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Your way of thinking is, essentially, correct. When it comes to this $\tilde\lambda\Lambda^2$, there is this famous quote (citing from memory, don't remember which book it's from, but it's famous), "Even though it is infinitely large, we will assume that it is finite, and that is furthermore infinitely small." In most QFTs the perturbative series diverges ...


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Quantum dynamics is commonly known to be generated by self-adjoint operators. Therefore in order to properly define the dynamics of a system it is necessary to introduce a suitable self-adjoint Hamiltonian operator. In quantum field theories, this task is extremely difficult, because the formal operators that emerge quantizing a classical field theory ...


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In the second approach, you do have the option to treat the short distance physics in a variety of ways. Choosing point interactions will give you some particular values for $ \lambda _{ bare}$ and some particular rules about how to calculate $\delta \lambda $, and if you choose a different "UV completion" you will get a different value and a different rule. ...


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I believe that this may be what you’re looking for. I don’t claim to be an expert in renormalization, so all members of the Physics Stack Exchange Community are welcome to correct my answer. To make the relationship between the two more precise, let’s call the on-shell renormalization scheme the ‘on-shell subtraction scheme’ and the BPHZ renormalization ...


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Consider the partition function $$Z = \int D\phi ~ e^{-S_0 - S_I},$$ where $S_0$ is the Gaussian/free part and $S_I$ is the interaction part of the action. Within a perturbative framework we may aim to systematically include the contributions of fast modes to the (effective) action for slow modes. For this we expand in the interaction strength as $$Z = \int ...



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