A Wilsonian <renormalization group> step:

* integrates out fast modes $\Lambda/\zeta<|k|<\Lambda$;
* rescales $x'=x/\zeta$ and $k'=\zeta k$ to restore the cutoff;
* rescales the field to restore the chosen gradient-term normalization.

The resulting effective free energy defines the transformed couplings and their flow.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 303 / 2 / a / Solution

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A Wilsonian renormalization-group step integrates out short-distance modes, rescales coordinates to restore the cutoff, and rescales fields to normalize a chosen kinetic term.


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 Incoming links: Renormalization group