Papers in Physics, vol. 9, art. 090002 (2017) Received: 8 November 2016, Accepted: 4 January 2017 Edited by: D. Gomez Dumm Licence: Creative Commons Attribution 3.0 DOI: http://dx.doi.org/10.4279/PIP.090002 www.papersinphysics.org ISSN 1852-4249 An alternative derivation of the Dirac operator generating intrinsic Lagrangian local gauge invariance Brian Jonathan Wolk 1∗ This paper introduces an alternative formalism for deriving the Dirac operator and equa- tion. 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 electro- magnetic �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 [2�6]. In the usual quantum �eld theoretic development, local gauge invariance is thus made an external condi- tion of and on the Lagrangian [3�6]. 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 [2�4]. More satisfactory from a theoretic standpoint would be a formalism in which derivation of the Dirac operator equation is associated with a La- grangian exhibiting internal local gauge symmetry. Such a formalism would alleviate both the need to impose local gauge invariance as an external man- date 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. The standard approach One consequence of the standard approach [1�6, 8, 20] in deriving the Dirac operator �∂ ≡ γα∂α = γ0∂/∂t−γ·∇, with γ = (γ1,γ2,γ3), which is related to the d'Alembertian operator � ≡ ∂µ∂µ associated with the Klein-Gordon equation �ψ = −M2ψ [2� 6,20], is that Cli�ord-Dirac elements {γµ} arise as necessary structures of the Dirac operator �∂, with the following properties [2�6,8,10,21] γα = gαδγδ; γµγ µ = 4 (1) γnγm = −γmγn(n 6= m) (γn) 2 = −1; ( γ0 )2 = 1 (γµ) † = γ0γµγ0 (γn) † = −γn � = �∂ 2 . Dirac's equation �∂ψ = ±iMψ follows for a fermionic �eld ψ such as the electron [2�6,21]. 090002-1 Papers in Physics, vol. 9, art. 090002 (2017) / B. Wolk ii. An alternate formalism Two conditions are set forth for developing an al- ternative formalism for deriving an operator, call it O, which operates on the wave function ψ for the subject fermionic particle and generates the equa- tion governing its evolution. The �rst condition is that since the wave func- tion ψ is a spinor, the Cli�ord elements must act, if at all, as operators on it [8, 13, 20]. Therefore, the applicable operator O should contain Cli�ord algebra elements. The second condition is that � should be deriv- able from O [2,4,6];1 there must exist a mapping z : O → �, and thus the governing equation itself must satisfy2 z(O)ψ = z(±iM)ψ −→ �ψ = −M2ψ. (2) To satisfy the d'Alembertian condition that z : O → �, the mapping must make use of the par- tial derivative operators, and so the operator ∂ ≡ (∂/∂t,∇) is de�ned. To meet the Cli�ord condi- tion that O contains Cli�ord elements, the oper- ator 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 equiv- alence between the ring of quaternions H with ba- sis (1, i,j,k) and R4 - the four-dimensional vec- tor space over the real numbers: {q ∈ H : q = u01 + bi + cj + dk|u0,b,c,d ∈ R} [10�12,14], with i2 = j2 = k2 = −1. The quaternion q can then be divided into its scalar and vector portions: {q = (u0,u) |u0 ∈ R,u ∈ R3} [11,12,14]. 1A condition also imposed by Dirac. Ref. [2], p. 86. 