Four-vector Dipole Moment: Electric & Magnetic

[arpon]

What is the four-vector related to electric and magnetic dipole moment?

 

[sweet springs]

I have never think about that. How about electromagnetic tensor $F^{\mu\nu}$ represented by $\mathbf{P}$ and $\mathbf{M}$ instead of $\mathbf{E}$ and $\mathbf{B}$ ? I am not sure at all $\epsilon$ and $\mu$ are constant in Lorentz transformation.
Best.

 

[Orodruin]

I have never think about that. How about electromagnetic tensor $F^{\mu\nu}$ represented by $\mathbf{P}$ and $\mathbf{M}$ instead of $\mathbf{E}$ and $\mathbf{B}$ ? I am not sure at all $\epsilon$ and $\mu$ are constant in Lorentz transformation.
Best.

It is not the field tensor $F^{\mu\nu}$, it is a separate anti-symmetric rank 2 tensor $M^{\mu\nu}$. It is true that it is constructed from the electric and magnetic dipole moments in the same way $F$ is constructed from the electric and magnetic fields. You can then put an interaction term proportional to $F_{\mu\nu}M^{\mu\nu}$ into the Lagrangian density, effectively describing the dipole interactions.

Edit: So to answer the OP. There is no 4-vector describing the dipole moments. The dipole moments together form an anti-symmetric rank 2 tensor. If you only consider the $SO(3)$ subgroup of spatial rotations, the 6-dimensional anti-symmetric tensor representation splits into the two 3-dimensional vector representations.

Edit 2: Well, actually one vector representation and one pseudo vector representation ...

Edit 3: Well, actually, if restricted to $SO(3)$ the vector and pseudo vector representations are the same ... It makes a difference if restricted to $O(3)$.