The Generating Functional

With the Lorentzian source term $+\int J\phi$, a normalized generating functional packages vacuum time-ordered correlation functions into one object: each functional derivative inserts a field and contributes one factor of $i$, so the $n$-point function is $(-i)^n$ times the $n$th derivative at zero source. This conclusion is meaningful only after the state, ordering, boundary prescription, regulator, source sign, and normalization have been fixed.

This page gives both the operator and regulated path-integral definitions, derives the insertion rule, and evaluates the functional exactly for the free real scalar. The result is an in–out/Feynman object. Retarded response, closed-time-path expectation values, connected functions, Wick factorization, effective actions, and Schwinger–Dyson identities require additional constructions developed elsewhere.

Required background. Gaussian Fields and Sources supplies the regulated completion of the square and the distinction between an inverse kernel and a formal continuum expression.

Helpful background. Characteristic Functions, Moments, Cumulants, and Generating Functionals supplies the parallel finite-dimensional language of moments and cumulants.

The normalized in–out source functional

Work in four-dimensional Minkowski spacetime and let $J$ be an external c-number test source. Write

$$J\cdot\phi \equiv \int \mathrm d^4x\,J(x)\phi(x), \qquad S_J[\phi]=S[\phi]+J\cdot\phi.$$

The test-source language matters because quantum fields and their correlation functions are distributions. Pointwise functional derivatives are shorthand for the corresponding smeared identities.

For a normalized vacuum $|\Omega\rangle$, the operator definition is

$$Z_{\mathrm{op}}[J] = \frac{ \langle\Omega| \mathrm T\exp\!\left(iJ\cdot\widehat\phi\right) |\Omega\rangle }{\langle\Omega|\Omega\rangle}, \qquad Z_{\mathrm{op}}[0]=1.$$

The symbol $\mathrm T$ is part of the definition. Differentiation will expose this ordering; it does not create it.

In a regulated functional description, let $\mathcal C_F$ denote the integration cycle together with the vacuum/Feynman boundary prescription. Define

$$\mathcal Z[J] = \int_{\mathcal C_F}\mathcal D\phi\, \exp\!\left(iS[\phi]+iJ\cdot\phi\right),$$

and, when $\mathcal Z[0]\ne0$,

$$Z[J] \equiv \frac{\mathcal Z[J]}{\mathcal Z[0]}, \qquad Z[0]=1.$$

The continuum notation summarizes a regulated construction and its proposed limit; it is not a flat infinite-dimensional Lebesgue measure. Matching $Z[J]$ with the operator functional requires the same action, regulator, field normalization, state, ordering, boundary data, cycle, and zero-source normalization. Weinberg 1995, § 9.1, pp. 378–383 derives the fixed-endpoint phase-space integral by time slicing and shows how ordered insertions become a time-ordered product. The additional regulator, state, cycle, field-normalization, and zero-source matching conditions are imposed here.

Source differentiation inserts ordered fields

Expand the ordered exponential as a formal power series, or as a convergent finite-regulator expression where justified:

$$\begin{aligned} \mathrm T e^{iJ\cdot\widehat\phi} &= \sum_{n=0}^{\infty}\frac{i^n}{n!} \int\prod_{r=1}^{n} \left[\mathrm d^4x_r\,J(x_r)\right] \\ &\qquad\times \mathrm T\!\left\{ \widehat\phi(x_1)\cdots\widehat\phi(x_n) \right\}. \end{aligned}$$

Using

$$\frac{\delta J(y)}{\delta J(x)} =\delta^{(4)}(x-y),$$

each derivative removes one source and leaves one ordered field insertion. Hence

$$\begin{aligned} &\left. \frac{\delta^n Z[J]} {\delta J(x_1)\cdots\delta J(x_n)} \right|_{J=0} \\ &\qquad= i^n \langle\Omega| \mathrm T\!\left\{ \widehat\phi(x_1)\cdots\widehat\phi(x_n) \right\} |\Omega\rangle. \end{aligned}$$

