ARTÍCULOS

DOI: http://dx.doi.org/10.4279/PIP.090002

An alternative derivation of the Dirac operator generating intrinsic Lagrangian local gauge invariance

 

Brian Jonathan Wolk¹*

*E-mail: attorneywolk@gmail.com
¹ 3551 Blairstone Road, Tallahassee, FL 32301 Suite 105, USA.

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This paper introduces an alternative formalism for deriving the Dirac operator and equa-tion. The use of this formalism concomitantly generates a sepárate 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 electro-magnetic 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.

Keywords Gauge Symmetry; Quantum Electrodynamics; Local Gauge Invariance; Dirac Equation; Quantum Field Theory

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I. Introduction

The Dirac equation [1] arises from a Lagrangian which lacks local gauge symmetry [2-6]. In the usual quantum field theoretic development, local gauge invariance is thus made an external condi-tion of and on the Lagrangian [3-6]. Introduction of a vector field A that couples to the Dirac field must then be introduced in order to satisfy the imposed local symmetry constraint [2-4].

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 mándate as well as the need to invent and introduce a vector field to satisfy the constraint. Symmetry would exist ab initio. This paper presents such an approach and derivation.

II. Alternative formalism

i. The standard approach

Dirac's equation 0ip = ±iMip follows for a fermionic field tp such as the electrón [2-6,21].

ii. An altérnate formalism

Two conditions are set forth for developing an al-ternative formalism for deriving an operator, cali it O, which operates on the wave function ip for the subject fermionic particle and generates the equa-tion governing its evolution.

The first condition is that since the wave function ip is a spinor, the Clifford elements must act, if at all, as operators on it [8,13,20]. Therefore, the applicable operator O should contain Clifford algebra elements.

The second condition is that D should be deriv-able from O [2,4,6];* there must exist a mapping F ■ O -> D, and thus the governing equation itself must satisfy²

To satisfy the d'Alembertian condition that F ■ O-íD, the mapping must make use of the par-tial derivative operators, and so the operator d = (d/dt, V) is defined. To meet the Clifford condition that O contains Clifford elements, the operator r¡ = (70,7) is put. Written explicitly, these fundamental operators are

We wish to use these fundamental operators in constructing O. To do so, use is made of the equiv-alence between the ring of quaternions H with ba-sis (l,i,j,k) and R⁴ - the four-dimensional vector space over the real numbers: {q G H : q = m₀1 + bi + cj + dk\u₀, b,c,d G R} [10-12,14], with i² = j² = k² = -1. The quaternion q can then be divided into its scalar and vector portions: {q = («o, u) \u₀ G R, u G R³} [11,12,14].

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 Clifford elements being [11,13]

The 7_M are then the first-order, primary entities [8,10,20] from which the quaternionic basis is con-structed.³

To genérate a new operator using the fundamental operators, the product r/d is taken. The prod-uct of two quaternionic operators v = (vo,v) and w = (wo,w) may be written as a product of their scalar and vector components in the R⁴ represen-tation using the formula

Setting F = [] gives a mapping F ■ [r)d]o -> -This mapping satisfies Eq. (2) [6,20]. The operator [r]d]o thus satisfies both the d'Alembertian and Clifford conditions. Putting [r)d]o = O and noting the obvious equivalence O = 0, the Dirac. operator is thus seen to be derived from the new formalism. Given Eq. (2), we have Oip = ±iMip as a possi-ble fermion field equation of motion. As any solu-tion to Oip = ±iMip 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 electrón.

III. The coupled operator

A new operator which is coupled to 0 is seen to arise within this formalism. This operator is the vector component of r¡d in Eq. (8), namely

Since the operators (0,^ ) are coupled, when 0

operates on some field so should X Inspection of Eq. (11) shows that Xs operation must be of a different sort and on a different yet coupled field. To see how,* operates and on what, some notation is first required.

represents a four-vector field, for which we can associate the Clifford field A = A"7^M> ^with A" = AÁ^X) being the field components of A. There is thus a component-wise bijection between J( and A.

A Clifford vector field is defined as 0 = C^M7_M, with each C^M being its own vector field. In this way, a general Clifford vector field operator is defined as _/A^/ = A^a7_a, with each component A^a being its own vector field operator.

