Connected Correlators and Cumulants

The logarithm of a normalized generating functional isolates connected correlators because exponentiating a cumulant functional reconstructs every way of partitioning the insertions into connected blocks. In the Lorentzian in–out convention used here, the precise statement is

$$G_{c,J}^{(n)}(x_1,\ldots,x_n) = (-i)^n\frac{\delta^n\log Z[J]} {\delta J(x_1)\cdots\delta J(x_n)} = (-i)^{n-1}\frac{\delta^nW[J]} {\delta J(x_1)\cdots\delta J(x_n)},$$

where $W[J]=-i\log Z[J]$. Normalization fixes the zeroth-order term and removes a source-independent vacuum factor, but positive-order derivatives of a logarithm are already insensitive to any nonzero source-independent multiplier. The result is algebraic: it does not by itself imply decay at large separation, causal response, or one-particle irreducibility.

Required background. The Generating Functional fixes the ordering, state, boundary prescription, regulator, source sign, and normalization of $Z[J]$.

Helpful background. Characteristic Functions, Cumulants, and Generating Functionals develops the same moment–cumulant combinatorics without Lorentzian phase factors.

The logarithm defines connected correlators

Work first with a finite regulator or with smeared fields, so all source derivatives are defined. Let

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

Near $J=0$, choose the logarithm with $\mathcal K[0]=0$, hence $W[0]=0$. At a genuine regulator this requires $Z[J]\neq0$ in the source neighborhood being differentiated. In perturbation theory, $Z[0]=1$ instead gives a unique formal logarithm order by order, without asserting convergence for finite $J$.

At nonzero source, define the full ordered moments by

$$G_J^{(n)}(x_1,\ldots,x_n) = (-i)^n\frac{1}{Z[J]} \frac{\delta^n Z[J]} {\delta J(x_1)\cdots\delta J(x_n)}.$$

The connected moments, or cumulants, are the derivatives of $\mathcal K$ displayed in the lead. At $J=0$ the subscript $J$ will be omitted. This definition is equivalent, at every fixed order, to the moment–cumulant inversion derived below; it therefore remains meaningful as a finite algebraic relation even when one does not treat $Z[J]$ as a convergent functional power series. The QFT connected-source construction and its explicit two-point subtraction are developed in Schwartz 2014, § 34.1.2, pp. 737–739.

The first two derivatives show the subtraction

The first source derivative gives the source-dependent mean field,

$$\bar\phi_J(x) = \frac{\delta W[J]}{\delta J(x)} =G_{c,J}^{(1)}(x) =G_J^{(1)}(x).$$

Differentiating once more subtracts the product of means:

$$\begin{aligned} \frac{\delta^2W[J]} {\delta J(x)\delta J(y)} &= i\left[ G_J^{(2)}(x,y) -G_J^{(1)}(x)G_J^{(1)}(y) \right], \\ G_{c,J}^{(2)}(x,y) &= G_J^{(2)}(x,y) -\bar\phi_J(x)\bar\phi_J(y) \\ &= -i\frac{\delta^2W[J]} {\delta J(x)\delta J(y)}. \end{aligned}$$

Thus $W^{(2)}=iG_{c,J}^{(2)}$, not $G_{c,J}^{(2)}$ itself. Moreover, this Hessian is the response of an in–out mean field to an in–out source. It carries Feynman boundary data and is not automatically a retarded, real, or positive response kernel.

Moments are partitions of connected blocks

Write $\delta_a\equiv\delta/\delta J(x_a)$ and, for a block $B$ of labels, $\delta_B\equiv\prod_{a\in B}\delta_a$. Repeated differentiation of $Z=e^{\mathcal K}$ gives the functional exponential formula

$$\frac{1}{Z[J]}\delta_1\cdots\delta_n Z[J] = \sum_{\pi\in\Pi_n} \prod_{B\in\pi}\delta_B\mathcal K[J],$$

where $\Pi_n$ is the set of partitions of $\{1,\ldots,n\}$. A partition records which labels belong to the same connected block. Because the block sizes add to $n$, the Lorentzian phases in the definitions cancel block by block, leaving the factor-free identity

$$G_J^{(n)}(1,\ldots,n) = \sum_{\pi\in\Pi_n} \prod_{B\in\pi}G_{c,J}^{(|B|)}(B).$$

Möbius inversion on the partition lattice gives the converse relation,

$$G_{c,J}^{(n)}(1,\ldots,n) = \sum_{\pi\in\Pi_n} (-1)^{|\pi|-1}(|\pi|-1)! \prod_{B\in\pi}G_J^{(|B|)}(B).$$

This is the functional form of the ordinary cumulant expansion: the logarithm selects the one-block coefficient, while exponentiation reconstructs every partition. The general cluster combinatorics and its QFT functional specialization are given in Zinn-Justin 2021, § 1.2.2, p. 4; §§ 7.3–7.3.1, pp. 129–131.

