Identifying a pseudovector

[snorkack]

Main Question or Discussion Point

The difference between a vector and a pseudovector is that while a vector changes direction on space reflection, a pseudovector does not.

However, a space reflection cannot be done physically, or can it?

Electric current and magnetic field are connected by a hand rule. This means that they cannot be both vectors, otherwise the hand would be reflected on space reflection. And neither can they be both pseudovectors.

How do you actually find out which of the two is a pseudovector - how do you prove that magnetic field is a pseudovector but current is a real vector, and not vice versa?

 

[phyzguy]

The very fact that you need a "hand rule" shows that the magnetic field is a pseudo-vector. While electric current is a true vector, the magnetic field is actually the space part of the EM field tensor, which is a rank 2 antisymmetric tensor. If you write out the EM field tensor, you will see that the E field is the time-space part of it, and the B field is the space-space part of it. Since the B-field includes two space indices, it does not change sign when you change the sign of the space coordinates, while the E-field, which contains only one space coordinate, does change sign. So under spatial transformations, the E field behaves as an ordinary vector, while the B field behaves as a pseudo-vector. Under Lorentz transformations, the E and B fields mix, so neither one transforms as a vector and you need to use the complete EM field tensor.