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.


Introduction
The Dirac equation [1] arises from a Lagrangian which lacks local gauge symmetry [26].In the usual quantum eld theoretic development, local gauge invariance is thus made an external condition of and on the Lagrangian [36].Introduction of a vector eld A µ that couples to the Dirac eld ψ must then 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 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.* E-mail: attorneywolk@gmail.com 1 3551 Blairstone Road, Tallahassee, FL 32301 Suite 105, USA.

II.
Alternative formalism i.
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 Cliord elements must act, if at all, as operators on it [8,13,20].Therefore, the applicable operator O should contain Cliord algebra elements.
The second condition is that should be derivable from O [2, 4, 6]; 1 there must exist a mapping : O → , and thus the governing equation itself must satisfy 2 (2) To satisfy the d'Alembertian condition that : O → , the mapping must make use of the partial derivative operators, and so the operator ∂ ≡ (∂/∂t, ∇) is dened.To meet the Cliord condition that O contains Cliord elements, the operator η ≡ (γ 0 , γ) is put.Written explicitly, these fundamental operators are and 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 R 4 -the four-dimensional vector space over the real numbers: {q ∈ H : q = u 0 1 + bi + cj + dk|u 0 , b, c, d ∈ R} [1012, 14], with i 2 = j 2 = k 2 = −1.The quaternion q can then be divided into its scalar and vector portions: [11,12,14].
1 A condition also imposed by Dirac.Ref.
[2], p. 86. 2 Putting (±iM ) = −M 2 in Eq. (2) presumes that the form and domain of the mapping is known.The more general relation would be (f [M ]) = −M 2 , with f [M ] = ±iM to be subsequently deduced.But once the form of O is discovered, the form of becomes evident, namely = [] 2 , and deducing f [M ] becomes trivial.Additionally however, given the fact that the RHS of Eq. (2) involves a square, one can intuit the correct form for in the rst instance.
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] The γ µ are then the rst-order, primary entities [8,10,20] from which the quaternionic basis is constructed. 3o generate a new operator using the fundamental operators, the product η∂ is taken.The product of two quaternionic operators v = (v 0 , v) and w = (w 0 , w) may be written as a product of their scalar and vector components in the R 4 representation using the formula producing the operator The operator η∂ is composed of two coupled operators (and thus will operate on two coupled elds).Its rst component operator is Given Eq. (2), we have Oψ = ±iM ψ as a possible fermion eld equation of motion.As any solution to Oψ = ±iM ψ is also a solution to the Klein-Gordon equation [2,6,21], this equation is naturally postulated as governing a fermionic particle such as the electron.

