Cluster decomposition and the connected S-matrix

Intuition Beginner

Run two experiments far apart — one in a lab on Earth, one in a lab on the Moon. Common sense says the outcome on Earth should not depend on what happens on the Moon. The cluster decomposition principle is the precise version of this common-sense demand inside quantum theory: when a process splits into two groups of particles separated by a large distance, the probability for the whole thing should be the product of the probabilities for each group on its own. Distant experiments give uncorrelated results.

This sounds almost too obvious to state. The surprise is what it forces. The object that records the outcome of a scattering experiment is the S-matrix, a big table whose entry $S_{\beta \alpha }$ is the amplitude to start in a state of incoming particles $\alpha$ and end in a state of outgoing particles $\beta$. Cluster decomposition says: when the particles fall into two far-apart bunches, this amplitude factorises into a product, one factor per bunch.

To make the demand sharp, physicists split every amplitude into a "connected" part and the rest. The connected part is the piece where every particle actually takes part in a single linked interaction — nothing just flies past untouched. A general amplitude is then a sum over all the ways of grouping the particles into connected clusters. Cluster decomposition becomes the statement that the connected part, the genuinely linked piece, dies away when the clusters move apart.

The payoff is the centrepiece of this unit. Demanding cluster decomposition — that bland statement about distant labs — turns out to pin down the entire architecture of quantum field theory. The interaction has to be built out of the creation and annihilation operators of the prior unit, assembled in one specific way. Far-apart labs do not influence each other only because nature builds its forces from local fields. That is the answer to the question "why fields?"

Visual Beginner

Picture the scattering amplitude for several particles as a blob with lines coming in and going out. A "connected" blob is one you cannot cut into two pieces without snipping a line — every incoming and outgoing particle is tied into the same tangle. A "disconnected" amplitude is two separate blobs sitting side by side, with no line crossing between them.

Now imagine grabbing one blob and carrying it a great distance away from the other. Cluster decomposition is the picture that, as the separation grows, any leftover lines linking the two blobs fade and vanish, leaving exactly two independent blobs. The amplitude for the pair becomes the product of the two single-blob amplitudes.

A single book-keeping mark on each blob tells the whole story: a momentum-conservation tag, the requirement that the momenta flowing in match the momenta flowing out. A connected blob carries exactly one such tag — the total momentum of that one tangle is conserved. Two side-by-side blobs carry two tags, one each. Counting these tags is the sharp, countable version of "connected versus disconnected," and the next tiers turn that count into a theorem.

Worked example Beginner

Take the simplest setting where the structure shows itself: four particles, labelled 1, 2, 3, 4, which can scatter. Write the full amplitude $S_4$ for all four. There are several ways the four can organise into clusters. They might all interact together in one connected lump. Or particles 1 and 2 might interact among themselves while 3 and 4 do their own separate thing. Or any other pairing.

The full amplitude is the sum over every such organisation. Schematically,

$$S_4=S_4^c+(S_{12}^c{S}_{34}^c+S_{13}^c{S}_{24}^c+S_{14}^c{S}_{23}^c)+\cdots ,$$

where $S^c$ denotes a connected piece, the superscript $c$ marking "everything in this group is genuinely linked." The first term is the fully connected four-particle amplitude. The bracketed terms are the ways the four split into two linked pairs. The dots cover splits with single spectator particles.

Now separate particles 1, 2 from 3, 4 by a huge distance. Each connected piece dies off as the particles inside it are pulled away from each other, so the only surviving terms are the ones where 1, 2 stay together and 3, 4 stay together. The fully connected $S_4^c$ dies (it links across the gap). The surviving piece is $S_{12}^c{S}_{34}^c$, plus the pieces where particles pass through without interacting. The amplitude factorises: the 1-2 lab and the 3-4 lab give a product. This is cluster decomposition in action, and the recursive sum-over-clusters is exactly the definition that makes the factorisation automatic once the connected pieces fall off.

Check your understanding Beginner

Formal definition Intermediate+

Work in the Fock space $\mathcal{F}$ of the prior unit 12.13.01, with creation and annihilation operators $a^{†}(\mathbf{p})$, $a(\mathbf{p})$ for single-particle momenta $\mathbf{p}$, and let $S$ be the S-matrix — the unitary operator on $\mathcal{F}$ relating the in- and out-states of scattering theory 12.06.04, with momentum-basis matrix elements $S_{\beta \alpha }=⟨{\beta }_{\mathrm{out}}|{\alpha }_{\mathrm{in}}⟩$. Here $\alpha =({\mathbf{p}}_1,\dots ,{\mathbf{p}}_M)$ and $\beta =({\mathbf{p}}_1^{\prime },\dots ,{\mathbf{p}}_N^{\prime })$ are multi-particle labels.

