Cromlech, menhirs and celestial sphere: an unusual representation of the Lorentz group (original) (raw)
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arXiv:math-ph/0108022, 2001
Finite-dimensional representations of the proper orthochronous Lorentz group are studied in terms of spinor representations of the Clifford algebras. The Clifford algebras are understood as an 'algebraic covering' of a full system of the finite-dimensional representations of the Lorentz group. Space-time discrete symmetries P , T and P T , represented by fundamental automorphisms of the Clifford algebras , are defined on all the representation spaces. Real, complex, quaternionic and octonionic representations of the Lorentz group are considered. Physical fields of the different types are formulated within such representations. The Atiyah-Bott-Shapiro periodicity is defined on the Lorentz group. It is shown that modulo 2 and modulo 8 periodicities of the Clifford algebras allow to take a new look at the de Broglie-Jordan neutrino theory of light and the Gell-Mann-Ne'emann eightfold way in particle physics. On the representation spaces the charge conjugation C is represented by a pseudoautomorphism of the complex Clifford algebra. Quotient representations of the Lorentz group are introduced. It is shown that quotient representations are the most suitable for description of the massless physical fields. By way of example, neutrino field is described via the simplest quotient representation. Weyl-Hestenes equations for neutrino field are given.
Relativistic Spherical Functions on the Lorentz Group
Journal of Physics A: Mathematical and General. – 2006. – Vol. 39, № 4. – P. 805–822. , 2006
Matrix elements of irreducible representations of the Lorentz group are calculated on the basis of complex angular momentum. It is shown that Laplace-Beltrami operators , defined in this basis, give rise to Fuchsian differential equations. An explicit form of the matrix elements of the Lorentz group has been found via the addition theorem for generalized spherical functions. Different expressions of the matrix elements are given in terms of hypergeometric functions both for finite-dimensional and unitary representations of the principal and supplementary series of the Lorentz group.
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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.
Journal of Mathematical Physics, 1994
Let TX=CX Y, be equipped with the Hermitian form lt~2-~~x~~2, where teC is a complex number and XE Y, is a vector in an abstract complex inner product space V,. The abstract complex Lorentz group (with real metric) is the invariance group of this form. A novel formalism to deal with the abstract real Lorentz transformation group has recently been developed by Ungar [Am. J. Phys. 59, 824-834 (1991); Am. J. Phys. 60, 815-828 (1992)], allowing one to solve in an abstract context previously poorly understood problems in one time and three space dimensions. Intrigued by the success of the abstract real Lorentz group formalism, resulting in the understanding of Thomas gyration in its abstract context, a formalism to deal with the abstract complex Lorentz group is proposed in this article. The extension from the real to the complex Lorentz group is not trivial. Complex Lorentz groups involve two interacting gyrations, a complex-time gyration and a complex-space gyration, as opposed to the real case which involves a single gyration, that is, a real-space gyration called Thomas gyration. The proposed formalism allows one to manipulate the seemingly involved abstract complex Lorentz group in a way analogous to the way one commonly manipulates Galilei transformation groups. Thus, for instance, equipped with the proposed formalism, one can readily (i) compose abstract complex Lorentz transformations, and (ii) determine those which link any two given events by manipulations analogous to Galilei group manipulations. When the abstract complex inner product space Y, associated with the underlying abstract complex Minkowski space is realized by a finite-dimensional complex Hilbert space of dimension n, the abstract complex Lorentz group studied in this article reduces to the group U(1,n) or SU(l,n), depending on whether unitary or special unitary transformations are being considered.
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In studying Lorentz-invariant wave equations, it is essential that we put our understanding of the Lorentz group on firm ground. We first define the Lorentz transformation as any transformation that keeps the 4-vector inner product invariant, and proceed to classify such transformations according to the determinant of the transformation matrix and the sign of the time component. We then introduce the generators of the Lorentz group by which any Lorentz transformation continuously connected to the identity can be written in an exponential form. The generators of the Lorentz group will later play a critical role in finding the transformation property of the Dirac spinors.
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Advances in Applied Clifford Algebras, 1999
A real representation of Dirac algebra, using η=diag(−1,1,1,1) as standard metric is discussed. Among other interesting properties it allows to define a generalization of Lorentz transformations. Ordinary boosts and rotations are subsets The additional transformations are shown to describe transformations to displaced systems, rotating systems, “charged systems”, and others. Poincaré transformations are shown to be approximations of these generalized Lorentz transformations. Appendix D gives an interpretation.
Geometric Algebra for Special Relativity and Manifold Geometry
This thesis is a study of geometric algebra and its applications to relativistic physics. Geometric algebra (or real Clifford algebra) serves as an efficient language for describing rotations in vector spaces of arbitrary metric signature, including Lorentzian spacetime. By adopting the rotor formalism of geometric algebra, we derive an explicit BCHD formula for composing Lorentz transformations in terms of their generators — much more easily than with traditional matrix representations. This is used to straightforwardly derive the composition law for Lorentz boosts and the concomitant Wigner angle. Later, we include a gentle introduction to differential geometry, noting how the Lie derivative and covariant derivative assume compact forms when expressed with geometric algebra. Curvature is studied as an obstruction to the integrability of the parallel transport equations, and we present a surface-ordered Stokes’ theorem relating the ‘enclosed curvature’ in a surface to the holonomy ...