The Pauli algebra approach to special relativity

We exhibit expressions, in terms of Pauli matrices, which directly generate Lorentz transformations in Minkowski space.

This paper describes a particularly didactic and transparent derivation of basic properties of the Lorentz group. The generators for rotations and boosts along an arbitrary direction, as well as their commutation relations, are written as functions of the unit vectors that define the axis of rotation or the direction of the boost (an approach that can be compared with the one that in electrodynamics, works with the electric and magnetic fields instead of the Maxwell stress tensor). For finite values of the angle of rotation or the boost's velocity (collectively denoted by V), the existence of an exponential expansion for the coordinate transformation's matrix, M, in terms of GV with G being the generator, requires that the matrix's derivative with respect to V be equal to GM. This condition can only be satisfied if the transformation is additive as it is indeed the case for rotations, but not for velocities. If it is assumed, however, that for boosts such an expansion exists, with V = V(v), v being the boodt's velocity, and if the above condition is imposed on the boost's matrix, then its expression in terms of hyperbolic cosh(V) and sinh(V) is recovered with V(= tanh −1 (v)).

International Journal of Theoretical Physics, 1985

By using the principle of relativity alone (no assumptions about signals or light) it is shown that a relativisitic group Of linear transformations of a spacetime plane is, if infinite, either Galilean, Lorentzian or rotational. Th e largest such finite group is a Klein 4-group, generate d by space-reversal and time-reversal. In the infinite case an invariant of the group, denoted c, appears. When c is real, nonzero, noninfinite, then the group is a Lorentz group and c is identified with the speed of light. Lorentz transformations are represented through an algebra D of iterants that provides a link among C!ifford algebras, the Pauli algebra and Herman Bondi's K-calculus.

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.

Trends in Mathematics, 2013

The present paper is aimed at proving necessary and sufficient conditions on the quaternionic-valued coefficients of a first-order linear operator to be associated to the generalized Cauchy-Riemann operator in quarternionic analysis and explicitly we give the description of all its nontrivial first-order symmetries.

A complex vector is a sum of a vector and a bivector and forms a natural extension of a vector. The complex vectors have certain special geometric properties and considered as algebraic entities. These represent rotations along with specified orientation and direction in space. It has been shown that the association of complex vector with its conjugate generates complex vector space and the corresponding basis elements defined from the complex vector and its conjugate form a closed complex four dimensional linear space. The complexification process in complex vector space allows generating higher n-dimensional geometric algebra from (n-1)-dimensional algebra by considering the unit pseudoscalar identification with square root of minus one. The spacetime algebra can be generated from the geometric algebra by considering a vector equal to square root of plus one. The applications of complex vector algebra are discussed mainly in the electromagnetic theory and in the dynamics of an elementary particle with extended structure. Complex vector formalism simplifies the expressions and elucidates geometrical understanding of the basic concepts. The analysis shows that the existence of spin transforms a classical oscillator into a quantum oscillator. In conclusion the classical mechanics combined with zeropoint field leads to quantum mechanics.

Reports on Mathematical Physics, 2014

Symmetry, 2022

There are a total of 64 possible multiplication rules that can be defined starting with the generalized imaginary units first introduced by Hamilton. Of these sixty-four choices, only eight lead to non-commutative division algebras: two are associated to the left- and right-chirality quaternions, and the other six are generalizations of the split-quaternion concept first introduced by Cockle. We show that the 4×4 matrix representations of both the left- and right-chirality versions of the generalized split-quaternions are algebraically isomorphic and can be related to each other by 4×4 permutation matrices of the C2×C2 group. As examples of applications of the generalized quaternion concept, we first show that the left- and right-chirality quaternions can be used to describe Lorentz transformations with a constant velocity in an arbitrary spatial direction. Then, it is shown how each of the generalized split-quaternion algebras can be used to solve the problem of quantum-mechanical ...

arXiv:math-ph/0203059, 2002

A group theoretical description of basic discrete symmetries (space inversion P, time reversal T and charge conjugation C) is given. Discrete subgroups of orthogonal groups of multidimensional spaces over the fields of real and complex numbers are considered in terms of fundamental automorphisms of Clifford algebras. In accordance with a division ring structure, a complete classification of automorphism groups is established for the Clifford algebras over the fields of real and complex numbers. Finite-dimensional representations of the proper orthochronous Lorentz group are studied in terms of spinor representations of the Clifford algebras. Real, complex, quaternionic and octonionic representations of the Lorentz group are considered. The Atiayh-Bott-Shapiro periodicity is defined on the Lorentz group. Quotient representations of the Lorentz group are introduced. It is shown that quotient representations are the most suitable for description of massless physical fields. An algebraic construction of basic physical fields is presented.

relativityworkshop.com, 2018

In this article we re-analyse the orthogonal endomorphisms structure. This is because of we prove previously (in Part I) that skew-adjoint en-domorphism structure holds invariant in an orthogonal transformation. After that we infer the Lorentz boost and 2-dimensional euclidian rotation highlighting their geometrical structure in the context of annihilating polynomials of the mentioned skew-adjoint endomorphisms, namely electromagnetic field associated endomorphisms, and orthogonal endomorphism as well. This has meaningful involvements inside the structure of the base of special theory of relativity.