Modern Foundations — From SR + QM to the QFT Postulates

This is the Wigner–Weinberg derivation of relativistic quantum field theory: starting from special relativity, ordinary quantum mechanics, and the cluster-decomposition principle, derive the framework whose end-product is the QFT postulates. It is the modern counterpart to the historical / canonical-quantization route, and the conceptual prequel to every specific theory built on the resulting postulates — QED, QCD, electroweak theory, the Standard Model, and any other theory obtained by adding a choice of field content, gauge symmetry, and renormalizable Lagrangian.

The companion in the historical direction is QED/historical.md, which goes the other way (single-particle Dirac equation → quantization → field theory). Where the historical route postulates relativistic wave equations and derives gauge invariance later as a consequence, the modern route here postulates relativistic invariance + cluster decomposition and derives the wave-equation structure, the spin–statistics theorem, the field operator framework, and gauge invariance for massless particles all as theorems.

To keep the logical structure transparent we tag every step as one of:

• (Postulate) — a primitive assumption (an empirical input or general principle).
• (Definition) — mathematical notation introduced for use later.
• (Theorem) — a result that follows from previous postulates plus standard mathematics.
• (Heuristic) — physically motivated step whose rigorous justification is deferred.

The modern route rests on three primitive postulates:

Everything in this document — fields as operator-valued distributions, microcausality, spin–statistics, $CPT$, and the connection to gauge invariance — is derived from M1 + M2 + M3 plus standard mathematical machinery (group representation theory, distribution theory, $C^{\ast }$-algebras).

Reference. Weinberg, The Quantum Theory of Fields, Vol. 1 (1995), Chs. 2–5, is the canonical reference. This document is a structural summary of that derivation, with cross-references to where each ingredient lives in this repository.

0. Preliminaries

The mathematical machinery used below — Lorentz / Poincaré groups, Wigner's classification, Lie algebras, irreducible unitary representations, cluster decomposition, $C^{\ast }$-algebras — is collected in QFT/preliminaries.md. This section names the specific ingredients we will rely on, without re-deriving them.

0.1 The Poincaré group and its Lie algebra

The Poincaré group $\mathcal{P}={\mathbb{R}}^{1,3}⋊\mathcal{L}$ is the semidirect product of spacetime translations and Lorentz transformations. Its Lie algebra is generated by:

• 4 translation generators $P^{\mu }$ (energy-momentum),
• 6 Lorentz generators $J^{\mu \nu }$ (3 rotations $J^i=\frac{1}{2}{ϵ}^{ijk}{J}^{jk}$ + 3 boosts $K^i=J^{0i}$).

Commutation relations:

$$[P^{\mu },P^{\nu }]=0,[J^{\mu \nu },P^{\rho }]=i({\eta }^{\nu \rho }{P}^{\mu }-{\eta }^{\mu \rho }{P}^{\nu }),[J^{\mu \nu },J^{\rho \sigma }]=i({\eta }^{\nu \rho }{J}^{\mu \sigma }-\cdots ).$$

The Casimir invariants are

$$P^2=P^{\mu }{P}_{\mu }({\text{mass}}^2),W^2=W^{\mu }{W}_{\mu }(\text{spin}),$$

with $W^{\mu }=-\frac{1}{2}{ϵ}^{\mu \nu \rho \sigma }{J}_{\nu \rho }{P}_{\sigma }$ the Pauli–Lubanski vector.

0.2 Cluster decomposition principle (M3, restated)

Let $|p_1,\dots ,p_n;{\alpha }_1,\dots ,{\alpha }_n⟩$ denote a multi-particle scattering state with momenta $p_i$ and other quantum numbers ${\alpha }_i$. The connected part of the $S$-matrix, $S_C$, is defined recursively by removing all "factorizable" pieces. Cluster decomposition demands

$$S_C(\{p_i,{\alpha }_i\})\to 0\text{as any subset of momenta is translated to spacelike infinity from the rest.}$$

Equivalently (and more usefully): the full $S$-matrix factorizes when the experiment splits into well-separated, non-overlapping subexperiments. This is the relativistic generalization of "labs in different cities don't influence each other" and is the seed of locality.

0.3 The minimal additional inputs (where empiricism enters)

On top of M1+M2+M3, the framework needs empirical inputs to specify which QFT we live in:

• A list of species (one-particle representations) observed in nature.
• For each, mass $m$ and spin $j$ (or, if massless, helicity).
• A choice of interaction consistent with M1+M2+M3.

The first two are pinned down by Wigner's classification (§1 below); the third is constrained but not fixed by the framework, and is where QED, QCD, electroweak theory, etc. diverge as different specializations of the same postulates.

1. The State Space

This section builds the state space of a relativistic quantum theory step by step, tagging each step as (Postulate), (Definition), (Theorem), or (Empirical input) so the assumption budget is transparent. The goal is to expose which assumptions feed into the Fock-space construction — several of them are silently invoked in textbook treatments.

1.1 Defining "particle" (Definition + empirical input)

The Wigner-Weinberg framework rests on a specific operational identification:

Definition. A particle in a relativistic quantum theory is an irreducible unitary representation of the proper orthochronous Poincaré group ${\mathcal{P}}_{+}^{\uparrow }$ acting on a complex separable Hilbert space.

