One-loop scalar effective action (${\Gamma }_1[\varphi ]$)

In Euclidean signature, for a canonically normalized real scalar field with potential $V$, the one-loop contribution to the quantum effective action is ${\Gamma }_1[\varphi ]=\frac{1}{2}\mathrm{Tr}\log [-{\partial }^2+V^{\prime \prime }(\varphi )]$, with a suitable regularization in quantum field theory. Expanding this functional determinant generates one-particle-irreducible Feynman diagrams with arbitrarily many external fields.

Expanding an action about a classical stationary field and retaining its second variation gives a Gaussian path integral. Its nonzero fluctuation modes determine a functional determinant and a one-loop soliton mass correction. Symmetry-generated zero modes in field theory must be removed from that determinant and integrated as collective coordinates.

Expanding the Hamiltonian about a static classical solution removes the linear term by its Euler-Lagrange field equation. For a canonical real scalar field with potential $U$, the quadratic fluctuation operator is $\mathcal{H}=-{\partial }_X^2+U^{\prime \prime }({\phi }_{\mathrm{cl}})$. Positive-frequency normal modes become quantum harmonic oscillators under canonical quantization, yielding the Gaussian contribution to a one-loop soliton mass correction. A negative eigenvalue signals an instability, while a physical zero mode in field theory must be replaced by a collective coordinate.