An alternative derivation of the Dirac operator generating intrinsic Lagrangian local gauge invariance
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 Cli ord eld, this coupled operator produces eld components which are formally equivalent to the eld components of Maxwell’s electromagnetic eld tensor. Consequently, the Lagrangian of the associated coupled eld exhibits internal local gauge symmetry. The coupled eld Lagrangian is seen to be equivalent to the Lagrangian of Quantum Electrodynamics.
I.
Introduction
The Dirac equation [1] arises from a Lagrangian which lacks
local gauge symmetry. In the usual quantum eld theoretic development,
local gauge invariance is thus made an external condition of and on the
Lagrangian. Introduction of a vector eld Aµ that couples to the
Dirac eld ψ must
then be introduced in order to satisfy the imposed local symmetry constraint.
More satisfactory from a theoretic standpoint would be a formalism in which derivation of the 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. Symmetry would exist ab initio. This paper presents such an approach and derivation.
II.
Alternative formalism
i. The
standard approach
One consequence of the standard approach [1 6, 8,20] in
deriving the Dirac operator 
γ0∂/∂t−γ·∇, with γ = (γ1,γ2,γ3), which is related to the d’Alembertian
operator ≡ ∂µ∂µ associated
with the Klein-Gordon equation,
is that Cli ord-Dirac elements {γµ} arise as necessary
structures of the Dirac operator ∂, with the following properties (1)
ii. An
alternate formalism
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 rst condition is that since the wave
function ψ is a
spinor, the Cli ord elements must act, if at all, as operators on it. Therefore, the applicable operator O should
contain Cli ord algebra elements.
The second condition is that should be derivable from O;[1] there must exist
a mapping z : O → , and thus the governing
equation itself must satisfy[2]
To satisfy the d’Alembertian condition
that z : O → , the mapping must make use of the
partial derivative operators, and so the operator ∂ ≡ (∂/∂t,∇) is de ned. To meet the
Cli ord condition that O contains Cli ord elements, the
operator η ≡ (γ0,γ) is put. Written explicitly,
these fundamental operators are
∂ = ∂/∂t + i∂/∂x + j∂/∂y + k∂/∂z, (3)
and
η = γ0 + iγ1 + jγ2 + kγ3. (4)
We wish to use
these fundamental operators in constructing O. To do
so, use is made of the equivalence between the ring of quaternions H with basis
(1,i,j,k)
and R4 - the four-dimensional
vector space over the real numbers: {q ∈ H :
q = u01 + bi + cj + dk|u0,b,c,d ∈ R} , with i2 = j2
= k2 = −1.
The quaternion q
can then be divided into its scalar and vector portions: {q = (u0,u)|u0 ∈ R,u ∈ R[3]}.
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 Cli ord elements being [11,13]
1 = γ0γ0, i = γ2γ3, j = γ3γ1, k
= γ1γ2. (5)
The γµ are then the rst-order, primary entities 8, from which the quaternionic basis is constructed.3
To generate a new operator using the
fundamental operators, the product η∂ is taken. The product of two quaternionic operators v = (v0,v) and w = (w0,w) may be written as a product of their scalar and vector
components in the R4 representation using the
formula
(v0,v)(w0,w) = (v0w0 −~v · w~,v0w~ +~vw0
+~v × w~),
(6)
where v → ~v,
and w → w~ . This
gives
η∂ ≡ (γ0,γ)(∂/∂t,∇), (7)
producing the operator
η∂ ≡([η∂]0,[η∂]∧) (8)
=(γ0∂/∂t − γ · ∇,γ0∇ + γ∂/∂t + γ × ∇).
The operator η∂ is composed of two coupled
operators (and thus will operate on two coupled elds). Its rst component
operator is
[η∂]0 = γ0∂/∂t − γ · ∇. (9)
Setting z = []2 gives
a mapping
.
This mapping satis es Eq. (2) . The operator [η∂]0 thus satis es
both the d’Alembertian and Cli ord conditions. Putting [η∂]0 = O and noting
the obvious equivalence
, the Dirac operator is thus
seen to be derived from the new formalism. Given Eq. (2),
we have Oψ = ±iMψ as a possible fermion eld
equation of motion. As any solution to Oψ = ±iMψ
is also a solution to the KleinGordon equation, this
equation is naturally postulated as governing a fermionic particle such as the electron.
III.
Conclusion
Local gauge symmetry plays the central, dominant role in modern eld theory. That being the case, it would be preferable that the intrinsic structure of fundamental physical theories exhibit this symmetry ab initio. Therefore, a formalism which produces the Dirac operator equation exhibiting inherent local gauge invariance while also jettisoning the need for invention of an auxiliary vector eld in order to satisfy an imposed symmetry constraint is more satisfying from a theoretic standpoint. This paper’s formalism achieves such an internal local symmetry, and in doing so naturally generates the fundamental equations of Quantum Electrodynamics. Such a uni ed description of these basic equations and their processes may also lead to a deeper understanding of the origin of these phenomena.
Brian Jonathan Wolk
Edited by: D. Gomez Dumm
Licence: Creative Commons
Attribution 3.0
DOI: http://dx.doi.org/10.4279/PIP.090002