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In Section 12.1 of Peskin & Schroeder they motivate Wilson's approach to renormalization by asking how a quantum field theory changes after changing the momentum scale. To answer this they start with a momentum scale $\Lambda$ and integrate out a single momentum shell in the path integral to get to a momentum scale $\Lambda'=b\lambda$ with $b<1$. The result is a new Lagrangian and hence a new action. I understand this part so far.

What confuses me is they proceed by rescaling the momenta: $$k' = \frac{k}{b}, \quad x' = bx, \tag{12.19}$$ where $b$ is chosen so that the cutoff $\Lambda'$ is restored back to $\Lambda$. They claim this is done so that we can compare the modified path integral (with the momentum shell integrated out) to the original one. However doesn't the rescaling (12.19) undo getting rid of all momentum modes $\Lambda' < |k| < \Lambda$, since the path integral will now include momenta in this range?

They do not mention it explicitly, but I suspect even with the rescaling that somehow the information in modes $\Lambda' < |k| < \Lambda$ that was in the original path integral remains lost and is not restored, is this true? What is the physical idea and motivation behind rescaling after integrating out high momentum modes?

Qmechanic
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CBBAM
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1 Answers1

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OP wrote:

However doesn't the rescaling (12.19) undo getting rid of all momentum modes $\Lambda' < |k| < \Lambda$, since the path integral will now include momenta in this range?

No, the rescaling eq. (12.19) is a relabelling, so that we can make a more careful comparison of the new functional integral $$Z~=~\int\!\left[\prod_{|k|<b\Lambda}d\hat{\phi}(k)\right]e^{-S_{\rm eff}[\phi]} \tag{12.6}$$ and the one we started with $$Z~=~\int\!\left[\prod_{|k|<\Lambda}d\hat{\phi}(k)\right]e^{-S[\phi]}, \tag{12.3}$$ as P&S write above eq. (12.19). The effective action $S_{\rm eff}[\phi]$ (and its coupling constants) in $Z$ is typically changed under the RG flow.

It should be mentioned that not all approaches to Wilsonian RG flow include the scaling (12.19), cf. e.g. my Phys.SE answer here.

Qmechanic
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