Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 304 / 2 / a

Split the field into disjoint ranges of Fourier modes,The Wilsonian effective action is defined byWriting $\Delta S[\varphi ,{\varphi }^{+}]=S_{{\Lambda }_0}[\varphi +{\varphi }^{+}]-S_{{\Lambda }_0}[\varphi ]$ immediately givesQuadratic cross terms vanish because the momentum supports do not overlap. For $h_0\ne 0$,where

When $h_0=0$, the cubic contribution is absent andAlthough terms odd in ${\varphi }^{+}$ occur for a fixed low field, the simultaneous transformation $(\varphi ,{\varphi }^{+})\mapsto (-\varphi ,-{\varphi }^{+})$ shows that integrating out the shell preserves the original ${\mathbb{Z}}_2$ discrete symmetry of the effective action.