2Putting z (±iM) = −M2 in Eq. (2) presumes that the form and domain of the mapping z is known. The more general relation would be z (f [M]) = −M2, with f [M] = ±iM to be subsequently deduced. But once the form of O is discovered, the form of z becomes evident, namely z = []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 z 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 Cli�ord elements being [11,13] 1 = γ0γ0, i = γ2γ3, j = γ3γ1, k = γ1γ2. (5) The γµ are then the �rst-order, primary entities [8,10,20] from which the quaternionic basis is con- structed.3 To generate a new operator using the fundamen- tal operators, the product η∂ is taken. The prod- uct 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 represen- tation using the formula (v0,v)(w0,w) = (v0w0 −~v · ~w,v0 ~w + ~vw0 + ~v× ~w), (6) where v →~v, and w → ~w [10�14]. 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 z : [η∂]0 → �. This mapping satis�es Eq. (2) [6,20]. The opera- tor [η∂]0 thus satis�es both the d'Alembertian and Cli�ord conditions. Putting [η∂]0 = O and noting the obvious equivalence O ≡ �∂, the Dirac operator is thus seen to be derived from the new formalism. Given Eq. (2), we have Oψ = ±iMψ as a possi- ble fermion �eld equation of motion. As any solu- tion to Oψ = ±iMψ 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. 3Note is made that η = γ0γ0γ0 + γ1γ2γ3 + γ1γ2γ3 + γ1γ2γ3. 090002-2 Papers in Physics, vol. 9, art. 090002 (2017) / B. Wolk 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 [η∂]∧ = γ0∇ + γ∂/∂t + γ ×∇. (10) To maintain consistency with the formalism used with �∂, the operator [η∂]∧ is also written in the γµ- basis. Designating this operator as ��c we have [η∂]∧ →��c = γ0(̂i∂/∂x + ĵ∂/∂y + k̂∂/∂z) + γ1(̂i∂/∂t− ĵ∂/∂z + k̂∂/∂y) + γ2(̂i∂/∂z + ĵ∂/∂t− k̂∂/∂x) + γ3(−î∂/∂y + ĵ∂/∂x + k̂∂/∂t). (11) Since the operators ( �∂,��c ) are coupled, when �∂ operates on some �eld so should ��c. Inspection of Eq. (11) shows that ��c's operation must be of a di�erent sort and on a di�erent yet coupled �eld. To see how ��c operates and on what, some notation is �rst required. A = A0(x) + A1(x)̂i + A2(x)ĵ + A3(x)k̂ repre- sents a four-vector �eld, for which we can associate the Cli�ord �eld �A = Aµγ µ, with Aµ ≡ Aµ(x) being the �eld components of A. There is thus a component-wise bijection between �A and A. A Cli�ord vector �eld is de�ned as �C = Cµγµ, with each Cµ being its own vector �eld. In this way, a general Cli�ord vector �eld operator is de�ned as ��4 = 4 αγα, with each component 4α being its own vector �eld operator. In standard vector analysis, vector �eld operators operate on scalar �elds [15]. Following suit, in order for a Cli�ord vector �eld operator's (��4) component vector �eld operators (4α) to operate on the scalar �elds Aµ of a Cli�ord �eld �A, an operation · must be de�ned such that ��4·�A = 4 µγµγ µAµ = 4µAµ = 4µAνgµν. (12) Using this formalism, the components c α of ��c are given by Eq. (11).4 Choosing a Cli�ord �eld of the general form 4For instance, c 0 = ∇. �Φ ≡ γµΦµ = γ0Φ0 + γ1Φ1 + γ2Φ2 + γ3Φ3, (13) with Φµ ≡ Φµ(x), and operating on�Φ with ��c gives ��c ·�Φ = (̂i∂/∂x + ĵ∂/∂y + k̂∂/∂z)Φ0 + (̂i∂/∂t− ĵ∂/∂z + k̂∂/∂y)Φ1 + (̂i∂/∂z + ĵ∂/∂t− k̂∂/∂x)Φ2 + (−î∂/∂y + ĵ∂/∂x + k̂∂/∂t)Φ3. (14) We have then the coupled �eld (ψ, Φµ) through action of the operator η∂. Unlike ψ, the Φµ are not 4-element column matrices and are not spinor �elds, since operating through in Eq. (14) excises the Cli�ord elements. Rearranging terms give the following set of six vector �eld components: (∂Φ0/∂x + ∂Φ1/∂t)̂i (∂Φ0/∂y + ∂Φ2/∂t)ĵ (∂Φ0/∂z + ∂Φ3/∂t)k̂ (∂Φ2/∂z −∂Φ3/∂y)̂i (∂Φ3/∂x−∂Φ1/∂z)ĵ (∂Φ1/∂y −∂Φ2/∂x)k̂ (15) These equations can be identi�ed