In standard vector analysis, vector field operators opérate on scalar fields [15]. Following suit, in order for a Clifford vector field operator's {j£) component vector field operators (A^a) to opérate on the scalar fields A^¹ ofa Clifford field /(, an operation . must be defined such that

Using this formalism, the components ▼ of F are given by Eq. (II).⁴ Choosing a Clifford field of the general form

We have then the coupled field (V>,<I>_M) through action of the operator r¡d. Unlike ip, the $_M are not 4-element column matrices and are not spinor fields, since operating through in Eq. (14) excises the Clifford elements. Rearranging terms give the following set of six vector field components:

with $ = ($i,⁽I>2,^cI^>3)- These equations repre-sent the six independent components of an anti-symmetric field tensor H, which has generated. There is thus a one-to-one and onto correspondence: {±X -0V-)- H}. Therefore, H can be written as the curl of the Clifford scalar field components

H is then formally equivalent to the electromagnetic field tensor [6, 16, 19, 22]. Using the component-wise bijection stated above: {X-f-> A}, )2-3 the components of 0 are identified with the components of the electromagnetic potential vector A: A^ = $_M. This being the case, A^⁵ represents a massless vector field (the photon) abiding by the gauge invariance condition [2,3,6,9,17-19,22]

i. The coupled locally gauge symmetric La-grangian

The gauge invariance condition, Eq. (19), can be exploited to impose an additional constraint on the potential A^, namely the Lorenz condition <9_MA^M = 0 [2,6].⁶ With the aid of the Lorenz gauge, the Lagrangian for the field A^ with source J^M [2,6,18] can be written as

While exhibiting global gauge invariance, the Dirac Lagrangian C^¡ is not locally gauge invariant [2-6]. The usual quantum field theoretic approach is to mándate local gauge symmetry [3, 6], thereby requiring subsequent introduction of a new vector field A^ in order to meet this mándate [2-6]. The current formalism does not require such a method. The Lagrangian for the coupled field is thus

where cetp^^tp = J^M is the quantum field current density satisfying the conservation equation [2,6,7]

This is an important result; for the conservation equation is a consequence of the intrinsic

⁵Where A_fl 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 £(^,a )> since J^M is simply the Noether current corresponding to the local phase transformation tp -> e^ia<-^x'tp concomitant with Eq. (19) as part of the local gauge invariance transformation [21]. As the Ward identity, given by k^M^¹ (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 (e-¡/>7^M-¡/>)v4_M of £■{■<]),A ) arises naturally in this formalism. An intrinsically coupled field must have a coupling pa-rameter - in this case e, the electric charge - and a Lagrangian interaction term [2,3,6]. Further, in rel-ativistic quantum mechanics, the probability current -¡/>7^M-¡/> takes the role of the conserved current J^M of the wave function ip [2,7,21]. It is natural then to intégrate 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 im-posed local gauge symmetry [2,6,21,22].

£(tp,A ) is locally gauge invariant [2,3,6,7,22]. The alternative formalism thus produces a coupled field (-(/>, A¡j) which is represented by an internally local gauge symmetric Lagrangian. There is no need then to either mándate local gauge invariance or thereafter to introduce an external field to meet the mándate, as both are inherent to the formalism; symmetry exists from inception.

Lastly, it is seen that -C(^,a ) = £qed, the Lagrangian of Quantum Electrodynamics.⁹ In canonically quantizing the theory this equivalence of La-grangians is conditioned on modification of the Lorenz condition relied on above in generating [5] Ca . For the canonically quantized formalism, Gupta-Bleuler's weak Lorenz condition given by <9_MA^M+ 1Ψ) = 0 replaces the Lorenz condition, in which A^+ acts as the photon lowering quantum ^ ' field operator and |Ψ) represents a ket of any num-ber of photons [2,21,22].¹⁰ It follows from this conditioned equivalence that the new formalism gener- 171 ates all of electrodynamics and specifies the current produced by the subject Dirac fields [2,3,6,21].ⁿ

IV. Conclusión

Local gauge symmetry plays the central, dominant role in modern field theory [22]. That being the [9] case, it would be preferable that the intrinsic structure of fundamental physical theories exhibit this produces the Dirac operator equation exhibiting in-herent local gauge invariance while also jettisoning the need for invention of an auxiliary vector field in order to satisfy an imposed symmetry constraint is more satisfying from a theoretic standpoint. This [12J paper's formalism achieves such an internal local symmetry, and in doing so naturally generates the"i fundamental equations of Quantum Electrodynamics. Such a unified description of these basic equations and their processes may also lead to a deeper understanding of the origin of these phenomena. [14]

 

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