- #1
Phrak
- 4,266
- 7
There are many conserved charges in electromagnetism besides electric charge.
The electric charge and current combine to form a Lorentz covariant vector, \ J^{\mu} = (\rho, \textbf{J}).
This vector is a derivative of the the Maxwell tensor, F^{\mu\nu}. (More specifically, a derivative of \ \epsilon_{\rho\sigma\mu\nu} F^{\mu\nu}.)
The Maxwell tensor can be defined in terms of the electric and magnetic potentials, \ (\phi ,\textbf{A}), so that \ J^{\mu} is also a function of (\phi,\textbf{A})
Without distraction by the mathematical details, the electric charge, is a second derivative of the potential field, \ A^{\nu}=(\phi,\textbf{A}) :
Applying the same function, f, it's immediately apparent that a quantity K, is conserved as well,
I can't imagine the physical significance of the charge, \ K^0 if there is one. (\rho,\textbf{J}) would be required to be twice differentiable over space and time.
Here appears to be this persistent stuff, whatever it is. It it, in general, nonzero. It never goes away, but we don't seem to notice it. Does it have a name?
BTW, there should be an infinite sequence of these 'charges,' f^{n}(A^{\mu}), n=0,1,2,...
The electric charge and current combine to form a Lorentz covariant vector, \ J^{\mu} = (\rho, \textbf{J}).
This vector is a derivative of the the Maxwell tensor, F^{\mu\nu}. (More specifically, a derivative of \ \epsilon_{\rho\sigma\mu\nu} F^{\mu\nu}.)
The Maxwell tensor can be defined in terms of the electric and magnetic potentials, \ (\phi ,\textbf{A}), so that \ J^{\mu} is also a function of (\phi,\textbf{A})
Without distraction by the mathematical details, the electric charge, is a second derivative of the potential field, \ A^{\nu}=(\phi,\textbf{A}) :
J^{\mu} = f(A^{\nu})
Applying the same function, f, it's immediately apparent that a quantity K, is conserved as well,
K^{\mu} = f(J^{\nu})
I can't imagine the physical significance of the charge, \ K^0 if there is one. (\rho,\textbf{J}) would be required to be twice differentiable over space and time.
Here appears to be this persistent stuff, whatever it is. It it, in general, nonzero. It never goes away, but we don't seem to notice it. Does it have a name?
BTW, there should be an infinite sequence of these 'charges,' f^{n}(A^{\mu}), n=0,1,2,...
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