 Solution

Couple a real source through $H_J=H_0-\int d^{D}xJ(x)\varphi (x)$. Its Fourier components obey $\tilde{J}(-p)=\tilde{J}(p{)}^{\ast }$, and the source term is $-{\int }_p\tilde{J}(p)\tilde{\varphi }(p{)}^{\ast }$. Complete the square using the positive quadratic kernel from the previous part:Translation of the integration variables in the regulated Gaussian functional integral therefore givesThis also fixes the sign: a real linear source increases the Gaussian generating function by a positive quadratic exponent, unlike an imaginary Fourier-characteristic-function source.

To specify $Z_G$ rather than conceal an infinite normalization, first impose finite volume and an ultraviolet cutoff, and choose orthonormal real coordinates for the independent field modes. If their number is $N_R$ and the eigenvalues of the quadratic form are ${\Delta }_j>0$, the measure ${\prod }_{j=1}^{N_R}d{\varphi }_j$ givesA measure ${\prod }_{j}d{\varphi }_j/\sqrt{2\pi }$ removes the prefactor. In the continuum one may express this as $\log Z_G=-\frac{1}{2}\mathrm{Tr}\log \Delta$ plus a source-independent measure constant, with the same regulator understood. The real-mode prescription prevents double counting the conjugate Fourier pairs. All normalized correlation functions follow from $Z_0[J]/Z_G$ and do not depend on that constant.