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

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.

Introduce the renormalization-group beta functions and field anomalous dimensionIndependence of physics from the arbitrary sliding scale gives the functional Callan-Symanzik equationup to the equivalent sign convention obtained by defining ${\gamma }_{\varphi }$ with a minus sign. For an operator $O_i$ of dimension $d_i$ containing $n_i$ fields, differentiating its coefficient shows the canonical and wave-function pieceswhere ${\beta }_i^{\mathrm{vertex}}$ contains mixing among local field operators and genuine corrections at higher loop order. Thus $d_i<4$, $d_i=4$, and $d_i>4$ canonically produce relevant operators, marginal operators, and irrelevant operators, respectively, before anomalous corrections.

Applying four functional derivatives to the functional equation gives the homogeneous equation for the renormalized four-point one-particle-irreducible correlation function:If the opposite convention for ${\gamma }_{\varphi }$ is used, the last sign changes. The factor four counts the four external renormalized fields.

A continuum limit of a quantum field theory sends the ultraviolet cutoff ${\Lambda }_0$ to infinity while holding chosen physical masses and amplitudes fixed. In cutoff units this requires a diverging correlation length, so the bare couplings must be tuned onto the critical surface of an ultraviolet renormalization-group fixed point. Each relevant direction of a fixed point requires one coordinate fixed by measurement; predictivity requires only finitely many such directions, and irrelevant microscopic details disappear.

Several behaviors are possible. In an asymptotically free theory the trajectory approaches a Gaussian fixed point in the ultraviolet and interactions vanish logarithmically. In an asymptotically safe quantum field theory it approaches a non-Gaussian fixed point with finitely many relevant directions. A theory with a Landau pole may possess only a trivial quantum field theory as its continuum limit; retaining a nonzero interaction then requires a finite cutoff. Finally, if no ultraviolet-complete trajectory exists, the model remains an effective field theory valid only below a physical cutoff.