Therefore the normalized vacuum $n$-point function is

$$\boxed{ \begin{aligned} G^{(n)}(x_1,\ldots,x_n) &= \langle\Omega| \mathrm T\!\left\{ \widehat\phi(x_1)\cdots\widehat\phi(x_n) \right\} |\Omega\rangle \\ &= \left.(-i)^n \frac{\delta^nZ[J]} {\delta J(x_1)\cdots\delta J(x_n)} \right|_{J=0}. \end{aligned} }$$

This derivation, including the plus-source convention and division by the zero-source vacuum amplitude, is given in Schwartz 2014, § 14.3, pp. 261–262. Schwartz calls the unnormalized functional $Z[J]$ and retains an explicit factor $Z[0]^{-1}$; the normalized notation on this page absorbs that factor. A source convention with $-iJ\cdot\phi$ instead would reverse every odd-derivative sign; the formula cannot be imported without translating the source.

Away from zero source, normalization by the source-dependent functional is also required. Define

$$\langle\mathcal O\rangle_J = \frac{ \displaystyle \int_{\mathcal C_F}\mathcal D\phi\, \mathcal O[\phi]e^{iS[\phi]+iJ\cdot\phi} }{\mathcal Z[J]}.$$

Then

$$\frac{1}{Z[J]} \frac{\delta^nZ[J]} {\delta J(x_1)\cdots\delta J(x_n)} = i^n \left\langle \mathrm T\phi(x_1)\cdots\phi(x_n) \right\rangle_J.$$

Omitting $Z[J]^{-1}$ would give an unnormalized insertion. At $J=0$, the chosen normalization makes this factor one.

For two fields, the sign check is especially useful. With

$$D_F(x,y) \equiv \langle\Omega| \mathrm T\phi(x)\phi(y) |\Omega\rangle,$$

one has

$$\left. \frac{\delta^2Z} {\delta J(x)\delta J(y)} \right|_{J=0} =i^2D_F(x,y) =-D_F(x,y).$$

Multiplying by $(-i)^2=-1$ returns $D_F$. These two minus signs are an efficient round-trip test of the declared convention.

The exact regulated free-scalar functional

At a finite regulator, collect the retained real field variables into a vector $\phi$. After integration by parts, write the free action with its damping prescription as

$$S_{0,\epsilon}[\phi] =-\frac12\phi^{\mathsf T}P_\epsilon\phi, \qquad P_\epsilon=P-i\epsilon B, \qquad P=P^{\mathsf T}, \qquad B=B^{\mathsf T}>0,$$

and suppose $P_\epsilon$ is invertible on the declared regulated space. Then $P_\epsilon$ and its inverse are symmetric. Set

$$G_\epsilon=P_\epsilon^{-1}.$$

The finite-dimensional completion of the square is exact:

$$\begin{aligned} -\frac12\phi^{\mathsf T}P_\epsilon\phi +J^{\mathsf T}\phi &= -\frac12 (\phi-G_\epsilon J)^{\mathsf T} P_\epsilon (\phi-G_\epsilon J) \\ &\qquad+ \frac12J^{\mathsf T}G_\epsilon J. \end{aligned}$$

Provided the contour may be translated by $G_\epsilon J$, the remaining Gaussian integral is the same source-independent factor in numerator and denominator. Thus

$$Z_{0,\epsilon}[J] = \exp\!\left( \frac{i}{2}J^{\mathsf T}G_\epsilon J \right).$$

Take the Feynman boundary-value limit on the same regulated family and define

$$G_F =\lim_{\epsilon\to0^+}G_\epsilon, \qquad G_F=P_F^{-1}=iD_F.$$

With

$$J\cdot G_F\cdot J \equiv \int\mathrm d^4x\,\mathrm d^4y\, J(x)G_F(x,y)J(y),$$

the exact normalized free functional is

$$\boxed{ Z_0[J] = \exp\!\left( \frac{i}{2}J\cdot G_F\cdot J \right) = \exp\!\left( -\frac12J\cdot D_F\cdot J \right). }$$

