The Maxwell equations in the Lorenz gauge

doug  <sweetser@world.com>

Introduction
Electric and magnetic fields
Electrostatics and magnetostatics
The Maxwell equations in the Lorenz gauge
Implications

Introduction

The Maxwell equations, along with the Lorentz force, govern all classical and quantum electromagnetic phenomena.  Books like Jackson's "Classical Electrodynamics" demonstrate the many implications of these equations.  Yet one might wonder why these particular equations are mathematically necessary.

In this notebook, quaternion-valued, nonhomogeneous, second-order differential equations are explored that relate to electromagnetism.  The operators used to generate these differential equations are not in general a linear combination of scalar and vector operators (tensors of rank zero and one).  Instead, the operators have the structure of a field, so that scalar and vector operators can mix.  The four Maxwell equations will be shown to be a particular case of a quaternion wave equation.

A quaternion will be represented in a 2-by-2 matrix as a scalar and a 3-vector.

q[s_, v3_] :=(({{s, -v3}, {v3, s}}))

The quaternion-valued potential is

q[ϕ, (Overscript[A, ⇀])]

This will turn out to be the standard electromagnetic 4 potential written as a quaternion.

The quaternion-valued differential operator employed here is

q[dt, (Overscript[▽, ⇀])];

In order to have a list of operators act on quaternion potential, a number of special functions had to be developed (not shown here).  The function "op" will take a list (or lists) of up to two operators and have those operators act on a quaternion.  This operator function only works symbolically.

Electric and magnetic fields

Make a connection to the electric and magnetic fields by applying the quaternion differential operator once on the quaternion potential.

op[{q[dt, (Overscript[▽, ⇀])]}, 
        q[ϕ[t], (Overscript[A, ⇀])[t]]].{1, 0}{-(Overscript[▽, ⇀])·(Overscript[A, ⇀])[
              t] + ∂ϕ[t]/∂t, 
        (Overscript[▽, ⇀]) X  (Overscript[A, ⇀])[t] + 
          (Overscript[▽, ⇀]), 
        ϕ[t] + ∂(Overscript[A, ⇀])[t]/∂t}

((∂/∂)t, (Overscript[▽, ⇀])) 
        (ϕ, (Overscript[A, ⇀])) = 
      (-(Overscript[▽, ⇀])·(Overscript[A
                , ⇀]) + ∂(ϕ/∂)t, 
          (Overscript[▽, ⇀]) X  (Overscript[A, ⇀]) + 
            (Overscript[▽, ⇀]), 
          ϕ + ∂((Overscript[A, ⇀])/∂)t)

The vector term is B - E.  Notice that the magnetic field would change signs if it had been formed by a differentiation on the right side, unlike the E field.  The scalar term has the form of a gauge relation and would change under a gauge transformation, unlike the vector term.

Electrostatics and Magnetostatics

For electrostatics, set the time derivative and the vector potential to zero.  Choose a scalar source term in Gaussian units, setting c = 1.

\({4\ π \ ρ, 0\&⇀}\  == \ 
      op[{q[0, ▽\&⇀], 
            q[0, ▽\&⇀]}, 
          q[ϕ, 0\&⇀]].{1, 0}
									
       == {\(-\((
            ▽\&⇀·\((
                ▽\&⇀, ϕ)
                \))\)\), 
        \(▽\&⇀\) X 
          \((▽\&⇀, ϕ)\)}
        \)

(4 π  ρ, (Overscript[0, ⇀]))  =  
      (((0, (Overscript[▽, ⇀])))^2) 
          (ϕ, (Overscript[0, ⇀]))  =  
        (-(Overscript[▽, ⇀])·(Overscript[▽, ⇀]), ϕ, 
            (Overscript[▽, ⇀]) X 
               (Overscript[▽, ⇀]), ϕ)

This is Gauss' law and Faraday's law for a static electric field.

For magnetostatics, set the time derivative and the scalar potential to zero.  Choose a relevant vector source term.

