Reduction of the electromagnetic vector potential to the irreducible representations of the inhomogeneous Lorentz group and manifestly covariant quantization with a positive-definite metric for the hilbert space

1966 ◽  
Vol 42 (4) ◽  
pp. 757-781 ◽  
Author(s):  
H. E. Moses
Keyword(s):  
Hilbert Space ◽  
Lorentz Group ◽  
Vector Potential ◽  
Positive Definite ◽  
Irreducible Representations ◽  
Covariant Quantization ◽  
Electromagnetic Vector Potential ◽  
Inhomogeneous Lorentz Group

Canonical representations for massless particles and zero-mass limits of the helicity representation

1965 ◽  
Vol 285 (1401) ◽  
pp. 287-296 ◽  
Keyword(s):  
Lorentz Group ◽  
Irreducible Representations ◽  
Two Dimensional ◽  
Euclidean Group ◽  
Positive Mass ◽  
Continuous Spin ◽  
Massless Particles ◽  
Zero Mass ◽  
Canonical Representations ◽  
Inhomogeneous Lorentz Group

The global forms of the unitary irreducible representations of the inhomogeneous Lorentz group corresponding to zero mass and finite or continuous spin are constructed by means of the little-group technique from those of the two-dimensional Euclidean group, and it is shown that these representations may be derived from the helicity representation for positive mass by taking suitable limits.

scholarly journals DERIVATION OF THE SOURCE-FREE MAXWELL AND GRAVITATIONAL RADIATION EQUATIONS BY GROUP THEORETICAL METHODS

2006 ◽  
Vol 15 (04) ◽  
pp. 505-519 ◽  
Author(s):  
JAIRZINHO RAMOS ◽  
ROBERT GILMORE
Keyword(s):  
Lorentz Group ◽  
Gravitational Radiation ◽  
Maxwell Equations ◽  
Free Field ◽  
Flat Space ◽  
Irreducible Representations ◽  
Theoretical Methods ◽  
Covariant Representations ◽  
Divergence Equations ◽  
Inhomogeneous Lorentz Group

We derive source-free Maxwell-like equations in flat space–time for any helicity j by comparing the transformation properties of the 2(2j+1) states that carry the manifestly covariant representations of the inhomogeneous Lorentz group with the transformation properties of the two helicity j states that carry the irreducible representations of this group. The set of constraints so derived involves a pair of curl equations and a pair of divergence equations. These reduce to the free-field Maxwell equations for j = 1 and the analogous equations coupling the gravito-electric and the gravito-magnetic fields for j = 2.

scholarly journals The Topological Origin of Quantum Randomness

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 581
Author(s):  
Stefan Heusler ◽  
Paul Schlummer ◽  
Malte S. Ubben
Keyword(s):  
High School ◽  
Hilbert Space ◽  
Group Theory ◽  
Lorentz Group ◽  
Quantum Physics ◽  
Time Development ◽  
Mathematical Structures ◽  
Stochastic Nature ◽  
Quantum Randomness ◽  
And Topology

What is the origin of quantum randomness? Why does the deterministic, unitary time development in Hilbert space (the ‘4π-realm’) lead to a probabilistic behaviour of observables in space-time (the ‘2π-realm’)? We propose a simple topological model for quantum randomness. Following Kauffmann, we elaborate the mathematical structures that follow from a distinction(A,B) using group theory and topology. Crucially, the 2:1-mapping from SL(2,C) to the Lorentz group SO(3,1) turns out to be responsible for the stochastic nature of observables in quantum physics, as this 2:1-mapping breaks down during interactions. Entanglement leads to a change of topology, such that a distinction between A and B becomes impossible. In this sense, entanglement is the counterpart of a distinction (A,B). While the mathematical formalism involved in our argument based on virtual Dehn twists and torus splitting is non-trivial, the resulting haptic model is so simple that we think it might be suitable for undergraduate courses and maybe even for High school classes.

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