Gaussian Fields and Sources

At a finite regulator, a free Euclidean field is a finite collection of coupled Gaussian variables. If its real symmetric kernel is positive definite, completing the square determines the integral exactly: the inverse kernel shifts the mean and gives the covariance, while the determinant fixes the source-independent normalization. Dividing by the zero-source value removes that determinant only when the kernel, domain, reference measure, and boundary data are unchanged.

The Lorentzian expression is different in kind. Its weight is oscillatory, so an inverse kernel and a square-root phase exist only after a contour or $i0$ boundary prescription has been selected. This page derives the Euclidean result as an ordinary finite integral, applies it to the regulated free scalar, and then translates the result to the Feynman prescription without treating a formal continuum product measure as already defined.

Required background. Regulated Bosonic Field Integrals supplies the finite variables, reference measure, lattice kernel, and zero-mode discipline used below. The Klein–Gordon Field and Its Modes supplies the free-scalar spectrum and periodic-box mode organization used to interpret the inverse kernel.

Helpful background. Gaussian Vectors, Processes, Random Distributions, and Wick Structure supplies general Gaussian probability language. Characteristic Functions, Moments, Cumulants, and Generating Functionals distinguishes moments from cumulants and generating objects. Neither page is needed for the finite calculation carried out here.

Completing the square exposes the inverse kernel

Let $x\in\mathbb R^N$, let $J\in\mathbb R^N$, and let

$$K_E=K_E^{\mathsf T}>0.$$

The inequality means $v^{\mathsf T}K_Ev>0$ for every nonzero real vector $v$. With dimensionless coordinates and the reference measure

$$\mathrm d\mu_0(x) = \prod_{a=1}^N\frac{\mathrm d x_a}{\sqrt{2\pi}},$$

define the unnormalized Euclidean source integral

$$\begin{aligned} \mathcal Z_E[J;K_E] &= \int_{\mathbb R^N}\mathrm d\mu_0(x) \\ &\quad\times \exp\!\left(-\frac12x^{\mathsf T}K_Ex\right) \\ &\quad\times \exp\!\left(J^{\mathsf T}x\right). \end{aligned}$$

Set $C_E=K_E^{-1}$. Symmetry of $K_E$ gives the exact identity

$$\begin{aligned} &-\frac12x^{\mathsf T}K_Ex+J^{\mathsf T}x \\ &\quad= -\frac12(x-C_EJ)^{\mathsf T}K_E(x-C_EJ) \\ &\qquad+ \frac12J^{\mathsf T}C_EJ. \end{aligned}$$

The translation $y=x-C_EJ$ preserves both $\mathbb R^N$ and its Lebesgue measure. Orthogonally diagonalizing $K_E$ then reduces the remaining integral to $N$ one-dimensional Gaussians. Hence

$$\begin{gathered} \boxed{ \begin{aligned} \mathcal Z_E[J;K_E] &=(\det K_E)^{-1/2} \\ &\quad\times \exp\!\left(\frac12J^{\mathsf T}C_EJ\right) \end{aligned} } \\ C_E=K_E^{-1}. \end{gathered}$$

The positive square root is fixed by $K_E>0$. The derivation uses the whole real integration domain: on a bounded region, a half-line, or a source-dependent cycle, translating the variable also translates the domain and the boxed formula need not follow. The finite Gaussian formula and its source derivatives are developed in Zinn-Justin 2021, § 1.1, pp. 1–3, and §§ 7.1–7.3, pp. 126–130.

Normalization separates determinants from source dependence

The normalized source functional is

$$\begin{aligned} Z_E[J;K_E] &\equiv \frac{\mathcal Z_E[J;K_E]} {\mathcal Z_E[0;K_E]} \\ &= \exp\!\left(\frac12J^{\mathsf T}C_EJ\right), \\ Z_E[0;K_E]&=1. \end{aligned}$$