\({0, \ \(4\ π\ \(j\& ⇀ \)\)\/c}\  == \ 
      op[{q[0, \(▽\& ⇀ \)], 
            q[0, \(▽\& ⇀ \)]}, 
          q[0, \(A\& ⇀ \)]].{1, 0}
										\ 
       == {\(-\(▽\& ⇀ \)\)·\((
              \(▽\& ⇀ \) X 
                \( A\& ⇀ \))\), 
          \(▽\& ⇀ \) X \((
              \(▽\& ⇀ \) X 
                \( A\& ⇀ \))\) - 
            \(▽\& ⇀ \), 
          \((\(▽\& ⇀ \)·\(A\& 
                  ⇀ \))\)}\)

(0,  4 π (Overscript[j, ⇀]))  = 
      (((0, (Overscript[▽, ⇀])))^2) 
          (0, (Overscript[A, ⇀])) = 
        (-(Overscript[▽, ⇀])·(Overscript[▽, ⇀])  X  (Overscript[A, ⇀]), 
            (Overscript[▽, ⇀]) X 
                 (Overscript[▽, ⇀]) X 
                 (Overscript[A, ⇀]) - 
              (Overscript[▽, ⇀]), 
            (Overscript[▽, ⇀])·(Overscript[A, ⇀]))

The first term says there are no magnetic charges.  In the Coulomb gauge, the second term is Ampere's law.

The Maxwell equations in the Lorenz gauge

Form a quaternion wave equation by having the differential act twice on the potential.

op[{q[dt,   (Overscript[▽, ⇀])], 
          q[dt,   (Overscript[▽, ⇀])]}, 
        q[ϕ[t], (Overscript[A, ⇀])[t]]].{1, 0}\({\(-▽\&⇀\)·\((
              \(▽\&⇀\) X 
                 A\&⇀[t])\) - 
          ▽\&⇀·▽\&⇀, 
        ϕ[t] - 
          2\ ▽\&⇀·∂A
                      \&⇀[t]\/∂t + 
          ∂\^2 ϕ[t]\/∂t\^2, 
		
        \(▽\&⇀\) X 
            \((\(▽\&⇀\) X 
                 A\&⇀[t])\) + 
          \(▽\&⇀\) X 
             ▽\&⇀, 
        ϕ[t] + 
          2\ \(▽\&⇀\) X 
            ∂A\&⇀[t]\/∂t - 
          \((▽\&⇀, 
              ▽\&⇀·A
                    \&⇀[t])\) + 
		
          2\ ▽\&⇀, 
        ∂ϕ[t]\/∂t + 
          ∂\^2 A\&⇀[t]\/∂t\^2}\)

\(\(\((∂\/∂t, \ \ 
              ▽\&⇀)\)\^2\) 
        \((ϕ, A\&⇀)\) = \ 
      \((∂\^2 ϕ\/∂t\^2 - 
            ▽\&⇀·▽\&⇀, 
          ϕ - 
            2\ ▽\&⇀·∂A
                      \&⇀\/∂t - 
            ▽\&⇀·▽\&⇀  X  A\&⇀, 
	
          ∂\^2 A\&⇀\/∂t\^2\  + \ 
            \(▽\&⇀\) X 
              \( ▽\&⇀\) X 
               A\&⇀ - 
            ▽\&⇀, 
          ▽\&⇀·A\&⇀ + 
            \(▽\&⇀\) X 
               ▽\&⇀, 
          ϕ + 
            2\ \(▽\&⇀\) X 
              ∂A\&⇀\/∂t + \ 
            2\ ▽\&⇀, 
          ∂ϕ\/∂t)\)\)

The quaternion wave equation appears to contain terms from the Maxwell equations and others generated by a change in reference frame.

Does an operator exist which will only generate the four Maxwell equations?  Such a quaternion operator must exist due to the properties of a field.  Fortunately, it just involves the sum of the differential and its conjugate both acting twice on the potential.

