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In physics there is no general criterion on how to write down suitable Lagrangians, rather than a posteriori check on the equations of motions: all the Lagrangians generating the same dynamics are equally correct. For example, as an exercise, you may try to write down all the possible Lagrangians giving you back $F_j = m \ddot{x}_j$. This said, to directly ...


4

It's not quite correct to take $\Lambda \rightarrow \infty$, even at the end of the calculation. That comes from ancient mistaken notions that the field theory under consideration needs to describe physics upto arbitrarily small distances. The modern way to think about this is that you're making your theory agnostic of the value of $\Lambda$. A theory is ...


2

If physics isn't an issue, you can add arbitrarily many terms. Once the physics comes in though, you will encounter a few restrictions : As said by Gennaro, it is assumed that the Poincaré symmetry applies. Higher derivative terms (second derivatives and above) are generally bad news. They can cause vacuum instability (energies can be arbitrarily ...


2

Yes, $\mu$ can be anything. Usually in renormalization, we measure (or define) the coupling constant $g$ at scale $\mu$, and then use this information to predict the coupling constant $g'$ at another scale $\mu'$. We require that $g'$ at $\mu'$ is independent of $\mu$. What I mean is that you should get the same $g'$ at $\mu'$ even if you use another $g$ ...


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Your questions are closely related :) The overall factors $(\not{p} - m)$ are indeed responsible for amputating the external legs. I don't have a copy of Schwartz on hand so I can't comment on what he might have meant, but the correlation functions do have propagators on the external legs before putting them on shell by applying LSZ. The mass that goes ...


2

There are a couple of points to be precise: in four spacetime dimensions there is no scalar relativistic interacting field theory that can be rigorously defined (i.e. in which the unitary dynamics can be constructed), at least for the moment. This does not mean it is not possible, but we have not the mathematical tools to do it. The physical calculations ...



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