Abstract
This paper introduces an alternative formalism for deriving the Dirac operator and equation. The use of this formalism concomitantly generates a separate operator coupled to the Dirac operator. When operating on a Clifford field, this coupled operator produces field components which are formally equivalent to the field components of Maxwell's electromagnetic field tensor. Consequently, the Lagrangian of the associated coupled field exhibits internal local gauge symmetry. The coupled field Lagrangian is seen to be equivalent to the Lagrangian of Quantum Electrodynamics.
 Received: 8 November 2016, Accepted: 4 January 2017; Edited by: D. Gomez Dumm; DOI: http://dx.doi.org/10.4279/PIP.090002
 Cite as: B J Wolk, Papers in Physics 9, 090002 (2017)
 This paper, by B J Wolk, is licensed under the Creative Commons Attribution License 3.0.
Highlights
Introduction of a vector eld Aμ that couples to the Dirac eld ψ must be introduced in order to satisfy the imposed local symmetry constraint [24]
More satisfactory from a theoretic standpoint would be a formalism in which derivation of the
Two conditions are set forth for developing an alternative formalism for deriving an operator, call it O, which operates on the wave function ψ for the subject fermionic particle and generates the equation governing its evolution
Summary
The Dirac equation [1] arises from a Lagrangian which lacks local gauge symmetry [26]. Dirac operator equation is associated with a Lagrangian exhibiting internal local gauge symmetry Such a formalism would alleviate both the need to impose local gauge invariance as an external mandate as well as the need to invent and introduce a vector eld to satisfy the constraint. Two conditions are set forth for developing an alternative formalism for deriving an operator, call it O, which operates on the wave function ψ for the subject fermionic particle and generates the equation governing its evolution. The second condition is that should be derivable from O [2, 4, 6];1 there must exist a mapping In this way, the operators given in (3) and (4) can be conceived as quaternionic operators, with the relations between the quaternionic basis elements and the Cliord elements being [11, 13].
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