Definition (cluster decomposition principle). The S-matrix satisfies the cluster decomposition principle if, whenever the particles of $\alpha$ and $\beta$ are partitioned into clusters $\alpha ={\alpha }_1\cup \cdots \cup {\alpha }_k$, $\beta ={\beta }_1\cup \cdots \cup {\beta }_k$ such that the particles of cluster $j$ are confined to a spatial region $R_j$ and the regions $R_1,\dots ,R_k$ are separated from one another by a distance $L\to \infty$, the S-matrix element factorises:

$$S_{\beta \alpha }⟶\prod _{j=1}^k{S}_{{\beta }_j{\alpha }_j}.$$

Definition (connected S-matrix). The connected part $S_{\beta \alpha }^c$ is defined recursively by subtracting all ways of partitioning the particles into proper clusters:

$$S_{\beta \alpha }=\sum _{\text{partitions}P}\prod _{({\alpha }_i,{\beta }_i)\in P}{S}_{{\beta }_i{\alpha }_i}^c,$$

the sum running over all ways of grouping the combined in/out particles into clusters, with the product over the clusters of the partition. This is a Möbius inversion over the lattice of set partitions: it defines $S^c$ uniquely from $S$, the connected amplitude being the part of $S$ left after all genuinely factorised contributions are removed. The fully connected $n$-particle amplitude $S^c$ vanishes whenever any of its particles is carried far from the rest.

Coefficient form of the interaction. Write the interaction Hamiltonian as a sum of normal-ordered monomials in creation and annihilation operators,

$$H=\sum _{N,M}\int d^3{q}_1^{\prime }\cdots d^3{q}_N^{\prime }{d}^3{q}_1\cdots d^3{q}_M{a}^{†}({\mathbf{q}}_1^{\prime })\cdots a^{†}({\mathbf{q}}_N^{\prime })a({\mathbf{q}}_1)\cdots a({\mathbf{q}}_M)h_{NM}({\mathbf{q}}_1^{\prime },\dots ;{\mathbf{q}}_1,\dots ),$$

with all creation operators to the left (normal ordering). The coefficient functions $h_{NM}$ are the data of the theory. The single-delta condition is the requirement that each $h_{NM}$ contains exactly one momentum-conservation factor,

$$h_{NM}={\delta }^3({\sum }_i{\mathbf{q}}_i^{\prime }-{\sum }_j{\mathbf{q}}_j){\tilde{h}}_{NM},$$

with ${\tilde{h}}_{NM}$ a smooth (non-singular) function carrying no further delta functions. The cluster property of the S-matrix is governed entirely by this condition on $H$.

Counterexamples to common slips

• Cluster decomposition is not the same as locality of fields, though it is closely tied to it. Cluster decomposition is an S-matrix statement (factorisation across large separations); microcausality $[\mathcal{H}(x),\mathcal{H}(y)]=0$ at spacelike separation is a stronger, off-shell field statement. The former is necessary for the latter to be a sensible way to enforce it; the two are distinct demands.

• A connected amplitude carrying two momentum-conservation deltas is not connected. Two deltas mean two independently-conserved momentum flows, which is the signature of a disconnected (factorised) contribution. The whole content of the central theorem is that "connected" equals "exactly one delta."

• The single-delta condition forbids coefficient functions ${\tilde{h}}_{NM}$ with poles or further delta singularities. A factor like $1/({\mathbf{q}}_1^2-{\mathbf{q}}_2^2)$ in $\tilde{h}$ would correspond to a particle propagating freely over arbitrary distance, spoiling the fall-off and hence cluster decomposition. Smoothness of $\tilde{h}$ is the precise non-singularity requirement.

Key theorem with proof Intermediate+

Theorem (Weinberg's cluster-decomposition theorem). The S-matrix generated by an interaction $H$ of the normal-ordered polynomial form above satisfies the cluster decomposition principle if and only if each coefficient function $h_{NM}$ contains a single momentum-conservation delta function ${\delta }^3(\sum {\mathbf{q}}^{\prime }-\sum \mathbf{q})$ multiplying a smooth function ${\tilde{h}}_{NM}$ — equivalently, if and only if the connected part $S_{\beta \alpha }^c$ contains exactly one overall delta ${\delta }^3(\sum {\mathbf{p}}_{\mathrm{out}}-\sum {\mathbf{p}}_{\mathrm{in}})$ and no others.

