A. Zee, Quantum Field Theory in a Nutshell · Problem III.2.1

Field Theory Redux

Physics — QFT

Problem

Consider the

d

-dimensional scalar field theory

S = \int d^d x(\frac{1}{2}(\partial\phi)^2 + \frac{1}{2}m^2\phi^2 + \lambda\phi^4 + \cdots + \lambda_n\phi^n + \cdots)

. Show that

[\phi] = (d - 2)/2

and

[\lambda_n] = n(2 - d)/2 + d

. Note that

\phi

is dimensionless for

d = 2