Thus the determinant cancels when the numerator and denominator use the same kernel and the same integration problem. It does not disappear from a comparison of two quadratic theories. For two positive kernels $K_0$ and $K_1$ acting on the same $N$ coordinates with the same reference measure,

$$\frac{\mathcal Z_E[0;K_1]} {\mathcal Z_E[0;K_0]} = \left(\frac{\det K_0}{\det K_1}\right)^{1/2},$$

and, for a common source convention, set $\Delta C=K_1^{-1}-K_0^{-1}$. Then

$$\begin{aligned} \frac{\mathcal Z_E[J;K_1]} {\mathcal Z_E[J;K_0]} &= \left(\frac{\det K_0}{\det K_1}\right)^{1/2} \\ &\quad\times \exp\!\left[ \frac12J^{\mathsf T}\Delta C J \right]. \end{aligned}$$

At finite $N$ one may equivalently write

$$\begin{aligned} \log\mathcal Z_E[0;K_E] &=-\frac12\log\det K_E \\ &=-\frac12\operatorname{Tr}\log K_E. \end{aligned}$$

These equalities do not define an absolute continuum determinant. Changing the number of variables, the coordinate units, the reference-measure scale, the boundary conditions, or the projected subspace changes the comparison. A useful determinant ratio therefore begins by declaring a common regulated space and measure.

Source derivatives distinguish moments from covariance

For an observable $F(x)$, define its expectation in the source-deformed Gaussian by

$$\begin{aligned} \langle F\rangle_J &= \frac{1}{\mathcal Z_E[J;K_E]} \int\mathrm d\mu_0(x)\,F(x) \\ &\quad\times e^{-x^{\mathsf T}K_Ex/2+J^{\mathsf T}x}. \end{aligned}$$

Differentiating under this convergent finite integral gives

$$\langle x_i\rangle_J = \frac{\partial\log Z_E}{\partial J_i} =(C_EJ)_i.$$

The second derivative of the logarithm is the connected covariance,

$$\begin{aligned} \frac{\partial^2\log Z_E} {\partial J_i\partial J_j} &= \langle x_ix_j\rangle_J -\langle x_i\rangle_J\langle x_j\rangle_J \\ &=(C_E)_{ij}. \end{aligned}$$

By contrast, two derivatives of $Z_E$ give the raw second moment:

$$\frac{1}{Z_E} \frac{\partial^2Z_E}{\partial J_i\partial J_j} = (C_E)_{ij} +(C_EJ)_i(C_EJ)_j.$$

At $J=0$ the mean vanishes, so the raw second moment and covariance agree. Away from $J=0$ they differ by the product of means. If

$$W_E[J]\equiv\log Z_E[J],$$

then $W_E=J^{\mathsf T}C_EJ/2$ for this quadratic problem: all its derivatives above second order vanish. The full functional still contains higher Gaussian moments. Writing $\partial_i=\partial/\partial J_i$, for example,

$$\begin{aligned} \left. \partial_i\partial_j\partial_k\partial_\ell Z_E \right|_{J=0} &= C_{E,ij}C_{E,k\ell} \\ &\quad+ C_{E,ik}C_{E,j\ell} \\ &\quad+ C_{E,i\ell}C_{E,jk}. \end{aligned}$$

This last identity is the free Gaussian check, not a development of the general connected hierarchy or Wick expansion; those belong to the correlator chapter.

A two-coordinate source checks every factor

Consider the positive matrix and source

$$K= \begin{pmatrix} 3&1\\ 1&2 \end{pmatrix}, \qquad J= \begin{pmatrix} j_1\\j_2 \end{pmatrix}.$$

Its leading principal minors are $3$ and $5$, so it is positive definite. Direct inversion gives

$$\begin{aligned} \det K&=5, \\ C=K^{-1} &= \frac15 \begin{pmatrix} 2&-1\\ -1&3 \end{pmatrix}. \end{aligned}$$

Write

$$Q(j_1,j_2) \equiv 2j_1^2-2j_1j_2+3j_2^2.$$

With the reference measure $\mathrm d x_1\mathrm d x_2/(2\pi)$,

$$\begin{aligned} \mathcal Z_E[J;K] &= \frac{1}{\sqrt5} \exp\!\left(\frac{Q(j_1,j_2)}{10}\right), \\ Z_E[J;K] &= \exp\!\left(\frac{Q(j_1,j_2)}{10}\right). \end{aligned}$$

If the measure were instead plain $\mathrm d x_1\mathrm d x_2$, the zero-source integral would be $2\pi/\sqrt5$. The distinction is entirely the declared reference measure.

