= Infrared-regulated phase covariance
{title2=$G_L(r)=\bar K^{-1}\int_{q\gtrsim L^{-1}}\frac{d^dq}{(2\pi)^d}\frac{e^{iq\cdot x}}{q^2}$}
Fix the uniform <order-parameter phase> and retain finite system size $L$ before evaluating a massless <Gaussian functional integral>. In $d\le2$, its coincident and separated <order-parameter phase> <covariances> each acquire a divergent additive constant as $L\to\infty$. Their difference $2[G_L(0)-G_L(r)]$ instead has a finite fixed-separation limit for $d>0$. This is why physical <XY model> <correlations> are obtained from <Gaussian phase averaging> of angle differences. An <ultraviolet cutoff> also remains necessary for coincident-point quantities; infrared cancellation does not remove it. The logarithmic two-dimensional <Green function> follows from the $d\to2$ power-law expression only after subtracting its constant pole.
Back to article page
