
Cartan Theory

I examine the groups which underly classical mechanics, non-relativistic quantum mechanics, special relativity, relativistic quantum mechanics, quantum electrodynamics, quantum flavourdynamics, quantum chromodynamics, and general relativity. This examination includes the rotations SO(2) and SO(3), the Pauli algebra, the Lorentz transformations, the Dirac algebra, and the U(1), SU(2), and SU(3) gauge transformations. I argue that general relativity must be generalized to Einstein-Cartan theory, so that Dirac spinors can be described within the framework of gravitation theory.

scholarly journals SEDEONIC GENERALIZATION OF RELATIVISTIC QUANTUM MECHANICS

2009 ◽  
Vol 24 (32) ◽  
pp. 6237-6254 ◽  
Author(s):  
VICTOR L. MIRONOV ◽  
SERGEY V. MIRONOV
Keyword(s):  
Quantum Mechanics ◽  
Relativistic Quantum ◽  
Space Time ◽  
Second Order ◽  
Wave Functions ◽  
Relativistic Quantum Mechanics ◽  
Noncommutative Space ◽  
Einstein Relation ◽  
First Order ◽  
Energy And Momentum

We represent sixteen-component values "sedeons," generating associative noncommutative space–time algebra. We demonstrate a generalization of relativistic quantum mechanics using sedeonic wave functions and sedeonic space–time operators. It is shown that the sedeonic second-order equation for the sedeonic wave function, obtained from the Einstein relation for energy and momentum, describes particles with spin 1/2. We showed that the sedeonic second-order wave equation can be reformulated in the form of the system of the first-order Maxwell-like equations for the massive fields. We proposed the sedeonic first-order equations analogous to the Dirac equation, which differ in space–time properties and describe several types of massive and massless particles. In particular we proposed four different equations, which could describe four types of neutrinos.

BILINEAR FORM AND SOLITON SOLUTIONS FOR THE COUPLED NONLINEAR KLEIN–GORDON EQUATIONS

2012 ◽  
Vol 26 (15) ◽  
pp. 1250057
Author(s):  
HE LI ◽  
XIANG-HUA MENG ◽  
BO TIAN
Keyword(s):  
Field Theory ◽  
Bilinear Form ◽  
Gordon Equation ◽  
Relativistic Quantum ◽  
Soliton Solutions ◽  
Relativistic Quantum Mechanics ◽  
Klein Gordon ◽  
Truncated Painlevé Expansion ◽  
Klein Gordon Equation ◽  
Singular Manifolds

With the coupling of a scalar field, a generalization of the nonlinear Klein–Gordon equation which arises in the relativistic quantum mechanics and field theory, i.e., the coupled nonlinear Klein–Gordon equations, is investigated via the Hirota method. With the truncated Painlevé expansion at the constant level term with two singular manifolds, the coupled nonlinear Klein–Gordon equations are transformed to a bilinear form. Starting from the bilinear form, with symbolic computation, we obtain the N-soliton solutions for the coupled nonlinear Klein–Gordon equations.

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