This is a definition, not a theorem. The next three subsections unpack what each clause means in concrete physical terms; the definitional content and its empirical commitments are summarised in §1.1.4. Two pieces of empirical input are baked in:

• (Empirical-1) Identification of "particle" with "Poincaré irrep". Justified physically: an elementary excitation should be characterized by quantum numbers (mass, spin) that are invariant under choice of inertial frame. Irreducibility = "cannot be split into uncoupled subspecies" = elementarity. Mathematically tight, but the physical claim that nature's elementary objects are organized this way is empirical.
• (Empirical-2) Hilbert space is complex and separable. Inherited from M2 (standard QM). Real-Hilbert-space and quaternionic-Hilbert-space alternatives exist mathematically but are not used in the Standard Model.

1.1.1 What "representation" means here

A unitary representation of ${\mathcal{P}}_{+}^{\uparrow }$ on a Hilbert space $\mathcal{H}$ is a group homomorphism

$$U:{\mathcal{P}}_{+}^{\uparrow }\to \mathcal{U}(\mathcal{H}),(\Lambda ,a)\mapsto U(\Lambda ,a),$$

where each $U(\Lambda ,a)$ is a unitary operator on $\mathcal{H}$ and $U({\Lambda }_1,a_1)U({\Lambda }_2,a_2)=U(({\Lambda }_1,a_1)({\Lambda }_2,a_2))$. Equivalently: every Poincaré symmetry — boost, rotation, translation — is realized as a concrete unitary operator moving state vectors around inside $\mathcal{H}$. Three adjectives carry the content:

• Unitary, because Poincaré transformations are symmetries of the probabilistic structure of QM: $|⟨\psi |\varphi ⟩{|}^2$ must be invariant under change of inertial frame.
• Irreducible, meaning no proper $U$-invariant subspace exists: there is no non-trivial ${\mathcal{H}}^{\prime }\subset \mathcal{H}$ closed under all $U(\Lambda ,a)$. Physically: the representation does not split into uncoupled subsectors that the Poincaré group keeps separate — anything bigger than an irrep would be several species in one Hilbert space.
• Projective allowed, i.e. equality up to a phase, because physical states are rays not vectors. The projective representations of ${\mathcal{P}}_{+}^{\uparrow }$ correspond to ordinary representations of its universal cover ${\mathbb{R}}^{1,3}⋊SL(2,\mathbb{C})$ (see math/group-theory.md § Group actions and representations and § Connectedness and discrete components). This is what allows half-integer-spin particles.

1.1.2 "Particle = irrep" is a statement about the Hilbert space, not a single state

The most common point of confusion. The definition does not say "a particle is a state vector $|\psi ⟩$"; it says

"a particle species" $⟷$ the entire one-particle Hilbert space ${\mathcal{H}}_1$ together with the ${\mathcal{P}}_{+}^{\uparrow }$ action on it.

The single space ${\mathcal{H}}_1$ contains all possible states of one specimen of that species (any momentum, any spin orientation, any superposition). The Poincaré group shuffles them around — boosts change momentum, rotations rotate spin, translations multiply by a phase $e^{ip\cdot a}$. Irreducibility says you can reach every state in ${\mathcal{H}}_1$ from any other by some Poincaré action (plus superposition); that is what makes the species one species, with no leftover "different kind of electron" sub-Hilbert-space.

So "what is a particle?" has two answers depending on which question you are asking: the kind of particle is an irrep; a particle in a state is a vector in that irrep's Hilbert space.

1.1.3 The natural basis: momentum and spin, not position

Wigner's classification (§1.2) hands you more than the existence of ${\mathcal{H}}_1$ — it hands you a canonical basis labelled by simultaneous eigenstates of the translation generators $P^{\mu }$ together with a little-group component $\sigma$:

$$|p,\sigma ⟩,P^{\mu }|p,\sigma ⟩=p^{\mu }|p,\sigma ⟩,p^2=m^2,$$

where $\sigma$ runs over the spin $z$-component $s_z\in \{-j,\dots ,+j\}$ (massive case) or the helicity $h$ (massless case). A general one-particle state is

$$|\psi ⟩=\sum _{\sigma }\int \frac{d^{3}p}{(2\pi {)}^{3}2E_p}{\psi }_{\sigma }(p)|p,\sigma ⟩,$$

with the Lorentz-invariant momentum measure. The momentum-space wavefunction ${\psi }_{\sigma }(p)$ is the natural description because momentum and spin component are Casimir-respecting quantum numbers handed to you by the group action itself.

Why momentum and not position? Three reasons, in order of severity:

1. Position is not a generator of the Poincaré algebra. The algebra has $P^{\mu }$ (translation generators ↔ momentum) and $J^{\mu \nu }$ (Lorentz generators ↔ angular momentum / boosts). There is no $X^{\mu }$. Momentum is intrinsic; position would have to be defined on top of ${\mathcal{H}}_1$, and no canonical choice exists.
2. There is no Lorentz-covariant position operator. The best-known attempt — the Newton–Wigner position operator — is self-adjoint on ${\mathcal{H}}_1$ but
   • is not Lorentz-covariant: a state localized at $\mathbf{x}$ in one frame is not localized in any boosted frame;
   • has eigenstates with non-local tails of width $\sim 1/m$ (the Compton wavelength);
   • fails entirely for massless particles of helicity $\ge 1$ (no localized photon states — the Newton–Wigner / Hegerfeldt obstructions).
3. Causality is in tension with localization. Any state strictly localized inside a spatial region at $t=0$ instantaneously develops support everywhere by $t>0$ (Hegerfeldt's theorem). For single-particle states there is no finite-signal-velocity rescue.

