Gauge transformations of the Maxwell equations

doug <sweetser@world.com>

Introduction
A quaternion gauge transformation
Implications

Introduction

A gauge defines how one measures the electromagnetic 4-potential, phi and A.  Within limitations, one is free to choose a standard for the 4-potential.  For example, if one decides that the divergence of A should always be zero, this is called choosing the Coulomb gauge.  Gauge transformations are a very powerful technique for solving practical problems.  Gauge symmetry also has profound implications theoretically, particularly for quantum field theory.  Spacetime-dependent gauge symmetry leads to the gauge fields that form the symmetry of the forces in the standard model, U(1)xSU(2)xSU(3).

In this notebook, I will explore gauge transformations with quaternions.  This work suggests two types of constraints on those transformations.  The first is the continuity equation for the scalar field.  The second is a "continuity-like" equation for the vector field.  This second constraint is relevant only under relativistic conditions.

A quaternion gauge transformation

The E and B fields can be generated from the 4-potential phi and A with the following quaternion equation:

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

The vector term is B - E.  The scalar term has the form of a gauge relation.  The typical scalar field used in a gauge transformation involves the gradient and the time derivative of the scalar field.  For a quaternion field, there is also a 3-vector component.  Consider the analogous quaternion field L (small "g" is a scalar field, and big "G" is a 3-vector field).

L = {
        g'[t] +  
          (Overscript[▽, ⇀])·(Overscript[G, ⇀])[
              t], -(
              (Overscript[▽, ⇀]), g[t])
               +  (Overscript[G, ⇀])'[t]}

L = 
      (g' +  (Overscript[▽, ⇀])·(Overscript[G
                , ⇀]), 
          -(Overscript[▽, ⇀]), 
          g +  (Overscript[G, ⇀])')

The terms with the small g are the typical scalar field gauge transformation.  The terms with the big G are my guess at a reasonable extension from a scalar to a quaternion field.  Big G is handled in a manner that parallels small g, namely there is a time derivative and del operation that transforms vector big G into the scalar div G, just like scalar small g was transformed to vector Grad g.  Transform the quaternion potential with L.

\(\((op[{q[dt, ▽\&⇀]}, 
            q[ϕ[t], A\&⇀[t]]]\  + 
\ \ 
          op[{q[dt, ▽\&⇀]}, 
            q[\(g'\)[t], 
              \(-\((▽\&⇀, 
                    g[t])\)\)]]\  + 
\ \ 
          op[{q[dt, ▽\&⇀]}, 
            q[\ ▽\&⇀·G
                    \&⇀[t], \(G\&⇀'\)[t]]])\).{1, 0}
        \)\({\(-▽\&⇀\)·A\&⇀[
              t] + ∂ϕ[t]\/∂t + \ 
          ▽\&⇀·\((
              ▽\&⇀, g[t])\) + 
          ∂\^2 g[t]\/∂t\^2, 
	
        \(+\ ▽\&⇀\) X  A\&⇀[t] + 
          \((▽\&⇀, ϕ[t])
            \) + ∂A\&⇀[t]\/∂t
\  - 
          \(▽\&⇀\) X 
             ▽\&⇀, 
        g[t] + \(▽\&⇀\) X 
            ∂G\&⇀[t]\/∂t + 
          ▽\&⇀, 
        \((▽\&⇀·G\&⇀[
                t])\) + ∂\^2 G\&⇀[t]\/∂t\^2}
      \)

\(\((∂\/∂t, ▽\&⇀)\) 
        \((\((ϕ, A\&⇀)\)\  + 
            \((g', \(-▽\&⇀\), 
                g)\)\  + 
            \((▽\&⇀·G
                  \&⇀, G\&⇀')\))\)
 = 
      \((∂ϕ\/∂t - 
            ▽\&⇀·A
                \&⇀\  + ∂\^2 g\/∂t\^2 + \ 
            ▽\&⇀·▽\&⇀, g, 
	
          ∂A\&⇀\/∂t + \ 
            \(▽\&⇀\) X  A\&⇀ + 
            ▽\&⇀, 
          ϕ + ∂\^2 G\&⇀\/∂t\^2 + 
            ▽\&⇀, 
          ▽\&⇀·G\&⇀ + 
            \(▽\&⇀\) X 
              ∂G\&⇀\/∂t - 
            \(▽\&⇀\) X 
               ▽\&⇀, g)\)\)

An equation should not be altered by a gauge transformation, but that does not appear to be the case with the quaternion field L at first glance!  There are 5 extra terms here (one of the extra terms, curl grad g, is always zero).

