Relativistic symmetry

2021 ◽  
pp. 8-30
Author(s):  
Iosif L. Buchbinder ◽  
Ilya L. Shapiro
Keyword(s):  
Group Theory ◽  
Lorentz Group ◽  
Irreducible Representations ◽  
Lorentz Transformations ◽  
Not Given ◽  
Poincare Group ◽  
Relativistic Symmetry ◽  
Tensor Representations ◽  
The Subject ◽  
Spinor Representations

This chapter discusses relativistic symmetry, starting from the Lorentz transformations. Basic notions of group theory are introduced before a more detailed discussion of the Lorentz and Poincaré groups is given. Tensor representations and spinor representations of the Lorentz group are described, although full proofs of the theorems are not given. The chapter ends with the irreducible representations of the Poincaré group. This chapter provides all of the necessary notions for group theory, although it is not intended to replace a textbook on the subject.

Group structure of superluminal Lorentz transformations and their representation

1989 ◽  
Vol 67 (7) ◽  
pp. 645-648
Author(s):  
H. C. Chandola ◽  
B. S. Rajput
Keyword(s):  
Lorentz Group ◽  
Group Structure ◽  
Lorentz Transformations ◽  
Poincaré Group ◽  
Poincare Group ◽  
Inertial Frames ◽  
Homogeneous Lorentz Group

Constructing the generalized superluminal Lorentz transformations for inertial frames with nonpreferred orientation, we have investigated their group theoretical and commutation properties. The extended homogeneous Lorentz group in terms of these transformations has been constructed and its generators have been derived; these have been shown to generate the ray representation of the resulting Poincaré group.

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.

Saving and Spending

2012 ◽  
Author(s):  
Lendol Calder
Keyword(s):  
Economic Development ◽  
Consumer Culture ◽  
Core Competencies ◽  
Economic Life ◽  
Business Success ◽  
Money Management ◽  
Not Given ◽  
History Of ◽  
The Subject ◽  
The Rich

Monetization, which describes the process whereby money became the dominant means of exchange in developing commercial societies, is an economic development whose profound social, political, and cultural consequences are not yet well understood. The monetization of household economic life elevated practices that once affected only the wealthy – Fan Li's ‘golden rules for business success’ – to core competencies of living, mandatory for everyone. Reflecting on the scholarship that has examined saving and spending, this article examines consumption and why historians of consumer culture have not given the financial affairs of consumers the attention the subject deserves. The historical work that has been done, though sparse, amply demonstrates the rich potential of the financial arts for generating significant problem areas for research. Few other subjects in the glittering universe of consumption lead more directly to the largest questions we can ask about desire, virtue, and the construction of the modern self. The article also considers the history of thrift, money management, and financialization.

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.

On the Dimension of an Irreducible Tensor Representation of the General Linear Group GL(d)

1970 ◽  
Vol 13 (3) ◽  
pp. 389-390
Author(s):  
J. A. J. Matthews ◽  
G. de B. Robinson
Keyword(s):  
Linear Group ◽  
General Linear Group ◽  
Irreducible Representations ◽  
Young Diagrams ◽  
Tensor Representation ◽  
Irreducible Tensor ◽  
Tensor Representations ◽  
Identity Representation ◽  
Raising Operator ◽  
Image Position

As has long been known, the irreducible tensor representations of GL(d) of rank n may be labeled by means of the irreducible representations of Sn, i.e., by means of the Young diagrams [λ], where λ1 + λ2 + … λr = n. We denote such a tensor representation by 〈λ〉. Using Young's raising operator Rij we can write [1, p. 42]1.1where the dot denotes the inducing process. For example, [3] . [2] is that representation of S5 induced by the identity representation of its subgroup S3 × S2.

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