The way QFT resolves this is to stop localizing particles and localize fields instead. Field operators $\varphi (x),\psi (x),A_{\mu }(x)$ (built in §2) live at spacetime points and satisfy microcausality $[\varphi (x),\varphi (y)]=0$ at spacelike separation (§3). The matrix element

$$⟨0|\varphi (x)|p,\sigma ⟩=u_{\sigma }(p)e^{-ip\cdot x}$$

looks like a position-space wavefunction but is genuinely the matrix element of an operator-valued distribution between two states — see QFT/preliminaries.md § States vs. Fields. Position-dependence re-enters at the operator level, not the state level. This inversion is exactly what makes relativistic quantum theory a field theory rather than a particle theory: particles are irreps (defined by momentum-space data), and the spacetime point $x$ is an argument of the operator, not a label on a state.

1.1.4 Summary

• "Representation" = a unitary action of ${\mathcal{P}}_{+}^{\uparrow }$ on ${\mathcal{H}}_1$. Irreducibility = no invariant sub-Hilbert-space = elementarity.
• A particle is a state only in the loose sense that states of that species live in ${\mathcal{H}}_1$. The species itself is the entire irrep, labelled by $(m,j)$ or $(0,h)$.
• The natural basis is momentum + spin component, not position. There is no good relativistic position operator, and any attempt at one breaks Lorentz covariance or causality. Position re-enters as the argument of field operators in §2, not as a label of single-particle states.

With this definition fixed, the rest of §1 unfolds with very few additional postulates.

1.2 One-particle states (Theorem — Wigner classification)

Premise: M1 (special relativity) + M2 (QM) + the §1.1 definition of particle.

Wigner's theorem (1939). The irreducible unitary representations of ${\mathcal{P}}_{+}^{\uparrow }$ are classified by two Casimirs:

• Mass-squared $P^2=m^2$ with $m\ge 0$.
• For $m>0$: spin $j\in \{0,\frac{1}{2},1,\frac{3}{2},\dots \}$ (irrep of the little group $SO(3)$).
• For $m=0$: helicity $h\in \{0,\pm \frac{1}{2},\pm 1,\pm \frac{3}{2},\dots \}$ (irrep of the little group $ISO(2)$, with $h$ quantized in integer/half-integer units by $4\pi$ rotation closure).

Each irrep is realized on a one-particle Hilbert space ${\mathcal{H}}_1$. The full derivation lives in QFT/preliminaries.md § Wigner's Classification.

Conclusion. ${\mathcal{H}}_1$ is forced, not chosen. Specifying a species means picking $(m,j)$ or $(0,h)$.

Empirical-3 (caveat). Continuous-spin representations of the massless little group exist mathematically but are not observed in nature. Their absence is empirical, not a theorem.

1.3 Identical particles and (anti)symmetrization (Postulate — symmetrization)

Premise: $n$ copies of ${\mathcal{H}}_1$ for the same species, plus the empirical fact that identical particles are truly indistinguishable.

The naive $n$-fold tensor product ${\mathcal{H}}_1^{\otimes n}$ contains many states distinguished by labels on the particles (which particle has which momentum). Indistinguishability requires that the physical state space be only the part of ${\mathcal{H}}_1^{\otimes n}$ on which the symmetric group $S_n$ acts trivially up to a phase:

$${\mathcal{H}}_n^{\text{sym}}={\mathrm{Sym}}^n({\mathcal{H}}_1)\text{(bosons)},{\mathcal{H}}_n^{\text{anti}}={\Lambda }^n({\mathcal{H}}_1)\text{(fermions)}.$$

(Postulate — symmetrization) Multi-particle states of identical species lie in either ${\mathrm{Sym}}^n({\mathcal{H}}_1)$ or ${\Lambda }^n({\mathcal{H}}_1)$, never in any other subspace. This is an additional assumption beyond M1+M2+M3 in $d\ge 4$ spacetime dimensions. (In $2+1$d it can be relaxed to give anyons — irreps of the braid group with arbitrary phases.)

Note on bosons vs. fermions. Which of ${\mathrm{Sym}}^n$ vs ${\Lambda }^n$ each species uses is not postulated here — it is determined later by the spin–statistics theorem (§4) from M1+M2+M3 + the field construction of §2. The symmetrization postulate only fixes the binary structure, not the assignment.

1.4 Variable particle number → Fock space (Empirical input + Definition)

Premise: the empirical observation that physical processes change the number of particles.

In non-relativistic QM, particle number is conserved and the state space is a single ${\mathcal{H}}_n$ for fixed $n$. In relativistic QFT this fails:

• $E=m{c}^2$ permits pair creation/annihilation when sufficient energy is available.
• Decays ($n\to 1$ becomes $n\to 2,3,\dots$).
• Scattering with different in/out particle counts.

(Empirical-4) Variable particle number is an empirical input of relativistic QFT, justified by relativity ($E=m{c}^2$) and observation. It is not derivable from M1+M2+M3 alone — it is a fact about nature that the formalism must accommodate.

Definition. Given variable $n$, the natural state space is the Fock space direct sum over all sectors:

$$\mathcal{F}=\underset{n=0}{\overset{\infty }{⨁}}{\mathcal{H}}_n,$$

with ${\mathcal{H}}_0=\mathbb{C}|0⟩$ the one-dimensional vacuum sector and ${\mathcal{H}}_n$ the appropriately (anti)symmetrized $n$-particle sector from §1.3.

Inputs combined. Constructing $\mathcal{F}$ thus requires:

• M1 + M2 + §1.1 definition (gives ${\mathcal{H}}_1$);
• §1.3 symmetrization postulate (gives ${\mathcal{H}}_n$ for each $n$);
• §1.4 variable-number empirical input (motivates ${⨁}_n$).

The detailed structure of $\mathcal{F}$ — sectors, vacuum, ladder action, etc. — is in fock-space-inventory.md.

