scholarly journals Finite dimensional representations of the quantum Lorentz group

1992 ◽  
Vol 144 (3) ◽  
pp. 557-580 ◽  
Author(s):  
Mitsuhiro Takeuchi
Keyword(s):  
Lorentz Group ◽  
Finite Dimensional

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.

scholarly journals DIFFERENTIAL CALCULUS AND CONNECTIONS ON A QUANTUM PLANE AT A CUBIC ROOT OF UNITY

2000 ◽  
Vol 12 (02) ◽  
pp. 227-285 ◽  
Author(s):  
R. COQUEREAUX ◽  
A. O. GARCÍA ◽  
R. TRINCHERO
Keyword(s):  
Representation Theory ◽  
Quantum Group ◽  
Lorentz Group ◽  
Differential Calculus ◽  
Differential Algebra ◽  
Space Time ◽  
Quantum Plane ◽  
Root Of Unity ◽  
Finite Dimensional

We consider the algebra of N×N matrices as a reduced quantum plane on which a finite-dimensional quantum group ℋ acts. This quantum group is a quotient of [Formula: see text], q being an Nth root of unity. Most of the time we shall take N=3; in that case dim(ℋ)=27. We recall the properties of this action and introduce a differential calculus for this algebra: it is a quotient of the Wess–Zumino complex. The quantum group ℋ also acts on the corresponding differential algebra and we study its decomposition in terms of the representation theory of ℋ. We also investigate the properties of connections, in the sense of non commutative geometry, that are taken as 1-forms belonging to this differential algebra. By tensoring this differential calculus with usual forms over space-time, one can construct generalized connections with covariance properties with respect to the usual Lorentz group and with respect to a finite-dimensional quantum group.

SPINOR OPERATORS

1992 ◽  
Vol 07 (26) ◽  
pp. 6537-6553
Author(s):  
J. DE LA JARA F. ◽  
R. TABENSKY R.
Keyword(s):  
Lorentz Group ◽  
Finite Dimensional ◽  
The Family ◽  
Grassmann Variables

We define and study spinor operators for finite dimensional representations of the Lorentz group. The family of all such operators is given. Grassmann variables appear as one of their simplest realizations. Besides we establish a relation between spinor operators and four-vector operators, which simplifies greatly the theory of vector operators.

scholarly journals A natural extension of the conformal Lorentz group in a field theory context

2016 ◽  
Vol 31 (28n29) ◽  
pp. 1645041 ◽  
Author(s):  
András László
Keyword(s):  
Field Theory ◽  
Automorphism Group ◽  
Lorentz Group ◽  
Natural Extension ◽  
Mathematical Structure ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Finite Dimensional ◽  
Semisimple Part ◽  
Unital Associative Algebra

In this paper a finite dimensional unital associative algebra is presented, and its group of algebra automorphisms is detailed. The studied algebra can physically be understood as the creation operator algebra in a formal quantum field theory at fixed momentum for a spin 1/2 particle along with its antiparticle. It is shown that the essential part of the corresponding automorphism group can naturally be related to the conformal Lorentz group. In addition, the non-semisimple part of the automorphism group can be understood as “dressing” of the pure one-particle states. The studied mathematical structure may help in constructing quantum field theories in a non-perturbative manner. In addition, it provides a simple example of circumventing Coleman–Mandula theorem using non-semisimple groups, without SUSY.

Non-Linear Dynamics of Cardiovascular Variability Signals

1994 ◽  
Vol 33 (01) ◽  
pp. 81-84 ◽  
Author(s):  
S. Cerutti ◽  
S. Guzzetti ◽  
R. Parola ◽  
M.G. Signorini
Keyword(s):  
Dimensional Space ◽  
Severe Heart Failure ◽  
Parametric Models ◽  
Deterministic Systems ◽  
Finite Dimensional ◽  
Return Maps ◽  
Pathological Conditions ◽  
Non Linear ◽  
Space State

Abstract:Long-term regulation of beat-to-beat variability involves several different kinds of controls. A linear approach performed by parametric models enhances the short-term regulation of the autonomic nervous system. Some non-linear long-term regulation can be assessed by the chaotic deterministic approach applied to the beat-to-beat variability of the discrete RR-interval series, extracted from the ECG. For chaotic deterministic systems, trajectories of the state vector describe a strange attractor characterized by a fractal of dimension D. Signals are supposed to be generated by a deterministic and finite dimensional but non-linear dynamic system with trajectories in a multi-dimensional space-state. We estimated the fractal dimension through the Grassberger and Procaccia algorithm and Self-Similarity approaches of the 24-h heart-rate variability (HRV) signal in different physiological and pathological conditions such as severe heart failure, or after heart transplantation. State-space representations through Return Maps are also obtained. Differences between physiological and pathological cases have been assessed and generally a decrease in the system complexity is correlated to pathological conditions.

A closer look at the stable completion

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Inverse Limit ◽  
Unit Vector ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Compact Sets ◽  
Linear Topology ◽  
Topological Embedding ◽  
Definable Set ◽  
Finite Dimensional ◽  
The One

This chapter introduces the concept of stable completion and provides a concrete representation of unit vector Mathematical Double-Struck Capital A superscript n in terms of spaces of semi-lattices, with particular emphasis on the frontier between the definable and the topological categories. It begins by constructing a topological embedding of unit vector Mathematical Double-Struck Capital A superscript n into the inverse limit of a system of spaces of semi-lattices L(Hsubscript d) endowed with the linear topology, where Hsubscript d are finite-dimensional vector spaces. The description is extended to the projective setting. The linear topology is then related to the one induced by the finite level morphism L(Hsubscript d). The chapter also considers the condition that if a definable set in L(Hsubscript d) is an intersection of relatively compact sets, then it is itself relatively compact.

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