The shifted mean and covariance are

$$\begin{aligned} \langle x\rangle_J &= \frac15 \begin{pmatrix} 2j_1-j_2\\ -j_1+3j_2 \end{pmatrix}, \\ \operatorname{Cov}_J(x)&=C. \end{aligned}$$

In particular, the negative mixed covariance $\langle x_1x_2\rangle_0=-1/5$ has the opposite sign from the positive off-diagonal entry of $K$. Multiplying $KC=I$ checks the inverse, differentiating $\log Z_E$ checks every covariance entry, and setting $J=0$ checks the normalization. These three operations provide an exact benchmark without numerical quadrature.

The regulated free scalar turns the inverse into a two-point kernel

Return to the periodic Euclidean lattice defined on Regulated Bosonic Field Integrals. Its $N_\Lambda$ dimensionless site fields $\varphi_n$ have action

$$S_{E,a}[\varphi] = \frac12\sum_{n,m}\varphi_n(K_a)_{nm}\varphi_m.$$

For $m>0$, the lattice kernel is positive and its momentum-space eigenvalues are

$$\lambda_a(k) = (am)^2 +4\sum_{\mu=1}^d \sin^2\!\left(\frac{k_\mu a}{2}\right)>0.$$

Couple a dimensionless site source through $j^{\mathsf T}\varphi=\sum_nj_n\varphi_n$. The normalized integral is then

$$\begin{aligned} Z_{E,a}[j] &= \exp\!\left(\frac12j^{\mathsf T}C_aj\right), \\ C_a&=K_a^{-1}. \end{aligned}$$

Consequently,

$$\begin{aligned} \langle\varphi_n\rangle_j &= \sum_m(C_a)_{nm}j_m, \\ \operatorname{Cov}_j(\varphi_n,\varphi_m) &=(C_a)_{nm}. \end{aligned}$$

Periodic translation invariance diagonalizes the covariance in the same Fourier basis as $K_a$:

$$(C_a)_{nm} = \frac{1}{N_\Lambda} \sum_{k\in\mathrm{BZ}} \frac{e^{ik\cdot(x_n-x_m)}}{\lambda_a(k)}.$$

The sum is real because the $k$ and $-k$ terms pair. Applying the regulated equation-of-motion kernel gives the decisive contact check

$$\sum_m(K_a)_{nm}(C_a)_{m\ell} = \delta_{n\ell}.$$

Thus the two-point function is not merely a matrix with the same eigenvalue denominator as the action: it is the inverse on the declared periodic lattice, including its boundary conditions and its complete set of regulated modes. The finite-lattice construction and its source derivatives follow the Euclidean development in Zinn-Justin 2021, §§ 2.5–2.6, pp. 27–32, and §§ 7.1–7.3, pp. 126–130.

The relation to continuum-normalized symbols must retain the powers of $a$. With

$$\varphi_n=a^{(d-2)/2}\phi(x_n),$$

the discretized continuum source term obeys

$$\begin{aligned} a^d\sum_nJ(x_n)\phi(x_n) &= \sum_nj_n\varphi_n, \\ j_n&=a^{(d+2)/2}J(x_n). \end{aligned}$$

and the field covariance scales as

$$\langle\phi(x_n)\phi(x_m)\rangle_{0,a} = a^{-(d-2)}(C_a)_{nm}.$$

These relations translate a finite result; they do not by themselves prove convergence as $a\to0$ or as the volume grows.

Euclidean and Lorentzian sources require different prescriptions

The Euclidean calculation used a positive kernel and the weight

$$\exp\!\left(-S_E+J^{\mathsf T}\phi\right).$$

Ordinary derivatives of $Z_E$ generate moments, and the free two-point kernel satisfies $K_EC_E=I$.

For a finite Lorentzian regulator, write the free scalar action after integration by parts as

$$\begin{aligned} S_L[\phi] &= -\frac12\phi^{\mathsf T}P\phi, \\ S_{L,J}&=S_L+J^{\mathsf T}\phi. \end{aligned}$$

The real-cycle integral with weight $e^{iS_{L,J}}$ is oscillatory. Suppose a Feynman damping prescription has already selected the boundary-value inverse

$$G_F = \lim_{\epsilon\to0^+} (P-i\epsilon B)^{-1}, \qquad B>0,$$

on the declared finite space. Completing the square on that prescribed cycle gives the normalized functional

$$\begin{aligned} Z_L[J] &= \exp\!\left(\frac{i}{2}J^{\mathsf T}G_FJ\right) \\ &= \exp\!\left(-\frac12J^{\mathsf T}D_FJ\right), \\ G_F&=iD_F. \end{aligned}$$