For example,

$$G_J^{(2)}(1,2) = G_{c,J}^{(2)}(1,2) +G_{c,J}^{(1)}(1)G_{c,J}^{(1)}(2),$$

and

$$\begin{aligned} G_J^{(3)}(1,2,3) &=G_{c,J}^{(3)}(1,2,3) \\ &\quad +G_{c,J}^{(2)}(1,2)G_{c,J}^{(1)}(3) \\ &\quad +G_{c,J}^{(2)}(1,3)G_{c,J}^{(1)}(2) \\ &\quad +G_{c,J}^{(2)}(2,3)G_{c,J}^{(1)}(1) \\ &\quad +G_{c,J}^{(1)}(1)G_{c,J}^{(1)}(2) G_{c,J}^{(1)}(3). \end{aligned}$$

No assumption that the one-point function vanishes entered this derivation. Products at coincident points require the usual regulated or renormalized interpretation; the cleanest statement is for distinct arguments or smeared fields.

There is also an exact factorization check. If two independent sectors have separate sources and

$$Z[J_A,J_B]=Z_A[J_A]Z_B[J_B],$$

then

$$W[J_A,J_B]=W_A[J_A]+W_B[J_B].$$

Every connected derivative involving at least one $J_A$ and one $J_B$ therefore vanishes. This is exact algebraic independence; it is not the claim that two regions of one interacting theory factorize merely because they are far apart.

The centered free scalar is an exact test

For the regulated centered free scalar from the preceding page,

$$\begin{aligned} Z_0[J] &= \exp\!\left( -\frac12J\mathbin{\cdot}D_F\mathbin{\cdot}J \right) \\ &= \exp\!\left( \frac{i}{2}J\mathbin{\cdot}G_F\mathbin{\cdot}J \right), \qquad G_F=iD_F. \end{aligned}$$

Consequently,

$$W_0[J] = \frac12J\mathbin{\cdot}G_F\mathbin{\cdot}J = \frac{i}{2}J\mathbin{\cdot}D_F\mathbin{\cdot}J.$$

The functional is quadratic, so at arbitrary source

$$\begin{aligned} G_{c,J}^{(1)}(x) &= \int\mathrm d^4y\,G_F(x,y)J(y), \\ G_{c,J}^{(2)}(x,y) &=-iG_F(x,y)=D_F(x,y), \\ G_{c,J}^{(n)}&=0, \qquad n\ge3. \end{aligned}$$

The last line does not make the higher full correlators vanish. At zero source the field is centered, and the four-point partition formula becomes

$$\begin{aligned} G^{(4)}(1,2,3,4) &=G_c^{(4)}(1,2,3,4) \\ &\quad+D_F(1,2)D_F(3,4) \\ &\quad+D_F(1,3)D_F(2,4) \\ &\quad+D_F(1,4)D_F(2,3). \end{aligned}$$

Here $G_c^{(4)}=0$, while the three products generally remain. The exact Gaussian source functional and its differentiation are given in Schwartz 2014, §§ 14.3.1–14.3.2, pp. 262–263; the quadratic connected functional and vanishing of its higher derivatives are explicit in Zinn-Justin 2021, § 7.3.1, p. 130. This example exhibits the linked-cluster structure: exponentiation assembles disconnected products from connected blocks. The general bosonic pairing rule and its fermionic signs belong to the next page, not to this illustration.

What normalization does—and does not—do

Suppose $\mathcal Z[0]\neq0$ and

$$\mathcal Z[J]=\mathcal Z[0]Z[J].$$

On compatible local logarithm branches,

$$\log\mathcal Z[J] = \log\mathcal Z[0]+\log Z[J].$$

Every positive-order source derivative is therefore insensitive to the nonzero source-independent factor $\mathcal Z[0]$. Normalization is nevertheless important: it sets the zeroth moment to one, gives $W[0]=0$, removes the source-independent vacuum contribution, and lets derivatives of $Z$ be read directly as normalized moments. If $\mathcal Z[0]=0$, the ratio is undefined; if the purported normalization depends on $J$, its derivatives change the correlators and it cannot be discarded.

The cancellation is meaningful only when numerator and denominator use the same action, regulator, field normalization, state, integration cycle or time contour, and boundary prescription. Changing any of those data in the denominator defines a different ratio rather than normalizing the original one.

