 Gaussian field theory

A Gaussian field theory has a quadratic free energy and therefore a Gaussian measure on its field modes. For independent real coordinates $x_{\alpha }$ with positive quadratic coefficients ${\lambda }_{\alpha }$, writeHere $T>0$ is temperature and $k_B$ is the Boltzmann constant. Each elementary Gaussian integral givesFor example, differentiating $\log Z_0$ with respect to ${\lambda }_{\alpha }$ gives $-⟨x_{\alpha }^2⟩/(2k_{B}T)$ and proves the variance formula.

For a real scalar field, paired complex Fourier modes obey ${\varphi }_{-q}={\varphi }_q^{\ast }$. If each independent complex pair contributes $J(q)|{\varphi }_q{|}^2$ to the physical free energy, then $⟨|{\varphi }_q{|}^2⟩=k_{B}T/J(q)$. The often-used inverse-kernel formula $J(q{)}^{-1}$ treats $J$ as the dimensionless kernel in $F_0/(k_{B}T)$. Unpaired real modes require the real-coordinate convention above. ^[ref]

The inverse quadratic kernel is the two-point correlation function; Wick theorem then determines higher correlation functions. This makes a Gaussian field theory a solvable reference for perturbation theory and renormalization group calculations. A zero or negative quadratic eigenvalue requires separate treatment because the corresponding unrestricted Gaussian integral is not finite.