With the chapter convention $e^{iS_L+iJ^{\mathsf T}\phi}$, ordered moments are generated by

$$\begin{aligned} \langle\mathrm T\,\phi_{i_1}\cdots\phi_{i_n}\rangle &= (-i)^n \\ &\quad\times \left. \frac{\partial^n Z_L[J]} {\partial J_{i_1}\cdots\partial J_{i_n}} \right|_{J=0}. \end{aligned}$$

For $n=2$, the two minus signs—one from $(-i)^2$ and one from differentiating $e^{-J^{\mathsf T}D_FJ/2}$—return $D_F$. For the translation-invariant free scalar, the usual damping choice reduces mode by mode to the standard $+i0$ denominator. In the continuum notation associated with the site’s $(+---)$ convention,

$$D_F(x-y) = \int\frac{\mathrm d^dp}{(2\pi)^d} \frac{i\,e^{-ip\cdot(x-y)}} {p^2-m^2+i0},$$

and

$$(\Box+m^2)D_F(x-y) = -i\delta^{(d)}(x-y).$$

The Euclidean covariance instead satisfies a delta-normalized elliptic equation. The Feynman, retarded, and advanced inverses obey different boundary conditions even when the same differential expression appears, so matching a denominator does not identify the physical kernel. The free source completion with an already selected Feynman inverse is treated in Schwartz 2014, § 14.3, pp. 261–263; p. 263 explicitly defers the derivation of the $i\epsilon$ prescription. The distinction among the resulting scalar kernels is developed on Scalar Propagators, Ordered Correlators, and Sources.

An ordered-exponential convention written with $-iJ\phi$ instead of $+iJ\phi$ amounts to relabeling $J\mapsto-J$. Odd source derivatives change sign under that relabeling; even vacuum correlators do not. More generally, Euclidean and Lorentzian formulas are related only after the continuation path, state or boundary condition, pole prescription, and square-root branch have been specified. Signs and factors of $i$ cannot be copied from one column to the other.

Singular kernels and changed contours define different problems

The exact formulas above have sharp failure conditions.

• A zero eigenvalue removes both the inverse and the Gaussian normalization. On the periodic lattice with $m=0$, the constant momentum mode has $\lambda_a(0)=0$, so the real integral diverges and $C_a$ does not exist on the full field space. Restricting the zero mode, fixing an average, adding a mass, or integrating it with nonquadratic terms are different definitions. A pseudoinverse or $\det'K$ is meaningful only after its projected subspace and measure are stated.
• A negative Euclidean direction is not repaired by finite dimensionality. The exponential grows on the real axis. Defining a complex contour changes the integration problem and may introduce orientation and square-root phases.
• An inverse depends on its domain and boundary conditions. Periodic, Dirichlet, Feynman, retarded, and advanced inverses need not agree even when their local differential expression does.
• A determinant depends on the reference measure. Rescaling a dimensionful coordinate changes the measure factor. Comparisons across different regulators or different numbers of modes require an explicit common normalization, not a bare ratio of formal determinants.
• Regulator limits are additional claims. Finite-volume, continuum, infinite-volume, and $i0$ limits can probe different singularities and need not commute. The finite calculation supplies the object to be studied; it does not establish those limits.

Common pitfalls

“Normalization means the determinant can always be dropped.” Dividing by $\mathcal Z_E[0;K_E]$ cancels the determinant for source derivatives at fixed $K_E$. It remains in comparisons that change the kernel, boundary data, domain, measure, or number of modes.

“Two derivatives always equal the covariance.” Two derivatives of $Z_E$, divided by $Z_E$, give the raw second moment. Two derivatives of $\log Z_E$ give the covariance; the two coincide only when the source-dependent mean vanishes.

“The inverse is fixed by the differential operator.” An inverse also requires a domain and boundary or support condition. In Lorentzian signature, the same Klein–Gordon expression admits Feynman, retarded, and advanced inverses with different physical meanings.