Caveat — free vs. interacting. $\mathcal{F}$ is the free Fock space, built from one-particle Wigner reps. The interacting Hilbert space of a non-trivial theory is not unitarily equivalent to $\mathcal{F}$ (Haag's theorem). $\mathcal{F}$ is the appropriate state space for asymptotic in/out states only — see §2.0.1 and QFT/remarks.md § Haag's theorem.

1.5 Cluster decomposition forces creation/annihilation operators (Theorem)

Premise: Fock space $\mathcal{F}$ from §1.4 + M3 (cluster decomposition).

Working with $n$-particle states sector by sector is unwieldy — every $n$ requires its own ${\mathcal{H}}_n$. Cluster decomposition singles out a much more economical formalism: build everything from operators that raise or lower particle number by one.

Definition. Creation operators $a^{†}(p,\sigma )$ take an $n$-particle state to an $(n+1)$-particle state by adjoining a particle of momentum $p$ and spin component $\sigma$. Annihilation operators $a(p,\sigma )$ are their adjoints. On Fock space they satisfy

$$[a(p,\sigma ),a^{†}(p^{\prime },{\sigma }^{\prime }){]}_{\pm }=(2\pi {)}^3(2E_p){\delta }^3(\mathbf{p}-{\mathbf{p}}^{\prime }){\delta }_{\sigma {\sigma }^{\prime }},$$

with the bosonic commutator or fermionic anticommutator according to the species statistics. (See QFT/preliminaries.md § Fock Space for the Terminology — ladder operators callout connecting these to the QM harmonic-oscillator $\hat{a},{\hat{a}}^{†}$.)

Theorem (Weinberg Vol. 1 §4.4). Given $\mathcal{F}$ from §1.4, any operator whose connected matrix elements satisfy cluster decomposition can be written as a polynomial in $a$ and $a^{†}$. Conversely, an interaction Hamiltonian not expressible in terms of these operators violates cluster decomposition (its connected part fails to vanish at large separation).

Conclusion. M3 + §1.4 ⇒ ladder-operator formalism is unique among the alternatives. Cluster decomposition does not derive Fock space itself (that came from §1.4); it derives the ladder-operator structure on top of an already-given Fock space.

1.6 Summary of §1: assumption budget

Three inputs beyond M1+M2+M3 are quietly invoked along the way: the operational definition of particle (§1.1), the symmetrization postulate (§1.3), and the empirical fact of variable particle number (§1.4). All three are universally accepted in standard QFT but are additional assumptions, not consequences of the core M1+M2+M3.

2. The Need for Local Fields

2.0 The S-matrix as the central object (Postulate / framing)

Before constructing fields, it is worth being explicit about what we are constructing them for. In the modern story, the S-matrix is closer to a primitive than the Hamiltonian is. This is the conceptual inversion relative to the historical route, and clarifies the logic of the rest of §2. The next three sub-subsections introduce the asymptotic-state framework that $S$ acts on (§2.0.1), define $S$ together with its three structural constraints (§2.0.2), and unpack what it means to take $S$ as the primary object (§2.0.3).

2.0.1 Asymptotic states (Definition)

Real experiments prepare configurations that are effectively free long before scattering and detect configurations that are effectively free long after. The mathematical objects representing these are asymptotic in-states $|i,\text{in}⟩$ and asymptotic out-states $|f,\text{out}⟩$.

Heuristic content. An asymptotic state is a multi-particle Fock-space configuration of the species classified by Wigner (§1.2) — a definite collection of momenta, spins, and species — whose interactions are negligible in the relevant time limit. The interacting Hilbert space $\mathcal{H}$ does not literally contain free states (interactions never fully turn off), but it contains states that behave as if free in the limits $t\to \mp \infty$.

Formal construction (deferred). Two equivalent formal definitions exist; both are technically delicate and we defer derivations to standard references:

• Møller operators. Define ${\Omega }_{\pm }\equiv {\lim }_{t\to \mp \infty }{e}^{iHt}{e}^{-i{H}_{0}t}$ (limits taken in a strong-operator-on-wavepackets sense). ${\Omega }_{+}$ maps a free Fock state to its dressed in-counterpart, ${\Omega }_{-}$ to its dressed out-counterpart: $$|i,\text{in}⟩={\Omega }_{+}|i,\text{free}⟩,|f,\text{out}⟩={\Omega }_{-}|f,\text{free}⟩.$$
• LSZ. Asymptotic states are extracted as residues of single-particle poles in the time-ordered correlators of interpolating fields (any local field with non-zero matrix element between the vacuum and a one-particle state). Unlike the Møller construction above, LSZ does not require a Hamiltonian — it takes correlators as input, which can come from a Lagrangian, the lattice, the conformal bootstrap, or any other source. See QFT/preliminaries.md § LSZ Reduction Formula for the formula and properties.

Asymptotic Hilbert spaces. The in-states span ${\mathcal{H}}_{\text{in}}\equiv \mathrm{ran}{\Omega }_{+}$; the out-states span ${\mathcal{H}}_{\text{out}}\equiv \mathrm{ran}{\Omega }_{-}$. Each is naturally a Fock space built on one-particle Wigner reps (§1.2–1.4). The two spaces are independent by construction, and may a priori be different from each other and from the full interacting $\mathcal{H}$.

The two-part structure of P10. QFT Postulate 10 packages two genuinely separate assumptions, plus a derived consequence:

So "$S$ exists as a unitary operator" is not an independent assumption — it is a theorem given (P10a) and (P10b). The substantive content of P10 is exactly (P10a) + (P10b), and the failure modes of $S$ in real theories trace back to whichever of these two is violated.

Where these are guaranteed vs. assumed.