\({4\ π\ ρ, \(4\ π\ j\&⇀\)\/c} == \ 

      Simplify[
	
        op[{q[dt, \ \ ▽\&⇀], 
                q[dt, \ \ ▽\&⇀]}, 

              				{q[dt, \(-▽\&⇀\)], 
                q[dt, \(-▽\&⇀\)]}, 

              					q[ϕ[t], A\&⇀[t]]].{1, 0}/2]

       == {\(-▽\&⇀\)·\((
                \(▽\&⇀\) X 
                   A\&⇀[t])\) - 
            ▽\&⇀·\((
                ▽\&⇀, 
                  ϕ[t])\) + ∂\^2 ϕ[t]\/∂t\^2, 

          				\ 
          \(▽\&⇀\) X 
              \((\(▽\&⇀\) X 
                   A\&⇀[t])\) + 
            \(▽\&⇀\) X 
              \((▽\&⇀, 
                  ϕ[t])\) - 
            ▽\&⇀, 
          \((▽\&⇀·A
                    \&⇀[t])\) + 
            ∂\^2 A\&⇀[t]\/∂t\^2}\)

\(\((\((∂\/∂t, \ \ 
                ▽\&⇀)\)\^2 + 
            \((∂\/∂t, 
                \(-\ ▽\&⇀\))\)\^2)\) 
        \((ϕ, A\&⇀)\)\/2
 = 
      \(\((∂\^2 ϕ\/∂t\^2 - 
              \(▽\&⇀\^2\) ϕ - 
              ▽\&⇀·▽\&⇀  X  A\&⇀, 
            ∂\^2 A\&⇀\/∂t\^2\  - 
              ▽\&⇀\(, \^2\)A
                \&⇀\  + 
              \(▽\&⇀\) X 
                 ▽\&⇀, ϕ)
          \) = 4\ π\ \((ρ, J\&⇀)\)\)\)

This one quaternion equation encompasses the four Maxwell equations in the Lorenz gauge in a vacuum. A different source term is required for different media. It should be noted that the operator is not the scalar D'Alembertian operator, which would only generate the source terms and not the homogenous equations.

Examine the terms that are generated by a change in reference frame.

\(Simplify[
	
      op[{q[dt, ▽\&⇀], 
              q[dt, ▽\&⇀]}, {
              q[\(-dt\), ▽\&⇀], 
              q[dt, \(-▽\&⇀\)]}, 
					
            q[ϕ[t], A\&⇀[t]]].{1, 0}/2]\){-2 
        (Overscript[▽, ⇀])·∂(Overscript[A
                  , ⇀])[t]/∂t, 
      2 ((Overscript[▽, ⇀]) X 
              ∂(Overscript[A, ⇀])[t]/∂t + 
            ((Overscript[▽, ⇀]), 
                ∂ϕ[t]/∂t))}

\(\((\((∂\/∂t, \ \ 
                ▽\&⇀)\)\^2 + 
            \((\(-\(∂\/∂t\)\), \ 
                \(-\ ▽\&⇀\))\)\^2)\) 
        \((ϕ, A\&⇀)\)\/2
 = \ 
      2 \((\(-\ ▽\&⇀\)·∂A
                    \&⇀\/∂t, 
            \(▽\&⇀\) X 
                ∂A\&⇀\/∂t + 
              ▽\&⇀, 
            ∂ϕ\/∂t)\)\)

The frame-dependent terms are all mixed derivatives formed from the scalar time derivative and vector del operator.

Implications

Examples of a quaternion-valued, nonhomogeneous, second-order differential equations have been explored. By careful selection of the operators and source terms, it has been shown that one representation of this equation gives the Maxwell equations. There may be other choices for fields, operators, and source terms which are relevant to physics. Since the operators also have the properties of a field, it should be possible in theory to generate any 4-dimensional differential equation or infinite series of 4-dimensional differential equations.