A New Formulation of Electrodynamics

A new formulation of electromagnetism based on linear differential commutator brackets is developed. Maxwell equations are derived, using these commutator brackets, from the vector potential $\vec{A}$, the scalar potential $\phi$ and the Lorentz gauge connecting them. With the same formalism, the continuity equation is written in terms of these new differential commutator brackets. Keywords: Mathematical formulation, Maxwell's equations

way that we are familiar today. In our present formulation, Maxwell equations are described by a set of two wave equations representing the evolution of the electric and magnetic fields. This is instead of having four equations. We aim in this paper to write down (derive) the physical equations by vanishing differential commutator brackets. We know that second order partial derivatives commute for space-space variables. We don't assume here this property is a priori for space and time. To guarantee this, we eliminate the time derivative of a quantity that is acted by a space (∇) derivative followed by a time derivative, and vice-versa. In expanding the differential commutator bracket, we don't commute time and space derivative, but rather eliminate the time derivative by the space derivative, and vice versa. This differential commutator bracket may enlighten us to quantize these physical quantities. By employing the differential commutator brackets of the vector A and scalar potential ϕ, we have derived Maxwell equations without invoking any a priori physical law. Maxwell arrives at his theory of electromagnetism by combing the Gauss, Faraday and Ampere laws. For mathematical consistency, he modified Ampere's law. He then came with the known Maxwell equations.

Relativistic prelude
From Lorentz transformations one obtain, We see that the commutator bracket where we have taken into account in the order of multiplication of the space and time differences, (△x , △t). This shows that the commutator is Lorentz invariant. This is a new invariant quantity in relativity. We, however, already knew that the square interval is Lorentz invariant, i.e., (△S) 2 = (△S ′ ) 2 [2]. It follows from Eq.(1) that the differential commutator bracket ∂ ∂t , ∇ = 0 is Lorentz invariant too, i.e., ∂ ∂t , ∇ = ∂ ∂t ′ , ∇ ′ . We know that the spatial second order derivatives of a function, f = f (x, y), is commutative, i.e., ∂ 2 f ∂x∂y = ∂ 2 f ∂y∂x . We wonder if the commutation of space and time derivatives are equally valid for all physical quantities. Motivated by this hypothesis, we propose the following differential commutator brackets to formulate the physical laws. In particular, we apply these differential commutator brackets, in this work to derive the continuity equation, Maxwell equations. 3

Differential commutators algebra
Define the three linear differential commutator brackets as follows: Equation (3) is correct since partial derivatives commute, i.e., ∂ 2 ∂t∂x ϕ = ∂ 2 ∂x∂t ϕ. For a scalar ψ and a vector G, one defines the three brackets as follows: 1 and for any vector F. The differential commutator brackets above satisfy the distribution rule whereÂ ,B ,Ĉ are ∇ , ∂ ∂t . It is evident that the differential commutator brackets identities follow the same ordinary vector identities. We call the three differential commutator brackets in Eq. (3) the grad-commutator bracket, the dot-commutator bracket and the cross-commutator bracket respectively. The prime idea here is to replace the time derivative of a quantity by the space derivative ∇ of another quantity, and vice-versa, so that the time derivative of a quantity is followed by a time derivative with which it commutes. We assume here that space and time derivatives don't commute. With this minimal assumption, we have shown here that all physical laws are determined by vanishing differential commutator bracket.
1 See the Appendix for other identities. 4

The continuity equation
Using quaternionic algebra [3], we have recently found that generalized continuity equations can be written as and Now consider the dot-commutator of ρJ Using Eqs.(11) -(13), one obtains For arbitrary ρ and J, Eq.(15) yields the two wave equations Equations (16) and (17) show that the charge and current density satisfy a wave equation travelling at speed of light in vacuum. It is remarkable to know that these two equations are already obtained in [3]. Hence, the current-charge density wave equations are equivalent to ∇ · B = 0 .
Similarly, differentiating Eq.(21) and using Eq.(20), one obtains These two equations state that the electromagnetic field propagates with speed of light in two cases: (i) charge and current free medium (vacuum), i.e., ρ = 0, J = 0, or and besides the familiar continuity equation in Eq.(11) are satisfied. Equation (23) and (24) resemble Einstein's general relativity equation where spacetimes geometry is induced by the distribution of matter present. We see here that the electromagnetic field is produced by any charge and current densities distribution (in space and time). Now define the electromagnetic vector F as Applying Eqs.(25), (26) (see [3]) in Eq.(29) yields  [3]. It is challenging to check wether any real fluid satisfies these equations or not.
We have recently shown that Schrodinger, Dirac and Klein -Gordon and diffusion equations are compatible with these generalized continuity equations [3]. Using Eqs. (19) and (20), the electric field dot-commutator bracket yields This is the familiar continuity equation. Hence, the continuity equation in the commutator bracket form can be written as Similar, using Eqs. (21) and (22), the magnetic field dot-commutator bracket yields The electric field cross-commutator bracket gives