III. The coupled operator
A new operator which is coupled to ¡ ∂ is seen to arise within this formalism.This operator is the vector component of η∂ in Eq. (8), namely To maintain consistency with the formalism used with ¡ ∂, the operator [η∂] ∧ is also written in the γ µbasis.Designating this operator as c we have  (11) shows that c's operation must be of a dierent sort and on a dierent yet coupled eld.
To see how c operates and on what, some notation is rst required.A = A 0 (x) + A 1 (x) î + A 2 (x) ĵ + A 3 (x) k represents a four-vector eld, for which we can associate the Cliord eld A = A µ γ µ , with A µ ≡ A µ (x) being the eld components of A. There is thus a component-wise bijection between A and A.
A Cliord vector eld is dened as C = C µ γ µ , with each C µ being its own vector eld.In this way, a general Cliord vector eld operator is dened as = α γ α , with each component α being its own vector eld operator.
In standard vector analysis, vector eld operators operate on scalar elds [15].Following suit, in order for a Cliord vector eld operator's ( ) component vector eld operators ( α ) to operate on the scalar elds A µ of a Cliord eld A, an operation • must be dened such that Using this formalism, the components c α of c are given by Eq. (11). 4 Choosing a Cliord eld of the general form 4 For instance, c 0 = ∇. with We have then the coupled eld (ψ, Φ µ ) through action of the operator η∂.Unlike ψ, the Φ µ are not 4-element column matrices and are not spinor elds, since operating through in Eq. ( 14) excises the Cliord elements.Rearranging terms give the following set of six vector eld components: These equations can be identied with the components of two vector elds and with Φ = (Φ 1 , Φ 2 , Φ 3 ).These equations represent the six independent components of an antisymmetric eld tensor H, which c • Φ has generated.There is thus a one-to-one and onto correspondence: {± c • Φ ↔ H}.Therefore, H can be written as the curl of the Cliord scalar eld components H is then formally equivalent to the electromagnetic eld tensor [6,16,19,22].Using the component-wise bijection stated above: { A ↔ A}, the components of Φ are identied with the components of the electromagnetic potential vector A: A µ ≡ Φ µ .This being the case, A µ 5 represents a massless vector eld (the photon) abiding by the gauge invariance condition [2, 3, 6, 9, 1719, 22] i.
The coupled locally gauge symmetric Lagrangian The gauge invariance condition, Eq. ( 19), can be exploited to impose an additional constraint on the potential A µ , namely the Lorenz condition 6 With the aid of the Lorenz gauge, the Lagrangian for the eld A µ with source J µ [2, 6,18] can be written as The Lagrangian for the Dirac eld ψ is given by [2, 6] While exhibiting global gauge invariance, the Dirac Lagrangian L ψ is not locally gauge invariant [2 6].The usual quantum eld theoretic approach is to mandate local gauge symmetry [3,6], thereby requiring subsequent introduction of a new vector eld A µ in order to meet this mandate [26].The current formalism does not require such a method.The Lagrangian for the coupled eld is thus where ceψγ µ ψ = J µ is the quantum eld current density satisfying the conservation equation [2,6,7] This is an important result; for the conservation equation is a consequence of the intrinsic 5 Where Aµ is now taken to represent the electromagnetic four-vector potential. 6This gauge condition is often incorrectly referred to as the Lorentz condition, vice the correct attribution as the Lorenz condition [23].
gauge symmetry of L (ψ,Aµ) , since J µ is simply the Noether current corresponding to the local phase transformation ψ → e iα(x) ψ concomitant with Eq. (19) as part of the local gauge invariance transformation [21].As the Ward identity, given by k µ M µ (k) = 0, is an expression which results from this current conservation, 7 it follows that the Ward identity is intrinsically manifest as well in the current formalism as a consequence of the inherent local gauge symmetry of the Lagrangian. 8 The form of the interaction term (eψγ µ ψ)A µ of L (ψ,Aµ) arises naturally in this formalism.An intrinsically coupled eld must have a coupling parameter -in this case e, the electric charge -and a Lagrangian interaction term [2,3,6].Further, in relativistic quantum mechanics, the probability current ψγ µ ψ takes the role of the conserved current J µ of the wave function ψ [2, 7,21].It is natural then to integrate the coupling parameter along with the probability current into the interaction term of Eq. (20).This results in the selfsame interaction term found via the standard derivation through imposed local gauge symmetry [2, 6,21,22].
L (ψ,Aµ) is locally gauge invariant [2, 3, 6, 7, 22].The alternative formalism thus produces a coupled eld (ψ, A µ ) which is represented by an internally local gauge symmetric Lagrangian.There is no need then to either mandate local gauge invariance or thereafter to introduce an external eld to meet the mandate, as both are inherent to the formalism; symmetry exists from inception.
Lastly, it is seen that L (ψ,Aµ) ≡ L QED , the Lagrangian of Quantum Electrodynamics. 9In canonically quantizing the theory this equivalence of Lagrangians is conditioned on modication of the 7 Ref.[21], sections 5.5 and 7.4.Where M (k) = µ(k)M µ (k) is the amplitude for some quantum electrodynamic process involving an external photon with momentum k. 8 Ref.
[2], section 13.2.4 (Local gauge invariance ←→ current conservation ←→ Ward identities). 9This paper does not contemplate the Yang-Mills generalization and extension of gauge invariance to non-abelian groups such as U(1) ⊗ SU(2) of the weak interaction or quantum chromodynamic's SU(3) [21,22], but only a formalism for an intrinsic local U(1) symmetry of QED.Therefore, such symmetries as the Becchi, Rouet, Stora and Tyutin (BRST) symmetry which is typically covered in quantization of nonabelian gauge theories is not addressed herein, but is left to the possible extension of this paper's formalism to such non-abelian generalizations with their associated invariant full eective Lagrangians [22].
Lorenz condition relied on above in generating L Aµ .For the canonically quantized formalism, Gupta-Bleuler's weak Lorenz condition given by ∂ µ A µ+ |Ψ = 0 replaces the Lorenz condition, in which A µ+ acts as the photon lowering quantum eld operator and |Ψ represents a ket of any number of photons [2, 21,22]. 10It follows from this conditioned equivalence that the new formalism generates all of electrodynamics and species the current produced by the subject Dirac elds [2, 3, 6, 21]. 11

IV. Conclusion
Local gauge symmetry plays the central, dominant role in modern eld theory [22].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 unied description of these basic equations and their processes may also lead to a deeper understanding of the origin of these phenomena.