Proof. The Dyson series 08.10.03 expands $S$ as a time-ordered exponential of $H$, so each S-matrix element is a sum of terms, each a momentum-space integral of products of the coefficient functions $h_{NM}$ contracted through the canonical commutation relations $[a(\mathbf{p}),a^{†}(\mathbf{q})]={\delta }^3(\mathbf{p}-\mathbf{q})$ of the prior unit 12.13.01. A contraction joining two operators produces one delta function tying two momenta together.

Track the delta functions. The connected part $S_{\beta \alpha }^c$ collects the terms in which every external line is tied through a chain of contractions into a single linked tangle. In such a term, the deltas from the contractions and from the coefficient functions chain together. If every $h_{NM}$ carries exactly one delta, the connected term retains exactly one overall delta — the others are used up routing internal momenta — leaving $S_{\beta \alpha }^c={\delta }^3(\sum {\mathbf{p}}_{\mathrm{out}}-\sum {\mathbf{p}}_{\mathrm{in}}){\tilde{S}}^c$ with ${\tilde{S}}^c$ smooth.

Now translate one cluster of particles by a displacement $\mathbf{a}$. Under the translation operator, each external momentum picks up a phase $e^{-i\mathbf{p}\cdot \mathbf{a}}$, and the connected smooth amplitude becomes

$${\tilde{S}}^c⟼\int d^3\mathbf{p}{e}^{-i\mathbf{p}\cdot \mathbf{a}}(\text{smooth in }\mathbf{p}).$$

By the Riemann-Lebesgue lemma, this oscillatory integral of a smooth function tends to zero as $|\mathbf{a}|\to \infty$. So the connected amplitude linking the two clusters dies off, and only the factorised partitions survive in the partition sum — which is precisely the factorisation $S_{\beta \alpha }\to {\prod }_j{S}_{{\beta }_j{\alpha }_j}$.

Conversely, suppose some $h_{NM}$ carried a second delta, say ${\delta }^3({\mathbf{q}}_1^{\prime }-{\mathbf{q}}_1)$ in addition to the overall one. Then the connected amplitude would contain a factor depending on a momentum difference that is not integrated against a smooth function — a piece independent of the relative separation. The Riemann-Lebesgue argument fails: this contribution does not die off, and the would-be connected amplitude does not vanish at large separation, violating cluster decomposition. Hence the single-delta condition is both sufficient and necessary. $\square$

Bridge. This theorem is the foundational reason quantum field theory has the structure it does: it builds toward the conclusion that the interaction Hamiltonian must be assembled from creation and annihilation operators in the specific normal-ordered polynomial form, because that is exactly the construction whose coefficient functions carry one delta and yield cluster-decomposable amplitudes. The single-delta condition is dual to the connectedness of the amplitude: one overall delta is the precise momentum-space signature of "all particles tied into one tangle." This pattern — a physical locality demand pinning down operator structure — appears again in the microcausality requirement $[\mathcal{H}(x),\mathcal{H}(y)]=0$ for spacelike separation, which is the off-shell field-theoretic way of enforcing the same cluster property. Putting these together, the central insight is that the answer to "why fields?" is forced rather than postulated: local fields built from $a$ and $a^{†}$ are the unique route to a Lorentz-invariant, cluster-decomposable S-matrix, and this is exactly the structural derivation that the Wightman cluster axiom of the free-scalar unit 12.05.04 takes as a starting axiom.

Exercises Intermediate+

A graded set covering the partition-sum definition, the delta-counting argument, the linked-cluster theorem, and the locality consequence.

