Electromagnetic tensor and restricted Lorentz group

[TrickyDicky]

How exactly is the EM field tensor related to the proper orthochronous Lorentz group?

 

[Matterwave]

What do you mean by "related". As a rank 2 tensor, the Faraday tensor transforms under a proper orthochronous Lorentz transformation just like any other rank 2 tensor:

$$F_{\mu'\nu'}=\Lambda^\sigma_{~\mu'}\Lambda^\tau_{~\nu'}F_{\sigma\tau}$$

 

[TrickyDicky]

What do you mean by "related". As a rank 2 tensor, the Faraday tensor transforms under a proper orthochronous Lorentz transformation just like any other rank 2 tensor:

$$F_{\mu'\nu'}=\Lambda^\sigma_{~\mu'}\Lambda^\tau_{~\nu'}F_{\sigma\tau}$$

I mean the specific representation of the group, why is it the direct sum of the (1,0) and (0,1) irreps, is it related with the tensor being a 2-form and its parity invariance?

 

[dextercioby]

It has to do with the parity invariance. Just like in the Dirac field case, it's the reason why we need the direct sum of elementary irreducible representations.

 

[sardar]

The electromagnetic tensor is a mathematical representation of the electromagnetic field, which describes the interaction between electric and magnetic fields. It is a 4x4 matrix that contains all the information about the strength, direction, and polarization of the EM field at a given point in space and time.

The proper orthochronous Lorentz group is a mathematical group that describes the transformations of space and time in special relativity. It includes rotations, boosts, and reflections that preserve the speed of light and the direction of time. This group is essential in understanding the behavior of physical systems in the presence of electromagnetic fields.

The relationship between the electromagnetic tensor and the proper orthochronous Lorentz group can be understood through the concept of gauge invariance. Gauge invariance refers to the fact that the physical laws governing the electromagnetic field remain unchanged under certain transformations. These transformations are precisely the ones described by the proper orthochronous Lorentz group.

In other words, the EM field tensor is invariant under the transformations of the proper orthochronous Lorentz group. This means that the same physical laws and equations that govern the electromagnetic field in one reference frame will also hold true in another reference frame that is moving at a constant velocity with respect to the first one. This is a fundamental principle of special relativity and is crucial in understanding the behavior of electromagnetic fields in different frames of reference.

In summary, the EM field tensor and the proper orthochronous Lorentz group are closely related through the concept of gauge invariance. The tensor is invariant under the transformations of the group, which allows us to understand the behavior of electromagnetic fields in different reference frames and make accurate predictions about their effects on physical systems.