with the compo- nents of two vector �elds X = −∇Φ0 −∂~Φ/∂t (16) and Y = ∇× ~Φ, (17) with ~Φ = (Φ1, Φ2, Φ3). These equations repre- sent the six independent components of an anti- symmetric �eld tensor H, which ��c ·�Φ has gener- ated. There is thus a one-to-one and onto corre- spondence: {±��c ·�Φ ↔ H}. Therefore, H can be written as the curl of the Cli�ord scalar �eld com- ponents Hµν ≡ ∂µΦν −∂νΦµ. (18) H is then formally equivalent to the electro- magnetic �eld tensor [6, 16, 19, 22]. Using the component-wise bijection stated above: {�A ↔ A}, 090002-3 Papers in Physics, vol. 9, art. 090002 (2017) / B. Wolk the components of �Φ are identi�ed with the com- ponents of the electromagnetic potential vector A: Aµ ≡ Φµ. This being the case, Aµ5 represents a massless vector �eld (the photon) abiding by the gauge invariance condition [2,3,6,9,17�19,22] Aµ −→ Aµ + ∂µλ. (19) 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 Aµ, namely the Lorenz condition ∂µA µ = 0 [2, 6].6 With the aid of the Lorenz gauge, the Lagrangian for the �eld Aµ with source J µ [2,6,18] can be written as LAµ = − 1 16π HµνHµν − 1 c JµAµ. (20) The Lagrangian for the Dirac �eld ψ is given by [2,6] Lψ = i~cψγµ∂µψ −mc2ψψ. (21) While exhibiting global gauge invariance, the Dirac Lagrangian Lψ 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 Aµ in order to meet this mandate [2�6]. The current formalism does not require such a method. The Lagrangian for the coupled �eld is thus L(ψ,Aµ) ≡Lψ + LAµ =[i~cψγµ∂µψ −mc2ψψ] − 1 16π HµνHµν − (eψγµψ)Aµ, (22) where ceψγµψ = Jµ is the quantum �eld current density satisfying the conservation equation [2,6,7] ∂µJ µ = 0. (23) This is an important result; for the conserva- tion equation is a consequence of the intrinsic 5Where 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 → eiα(x)ψ concomitant with Eq. (19) as part of the local gauge invariance trans- formation [21]. As the Ward identity, given by kµMµ (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 cur- rent formalism as a consequence of the inherent lo- cal gauge symmetry of the Lagrangian.8 The form of the interaction term (eψγµψ)Aµ of L(ψ,Aµ) arises naturally in this formalism. An in- trinsically coupled �eld 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 cur- rent ψγµψ takes the role of the conserved current Jµ 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 im- posed 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,Aµ) 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) ≡ LQED, the La- grangian of Quantum Electrodynamics.9 In canon- ically quantizing the theory this equivalence of La- grangians is conditioned on modi�cation of the 7Ref. [21], sections 5.5 and 7.4. Where M(k) = �µ(k)Mµ(k) is the amplitude for some quantum electrody- namic process involving an external photon with momentum k. 8Ref. [2], section 13.2.4 (Local gauge invariance ←→ cur- rent conservation ←→ Ward identities). 9This paper does not contemplate the Yang-Mills gener- alization and extension of gauge invariance to non-abelian groups such as U(1)⊗SU(2) of the weak interaction or quan- tum 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 non- abelian 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 e�ective Lagrangians [22]. 