Lean formalization Intermediate+

lean_status: none — Mathlib has the Hilbert-space and tensor-power scaffolding the prior unit's Fock space rests on, but nothing approaching a scattering-theoretic S-matrix. A formalisation here would require: the S-matrix as a unitary on the bosonic Fock space with distributional momentum-space matrix elements; the recursive partition-sum definition of the connected amplitude $S^c$ as a Möbius inversion over the lattice of set partitions (Mathlib has Finpartition and incidence-algebra Möbius machinery, but not the application to scattering amplitudes); the delta-counting bookkeeping that distinguishes one momentum-conservation factor from several; and the Riemann-Lebesgue large-separation limit (Mathlib has MeasureTheory.Integrable.tendsto_integral_smul_exp style results that could supply the decay, but not the surrounding Fock-space apparatus). The central theorem — that the single-delta condition on the interaction coefficient is equivalent to cluster decomposition — would chain all of these together. No part of the chain exists in Mathlib at this date; the unit is reviewer-attested. See the Mathlib gap analysis for the enumeration.

Advanced results Master

The connected operator and the cluster expansion. Beyond the matrix-element bookkeeping, the connected S-matrix can be packaged as an operator. Writing $S=1+iT$, one defines the connected operator $T^c$ whose kernel is $S^c$ with the no-scattering term removed. Weinberg's Ch. 4 organises the whole construction as a statement that the connected part of $\ln S$ — interpreting the partition sum as the combinatorial exponential of connected pieces — is a sum of operators each of the normal-ordered form $\int a^{†}\cdots a\cdots$ with a single delta. The exponential-of-connected structure is the operator-level statement of the linked-cluster theorem: the full S-matrix exponentiates its connected part, just as the vacuum amplitude exponentiates its connected vacuum diagrams.

Why creation and annihilation operators specifically. The deepest content of the theorem is the uniqueness of the construction. One asks: among all ways of building an interaction that (i) is Lorentz-invariant, (ii) conserves probability (unitary $S$), and (iii) satisfies cluster decomposition, what is forced? Weinberg's answer is that the interaction must be a polynomial in the $a,a^{†}$ of the one-particle Fock space 12.13.01, with smooth coefficients carrying a single delta — and, when Lorentz invariance is imposed in addition, these operators must be assembled into local fields $\psi (x)=\int (a{e}^{-ipx}+a^{†}{e}^{+ipx})$ transforming covariantly. The fields are not an extra postulate; they are the unique device that simultaneously secures relativistic invariance and cluster decomposition. This is the precise sense in which quantum field theory is derived from a handful of physical principles rather than assumed.

Relation to the Wightman cluster axiom. In the axiomatic framework, cluster decomposition appears as the statement that the vacuum is the unique translation-invariant state and that the Wightman functions factorise in the limit of large spacelike separation of their arguments, $⟨\Omega |\varphi (x_1)\cdots \varphi (x_n)\varphi (y_1+\mathbf{a})\cdots |\Omega ⟩\to ⟨\cdots ⟩⟨\cdots ⟩$ as $|\mathbf{a}|\to \infty$. The free scalar unit 12.05.04 lists this as one reconstruction axiom. The present theorem is the operational, S-matrix-level counterpart: the same demand, phrased on scattering amplitudes rather than vacuum correlation functions, and shown to fix the operator content of the interaction.

Synthesis. Cluster decomposition is the foundational reason the entire apparatus of creation and annihilation operators, normal-ordered interactions, and local fields hangs together: it is exactly the demand that distant experiments are uncorrelated, and it forces — does not merely permit — the single-delta normal-ordered form of the interaction. This is the central insight that putting Lorentz invariance and cluster decomposition together produces local quantum fields as the unique solution; the connectedness structure is dual to the momentum-space delta count, and the exponentiation of disconnected contributions generalises from the vacuum linked-cluster theorem to the full S-matrix. The bridge from the prior unit's Fock space 12.13.01 to interacting field theory is precisely this theorem, and it appears again in the microcausality requirement of the free-scalar unit 12.05.04, in the linked-cluster cumulant expansions of the stat-field framework 08.10.03, and in every textbook derivation of "why fields?" — the single most economical justification of the field concept in physics.

Full proof set Master

Proposition (Möbius-inversion form of the connected amplitude). The recursive partition-sum relation $S_{\beta \alpha }={\sum }_P{\prod }_{c\in P}{S}_c^c$ over the lattice $\Pi$ of set partitions of the combined in/out particle labels determines the connected amplitudes $S^c$ uniquely from $S$, via Möbius inversion on $\Pi$.

