I have a question regarding the constant terms in the lagrangian in field theory. Lets take the electromagnetic action in a vacuum for example: $$ S = \int \left( -\frac{1}{4} F_{\mu\nu} F^{\mu\nu} \right) d^4x. $$ If we remove the $-\frac{1}{4}$ won't the field stay the same? And if it stays the same why add it in the first place? This can also work for the einstein-hilbert action.
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