• Free theory: (a) and (b) are trivially satisfied, $S=1$.
• Constructive QFT in $d<4$: (a) and (b) are theorems (Haag–Ruelle 1962, given Wightman axioms + isolated mass shells).
• Realistic 4D QFT (QED, QCD): (a) and (b) are conjectural — no rigorous construction exists; they are physically motivated assumptions justified retroactively by the agreement of perturbative $S$-matrix elements with experiment.
• Modified asymptotics: When (a) or (b) fails, the framework is modified rather than abandoned — Faddeev–Kulish coherent states for QED soft photons; hadron Fock space for QCD; conformal-bootstrap data for CFTs without a mass gap.

For the rest of this document we take (P10a) + (P10b) for granted, consistent with how Wightman / Weinberg structure the framework.

2.0.2 The S-matrix and its three structural constraints (Definition)

Given asymptotic states from §2.0.1, the S-matrix is the unitary operator linking the in- and out-bases of $\mathcal{H}$:

$$\boxed{S:{\mathcal{H}}_{\text{in}}\to {\mathcal{H}}_{\text{out}},|f,\text{out}⟩=S|i,\text{in}⟩.}$$

Its matrix elements $⟨f|S|i⟩$ encode all observable scattering probabilities. Three structural constraints — each a direct consequence of one of the modern postulates M1, M2, M3 (plus P10) — pin down what $S$ can look like:

1. Unitarity (M2 ⇒ probability conservation): $$S^{†}S=S{S}^{†}=1.$$ Equivalent to "total probability of some outcome equals 1": ${\sum }_f|⟨f|S|i⟩{|}^2=⟨i|S^{†}S|i⟩=1$. The optical theorem ($\mathrm{Im}{\mathcal{M}}_{i\to i}\propto {\sum }_f|{\mathcal{M}}_{i\to f}{|}^2$) is the perturbative content of this identity.
2. Lorentz invariance (M1 ⇒ frame-independence): $$U(\Lambda ,a)SU(\Lambda ,a{)}^{-1}=S\forall (\Lambda ,a)\in {\mathcal{P}}_{+}^{\uparrow },$$ with $U$ the strongly continuous unitary representation of the Poincaré group on $\mathcal{H}$ (see QFT/preliminaries.md § Lorentz and Poincaré Groups). This is what forces the spacetime-translation ${\delta }^4(\sum p_{\text{in}}-\sum p_{\text{out}})$ to factor out of every matrix element, leading to the $\mathcal{M}$-residue construction (see Computational machinery below).
3. Cluster decomposition (M3 ⇒ no spooky long-distance correlations): if a multi-particle process splits into well-separated, non-overlapping subprocesses, the connected $S$-matrix factorizes: $$S_C(\{\text{combined process}\})⟶\prod _i{S}_C(\{{\text{subprocess}}_i\})\text{as subsets are translated to spacelike infinity from each other.}$$ This is what §1.5 used to force the ladder-operator structure, and §2.1 will use to force locality of the interaction density.

Together, (unitarity + Lorentz invariance + cluster decomposition) is the modern definition of "a relativistic quantum scattering theory". Specifying which $S$ — i.e. which species couple to which, and how — is the empirical content of any given theory (QED, QCD, …).

The transition operator $T=-i(S-1)$ and the invariant amplitude ${\mathcal{M}}_{fi}$ are introduced from $S$ exactly as in QED/historical.md §5.3.

2.0.3 The status of $S$ as primary (Framing)

What this means structurally. "Specifying a relativistic quantum theory" is operationally the same as "specifying a Lorentz-invariant, cluster-decomposable, unitary $S$-matrix". The Hamiltonian formalism, the Lagrangian formalism, and Feynman-diagram perturbation theory are all means to that end — concrete parameterizations of valid $S$-matrices, none more privileged than the others. (In its purest form, this idea is the S-matrix bootstrap / modern amplitudes program, where one writes down constraints — Lorentz, unitarity, analyticity, locality — directly on $S$ and never introduces a Lagrangian at all. This is most powerful for theories where Lagrangian descriptions are awkward or non-existent, e.g. higher-spin, certain CFTs, and the on-shell amplitudes literature.)

Why fields then appear. Demanding that $S$ be Lorentz invariant + cluster-decomposable + built from the ladder operators of §1.5 forces the construction in §2.1: the only consistent way to package the ladder operators into a Lorentz-covariant local interaction is via local fields $\psi (x),\varphi (x),A_{\mu }(x)$. Fields are construction tools for $S$, not the fundamental objects.

Comparison with the historical route. QED/historical.md §5.0 starts from a Hamiltonian $H=H_0+H_{\text{int}}$ and defines $S\equiv U_I(+\infty ,-\infty )$ as the interaction-picture evolution operator. There $S$ is derived from $H$. Here we run the logic the other way: $S$ is postulated to exist (P10), and the Hamiltonian (or Lagrangian) is one of several conventional ways to specify which $S$. Both routes converge at the level of the Dyson + Wick + Feynman-rules computation — the same machinery, motivated differently. The distinction is conceptual: "what is fundamental?" not "what do we actually compute?".