Proof. Order the lattice $\Pi$ of set partitions of the $n$ external labels by refinement, with the finest partition $\hat{0}$ (all singletons) at the bottom and the coarsest $\hat{1}$ (one block) at the top. The full amplitude with a fixed clustering corresponds to summing connected amplitudes over all partitions coarser than or equal to a given one; this is a sum of the form $F(\pi )={\sum }_{\sigma \ge \pi }f(\sigma )$, where $f(\sigma )={\prod }_{c\in \sigma }{S}_c^c$ is the product of connected amplitudes over the blocks of $\sigma$ and $F$ is the corresponding full amplitude. By the Möbius inversion formula on a finite lattice, this relation inverts to $f(\pi )={\sum }_{\sigma \ge \pi }\mu (\pi ,\sigma )F(\sigma )$, where $\mu$ is the Möbius function of $\Pi$. Setting $\pi =\hat{1}$ (the one-block partition) recovers the fully connected amplitude $S_{\hat{1}}^c=f(\hat{1})$ as an explicit alternating sum of products of full amplitudes over coarser partitions. Since the Möbius function of any finite lattice exists and is unique, $S^c$ is determined uniquely by $S$. The partition lattice ${\Pi }_n$ has the known Möbius value $\mu (\hat{0},\hat{1})=(-1{)}^{n-1}(n-1)!$, which fixes the leading subtraction coefficients. $\square$

Proposition (decay of the connected amplitude under translation). Let $S_{\beta \alpha }^c={\delta }^3(\sum {\mathbf{p}}_{\mathrm{out}}-\sum {\mathbf{p}}_{\mathrm{in}}){\tilde{S}}^c(\{\mathbf{p}\})$ with ${\tilde{S}}^c$ a smooth, integrable function of the momenta. Translating one cluster of particles by $\mathbf{a}$ multiplies its external momenta by phases $e^{-i\mathbf{p}\cdot \mathbf{a}}$, and the resulting connected contribution tends to zero as $|\mathbf{a}|\to \infty$.

Proof. Under the spatial translation generated by the total momentum operator, a one-particle state of momentum $\mathbf{p}$ acquires a phase $e^{-i\mathbf{p}\cdot \mathbf{a}}$. Applying this to the particles of one cluster, the connected amplitude integrated against test wave-packets becomes

$$I(\mathbf{a})=\int d^3{\mathbf{p}}_1\cdots d^3{\mathbf{p}}_m{e}^{-i({\mathbf{p}}_1+\cdots +{\mathbf{p}}_m)\cdot \mathbf{a}}g({\mathbf{p}}_1,\dots ,{\mathbf{p}}_m),$$

where $g$ is the product of ${\tilde{S}}^c$ with smooth normalisable wave-packet profiles, hence integrable. This is the Fourier transform of an $L^1$ function evaluated at argument $\mathbf{a}$. By the Riemann-Lebesgue lemma, the Fourier transform of an $L^1$ function vanishes at infinity, so $I(\mathbf{a})\to 0$ as $|\mathbf{a}|\to \infty$. The rate of decay is controlled by the smoothness of $g$: if $g\in C^k$ with integrable derivatives, $|I(\mathbf{a})|=O(|\mathbf{a}{|}^{-k})$ by repeated integration by parts. Thus a single-delta, smooth connected amplitude necessarily decouples distant clusters, while any non-smooth (extra-delta) piece would survive. $\square$

Connections Master

• Bosonic Fock space and second quantisation 12.13.01. Direct prerequisite. The creation and annihilation operators $a^{†},a$ and their canonical commutation relation $[a(\mathbf{p}),a^{†}(\mathbf{q})]={\delta }^3(\mathbf{p}-\mathbf{q})$ built there are exactly the building blocks the present theorem forces the interaction to be made from. The whole point of this unit is to show that the Fock-space operator structure of 12.13.01 is not an arbitrary choice but the unique answer to a physical demand: cluster decomposition selects the normal-ordered polynomial in $a,a^{†}$ as the form of any admissible interaction.

• Crossing symmetry and the S-matrix 12.06.04. Lateral prerequisite. That unit defines the S-matrix, the in- and out-states, and Lorentz invariance $U(\Lambda ,a)SU(\Lambda ,a{)}^{†}=S$; the present unit takes that S-matrix and asks what additional structure cluster decomposition imposes. The two together give the trio of constraints — unitarity, Lorentz invariance, cluster decomposition — from which Weinberg derives the field concept. The single-delta condition derived here refines the Lorentz-invariant S-matrix of 12.06.04 into a cluster-decomposable one.

