On the irreducible representations of the Lorentz group (original) (raw)
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Infinite Irreducible Representations of the Lorentz Group
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 1947
It is shown that corresponding to every pair of complex numbers κ , κ* for which 2( κ - κ* ) is real and integral, there exists, in general, one irreducible representation D κ, κ* , of the Lorentz group. However, if 4 κ , 4 κ* are both real and integral there are two representations D + κ, κ* and D - k, k* associated to the pair ( k, κ* ). All these representations are infinite except D - κ, κ* which is finite if 2 κ , 2 κ* are both integral. For suitable values of ( κ, κ* ), D κ, κ* or D + κ, κ* is unitary. U and B matrices similar to those given by Dirac (1936) and Fierz (1939) are introduced for these infinite representations. The extension of Dirac’s expansor formalism to cover half-integral spins is given. These new quantities, which are called expinors, bear the same relation to spinors as Dirac’s expansors to tensors. It is shown that they can be used to describe the spin properties of a particle in accordance with the principles of quantum mechanics.
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Journal of Physics A, 2008
In this paper we develop an algebraic technique for building relativistic models in the framework of the direct-interaction theories. The interacting mass operator M in the Bakamjian-Thomas construction is related to a quadratic Casimir operator C of non-compact group G. As a consequence the S matrix can be gained from an intertwining relation between Weyl-equivalent representation of G. The method is illustrated by explicit application to a model with SO(3, 1) dynamical symmetry.
The Representation Theory of the Lorentz Group
In this paper, we introduce the mathematical formalism of representation theory and its application to physics. In particular, we discuss the proper orthochronous Lorentz group SO + (1, 3) with the goal of classifying all finite dimensional irreducible representations of its Lie algebra. With this in hand, we discuss several representations of vital importance in the theory of special relativity.
A Bargmann-Wightman-Wigner-type quantum field theory
Physics Letters B, 1993
We show that the (j, 0) ⊕ (0, j) representation space associated with massive particles is a concrete realisation of a quantum field theory, envisaged many years ago by Bargmann, Wightman and Wigner, in which bosons and antibosons have opposite relative intrinsic parities. Demonstration of the result requires a careful ab initio study of the (j, 0) ⊕ (0, j) representation space for massive particles, introducing a wave equation with well defined transformation properties under C, P and T, and addressing the issue of nonlocality required of such a theory by the work of Lee and Wick. * This work was done under the auspices of the U. S. Department of Energy. 1 While most of the specific theoretical questions of hadronic structure and interactions must be decided within the framework of quantum chromodynamics, there remain certain aspects which depend only on the constraints imposed by Poincaré covariance. Many years ago, Wigner [1] provided the basic framework for the Poincaré covariant considerations. The essential elements of these considerations are the kinematical symmetries (continuous Poincaré symmetries and space time reflections) and the behaviour of quantum mechanical states under these transformations. From these follow certain general characteristics such as equal masses and relative intrinsic parities of particle and antiparticle pairs. Such an approach may therefore have utility in establishing the general framework of an effective field theory of hadrons, and such considerations have, in fact, motivated the pioneering work of Weinberg [2] on field theories in a specific Lorentz group representation, (j, 0) ⊕ (0, j), of spin-j particles, as well as recent extensions of this work [3].
Annales de la Fondation Louis de Broglie
Developing recently proposed constructions for the description of particles in the (1/2, 0) ⊕ (0, 1/2) representation space, we derive the second-order equations. The similar ones were proposed in the sixties and the seventies in order to understand the nature of various mass and spin states in the representations of the O(4, 2) group. We give some additional insights into this problem. The used procedure can be generalized for arbitrary number of lepton families.
Open Systems & Information Dynamics, 1993
In order to provide a general framework for applications of nonstandard analysis to quantum physics, the hyperfinite Heisenbeig group, which is a finite Heisenberg group in nonstandard universe, is formulated and its unitary representations are examined. The ordinary Schr\"odinger representation of the Heisenberg group is obtained by a suitable standardization of its internal representation. As an application, a nonstandard-analytical proof of noncommutative Parseval's identity based on the orthogonality relations for unitary representations of finite groups is shown.
Harmonic Analysis on the Quantum Lorentz Group
Communications in Mathematical Physics, 1999
This work begins with a review of complexi cation and reali cation of Hopf algebras. We emphasize the notion of multiplier Hopf algebras for the description of di erent classes of functions (compact supported, bounded, unbounded) on complex quantum groups and the construction of the associated left and right Haar measure. Using a continuation of 6j symbols of SUq (2) with complex spins, we give a new description of the unitary representations of SLq (2; C ) R and nd explicit expressions for the characters of SLq (2; C ) R . The major theorem of this article is the Plancherel theorem for the Quantum Lorentz Group.