Computational machinery (deferred). Once $S$ has been specified by a choice of Lagrangian / Hamiltonian, the entire computational chain

$$\text{Dyson series}\to \text{Wick’s theorem}\to \text{Feynman rules}\to T\to \mathcal{M}\to \overline{|\mathcal{M}{|}^2}\to d\sigma ,d\Gamma$$

is identical to the one in the historical route, and we do not reproduce it here. The full derivation lives in:

• QED/historical.md §5.0 — Perturbative machinery: S-matrix, Dyson series, Wick's theorem — Dyson series, Wick's theorem, generic Feynman-diagram structure.
• QED/historical.md §5.2 — The QED S-matrix and Feynman rules — momentum-space Feynman-rules table.
• QED/historical.md §5.3 — The Invariant Amplitude $\mathcal{M}$ — definition of $T=-i(S-1)$ and ${\mathcal{M}}_{fi}$ as the residue after stripping the spacetime-translation ${\delta }^4$.
• QED/historical.md §5.4 — The Squared Amplitude $|\mathcal{M}{|}^2$ — Casimir's trick, spin/polarization sums, physical interpretation, six-step worked-example pipeline.
• QFT/cross-sections.md — master formula $d\sigma =(1/F)\overline{|\mathcal{M}{|}^2}d{\Pi }_n$; the parallel decay-rate formula $d\Gamma =(1/2M)\overline{|\mathcal{M}{|}^2}d{\Pi }_n$ lives in decay-rates.md.
• QFT/preliminaries.md § Time-Ordering Operator — the convention used in the Dyson series.

These computational results carry over verbatim to the modern story; only their interpretive status differs (e.g. the Hamiltonian whose Dyson series is being expanded is, in the modern view, just a parameterization of $S$ rather than the primary object).

Aside — is $H$ derived from $S$? The modern viewpoint elevates $S$ above $H$, but does not provide a constructive recipe to extract $H$ from $S$. The relationship is more subtle:

• $H_0$ is fully determined by the species content (Wigner classification, §1). For free fields the Hamiltonian is forced: $$H_0=\sum _{\sigma }\int \frac{d^{3}p}{(2\pi {)}^3}{E}_p{a}^{†}(p,\sigma )a(p,\sigma ).$$
• $H_{\text{int}}$ is only existentially constructive. Weinberg Vol. 1 Ch. 3 shows that any Lorentz-invariant, cluster-decomposable, unitary $S$ admits some interaction-picture Hamiltonian density ${\mathcal{H}}_{\text{int}}(x)$ built from local fields satisfying $[{\mathcal{H}}_{\text{int}}(x),{\mathcal{H}}_{\text{int}}(y)]=0$ for $(x-y{)}^2<0$. But this ${\mathcal{H}}_{\text{int}}$ is not unique: many different Hamiltonians (differing by field redefinitions) produce the same $S$.
• Field-redefinition equivalence (LSZ). Two $H_{\text{int}}$'s related by a local field redefinition $\psi (x)\to \psi (x)+g[\psi (x)]$ give the same on-shell $S$-matrix. This is why effective field theories using different operator bases can describe identical physics, and why the choice of "fundamental" field is partly conventional.
• The bootstrap view. Modern programs (on-shell amplitudes, conformal bootstrap, double-copy constructions) compute $S$-matrix elements directly from Lorentz / unitarity / locality / analyticity constraints without writing down any $H$ — empirical evidence that $H$ is genuinely a parameterization, not a derivation target.

So the modern claim is not "$H$ is computed from $S$" but rather "$H$ exists, is non-unique, and is one of several equivalent parameterizations of the same $S$." The historical and modern routes thus differ on which is primary, but agree that they are equally complete descriptions of any given theory.

2.1 Lorentz invariance + cluster decomposition demand a special structure (Theorem)

To build a Lorentz-invariant interaction Hamiltonian $H_{\text{int}}$, the natural ansatz is

$$H_{\text{int}}=\int d^{3}x{\mathcal{H}}_{\text{int}}(\mathbf{x},t)$$

with ${\mathcal{H}}_{\text{int}}(x)$ a scalar density under the Poincaré group. But this Hamiltonian density must be built from creation/annihilation operators (by §1.5, to satisfy cluster decomposition), and must be a Lorentz scalar.

Naive ladder-operator combinations don't transform covariantly. Under a Lorentz transformation, individual $a^{†}(p,\sigma )$ transforms via Wigner's little-group rotation $W(\Lambda ,p)$, which depends on $p$ — not as an ordinary tensor field. Building a Lorentz scalar density out of these is non-trivial.

Solution: package the ladder operators into local fields. Define

$${\psi }_{\ell }^{+}(x)\equiv \sum _{\sigma }\int \frac{d^{3}p}{(2\pi {)}^3(2E_p)}{u}_{\ell }(p,\sigma )a(p,\sigma )e^{-ip\cdot x},$$ $${\psi }_{\ell }^{-}(x)\equiv \sum _{\sigma }\int \frac{d^{3}p}{(2\pi {)}^3(2E_p)}{v}_{\ell }(p,\sigma )a^{c†}(p,\sigma )e^{+ip\cdot x},$$

with mode functions $u_{\ell }(p,\sigma )$ and $v_{\ell }(p,\sigma )$ chosen so that the combined field

$${\psi }_{\ell }(x)\equiv {\psi }_{\ell }^{+}(x)+{\psi }_{\ell }^{-}(x)$$

transforms as a finite-dimensional representation of the Lorentz group:

$$U(\Lambda ,a){\psi }_{\ell }(x)U(\Lambda ,a{)}^{-1}=\sum _m{S}_{\ell m}({\Lambda }^{-1}){\psi }_m(\Lambda x+a).$$

The mode functions $u,v$ are uniquely determined by this requirement (up to normalization and basis choice) for each species; for spin $\frac{1}{2}$ they are the Dirac spinors of the historical route, for spin 1 the polarization vectors, etc.

