Time reversal transformation of electromagnetic four-potential

[Backpacker]

Consider the time-reversal Lorentz transformation given by the 4x4 matrix:

[itex]\Lambda_T = \begin{pmatrix}
-1 & 0 & 0 & 0\\
0 &1 & 0 & 0\\
0 & 0 & 1 & 0 \\
0 & 0 & 0 &1
\end{pmatrix}.
[/itex]

In my relativistic quantum mechanics lecture, we discussed how the electromagnetic 4-potential transforms under this particular Lorentz transformation. Without invoking any sort of mathematical argument, the prof argued that the four-potential transforms as
[itex]
\begin{align*}
A_0 (x^0,x^i)\longmapsto & A'_0 (x'^0,x'^i)=A_0 (-x^0,x^i)\\
A_j (x^0,x^i)\longmapsto & A'_j (x'^0,x'^i)=-A_j (-x^0,x^i)
\end{align*}
[/itex]
based on the idea that currents reverse under time-reversal.

Is there a good mathematical reasoning for this? It seems to me that since four-vectors transform as [itex]A\mapsto \Lambda A[/itex], the minus sign should be applied to [itex]A_0[/itex].

 

[dextercioby]

A coupling (scalar, not pseudoscalar) [itex] j^{\mu}A_{\mu} [/itex] is invariant, so that if j⁰ is invariant, then the space components of j and the ones of A have the same sign, namely -.

 

[Backpacker]

Ok, so I'm comfortable with the fact that the spatial components of the current are inverted under time reversal, i.e. ##{j'}^i=-j^i##.

But, why is ##j^\mu A_\mu## invariant? And what does ##j^\mu A_\mu## mean physically?

 

[Backpacker]

Hold on, I guess I see where some of my confusion is coming from.

In class, we showed that under a Lorentz transformation ##\Lambda##, the current (just like any good 4-vector) transforms as ##j'^\mu={\Lambda^\mu}_\nu j^\nu##. But I guess this is only for proper orthchronous transformations. Why is this different for improper transformations?

 

[paranormal]

I would like to provide a mathematical explanation for the transformation of the electromagnetic 4-potential under the time-reversal Lorentz transformation given by the 4x4 matrix \Lambda_T. First, let us define the electromagnetic four-potential as A=(A_0,A_1,A_2,A_3).

The time-reversal transformation given by \Lambda_T can be written as A'=\Lambda_T A. This means that the transformed four-potential is given by A'=(A_0',A_1',A_2',A_3')=(A_0,-A_1,-A_2,-A_3).

Now, let's consider the time-reversal transformation of the electric and magnetic fields, given by E'=-E and B'=-B. Using the definition of the electromagnetic fields in terms of the four-potential, E=-\nabla A_0-\frac{\partial A}{\partial t} and B=\nabla\times A, we can write the transformed fields as E'=-(-\nabla A_0-\frac{\partial A'}{\partial t})=-(-\nabla A_0+\frac{\partial A}{\partial t}) and B'=\nabla\times A'=-(-\nabla\times A).

From this, we can see that the time component of the four-potential, A_0, does not change under time-reversal, while the spatial components of the four-potential, A_i, change sign. This is consistent with the transformation given by the professor, as A_0'=-A_0 and A_j'=-A_j.

To summarize, the mathematical reasoning for the transformation of the electromagnetic four-potential under the time-reversal Lorentz transformation is based on the transformation of the electric and magnetic fields, which in turn is derived from the definition of the electromagnetic fields in terms of the four-potential. This provides a solid mathematical explanation for the professor's argument that the four-potential transforms as A\mapsto A'=(A_0,-A_1,-A_2,-A_3) under time-reversal.