P and T transformations of EM vector potential

[TriTertButoxy]

Help! What are the P and T transformation laws for the electromagnetic vector potential, $A_\mu$? and how are these consistent with the transformation laws of the electric and magnetic vectors that I am familiar with?

under P: E is odd, B is even
under T: E is even, B is odd

When I try to use the formulas,
$$\mathbf{E}=-\nabla\phi-\frac{\partial\mathbf{A}}{\partial t}$$
$$\mathbf{B}=\nabla\times\mathbf{A}$$,
I get incorrect results.

 

[gtoguy97]

The transformation laws for the electromagnetic vector potential A_\mu are as follows: Under P (parity): A_\mu is odd (i.e., A_\mu(-\mathbf{x},t)=-A_\mu(\mathbf{x},t)) Under T (time reversal): A_\mu is even (i.e., A_\mu(-\mathbf{x},-t)=A_\mu(\mathbf{x},t)). These transformation laws are consistent with the transformation laws of the electric and magnetic vectors that you are familiar with. Under parity, the electric field (E) is odd and the magnetic field (B) is even, while under time reversal, the electric field is even and the magnetic field is odd. These transformation laws can be seen by applying the formulas you provided to the electromagnetic vector potential A_\mu. For instance, under parity, we have \mathbf{E}=-\nabla\phi-\frac{\partial(-A_\mu(-\mathbf{x},t))}{\partial t}=-\nabla\phi+\frac{\partial A_\mu(\mathbf{x},t)}{\partial t}=-\nabla\phi-\frac{\partial\mathbf{A}}{\partial t}which shows that E is odd, and similarly, \mathbf{B}=\nabla\times(-A_\mu(-\mathbf{x},t))=\nabla\times A_\mu(\mathbf{x},t)=\nabla\times\mathbf{A}which shows that B is even.

 

[astrokreng]

The P and T transformations of the electromagnetic vector potential, A_\mu, follow the same laws as the electric and magnetic vectors, E and B. Under a parity (P) transformation, the electric field (E) is odd and the magnetic field (B) is even. This means that under P, the vector potential must also transform as an odd quantity. Similarly, under a time reversal (T) transformation, E is even and B is odd, so A_\mu must also transform as an even quantity.

To understand why your results using the given formulas do not match the expected transformations, it is important to remember that the vector potential is not a physically measurable quantity. It is a mathematical construct used to describe the behavior of the electric and magnetic fields. Therefore, it does not transform in the same way as the electric and magnetic fields themselves.

To correctly apply the P and T transformations to A_\mu, you can use the following formulas:

Under P:
A_0 \rightarrow -A_0
\mathbf{A} \rightarrow -\mathbf{A}

Under T:
A_0 \rightarrow A_0
\mathbf{A} \rightarrow -\mathbf{A}

Using these transformations, you should be able to obtain the correct results for the P and T transformations of the vector potential. It is also important to note that these transformations are consistent with the transformation laws of the electric and magnetic fields, as they are derived from the same underlying principles of symmetry and invariance.

In summary, the P and T transformations of the electromagnetic vector potential follow the same laws as the electric and magnetic fields. However, since the vector potential is a mathematical construct, it transforms differently from the physical fields themselves. By using the correct transformation formulas, you should be able to obtain the expected results.