Cromlech, menhirs and celestial sphere: an unusual representation of the Lorentz group (original) (raw)
Clifford Algebras and Lorentz Group
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.
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.
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.
Clifford-algebra based polydimensional relativity and relativistic dynamics
Foundations of Physics 31 (2001) 1185-1209, 2001
Starting from the geometric calculus based on Clifford algebra, the idea that physical quantities are Clifford aggregates (“polyvectors”) is explored. A generalized point particle action (“polyvector action”) is proposed. It is shown that the polyvector action, because of the presence of a scalar (more precisely a pseudoscalar) variable, can be reduced to the well known, unconstrained, Stueckelberg action which involves an invariant evolution parameter. It is pointed out that, starting from a different direction, DeWitt and Rovelli postulated the existence of a clock variable attached to particles which serve as a reference system for identification of spacetime points. The action they postulated is equivalent to the polyvector action. Relativistic dynamics (with an invariant evolution parameter) is thus shown to be based on even stronger theoretical and conceptual foundations than usually believed.
The Proper-Time Lorentz Group Demystified
2012
Proper velocities are measured by proper time as opposed to coordinate velocities, which are measured by coordinate time. The standard Lorentz transformation group, in which each transformation is expressed by a coordinate velocity and an orientation between two inertial frames, is well known. In contrast, the equivalent proper-time Lorentz transformation group, in which each transformation is expressed by a proper velocity and an orientation between two inertial frames is unknown. The dignity of special relativity theory requires that every possible means be explored for the solution of a problem so elegant and so celebrated. Fortunately, a so called gyro-formalism approach to special relativity enables the elusive proper-time Lorentz transformation group to be uncovered.
Relativity in Clifford's Geometric Algebras of Space and Spacetime
International Journal of Theoretical Physics, 2000
Of the various formalisms developed to treat relativistic phenomena, those based on Clifford's geometric algebra are especially well adapted for clear geometric interpretations and computational efficiency. Here we study relationships between formulations of special relativity in the spacetime algebra (STA) Cℓ 1,3 of the underlying Minkowski vector space, and in the algebra of physical space (APS) Cℓ 3 . STA lends itself to an absolute formulation of relativity, in which paths, fields, and other physical properties have observer-independent representations. Descriptions in APS are related by a one-to-one mapping of elements from APS to the even subalgebra STA + of STA. With this mapping, reversion in APS corresponds to hermitian conjugation in STA. The elements of STA + are all that is needed to calculate physically measurable quantities (called measurables) because only they entail the observer dependence inherent in any physical measurement. As a consequence, every relativistic physical process that can be modeled in STA also has a representation in APS, and vice versa. In the presence of two or more inertial observers, two versions of APS present themselves. In the absolute version, both the mapping to STA + and hermitian conjugation are observer dependent, and the proper basis vectors of any observer are persistent vectors that sweep out timelike planes in spacetime. To compare measurements by different inertial observers in APS, we express them in the proper algebraic basis of a single observer. This leads to the relative version of APS, which can be related to STA by assigning every inertial observer in STA to a single absolute frame in STA. The equivalence of inertial observers makes this permissible. The mapping and hermitian conjugation are then the same for all observers. Relative APS gives a covariant representation of relativistic physics with spacetime multivectors represented by multiparavectors in APS. We relate the two versions of APS as consistent models within the same algebra.
Derivation of the Complex Unit Sphere and its Hyperbolic Analogue
We present the derivation of the 6-dimensional Eulerian Lie group of the form SO(3,C). We describe our derivation process, which involves creation of finite group by using permutation matrices, and the exponentiation of the adjoint representation of the subset representing the generators of the finite group. We take clues from the 2-dimensional complex rotation matrix to present, what we believe, is a true representation of the Lie group for the six-dimensional complex unit sphere and proceed to study its dynamics. We also derive the 6-dimensional form of the hyperbolic Lie group representing the higher dimensional exponential, apply this to special relativity considerations, and show its relation to its Eulerian counterpart. With this approach, we discover a profound link with SO(3,C) and SO(3,3) and proceed to show the isomorphism to the Lie group SU(3). The following findings will likely prove useful in mathematical physics, complex analysis and applications in deriving higher dimensional forms of similar division algebras.