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The Fourier transform turns ${\nabla }^2$ into multiplication by $-q^2$. Each Cartesian component is an Ornstein-Uhlenbeck process:Since $r(q)>0$, the transient from the remote past vanishes, and the integrating-factor solution isUsing the stated Gaussian white noise covariance and assuming $t_1\ge t_2$,Interchanging the times gives the stationary two-time correlation functionAt equal times this is the equilibrium Gaussian field theory covariance, and the decay time is $r(q{)}^{-1}$.

LetEvolution from the earlier time to the later time multiplies the earlier field by the deterministic decay factor, plus fresh noise independent of that field. The unequal-time correlation function is thereforeEquivalently,When both observation times are many relaxation times after the quench, the second term vanishes and the stationary Ornstein-Uhlenbeck process covariance remains:

Choose the vacuum $\varphi =(v+\sigma ,{\pi }_1,\dots ,{\pi }_{N-1})$. The quadratic free energy contains a positive mass term for the radial field $\sigma$ but no mass term for any transverse field ${\pi }_a$:Thus the momentum-space correlation function is $⟨{\pi }_a(k){\pi }_b(-k)⟩={\delta }_{ab}/k^2$. Since a massive correlator has denominator $k^2+{\xi }^{-2}$, these Goldstone modes have ${\xi }^{-1}=0$ and hence infinite correlation length.

Canonical quantization imposesThe two-point correlation function has mode power $P_{\delta \varphi }=|f_k{|}^2/a^2$. After cosmological horizon exit, $|k\tau |\ll 1$, soThereforeThis is the scale-invariant inflationary power spectrum of a light canonical scalar.

The eliminated field obeys $\varphi =-c{p}^2/(2a)$, so its connected correlation function isFor the zero-mean Gaussian field of part (i), Wick theorem givesIt is therefore nonnegative and has the square of the Ornstein--Zernike spatial form. In particular, if $C_p(r)$ has exponential factor $e^{-r/{\xi }_p}$, then $C_{\varphi }(r)$ has $e^{-2r/{\xi }_p}$ and correlation length ${\xi }_p/2$, with the algebraic prefactor also squared.

The phase stiffness is the coefficient of $\frac{1}{2}\int d^{d}x(\nabla \theta {)}^2$ in the long-distance free energy. Its renormalized value determines the thermal phase fluctuations and algebraic correlation function of a two-dimensional O(N) model with $N=2$.

In a scalar field path integral with weight $\exp [i(S+\int J\varphi )/\hslash ]$, each internal propagator contributes $\hslash$, each interaction vertex contributes ${\hslash }^{-1}$, and each source insertion contributes ${\hslash }^{-1}$. A connected graph with $I$ internal lines and $V$ interaction vertices has loop order $L=I-V+1$. Its amputated amplitude, or its coefficient in $\log Z[J]$ with the source insertions included, is proportional to ${\hslash }^{I-V}={\hslash }^{L-1}$. An ordinary unamputated connected $n$-field correlation function instead has the additional $n$ external propagator factors and scales as ${\hslash }^{L+n-1}$. Multiplication by $-i\hslash$ to form the connected generating functional shifts the source-kernel counting to ${\hslash }^L$. These are compatible normalization conventions.

In a finite-dimensional integral, $u_i{e}^{b\cdot u}={\partial }_{b_i}{e}^{b\cdot u}$. Expand $e^{V(u)}$ as a formal perturbation series to obtain $\int du{e}^{-u^{T}Au/2+b\cdot u+V(u)}=e^{V({\partial }_b)}{I}_0(b)$. With a finite-mode regulator, a Minkowski source insertion satisfies $\varphi (x)e^{i\int J\varphi }=(1/i)\delta e^{i\int J\varphi }/\delta J(x)$; this gives the displayed interacting generating functional. The substitution is a perturbative differential-operator identity, not an assertion that the interaction exponential defines a convergent series. Divide the resulting functional by $Z[0]$ for normalized correlation functions.

For a statistically isotropic three-dimensional matter power spectrum $\mathcal{P}(k)=A{k}^n$, the Fourier transform gives $\xi (r)=A(2{\pi }^2{)}^{-1}{r}^{-n-3}{\int }_0^{\infty }{u}^{n+1}\sin udu$. The improper integral converges at zero precisely when $n>-3$ and at infinity when $n<-1$; it is absolutely convergent only when $n<-2$. Within this range its positive coefficient is $\Gamma (n+2)\sin [\pi (n+2)/2]$, interpreted continuously as $\pi /2$ at $n=-2$. Outside this range, cutoffs, smoothing or an explicitly stated distributional interpretation are necessary; dimensional scaling alone does not establish an ordinary positive power-law correlation function.

The renormalized phase stiffness has this discontinuity at the Berezinskii–Kosterlitz–Thouless transition. The algebraic correlation function has exponent $1/4$ at the transition from the ordered side.