• Free Klein-Gordon scalar quantum field 12.05.04. The free-scalar unit lists cluster decomposition as one of the Wightman reconstruction axioms — the factorisation of vacuum correlation functions at large spacelike separation. The present unit supplies the S-matrix-level derivation of that same demand and shows why a local interaction density $\mathcal{H}(x)$ with spacelike-commuting values satisfies it. This closes the loop between the axiomatic statement of cluster decomposition there and its operational content here.

• φ⁴ theory and the Dyson series 08.10.03. The Dyson time-ordered exponential of that unit is the computational engine behind the proof here: each S-matrix element is a Dyson-series term whose contractions produce the delta functions whose counting drives the central theorem. The linked-cluster / connected-diagram structure appears in 08.10.03 in its stat-field cumulant framing (connected correlators exponentiate); the present unit recasts the same connectedness combinatorics as the S-matrix factorisation property, the two being the cumulant and amplitude faces of one structure.

Historical & philosophical context Master

The cluster decomposition principle was given its sharp modern formulation and its decisive role by Steven Weinberg, who placed it at the foundation of his treatment of quantum field theory in The Quantum Theory of Fields, Vol. 1 (1995) ^{[Weinberg 1995]}. Weinberg's Chapter 4 makes cluster decomposition the central axiomatic move of the entire subject: rather than postulating fields and then checking they behave well, he postulates that distant experiments are uncorrelated — a demand he regarded as more physically transparent than Lorentz invariance itself — and derives the field structure as a consequence. The analysis rests on earlier work, notably the study of the cluster-decomposition properties of connected scattering amplitudes by Eyvind Wichmann and J. H. Crichton in 1963 ^{[Wichmann-Crichton 1963]}, which examined precisely how the connected parts of the S-matrix must behave for distant subsystems to decouple. Klaus Hepp's 1965 work ^{[Hepp 1965]} gave a rigorous account of the linked-cluster / connectedness structure in the Wightman framework.

Philosophically, the result is among the most striking "derivations of physics from principle" in theoretical physics. The naive expectation is that quantum field theory — with its infinite-dimensional Fock spaces, its creation and annihilation operators, its local fields — is a baroque structure adopted for its empirical success and convenience. Weinberg's argument inverts this: the structure is forced. Given that one wants a quantum theory (Hilbert space, unitary evolution), that it should respect special relativity (Lorentz-invariant S-matrix), and that it should be local in the minimal sense that distant experiments do not influence each other (cluster decomposition), the creation-and-annihilation-operator architecture and the local-field assembly are the unique solution. This is the cleanest available answer to the perennial question "why fields?" — and it reframes quantum field theory not as one model among many but as the essentially unique way to combine quantum mechanics, special relativity, and locality. The linked-cluster theorem on the vacuum side, with its exponentiation of disconnected diagrams, is the same connectedness principle operating in the vacuum sector, and it underlies the practical rule that only connected diagrams contribute to physical observables.

Bibliography Master

Primary literature (cite when used; not all currently in reference/):

• Weinberg, S., The Quantum Theory of Fields, Vol. 1: Foundations (Cambridge University Press, 1995). Ch. 4 — cluster decomposition principle, the connected S-matrix, the single-delta theorem, and the derivation of the creation-and-annihilation-operator structure of interactions.
• Wichmann, E. H. & Crichton, J. H., Phys. Rev. 132, 2788 (1963). Cluster decomposition properties of the S-matrix — the connected-amplitude analysis underlying Weinberg's Ch. 4.
• Hepp, K., Comm. Math. Phys. 1, 95 (1965). On the connection between the LSZ and Wightman quantum field theory — rigorous connectedness / linked-cluster structure.

Textbooks and lecture notes:

• Brown, L. S., Quantum Field Theory (Cambridge University Press, 1992). Ch. 6 — linked-cluster theorem, connected generating functionals, exponentiation of disconnected diagrams.
• Duncan, A., The Conceptual Framework of Quantum Field Theory (Oxford University Press, 2012). Ch. 6 — clustering and the structure of the S-matrix.
• Zee, A., Quantum Field Theory in a Nutshell, 2nd ed. (Princeton University Press, 2010). §I.4 — locality and "why fields."
• Coleman, S., Lectures of Sidney Coleman on Quantum Field Theory (World Scientific, 2018). Locality and cluster properties.
• Tong, D., Quantum Field Theory, DAMTP Cambridge lecture notes (raw/pdfs/qft/qft.pdf). §3 — interacting fields, Wick contractions, exponentiation of vacuum bubbles. [Have.]