“A zero mode can be silently omitted.” Removing a mode changes the integration space and its normalization. State the constraint or quotient, the remaining measure, and the meaning of any prime on an inverse or determinant.

Check your understanding

1. Shifted square. Starting from $-x^{\mathsf T}Kx/2+J^{\mathsf T}x$, determine the shift and the source-only term.

   Solution

   Put $C=K^{-1}$ and shift $x=y+CJ$. Expanding gives

   $$\begin{aligned} &-\frac12x^{\mathsf T}Kx+J^{\mathsf T}x \\ &\quad= -\frac12y^{\mathsf T}Ky +\frac12J^{\mathsf T}CJ. \end{aligned}$$

   A satisfactory check also notes that translating the integration variable is harmless here because the domain is all of $\mathbb R^N$.

2. Raw moment versus covariance. For the two-coordinate benchmark, compute $\langle x_1x_2\rangle_J$.

   Solution

   The covariance entry is $C_{12}=-1/5$, while

   $$\begin{aligned} \langle x_1\rangle_J &=\frac{2j_1-j_2}{5}, \\ \langle x_2\rangle_J &=\frac{-j_1+3j_2}{5}. \end{aligned}$$

   Therefore

   $$\begin{aligned} \langle x_1x_2\rangle_J &= -\frac15 +\langle x_1\rangle_J\langle x_2\rangle_J. \end{aligned}$$

   At $J=0$, the product of means vanishes and the raw moment equals the covariance.

3. Determinant comparison. Let $K_0=I_2$ and let $K_1$ be the benchmark matrix. Find the zero-source ratio using the normalized reference measure, then state one change that would invalidate the comparison.

   Solution

   Since $\det K_0=1$ and $\det K_1=5$,

   $$\frac{\mathcal Z_E[0;K_1]} {\mathcal Z_E[0;K_0]} =\frac1{\sqrt5}.$$

   The formula would no longer be this bare ratio if the two integrals used different coordinate scales, domains, reference measures, boundary conditions, or numbers of variables.

4. Lattice contact equation. Use the Fourier representation of $C_a$ to show that $K_aC_a=I$. What fails at $m=0$?

   Solution

   Acting with $K_a$ multiplies every Fourier mode by $\lambda_a(k)$, which cancels the factor $1/\lambda_a(k)$. Fourier completeness then gives $\delta_{n\ell}$. At $m=0$, $\lambda_a(0)=0$, so the constant mode cannot be inverted and the full real Gaussian is not normalizable.

5. Lorentzian factors. Starting from $Z_L[J]=\exp(-J^{\mathsf T}D_FJ/2)$, verify the two-point derivative rule.

   Solution

   At zero source,

   $$\left. \frac{\partial^2Z_L} {\partial J_i\partial J_j} \right|_{J=0} =-(D_F)_{ij}.$$

   Since $(-i)^2=-1$,

   $$\left. (-i)^2 \frac{\partial^2Z_L} {\partial J_i\partial J_j} \right|_{J=0} =(D_F)_{ij}.$$

   This check presupposes the Feynman boundary prescription; it does not select that prescription from the local operator alone.

Where to continue

• The Generating Functional develops normalized ordered source derivatives and the general connected hierarchy.
• Scalar Propagators, Ordered Correlators, and Sources distinguishes Feynman, retarded, advanced, Wightman, and commutator kernels.
• Lorentzian Boundary Conditions and the iε Prescription and Wick Rotation and Analytic Continuation treat the contour and continuation questions suppressed by the finite Euclidean derivation.
• Canonical–Functional Crosswalk for Regulated Systems compares the functional covariance with the canonical free-scalar answer after its additional prerequisites are in place.
• Gaussian Euclidean Fields as Measures asks the separate continuum-construction question on distribution spaces.

The finite result is exact and conditional: once the regulated variables, real Euclidean domain, positive kernel, reference measure, and source convention are fixed, $K_E^{-1}$ gives the shifted mean and covariance, $\det K_E$ gives the unnormalized Gaussian factor, and source derivatives generate all finite Gaussian moments. Changing signature replaces positivity by a boundary-value problem and requires every sign, factor of $i$, and determinant phase to be derived again.

References

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014. DOI.
• Zinn-Justin, Jean. Quantum Field Theory and Critical Phenomena. 5th ed. Oxford: Oxford University Press, 2021. DOI.