^{*}

^{*}

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, the scalar potential φ and the Lorentz gauge connecting them. With the same formalism, the continuity equation is written in terms of these new differential commutator brackets.

Maxwell equations are first order differential equations in space and time. They are compatible with Lorentz transformation which guarantees its applicability to any inertial frame. A symmetric space-time formulation of any theory will generally guarantee the universality of the theory. With this motivation, we adopt a differential commutator bracket involving first order space and time derivative operators to formulate the Maxwell equations and quantum mechanics. This is in addition to our recent quaternionic formulation of physical laws, where we have shown that many physical equations are found to emerge from a unified view of physical variables [

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, (). 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., [

Define the three linear differential commutator brackets as follows:

Equation (3) is correct, since partial derivatives commute, i.e.,. For a scalar and a vector, one defines the three brackets as follows:^{1}

and

It follows that

(7)

(8)

(9)

for any vector. The differential commutator brackets above satisfy the distribution rule

where are. It is evident that the differential commutator brackets identities follow the same ordinary vector identities. We call the three differential commutator brackets in Equation (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.

Using quaternionic algebra [

and

Now consider the dot-commutator of

Using Equations (11)-(13), one obtains

For arbitrary and, Equation (15) yields the two wave equations

and

Equations (16) and (17) show that the charge and current density satisfy a wave equation traveling at speed of light in vacuum. It is remarkable to know that these two equations are already obtained in [

Maxwell’s equations are first order differential equations in space and time of the electromagnetic field, viz.,

These equations show that charge () and current () densities are the sources of the electromagnetic field. Differentiating Equation (20) and using Equation (21), one obtains

Similarly, differentiating Equation (21) and using Equation (20), one obtains

These two equations state that the electromagnetic field propagates with speed of light in two cases:

1) charge and current free medium (vacuum), i.e., , or 2) if the two equations

and

besides the familiar continuity equation in Equation (11)

are satisfied. Equation (23) and (24) resemble Einstein's general relativity equation where space-times 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 as

Adding Equation (25) and Equation (26) according to Equation (28), one obtains

Applying Equations(25), (26) (see [

This is a wave equation propagating with speed of light in vacuum (). Hence, Maxwell wave equations can be written as a pure single wave equation of an electromagnetic sourceless complex vector field. We call Equations (25)-(27) the generalized continuity equations. We have recently obtained these generalized continuity equations by adopting quaternionic formalism for fluid mechanics [

(31)

This is the familiar continuity equation. Hence, the continuity equation in the commutator bracket form can be written as

Similar, using Equations (21) and (22), the magnetic field dot-commutator bracket yields

The electric field cross-commutator bracket gives

Using Equations (20) and (21), this yields

This equation is nothing but Equation (24) above. Similarly, the magnetic field cross-commutator bracket gives

Using Equations (20) and (21) this yields,

(37)

This equation is nothing but Equation (23) above. Hence, Equations (35) and (37), i.e.,

represent the combined Maxwell equations. In terms of the vector defined in Equation (33), the wave equation in Equation (35) can be written as

which is also evident from Equation (28).

The electric and magnetic fields are defined by the vector and the scalar potential as follows

These are related by the Lorentz gauge as

Comparing this equation with Equation (11) reveals that the continuity equation is nothing but a gauge condition. This means that a new current density can be found so that the equation is still intact. We have recently explored such a possibility which showed that it is true [

Using Equations (40) and (41), one obtains

This yields the wave equation of the vector field as

Similarly, the dot-commutator bracket of the vector

Using Equations (40) and (41), one obtains

This yields the wave equation of

The cross-commutator bracket of the scalar potential

Using Equation (40), one finds

This yields the Faraday’s equation,

It is interesting to arrive at this result by using the definition in Equation (40) only. Now consider the dotcommutator bracket of

Using Equations (40), (41) and the vector identities

Equation (51) yields

For arbitrary and, Equation (53) yields the two equations

and

Equations (54) and (55) are the Gauss and Ampere equations.

Similarly, the cross-commutator bracket of

Using Equations (40), (41) and the vector identity

Equation (56) yields

(58)

For arbitrary and, Equation (58) yields the two equations

and

Once again, Equations (59) and (60) are the Faraday and Ampere equations, respectively. Hence, the four Maxwell equations are completed. To sum up, Maxwell equations are the commutator brackets

In electromagnetism, the energy conservation equation for electromagnetic field is written as

where

The energy conservation equation of the electromagnetic field can be easily obtain using the following vector identity

Let now, so that Equation (64) becomes

Employing Equations (20), (21) and (63), Eq.(65) yields

which is the familiar energy conservation equation of the electromagnetic field [

By introducing three vanishing linear differential commutator brackets for scalar and vector fields, and and the Lorentz gauge connecting them, we have derived the Maxwell’s equations and the continuity equation without resort to any other physical equation. Using different vector identities, we have found that no any independent equation can be generated from the three differential commutators brackets.

Equations (14) and (15) are in the form of coupled wave equations known as inhomogeneous Helmholtz equations. We see that the current density J enters into these equations in a relatively complicated way, and for this reason these equations and are not readily soluble in general. This work is supported by the university of Khartoum research fund. We are grateful for this support.