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

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- Solution a

Solution

A Wilsonian renormalization group step:

integrates out fast modes $Λ/ ζ < ∣ k ∣ < Λ$ ;
rescales $x^{'} = x / ζ$ and $k^{'} = ζ k$ 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.

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