Mechanical models for Lorentz group representations

1993 ◽  
Vol 23 (2) ◽  
pp. 245-260 ◽  
Author(s):  
N. Mukunda
Keyword(s):  
Lorentz Group ◽  
Group Representations ◽  
Mechanical Models ◽  
Lorentz Group Representations

scholarly journals Lorentz Group Projector Technique for Decomposing Reducible Representations and Applications to High Spins

Universe ◽  
2019 ◽  
Vol 5 (8) ◽  
pp. 184 ◽  
Author(s):  
Victor Miguel Banda Guzmán ◽  
Mariana Kirchbach
Keyword(s):  
Lorentz Group ◽  
Degrees Of Freedom ◽  
Dirac Spinor ◽  
Product Spaces ◽  
Group Representations ◽  
Finite Dimensional Representation ◽  
Finite Dimensional ◽  
Irreducible Components ◽  
Carrier Space ◽  
Lorentz Group Representations

The momentum-independent Casimir operators of the homogeneous spin-Lorentz group are employed in the construction of covariant projector operators, which can decompose anyone of its reducible finite-dimensional representation spaces into irreducible components. One of the benefits from such operators is that any one of the finite-dimensional carrier spaces of the Lorentz group representations can be equipped with Lorentz vector indices because any such space can be embedded in a Lorentz tensor of a properly-designed rank and then be unambiguously found by a projector. In particular, all the carrier spaces of the single-spin-valued Lorentz group representations, which so far have been described as 2 ( 2 j + 1 ) column vectors, can now be described in terms of Lorentz tensors for bosons or Lorentz tensors with the Dirac spinor component, for fermions. This approach facilitates the construct of covariant interactions of high spins with external fields in so far as they can be obtained by simple contractions of the relevant S O ( 1 , 3 ) indices. Examples of Lorentz group projector operators for spins varying from 1 / 2 –2 and belonging to distinct product spaces are explicitly worked out. The decomposition of multiple-spin-valued product spaces into irreducible sectors suggests that not only the highest spin, but all the spins contained in an irreducible carrier space could correspond to physical degrees of freedom.

Elements of Group Theory

2021 ◽  
pp. 51-110
Author(s):  
J. Iliopoulos ◽  
T.N. Tomaras
Keyword(s):  
Group Theory ◽  
Lie Groups ◽  
Lorentz Group ◽  
Unitary Representations ◽  
Compact Groups ◽  
Mathematical Language ◽  
Group Representations ◽  
Infinite Groups ◽  
Infinite Dimensional ◽  
Symmetry Properties

The mathematical language which encodes the symmetry properties in physics is group theory. In this chapter we recall the main results. We introduce the concepts of finite and infinite groups, that of group representations and the Clebsch–Gordan decomposition. We study, in particular, Lie groups and Lie algebras and give the Cartan classification. Some simple examples include the groups U(1), SU(2) – and its connection to O(3) – and SU(3). We use the method of Young tableaux in order to find the properties of products of irreducible representations. Among the non-compact groups we focus on the Lorentz group, its relation with O(4) and SL(2,C), and its representations. We construct the space of physical states using the infinite-dimensional unitary representations of the Poincaré group.

scholarly journals CLASSIFYING REPORTED AND "MISSING" RESONANCES ACCORDING TO THEIR P and C PROPERTIES

2000 ◽  
Vol 15 (10) ◽  
pp. 1435-1451 ◽  
Author(s):  
M. KIRCHBACH
Keyword(s):  
Hilbert Space ◽  
Spin States ◽  
Particle Data Group ◽  
Group Representations ◽  
Opposite Parity ◽  
Quantum Numbers ◽  
Lower Spin ◽  
Missing Resonances ◽  
Lorentz Group Representations ◽  
Data Group

The Hilbert space ℋ3q of the three quarks with one excited quark is decomposed into Lorentz group representations. It is shown that the quantum numbers of the reported and "missing" resonances fall apart and populate distinct representations that differ by their parity or/and charge conjugation properties. In this way, reported and "missing" resonances become distinguishable. For example, resonances from the full listing reported by the Particle Data Group are accommodated by Rarita–Schwinger (RS) type representations [Formula: see text] with k=1, 3, and 5, the highest spin states being J=3/2-, 7/2+, and 11/2+, respectively. In contrast to this, most of the "missing" resonances fall into the opposite parity RS fields of highest-spins 5/2-, 5/2+, and 9/2+, respectively. Rarita–Schwinger fields with physical resonances as lower-spin components can be treated as a whole without imposing auxiliary conditions on them. Such fields do not suffer the Velo–Zwanziger problem but propagate causally in the presence of electromagnetic fields. The pathologies associated with RS fields arise basically because of the attempt to use them to describe isolated spin-J=k+½ states, rather than multispin-parity clusters. The positions of the observed RS clusters and their spacing are well explained trough the interplay between the rotational-like [Formula: see text]-rule and a Balmer-like [Formula: see text]-behavior.

Interactive Segmentation of Scanned Mechanical Models Based on Quadratic Surfaces Fitting

2019 ◽  
Vol 31 (7) ◽  
pp. 1210
Author(s):  
Zhongqi Wu ◽  
Jianwei Guo ◽  
Jun Xiao ◽  
Xiangyong Zeng ◽  
Ying Wang ◽  
...  
Keyword(s):  
Interactive Segmentation ◽  
Mechanical Models ◽  
Quadratic Surfaces
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