- #1

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I found a nice little derivation of the Proca (and Maxwell) equations by Gersten (1998, PRL, 12, 291-298 is the reference, but I got it from the web so what I have may not be the full paper). It starts is a similar way to the Dirac derivation

##E^2-p^2=0##

And now factor this as:

##(E 1_3 - \textbf{p} \cdot \textbf{S} ) (E 1_3 + \textbf{p} \cdot \textbf{S} ) \boldsymbol{\psi} - m^2 \boldsymbol{\psi} = 0##

where ##1_3## is the 3 x 3 identity matrix,

**p**is the 3-momentum,

**S**is a set of three 3 x 3 matrices to be determined, and ##\boldsymbol{\psi}## is a 3 component vector of spacetime functions.

To solve this, Gersten added a term:

##(E 1_3 - \textbf{p} \cdot \textbf{S} ) (E 1_3 + \textbf{p} \cdot \textbf{S} ) \boldsymbol{\psi} + \textbf{p} ( \textbf{p} \cdot \boldsymbol{\psi} ) - m^2 \boldsymbol{\psi} = 0##

and required that

##\textbf{p} \cdot \boldsymbol{\psi} = i m^2 \phi##

and

##(E 1_3 + \textbf{p} \cdot \textbf{S} ) \boldsymbol{\psi} = i m^2 \textbf{A}##

Now, I'm mainly okay with this. We are adding a term and supplying 4 functions to take up the slack for it that we can use to "fix" the correct solution. And I like the simplicity of it.

But, if you take this down to m = 0 (the Maxwell field) you get

##(E 1_3 - \textbf{p} \cdot \textbf{S} ) (E 1_3 + \textbf{p} \cdot \textbf{S} ) \boldsymbol{\psi} + \textbf{p} ( \textbf{p} \cdot \boldsymbol{\psi} ) = 0##

with

##\textbf{p} \cdot \boldsymbol{\psi} = 0##

and

##(E 1_3 + \textbf{p} \cdot \textbf{S} ) \boldsymbol{\psi} = 0##

Now we've lost all of our freedom to "fix" the solution. Clearly, this should generate a solution, but we should be able to pick any extra term to add to the energy equation (so long as its compatible with the second equation, of course.) How does this process guarantee the correct solution to the Maxwell equations?

Is there a way to a priori justify adding this term?

-Dan