The two extra scalar terms must always add up to zero.  These terms involve the time derivative of a scalar and the divergence of a vector.  That sounds like the continuity equation, so test out the following mapping: grad g -> J, dg/dt -> rho.

0 == (Overscript[▽, ⇀])·(
              (Overscript[▽, ⇀]), g[t]) + 
          (∂^2) g[t]/∂(t^2) /.
	{((Overscript[▽, ⇀]), g[t]) -> 
             J,Derivative[2][g] -> Derivative[1][ρ]}0 == (Overscript[▽, ⇀])·J + 
        ∂ρ[t]/∂t

(Overscript[▽, ⇀])·(Overscript[▽, ⇀]), g + (∂^2) (g/∂)(t^2) = 0\(if	▽\&⇀\),g ->  J,∂(g/∂)t -> ρ, then  (Overscript[▽, ⇀])·J+∂(ρ/∂)t = 0

This is the continuity equation.  It is noteworthy that the continuity equation arises naturally from an analysis of a quaternion gauge transformation.  The J must be irrotational (Curl J = Curl Grad g = 0).  This may be due to my choice for big G which lacked a curl, but I am uncertain on this point.

What about the three vector terms?  They reform the vector potential with the following mapping.

0 == (Overscript[▽, ⇀]) X 
             (Overscript[G, ⇀])'[t] + 
          (Overscript[▽, ⇀]) 
            ((Overscript[▽, ⇀])·(Overscript[G
                    , ⇀])[t]) + 
          (∂^2) (Overscript[G, ⇀])[t]/∂(t^2) /.
		{((Overscript[▽, ⇀])·(Overscript[G
                    , ⇀])[t]) -> ϕ[t], Derivative[2][(Overscript[G, ⇀])]->Derivative[1][(OverVector[A])],
				Derivative[1][(Overscript[G, ⇀])]->(OverVector[A])}0 == (Overscript[▽, ⇀]) X  (Overscript[A, ⇀])[t] + 
        (Overscript[▽, ⇀]) ϕ[t] + 
        ∂(Overscript[A, ⇀])[t]/∂t

(Overscript[▽, ⇀]) X 
          ∂((Overscript[G, ⇀])/∂)t + 
        (Overscript[▽, ⇀]) 
          ((Overscript[▽, ⇀])·(Overscript[G
                , ⇀])) + 
        (∂^2) ((Overscript[G, ⇀])/∂)(t^2)  = 0if   ∂((Overscript[G, ⇀])/∂)t->(OverVector[A]) ,(Overscript[▽, ⇀])·(Overscript[G, ⇀])->ϕ   then   (Overscript[▽, ⇀])X(Overscript[A, ⇀])+(Overscript[▽, ⇀]) ϕ+∂((Overscript[A, ⇀])/∂)t=0

This mapping proposes that the a gauge transformation will be valid only if B - E = 0.  In some sense, this is like a continuity equation for the vector field big G.  The time rate of change of A must balance the gradient phi and the curl of A, which all can be expressed in terms of the same vector field big G.  This is a new constraint, but it neatly parallels the scalar continuity equation, which applied only to the scalar field g with the mapping discussed here.

Implications

Every constraint has regions where is is applicable.  Newtonian physics works until the speed of light is approached.  Classical atomic physics works until the number of particles gets small.  The vector gauge continuity constraint is not relevant for classical electrodynamics.  The classical region of spacetime happens for intervals where

|Δt |   >>>  |Δx,  Δy, 
     Δz |    ==  classical physics

|Δt |   >>>  |Δx,  Δy, 
     Δz |    ==  classical physics

An analogous region for a gauge transformation is

g >>> (Overscript[G, ⇀])   ==  classical EM

g >>> (Overscript[G, ⇀])   ==  classical EM

In this region, the gauge transformation then simplifies to

ϕ  ->  ϕ'  =  
      ϕ  -  ∂g[t]/∂tA  ->  A'  =  A  +  (Overscript[▽, ⇀]) g

ϕ  ->  ϕ'  =  ϕ  -  ∂(g/∂)tA  ->  A'  =  A  +  (Overscript[▽, ⇀]) g

This is precisely the type of transformation that has been used so often before, and it is still part of the quaternion formulation of the Maxwell equations in the classical domain.

The question now concerns the validity of gauge freedom of the Maxwell equations under relativistic conditions, a cornerstone of quantum field theory.  I prefer to constrain the vector portion of gauge transformations under relativistic conditions with a continuity equation for the vector field that has symmetry with the scalar continuity equation.  This may help avoid the difficulties of regularization and renomalization, but that issue needs more investigation.