090002-4 Papers in Physics, vol. 9, art. 090002 (2017) / B. Wolk Lorenz condition relied on above in generating LAµ. For the canonically quantized formalism, Gupta-Bleuler's weak Lorenz condition given by ∂µA µ+ |Ψ〉 = 0 replaces the Lorenz condition, in which Aµ+ acts as the photon lowering quantum �eld operator and |Ψ〉 represents a ket of any num- ber of photons [2,21,22].10 It follows from this con- ditioned equivalence that the new formalism gener- ates all of electrodynamics and speci�es 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 struc- ture of fundamental physical theories exhibit this symmetry ab initio. Therefore, a formalism which produces the Dirac operator equation exhibiting in- herent 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 Electrodynam- ics. Such a uni�ed description of these basic equa- tions and their processes may also lead to a deeper understanding of the origin of these phenomena. [1] P A M Dirac, The quantum theory of the elec- tron, Proc. Roy. Soc. Lond. A117, 610 (1928); Ibid, part II, A118, 351 (1928). [2] R D Klauber, Student friendly quantum �eld theory, Sandtrove Press, Fair�eld, Iowa (2013). [3] S Weinberg, The quantum theory of �elds, Vol. I, Cambridge University Press, Cambridge (2005). [4] P J E Peebles, Quantum mechanics, Prince- ton University Press, Princeton, New Jersey (1992). 10This modi�cation functions as a necessary constraint on the longitudinal and scalar photons in any given quantum state, permitting their mutual cancellation when calculat- ing the Hamiltonian expectation value of the quantized �eld theory [2,22]. 11See, e.g., Ref. [6], p. 360 and Ref. [21], p. 78. [5] A Zee, Quantum �eld theory in a nutshell, Princeton University Press, Princeton, New Jersey (2003). [6] D Gri�ths, Introduction to elementary par- ticles, 2nd Rev. Ed., Wiley-VCH, Weinheim, Germany (2008). [7] W Pauli, Wave mechanics, Pauli Lectures on Physics, Vol. 5, Dover Publications Inc., Mi- neola, New York (1973). [8] R Penrose, W Rindler, Spinors and space- time, Vol. I, Cambridge University Press, Cambridge (1984); Ibid Vol. II (1986). [9] P A M Dirac, Directions in physics, John Wi- ley & Sons, New York (1978). [10] D Hestenes, G Sobczyk, Cli�ord algebra to ge- ometric calculus, Reidel, Dordrecht (1999). [11] J Dieudonne, Foundations of modern analysis, Academic Press, New York (1960). [12] O Veblen, J W Young, Projective geometry, Ginn & Co., Boston (1918). [13] P Lounesto, Cli�ord algebras and spinors, Cambridge University Press, Cambridge (2001). [14] T W Judson, Abstract algebra, PWS Publish- ing, Texas (1994). [15] G E Hay, Vector and tensor analysis, Dover Publications Inc. Mineola, New York (1953). [16] M Schwartz, Principles of electrodynamics, Dover Publications Inc., Mineola, New York (1972). [17] J R Lucas, P E Hodgson, Spacetime and elec- tromagnetism: An essay on the philosophy of the special theory of relativity, Clarendon Press, New York (1990). [18] W Pauli, Theory of relativity, Pergamon Press, Oxford (1958). [19] W Greiner, Classical electrodynamics, Springer-Verlag, New York (1998). [20] R Penrose, The road to reality:A complete guide to the laws of the universe, Vintage Books, New York (2004). 090002-5 Papers in Physics, vol. 9, art. 090002 (2017) / B. Wolk [21] M E Peskin, D V Schroeder, An intro- duction to quantum �eld theory, (Economy Edition), Westview Press, Reading, Mas- sachusetts (2016). [22] G Sterman, An introduction to quantum �eld theory, Cambridge University Press, Cam- bridge (1993). [23] J D Jackson, L B Okun, Historical roots of gauge invariance, Rev. Mod. Phys. 73, 663 (2001). 090002-6