A. Zee, Quantum Field Theory in a Nutshell · Problem VI.6.3

Field Theory Redux

Physics — QFT

Problem

Field theory can often be cast into apparently rather different forms by a change of variable. Show that by writing

U = e^{\frac{1}{2} gh}

we can change the action (7) to

S = \frac{2}{g^2} \int d^D x \, dt \left[ U^{-1} \frac{\partial}{\partial t} U - U^{-1} \nabla^2 U \right]^2

, a kind of nonlinear

\sigma

model.