A. Zee, Quantum Field Theory in a Nutshell · Problem IV.3.1

Field Theory Redux

Physics — QFT

Problem

Consider the effective potential in

(0 + 1)

-dimensional spacetime:

V_{\text{eff}}(\phi) = V(\phi) + \frac{\hbar}{2} \int \frac{dk_E}{(2\pi)} \log \frac{k_E^2 + V''(\phi)}{k_E^2} + O(\hbar^2)

No counterterm is needed since the integral is perfectly convergent. But

(0 + 1)

-dimensional field theory is just quantum mechanics. Evaluate the integral and show that

V_{\text{eff}}

is in complete accord with your knowledge of quantum mechanics.