For the translation-invariant vacuum,

$$D_F(x-y) = \int\frac{\mathrm d^4p}{(2\pi)^4} \frac{i\,e^{-ip\cdot(x-y)}} {p^2-m^2+i0}.$$

With the site’s Fourier convention, $P_F=\Box+m^2$ acts as the Feynman boundary-value operator, so

$$P_FG_F=I, \qquad P_FD_F=-iI.$$

The delta-normalized inverse is $G_F=iD_F$, not $D_F$ itself. The free completion and Gaussian source functional are developed in Schwartz 2014, §§ 14.3.1–14.3.2, pp. 262–263; dividing by its zero-source value gives the normalized formula used here. The vacuum boundary deformation and resulting $+i0$ Feynman denominator are derived in Schwartz 2014, § 14.4.1, pp. 265–266. Comparison with other scalar kernels belongs to Scalar Propagators, Ordered Correlators, and Sources.

Two further derivatives expose the difference between a source-deformed raw moment and its centered part. Define

$$\bar\phi_J(x) \equiv -i\,\frac{1}{Z_0[J]} \frac{\delta Z_0[J]}{\delta J(x)} = \int\mathrm d^4y\,G_F(x,y)J(y).$$

Then

$$P_F\bar\phi_J=J,$$

with Feynman rather than retarded boundary data, and

$$\left\langle \mathrm T\phi(x)\phi(y) \right\rangle_J = D_F(x,y) +\bar\phi_J(x)\bar\phi_J(y).$$

At $J=0$, the centered free vacuum gives exactly $D_F$. The Gaussian functional also contains all higher free moments, but the general pairing theorem and fermionic signs belong to Wick’s Theorem and Free Gaussian Factorization.

Normalization, signs, and boundary checks

Each line below detects a different category of error.

Check	Correct result	What a failure means
Engineering dimension	In four dimensions, $[\phi]=1$ and $[J]=3$, so $J\cdot\phi$ is dimensionless	The source normalization or spacetime measure is inconsistent
Zero source	$Z[0]=1$	The vacuum factor was not divided out, or the numerator and denominator are different problems
Two derivatives	$\delta^2Z/\delta J(x)\delta J(y)	_0=-D_F(x,y)$
Inverse kernel	$P_FG_F=I$ and $P_FD_F=-iI$	$D_F$ and $G_F=iD_F$ were conflated, or the boundary prescription changed
Nonzero source	$Z^{-1}\delta^nZ=i^n\langle\mathrm T\phi^n\rangle_J$	The source-deformed insertion was left unnormalized
Euclidean cross-check	$Z_E[J]=\exp(\tfrac12J\cdot C_E\cdot J)$ and ordinary derivatives generate moments	Lorentzian factors were copied into a positive Euclidean Gaussian, or conversely

Zero-source normalization has a limited but exact job. It cancels a source-independent vacuum factor or Gaussian determinant only when the numerator and denominator use the same action, kernel, regulator, domain, measure, contour, and boundary data. It does not remove regulator dependence from $D_F$, cancel determinant ratios between different theories, repair a zero mode, construct an interacting continuum measure, or turn a complex Lorentzian functional into a probability distribution.

What this source functional orders

The adjective “generating” does not mean that one source functional produces every correlator by changing notation.

Desired object	Defining extra structure	Is it generated by the $Z[J]$ above?
Vacuum time-ordered correlator	$\mathrm T$ ordering and Feynman/in–out boundary data	Yes
Wightman function	A specified unsorted operator order	No
Retarded response	A commutator multiplied by future-support step functions	No
Euclidean Schwinger function	A Euclidean functional and its own source and continuation conventions	No
In–in expectation value	A closed time path, initial state or density operator, and doubled branch sources	No

In particular, $\bar\phi_J=G_FJ$ is generally an in–out mean field and may be complex. It is not the retarded solution produced by an initial-value source. The distinctions among Feynman, Wightman, retarded, and advanced kernels are developed on Scalar Propagators, Ordered Correlators, and Sources; the in–out/in–in distinction belongs to In–Out versus In–In Expectation Values.

