Why Is There a -1/4 Factor in the Electromagnetic Lagrangian?

[captain]

why is there a factor of -1/4 in front of the two tensors? Is it formulated that way for convenience?

 

[Parlyne]

It's not convenience. The kinetic term for any real scalar field has a factor of 1/2. But, when you work out the explicit form of the EM lagrangian in terms of the photon field, you find that the contraction of the tensors gives four terms; but, each of the four is identical to one of the others. This means that, in terms of the field, there's an extra factor of 2 that needs to be accounted for, turning the 1/2 into a 1/4.

 

[astrokreng]

The factor of -1/4 in front of the two tensors in the electromagnetic Lagrangian is not just for convenience, but it is a fundamental part of the formulation. It is derived from the symmetries of the electromagnetic field and is necessary for the Lagrangian to be gauge invariant.

In the electromagnetic Lagrangian, the two tensors represent the electric and magnetic fields, and their interaction with each other. The -1/4 factor is a result of the tensor product of these two fields. It ensures that the Lagrangian is symmetric under gauge transformations, meaning that the physical laws described by the Lagrangian remain unchanged even if we change the reference frame or the gauge of the fields.

Furthermore, this factor also ensures that the equations of motion derived from the Lagrangian are consistent with the experimental observations of electromagnetic phenomena, such as the Maxwell's equations. In other words, it is a necessary component for the Lagrangian to accurately describe the behavior of the electromagnetic field.

In summary, the factor of -1/4 in front of the two tensors in the electromagnetic Lagrangian is not just a convenience, but it is a crucial part of the formulation that ensures the Lagrangian is gauge invariant and consistent with experimental observations.