Why can we always drop any constant term in a Lagrangian density in quantum field theory?
This issue is somehow related to the constant term being some kind of cosmological constant.
Can you please explain this issue?
Answer
You can drop a constant term in the Lagrangian because it doesn't affect the Euler-Lagrange equations, and therefore the equations of motion are the same.
A constant term does have an effect in the actual value of, say, the Hamiltonian (and the rest of Noether charges). But you can only measure differences in energies, and therefore a constant term in the Hamiltonian is again irrelevant (the same can be said about all the scalar Noether charges; if a charge has a Lorentz index then the constant term vanishes by symmetry). In other words, a constant term doesn't affect predictions of conserved operators.
If you include gravity, then a constant term becomes relevant, and you cannot drop it. But the quantisation program doesn't specify a canonical way to choose a constant term: we must resort to experiments to measure it (in other words, the cosmological constant, like most parameters in a theory, cannot be predicted but is an input to the model instead). Or put it another way: if you consider gravity, then the vacuum energy is no longer irrelevant, but it is still arbitrary: no theory can predict its value (as far as we now today).
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