Chapter 1 Lorentz Group and Lorentz Invariance (original) (raw)
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Concerning the Dirac \textgamma-Matrices Under a Lorentz Transformation of the Dirac Equation
We embolden the idea that the Dirac 4 × 4 γ-matrices are four-vectors where the space components (γ i) represent spin and the forth component (γ 0) should likewise represent the time component of spin in the usual four-vector formalism of the Special Theory of Relativity. With the γ-matrices as four-vectors, it is seen that the Dirac equation admits two kinds of wavefunctions-(1) the usual four component Dirac bispinor ψ and (2) a scalar four component bispinor φ. Realizing this, and knowing forehand of the existing mystery as to why Leptons and Neutrinos come in pairs, we seize the moment and make the suggestion that the pair (ψ, φ) can be used as a starting point to explain mystery of why in their three generations [(e ± , ν e), (µ ± , ν µ), (τ ± , ν τ)], Leptons and Neutrinos come in doublets. In this suggestion, the scalar-bispinor φ can be thought of as the Neutrino while the usual Dirac bispinor ψ can be thought of as the Lepton. "We have found it of paramount importance that in order to progress we must recognize our ignorance and leave room for doubt."
Reflections, spinors, and projections on a Minkowski space underlie Dirac's equation
Linear Algebra and Its Applications, 1996
Reflections and spinors on a Minkowski space are obtained by the methods of Cartan. The group of all such reflection transformations has as its subgroup the Poincare group of Lorentz transformations. Two types of spinors are shown to exist for a Minkowski space. Spinor reflections and Lorentz transformations exist for both types of spinors. Associated with these spinors are two Hermitian, orthogonal projection operators which together spectrally resolve the identity operator of a two-dimensional complex vector space. These spinors and their associated projection operators are applied to find the structure of a 4 X 4 matrix G which is equivalent to the relativistic conservation law of the energy and momentum of a single moving particle. These spinor-calculus procedures demonstrate that G is singular of rank 2, and as a consequence the solutions of Ge = 0 consists of all bispinor elements of the null space of G. This equation and its solutions are equivalent to those of Dirac's quantum-mechanical equation of an electron in the Fourier domain of frequency and wavenumber. Finally some properties of 2-spinors and 4-spinors found herein are shown to extend naturally to n dimensions. 1.
Generalized Lorentz transformations
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.
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)).
On the new invariance groups of the Dirac and Kemmer-Duffin-Petiau equations
1977
In works [1-6] the canonical-transformation method has been proposed for the investigations of the group properties of the differential equations of the quantum mechanics. This method essence in that the system of differential equation is first transformed to the diagonal or Jordan form and then the invariance algebra of the transformed equation is established. The explicit form of this algebra basis elements for the starting equations is found by the inverse transformation.
The necessity of Lorentz transforming the Dirac matrices is an ongoing issue with contradicting opinions. The Lorentz transformation of Dirac spinors is clear but for the Dirac adjoint, the combination of a spinor and the 'time-like' zeroth gamma-matrix, the situation is fussy again. In the Feynman slash objects, the gamma matrix four vector connects to the dynamic four vectors without really becoming one itself. The Feynman slash objects exist in 4-D Minkowsky space-time on the one hand, the gamma matrices are often taken as inert objects like the Minkowski metric itself on the other hand. To be short, a slumbering confusion exists in RQM's roots. In this paper, first a Pauli-level biquaternion environment equivalent to Minkowski space-time is presented. Then the Weyl-Dirac environment is produced as a PT doubling of the biquaternion Pauli-environment. It is the production process from basic elements that produces some clarification regarding the mentioned RQM foundational fussiness.
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.