A different local logarithm branch shifts $W$ only by a source-independent constant, so connected correlators do not change. A zero of $Z[J]$, however, obstructs continuing one branch through that point. Only a local branch near the source about which derivatives are taken is needed.

Connectedness is not clustering, 1PI, or causality

Several useful notions share similar language but answer different questions.

Notion	Criterion	What it requires beyond this page
Cumulant-connected	The one-block term left after subtracting every product associated with a nontrivial set partition	Only the algebra of moments and cumulants
Cluster decay	Connected correlations tend to zero when groups of arguments are separated	Vacuum, spectral, infrared, and regularity hypotheses appropriate to the theory
One-particle irreducible	A connected graph cannot be separated by cutting one internal propagator	The Legendre transform to $\Gamma$ and a diagrammatic expansion
Causal response	A disturbance affects only its causal future	Retarded or closed-time-path observables, not merely the in–out functional

Algebraic connectedness therefore implies no large-distance decay. Clustering, Vacuum Assumptions, and Long-Range Correlations states the hypotheses under which such decay follows; a relativistic scattering formulation of the separate cluster-decomposition principle appears in Weinberg 1995, § 4.3, pp. 177–181.

Nor does the logarithm make a connected contribution 1PI: a connected graph may still fall apart after one internal line is cut. And although perturbative disconnected diagrams exponentiate into collections of connected components, their graph symmetry factors and general counting proof are beyond the present algebraic derivation.

There is also a signature-dependent warning. For a positive Euclidean measure, $\log Z_E[J]$ has a covariance Hessian and corresponding convexity properties. The Lorentzian functional here uses $W=-i\log Z$; neither positivity nor convexity of $W^{(2)}$ follows.

Common pitfalls

Dropping the Lorentzian phases. The connected two-point function is $G_c^{(2)}=-iW^{(2)}$, not $W^{(2)}$. The phase-free partition identities appear only after the factors in the definitions of both full and connected correlators have been included.

Centering too early. The one-point blocks in the partition formulas vanish only for a centered state at the source under consideration. A nonzero background or nonzero source restores them.

Treating the logarithm as global. Derivatives near $J=0$ need a local or formal logarithm. Zeros of $Z$ can obstruct a single branch over a larger source domain.

Promoting the Gaussian result to an interacting theory. A quadratic $W_0$ has no connected correlators above second order. Interactions or a non-Gaussian state generally generate higher cumulants.

Check your understanding

Use the stated criteria to locate any error in source normalization, cumulant subtraction, or partition combinatorics.

Mode	Prompt	A satisfactory response	Repair or continuation
Retrieval	State the full and connected $n$-point definitions at nonzero source	Includes $Z[J]^{-1}$ for the full moment and gives $G_{c,J}^{(n)}=(-i)^{n-1}W^{(n)}$	The logarithm defines connected correlators
Derivation	Differentiate $W=-i\log Z$ twice	Obtains $G_{c,J}^{(2)}=G_J^{(2)}-G_J^{(1)}G_J^{(1)}=-iW^{(2)}$	The first two derivatives show the subtraction
Combinatorics	Reconstruct a full three-point function from connected blocks	Lists the one three-block, three pair–singleton partitions, and the all-singleton partition	Moments are partitions of connected blocks
Sign check	Start from $W_0=J\cdot G_F\cdot J/2$ with $G_F=iD_F$	Finds $G_c^{(2)}=-iG_F=D_F$	The centered free scalar is an exact test
Failure mode	Let $Z[J]$ cross zero along a source path	Says that one global logarithm branch fails, while local derivatives away from the zero can still be defined	What normalization does—and does not—do
Boundary	Decide whether $G_c^{(n)}$ must decay when points separate	Answers no and identifies the extra clustering hypotheses rather than inferring decay from cumulant connectedness	Connectedness is not clustering, 1PI, or causality

Where to continue

• Derive Gaussian pairings: Wick’s Theorem and Free Gaussian Factorization proves the general bosonic rule and tracks fermionic permutation signs.
• Pass from connected to 1PI data: The 1PI Effective Action and Mean-Field Equations develops the Legendre transform and inverse two-point kernel.
• Add spacetime separation: Clustering, Vacuum Assumptions, and Long-Range Correlations states the physical hypotheses behind cluster decay.
• Count perturbative components: Connected, Disconnected, and Vacuum Diagrams develops diagram classes, vacuum bubbles, and linked-cluster counting.
• Generate exact hierarchies: Schwinger–Dyson Identities derives source identities from regulated changes of integration variables.

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.
• Zinn-Justin, Jean. Quantum Field Theory and Critical Phenomena. 5th ed. Oxford: Oxford University Press, 2021. DOI.