The derivatives of $Z$ give full ordered moments, including disconnected products. The logarithm $W=-i\log Z$ and the moment–cumulant separation are the subject of Connected Correlators and Cumulants. This page does not use the free Gaussian formula as an interacting identity, derive Wick’s theorem, perform a Legendre transform, or derive a Schwinger–Dyson hierarchy.

Common pitfalls

Differentiation does not choose the ordering. The time-ordering symbol in the operator definition and the Feynman boundary construction in the path integral choose it. Differentiation only inserts fields into that already-defined object.

A normalized functional is not a probability distribution. In Lorentzian signature its weight is oscillatory and its values may be complex. The equation $Z[0]=1$ is a vacuum-normalization statement, not positivity.

The Feynman inverse is not an arbitrary inverse. Retarded, advanced, and Feynman kernels can invert the same local differential expression while satisfying different boundary conditions. Replacing one by another changes the generated object.

Connectedness is not automatic. Derivatives of $Z$ generate full moments. Taking a logarithm and deriving the connected partition relations is a separate step.

Check your understanding

Use the stated criteria to locate any error in source normalization, differentiation factors, signs, or boundary data.

Mode	Prompt	A satisfactory response	Repair or continuation
Retrieval	Define normalized $Z[J]$ and name the data hidden by a bare functional-integral symbol	Gives the zero-source ratio and names the regulator, state, ordering, boundary prescription, cycle, source sign, and normalization	Gaussian Fields and Sources
Derivation	Recover the factor multiplying the $n$th source derivative	Expands $e^{iJ\cdot\phi}$, obtains $i^n$ insertions, and solves for the correlator with $(-i)^n$	Source differentiation inserts ordered fields
Sign check	Differentiate $Z_0=e^{-J\cdot D_F\cdot J/2}$ twice at $J=0$	Obtains $-D_F$ from differentiation and then $D_F$ after multiplication by $(-i)^2$	The exact free-scalar functional
Failure mode	Set one eigenvalue of $P_\epsilon$ to zero before taking the inverse	Says that neither $G_\epsilon$ nor the displayed Gaussian normalization exists on the full space and names a declared projection, mass deformation, or separate zero-mode treatment	Gaussian Fields and Sources
Transfer	For a symmetric invertible finite matrix $G$, start from $Z(J)=e^{iJ^{\mathsf T}GJ/2}$	Finds $-iZ^{-1}\partial Z/\partial J=GJ$ and the raw second moment $-iG+(GJ)(GJ)^{\mathsf T}$; with $G=iD$, this becomes $D+(GJ)(GJ)^{\mathsf T}$	The exact free-scalar functional
Handoff	Decide where to compute a connected function or a causal expectation value	Sends connected moments to the next leaf and causal expectation values to the in–in/closed-time-path chapter rather than relabeling $Z$	Connected Correlators and Cumulants and In–Out versus In–In Expectation Values

Where to continue

• Separate connected data: Connected Correlators and Cumulants develops $W=-i\log Z$ and the moment–cumulant relations.
• Exploit the free Gaussian: Wick’s Theorem and Free Gaussian Factorization derives the bosonic pairing rule and fermionic permutation signs without extending them to arbitrary interacting correlators.
• Change the real-time question: Lorentzian, Euclidean, and In-In Formulations develops boundary prescriptions, continuation, causal response, and closed-time-path sources.
• Introduce renormalized sources: Renormalization and Effective Field Theory develops renormalized source terms, composite-operator mixing, and scheme dependence.

References

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014. DOI.
• Weinberg, Steven. The Quantum Theory of Fields, Volume I: Foundations. Cambridge: Cambridge University Press, 1995. First edition; 2005 paperback, 2012 printing consulted. DOI.