Two non-trivial consequences emerge automatically:

1. Antiparticles. The ${\psi }^{-}$ piece must contain $a^{c†}$ — creation operators for a different species (the antiparticle) with the same mass and spin and opposite charge. (For neutral self-conjugate fields, particle = antiparticle.) The existence of antiparticles is not postulated separately; it is forced by the requirement that $\psi$ transform locally and covariantly.

2. Microcausality. For the interaction Hamiltonian density ${\mathcal{H}}_{\text{int}}(x)$ to commute with itself at spacelike separation (which is necessary for Lorentz-invariant time-evolution and for cluster decomposition of the $S$-matrix), one finds that the fields themselves must (anti)commute at spacelike separation:

   $$[{\psi }_{\ell }(x),{\psi }_m(y){]}_{\pm }=0\text{for }(x-y{)}^2<0.$$

   This is microcausality — derived, not postulated.

2.2 Fields are operator-valued distributions (Theorem — heuristic)

The fields ${\psi }_{\ell }(x)$ defined above are not operators on $\mathcal{H}$ in the strict sense: ${\psi }_{\ell }(x)|0⟩$ has infinite norm (the integrand involves $e^{-ip\cdot x}$ at one spacetime point, with no damping). They are operator-valued tempered distributions: only the smeared objects

$${\psi }_{\ell }(f)=\int d^{4}xf(x){\psi }_{\ell }(x),f\in \mathcal{S}({\mathbb{M}}^4)$$

are bona fide (unbounded) operators. This is the source of the technical complication that distinguishes QFT from finite-DOF QM.

Why "heuristic"? The above is rigorous for free fields. For interacting fields in $d=4$, Haag's theorem shows the construction does not survive intact and the fields must be reconstructed via renormalization. See QFT/remarks.md § Haag's Theorem.

3. Microcausality and Locality

§2.1 already established microcausality

$$[{\psi }_{\ell }(x),{\psi }_m(y){]}_{\pm }=0,(x-y{)}^2<0$$

as a consequence of M1 + M3 plus the field construction. The (anti)commutator choice is fixed by spin–statistics (§4 below). Three points worth highlighting:

1. Microcausality is what makes locality precise. Cluster decomposition (M3) is a statement about scattering at large spatial separation; microcausality is the operator-level consequence in Hilbert space.
2. It is necessary for relativistic causality. Two operators that don't (anti)commute at spacelike separation could in principle propagate signals faster than light by repeated measurement.
3. The (anti)commutator must vanish, not just be small. This is a much stronger condition than thermodynamic locality and is what fails in (e.g.) string field theory, leading to its non-local behavior on small scales.

The full statement is QFT Postulate 6; the modern derivation just outlined is summarized in §2.1.

4. Spin–Statistics (Theorem)

In §1.3 we left the choice between ${\mathrm{Sym}}^n({\mathcal{H}}_1)$ (bosonic) and ${\Lambda }^n({\mathcal{H}}_1)$ (fermionic) open. The spin–statistics theorem fixes it:

Theorem (Pauli 1940; Lüders–Zumino 1958). In a Lorentz-invariant local QFT with positive energy and a unique vacuum, fields of integer spin must be quantized as bosons (commutators) and fields of half-integer spin as fermions (anticommutators). Any other choice produces either negative-norm states, violation of microcausality, or violation of the spectrum condition.

Proof sketch. Construct the field ${\psi }_{\ell }(x)$ for spin $j$ using both choices (commutators vs anticommutators) and compute $[\psi (x),{\psi }^{†}(y){]}_{\pm }$ at spacelike separation. The result has the form $(/\partial +m)\Delta (x-y)$ for spin $\frac{1}{2}$, ${\eta }_{\mu \nu }\Delta (x-y)-\cdots$ for spin 1, etc., where $\Delta (x-y)$ is the commutator function. For half-integer spin, anticommutators give the spacelike-vanishing combination; commutators give a result that does not vanish (and would violate microcausality). For integer spin it's the reverse.

So spin–statistics is not an independent postulate in the modern derivation — it is forced by M1 + M2 + M3 + the field construction of §2.

Corollary — Pauli exclusion. Anticommutators of fermion creation operators give $a^{†}{a}^{†}=0$, forbidding two identical fermions in the same state. The Pauli exclusion principle is a downstream consequence of M1 + M2 + M3, not an independent postulate.

5. Massless Particles and the Origin of Gauge Invariance (Theorem)

Massless particles have a richer little group than massive ones — $ISO(2)$ instead of $SO(3)$ — with consequences for the construction in §2.1.

5.1 Polarization vectors of massless spin 1 don't transform as 4-vectors

For a massive spin-1 particle, the polarization vectors ${ϵ}^{\mu }(p,\lambda )$ form a true 4-vector representation of the little group $SO(3)$ (3 transverse polarizations). For a massless spin-1 particle, the little group $ISO(2)$ acts on ${ϵ}^{\mu }(p,\lambda )$ with an inhomogeneous piece:

$${ϵ}^{\mu }(p,\lambda )\stackrel{\Lambda }{\to }{e}^{i\lambda \theta (\Lambda ,p)}[{ϵ}^{\mu }(\Lambda p,\lambda )+\beta (\Lambda ,p)p^{\mu }].$$

The $\beta p^{\mu }$ piece is not removable. So ${ϵ}^{\mu }$ is not a true 4-vector under Lorentz transformations.

5.2 Lorentz invariance forces gauge invariance

For an interaction term ${\mathcal{L}}_{\text{int}}=j^{\mu }(x)A_{\mu }(x)$ to be Lorentz-invariant despite the non-tensorial transformation of ${ϵ}^{\mu }$, the inhomogeneous piece $\beta (\Lambda ,p)p^{\mu }$ must drop out. This requires

$$p^{\mu }{j}_{\mu }(p)=0,$$

i.e. the current $j^{\mu }$ must be conserved: ${\partial }_{\mu }{j}^{\mu }=0$.

