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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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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 ...

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1.why is renormalization even necessary? It is because, experiments force us to do that. You suppose to get finite values i.e, for charge or for mass of a particle. 2.if we always get a single finite result, why do we care that there are infinities in the equations at the level of amplitudes? Actually we don't. Your integrals in the loop ...

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Actually there are no infinities in physics. These "infinities" come from integrating over loop momenta when the momentum goes to infinity. But we don't know what actually happens to particles when they have infinite momentum. Maybe they turn into strings, maybe loop quantum gravity and black holes become relevant, maybe there's infinitely small unicorns ...

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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 ...

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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 ...

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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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Now I know why the Logarithmic discretization are take place in Anderson Model for low temperatures. We want to discretize the energy band-width $[-D,D]$ such that we can perform a numerical calculation. But we want to answering questions of low temperature, and we need to have very careful to apply the thermodynamic limit $N\rightarrow \infty$ before the ...

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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 ...

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