The Clifford algebra of physical space and Dirac theory

@article{Vaz2016TheCA,
  title={The Clifford algebra of physical space and Dirac theory},
  author={Jayme Vaz},
  journal={European Journal of Physics},
  year={2016},
  volume={37},
  pages={055407}
}
  • J. Vaz
  • Published 11 July 2016
  • Physics
  • European Journal of Physics
The claim found in many textbooks that the Dirac equation cannot be written solely in terms of Pauli matrices is shown to not be completely true. It is only true as long as the term in the usual Dirac factorization of the Klein–Gordon equation is assumed to be the product of a square matrix β and a column matrix ψ. In this paper we show that there is another possibility besides this matrix product, in fact a possibility involving a matrix operation, and show that it leads to another possible… 
View via Publisher
repositorio.unicamp.br
9 Citations
Highly Influential Citations
1
Background Citations
2
Methods Citations
1

9 Citations

Dirac Theory in Euclidean 3D Geometric Algebra (Cl3)
This article is intended as an addition to the book [10], since, in the first edition, I was double minded whether to introduce the Dirac theory for young students. Now I am quite sure that it should…
The Clifford Algebra of Physical Space and Elko Spinors
  • J. Vaz
  • Mathematics, Physics
  • 2018
Elko spinors are eigenspinors of the charge conjugation operator. In this work we use the Clifford algebra of the physical space in order to formulate the theory of Elko spinors and use a procedure…
Operators and Field Equations in the Electroweak Sector of Particle Physics
  • G. McClellan
  • Physics, Mathematics
    Advances in Applied Clifford Algebras
  • 2021
This paper derives a linear, first-order, partial differential field equation (a Dirac-like equation) in the geometric calculus of the geometric algebra $${\mathcal {G}}_{4,1}$$ that has free…
DKP algebra, DKP equation, and differential forms
It is well known that the Clifford algebras and the Dirac equation have a representation in terms of differential forms known as the Kahler-Atiyah algebra and the Dirac-Kahler equation, respectively.…
Spinor Fields, Singular Structures, Charge Conjugation, ELKO and Neutrino Masses
In this paper, we consider the most general treatment of spinor fields, their kinematic classification and the ensuing dynamic polar reduction, for both classes of regular and singular spinors;…
On Paravectors and Their Associated Algebras
  • J. Vaz
  • Mathematics
    Advances in Applied Clifford Algebras
  • 2019
Some algebraic structures that can be defined on the spaces of paravectors and k-paravectors are studied. Firstly, a version of the exterior and interior products resembling those in the exterior…
On the generalized spinor classification: beyond the Lounesto’s classification
In this paper we advance into a generalized spinor classification, based on the so-called Lounesto’s classification. The program developed here is based on an existing freedom on the spinorial dual…
Unveiling a spinor field classification with non-Abelian gauge symmetries
Paravectors and the Geometry of 3D Euclidean Space
We introduce the concept of paravectors to describe the geometry of points in a three dimensional space. After defining a suitable product of paravectors, we introduce the concepts of biparavectors…

References

SHOWING 1-10 OF 42 REFERENCES
The Dirac Equation
Ever since its invention in 1929 the Dirac equation has played a fundamental role in various areas of modern physics and mathematics. Its applications are so widespread that a description of all…
Observables, operators, and complex numbers in the Dirac theory
The geometrical formulation of the Dirac theory with spacetime algebra is shown to be equivalent to the usual matrix formalism. Imaginary numbers in the Dirac theory are shown to be related to the…
Algebra of physical space and the geometric spacetime solution of Dirac's equation
The algebra of physical space (APS) is a name for the Clifford or geometric algebra, which can be closely associated with the geometry of special relativity and relativistic spacetime. For example,…
Applications of Clifford Algebras in Physics
Clifford’s geometric algebra is a powerful language for physics that clearly describes the geometric symmetries of both physical space and spacetime. Some of the power of the algebra arises from its…
The Paravector Model of Spacetime
The geometric (or Clifford) algebra Cl3 of three-dimensional Euclidean space is endowed with a natural complex structure on a four-dimensional space. If paravectors, which are formed sums of scalars…
Elko and mass dimension one field of spin one half: causality and Fermi statistics
We review how Elko arise as an extension of complex valued four-component Majorana spinors. This is followed by a discussion that constrains certain elements of phase freedom. A proof is reviewed…
Geometry of Paravector Space with Applications to Relativistic Physics
Clifford’s geometric algebra, in particular the algebra of physical space (APS), lubricates the paradigm shifts from the Newtonian worldview to the post-Newtonian theories of relativity and quantum…
The Complex Algebra of Physical Space: A Framework for Relativity
A complex and, equivalently, hyperbolic extension of the algebra of physical space (APS) is discussed that allows one to distinguish space-time vectors from paravectors of APS, while preserving the…
A Clifford Algebra Approach to the Classical Problem of a Charge in a Magnetic Monopole Field
  • J. Vaz
  • Mathematics, Physics
  • 2013
The motion of an electric charge in the field of a magnetic monopole is described by means of a Lagrangian model written in terms of the Clifford algebra of the physical space. The equations of…
Cli ord Algebras and the Classical Groups
The Clifford algebras of real quadratic forms and their complexifications are studied here in detail, and those parts which are immediately relevant to theoretical physics are seen in the proper…
...
...