By Noether's theorem, a conserved current corresponds to an internal symmetry. For an external photon line with polarization ${ϵ}^{\mu }(k,\lambda )$, the residual transformation ${ϵ}^{\mu }\to {ϵ}^{\mu }+\beta k^{\mu }$ must be a symmetry of the interaction — and that residual transformation is exactly $U(1)$ gauge invariance:

$$A_{\mu }\to A_{\mu }+{\partial }_{\mu }\alpha .$$

So gauge invariance of QED is a theorem in the modern derivation: it is what is required to make a massless spin-1 particle interact in a Lorentz-invariant way. The photon's gauge invariance is not a separate postulate; it is a consistency requirement of M1 + M2 + M3 applied to a massless spin-1 species.

Generalization. For multiple massless spin-1 particles forming a non-trivial multiplet under an internal symmetry $G$, the same argument forces the Yang–Mills structure: the connection 1-form, structure constants $f^{abc}$, and self-interactions are all uniquely determined. See QED for the abelian case ($U(1)$, one massless spin-1) and QCD for the non-abelian realization ($SU(3{)}_{\text{color}}$, eight massless spin-1 in the adjoint).

What this story does not derive. The existence of multiple massless spin-1 particles forming a $G$-multiplet is an empirical input, not a theorem. Likewise, massive gauge bosons (e.g. $W^{\pm },Z^0$) are permitted by Wigner classification without gauge invariance, but their longitudinal polarizations break perturbative unitarity at high energies — the resolution (the Higgs mechanism, with a scalar VEV breaking the gauge symmetry spontaneously) is a postulate added on top of the modern story, not a derivation. See QFT/remarks.md § Spontaneous Symmetry Breaking and the electroweak theory doc, where the Higgs mechanism is worked out in detail.

6. CPT Theorem (Theorem)

Theorem (Pauli 1955; Lüders 1957; Jost 1957). Every Lorentz-invariant local QFT with positive energy and a unique vacuum is invariant under the combined action $CPT$ (charge conjugation × parity × time reversal), where $C$ swaps particles and antiparticles, $P$ inverts spatial coordinates, and $T$ reverses time.

The $CPT$ theorem is a consequence of M1 + M2 + M3 + the field construction of §2 (specifically, the analytic-continuation properties of the Wightman functions in complexified Minkowski space). $C$, $P$, and $T$ individually are not required to be symmetries — and indeed are violated by the weak interaction — but their combined product must be.

For the proof structure see Streater–Wightman PCT, Spin and Statistics, and All That (1964); for the modern statement see Weinberg Vol. 1 §5.8.

7. Arrival at the QFT Postulates

We can now read off the QFT postulates as theorems / definitions / postulates of the modern derivation:

Two postulates remain genuinely additional in the modern story:

• P9 is more of a definitional convention — the Lagrangian formalism is a practical way to specify a Lorentz-invariant interaction, but other formalisms (Hamiltonian, path integral, S-matrix bootstrap) would also work.
• P10 is a real assumption about the dynamics: that the in/out spaces exist and equal $\mathcal{H}$. Examples (confinement in QCD) where individual asymptotic-particle interpretations break down show this is not automatic.

So the modern story compresses 10 postulates into 3 (M1, M2, M3) + 1 dynamical assumption (P10) + empirical inputs (field content, masses, couplings).

7.1 Reverse mapping: where do §1's non-M assumptions live in the QFT postulates?

The §1 derivation invokes three inputs beyond the core M1+M2+M3 (see §1.6 assumption budget). Tracing each back into QFT/postulates.md shows where they are explicit, where they are absorbed into other postulates, and where they are silently assumed:

So the 10 P-postulates are not literally a complete restatement of the modern derivation: variable particle number in particular is an unflagged empirical input in postulates.md. For a corresponding note on the postulates side, see QFT/postulates.md § Implicit Empirical Inputs.

8. Comparison with Other Routes

All three routes converge on the same physical content — the Wightman postulates and the Lagrangians of the Standard Model — but they organize the inputs differently. The modern route is the most economical (fewest primitive postulates) but requires the most mathematical machinery; the historical route is most intuitive but forces gauge invariance to be discovered later as a "lucky accident"; the Wightman axioms are the most rigorous but presuppose what other routes try to motivate.

9. What Comes Next

Given the postulates of §7, the next questions are:

• Specialize to a specific theory. Pick field content, internal symmetry, renormalizable Lagrangian. Worked examples:
  • QED — $U(1)$ with Dirac matter.
  • QCD — $SU(3{)}_{\text{color}}$ with quark matter.
  • Electroweak — $SU(2{)}_L\times U(1{)}_Y$ with chiral fermions, broken to $U(1{)}_{\text{EM}}$ by the Higgs mechanism.
  • Standard Model — the union $SU(3{)}_C\times SU(2{)}_L\times U(1{)}_Y$ with cross-sector content (anomaly cancellation, generations + GIM, CP problems).
  • GUTs, supersymmetric extensions, effective theories (forthcoming).
• Extract observables. S-matrix elements via LSZ + Feynman rules; cross sections via cross-sections.md, decay rates via decay-rates.md; the broader observable inventory is in observables.md.
• Address foundational caveats. Haag's theorem, rigorous construction in $d=4$, measurement in QFT — see QFT/remarks.md.
• Move beyond. EFT / Wilsonian framing; QFT as the universal IR description of relativistic systems; possible UV completion (string theory, asymptotic safety).