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

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The factor of -1/4 in the electromagnetic Lagrangian arises from the need to account for the contraction of the two tensors, which results in four identical terms. While the kinetic term for a real scalar field includes a factor of 1/2, the electromagnetic case requires an adjustment. This adjustment leads to an additional factor of 2, changing the 1/2 to 1/4. The formulation is not for convenience but is necessary to accurately represent the dynamics of the electromagnetic field. Understanding this factor is crucial for correctly interpreting the Lagrangian's structure.
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why is there a factor of -1/4 in front of the two tensors? Is it formulated that way for convenience?
 
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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.
 

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