= Gaussian field theory
{c}
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 $\lambda_\alpha$, write
$$
F_0=\frac12\sum_\alpha\lambda_\alpha x_\alpha^2,\qquad
Z_0=\int\prod_\alpha dx_\alpha\,e^{-F_0/(k_BT)}.
$$
Here $T>0$ is <temperature> and $k_B$ is the <Boltzmann constant>. Each elementary <Gaussian integral> gives
$$
Z_0=\prod_\alpha\sqrt{\frac{2\pi k_BT}{\lambda_\alpha}},\qquad
\langle x_\alpha x_\beta\rangle=\delta_{\alpha\beta}\frac{k_BT}{\lambda_\alpha}.
$$
For example, differentiating $\log Z_0$ with respect to $\lambda_\alpha$ gives $-\langle x_\alpha^2\rangle/(2k_BT)$ and proves the <variance> formula.
For a real <scalar field>, paired complex <Fourier modes> obey $\phi_{-q}=\phi_q^*$. If each independent complex pair contributes $J(q)|\phi_q|^2$ to the physical <free energy>, then $\langle|\phi_q|^2\rangle=k_BT/J(q)$. The often-used inverse-kernel formula $J(q)^{-1}$ treats $J$ as the dimensionless kernel in $F_0/(k_BT)$. Unpaired real modes require the real-coordinate convention above. <Statistical Field Theory by David Tong>[Chapter 2, Gaussian field integral and inverse-kernel two-point function; the physical-temperature factors are restored explicitly here]{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.
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