Definitions and Preliminaries

Familiarity with non-relativistic quantum mechanics (Hilbert spaces, Hermitian operators, the Born rule, etc.) is assumed — see the QM notes. This page collects the additional structures specific to QFT.

Minkowski Spacetime

QFT is formulated on Minkowski spacetime ${\mathbb{M}}^4=({\mathbb{R}}^4,\eta )$ with metric

$${\eta }_{\mu \nu }=\mathrm{diag}(+1,-1,-1,-1).$$

A spacetime point is denoted $x=(x^0,\mathbf{x})=(t,\mathbf{x})$ (with $c=1$). The invariant interval $x^2={\eta }_{\mu \nu }{x}^{\mu }{x}^{\nu }$ classifies separations as timelike ($x^2>0$), lightlike ($x^2=0$), or spacelike ($x^2<0$). Greek indices run over $0,1,2,3$ and are raised/lowered with $\eta$; repeated indices are summed (Einstein convention).

Group Theory Prerequisites

The Lorentz, Poincaré, and internal symmetry groups (e.g. $U(1)$, $SU(2)$, $SU(3)$) used throughout QFT are all instances of Lie groups with associated Lie algebras and representations. The abstract definitions — group axioms, subgroups and quotients, homomorphisms, representations (unitary, irreducible, projective), the standard matrix groups, Lie groups and Lie algebras (generators, structure constants, Casimirs), direct and semidirect products, compactness, connectedness and universal covers, and worked examples ($SO(2)$ and $SU(2)$) — are collected separately in math/group-theory.md. The rest of this page assumes that material as background.

Lorentz and Poincaré Groups

The Lorentz and Poincaré groups are the symmetry groups of special relativity, and are the geometric input of every relativistic quantum theory. They are 6- and 10-dimensional Lie groups respectively; this section catalogues their components, generators, algebra, representations, and the Casimir invariants used to label particles.

The Lorentz group

The Lorentz group $O(1,3)$ consists of linear transformations $\Lambda :{\mathbb{M}}^4\to {\mathbb{M}}^4$ preserving the metric:

$${\Lambda }^{\mu }{}_{\rho }{\Lambda }^{\nu }{}_{\sigma }{\eta }_{\mu \nu }={\eta }_{\rho \sigma }⟺{\Lambda }^T\eta \Lambda =\eta .$$

It is a 6-dimensional non-compact Lie group with four disconnected components, distinguished by two ${\mathbb{Z}}_2$-valued discrete invariants:

• $\det \Lambda =\pm 1$ — proper ($+1$) vs. improper ($-1$).
• $\mathrm{sign}({\Lambda }^0{}_0)=\pm 1$ — orthochronous ($+1$, preserves the direction of time) vs. non-orthochronous ($-1$).

The four components are connected to one another by the discrete operations:

The proper orthochronous Lorentz group ${\mathcal{L}}_{+}^{\uparrow }$ is the identity-component subgroup; the discrete symmetries $P$ and $T$ are separate (and may or may not be symmetries of a given physical theory — the weak interaction violates both).

The Poincaré group

Spacetime translations $x^{\mu }\to x^{\mu }+a^{\mu }$ commute with each other but not with Lorentz transformations: combining a Lorentz transformation followed by a translation gives $x^{\mu }\to {\Lambda }^{\mu }{}_{\nu }{x}^{\nu }+a^{\mu }$, and the order matters. This is captured by the semidirect product structure:

$$\mathcal{P}={\mathbb{R}}^{1,3}⋊\mathcal{L},$$

with group multiplication

$$({\Lambda }_1,a_1)\cdot ({\Lambda }_2,a_2)=({\Lambda }_1{\Lambda }_2,a_1+{\Lambda }_1{a}_2).$$

The four-dimensional translation subgroup ${\mathbb{R}}^{1,3}$ is normal in $\mathcal{P}$; the Lorentz subgroup $\mathcal{L}$ is not (boosting and translating do not commute). $\mathcal{P}$ is 10-dimensional (6 Lorentz + 4 translations) and has the same four-component structure as the Lorentz group.

The proper orthochronous Poincaré group ${\mathcal{P}}_{+}^{\uparrow }={\mathbb{R}}^{1,3}⋊{\mathcal{L}}_{+}^{\uparrow }$ is the connected identity component. It is the symmetry group of relativistic physics (excluding the discrete $C,P,T$ operations, which are treated separately).

Lie algebra and generators

The Lie algebra $\mathfrak{p}$ of ${\mathcal{P}}_{+}^{\uparrow }$ has 10 generators:

• 4 translation generators $P^{\mu }$ (energy–momentum), $\mu =0,1,2,3$.
• 6 Lorentz generators $J^{\mu \nu }=-J^{\nu \mu }$, conventionally split into:
  • 3 rotation generators $J^i\equiv \frac{1}{2}{ϵ}^{ijk}{J}^{jk}$ (angular momentum),
  • 3 boost generators $K^i\equiv J^{0i}$.

The defining commutation relations are

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

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

In rotation/boost form these read:

$$[J^i,J^j]=i{ϵ}^{ijk}{J}^k,[J^i,K^j]=i{ϵ}^{ijk}{K}^k,[K^i,K^j]=-i{ϵ}^{ijk}{J}^k,[P^0,J^i]=0,[P^0,K^i]=i{P}^i.$$

The combinations $J_{\pm }^i=\frac{1}{2}(J^i\pm i{K}^i)$ satisfy two independent $\mathfrak{su}(2)$ algebras:

$$[J_{+}^i,J_{+}^j]=i{ϵ}^{ijk}{J}_{+}^k,[J_{-}^i,J_{-}^j]=i{ϵ}^{ijk}{J}_{-}^k,[J_{+}^i,J_{-}^j]=0,$$

so the complexified Lorentz algebra factorizes as $\mathfrak{so}(1,3{)}_{\mathbb{C}}\cong \mathfrak{su}(2)\oplus \mathfrak{su}(2)$. Finite-dimensional representations are therefore labelled by two half-integers $(j_{+},j_{-})$ — see Representations below. (The $J_{\pm }^i$ are not Hermitian on their own, since $K^i$ is anti-Hermitian in unitary representations; the factorization is at the level of the complexified algebra.)

Universal cover $SL(2,\mathbb{C})$

The proper orthochronous Lorentz group ${\mathcal{L}}_{+}^{\uparrow }=S{O}^{+}(1,3)$ is doubly connected: there are loops in it (e.g. $2\pi$ rotations) that cannot be continuously contracted to a point, but all such loops contract after being traversed twice (a $4\pi$ rotation). Its universal cover is

$$SL(2,\mathbb{C})\stackrel{2:1}{\to }{\mathcal{L}}_{+}^{\uparrow },$$

where the kernel of the covering map is $\{\pm 1\}\subset SL(2,\mathbb{C})$. The Poincaré universal cover is correspondingly ${\mathbb{R}}^{1,3}⋊SL(2,\mathbb{C})$.

Why we care. Single-valued representations of $SL(2,\mathbb{C})$ are double-valued representations of $S{O}^{+}(1,3)$ — i.e. spinor representations. To accommodate spin-$\frac{1}{2}$ particles (electrons, quarks, neutrinos) the relevant covering group is $SL(2,\mathbb{C})$, not $S{O}^{+}(1,3)$. This is the deep reason a $2\pi$ rotation flips the sign of a spinor wavefunction: at the level of the fundamental physical group, "$2\pi$ rotation" is not the identity but $-1$.

Representations

Finite-dimensional irreducible representations of $SL(2,\mathbb{C})$ (= projective irreps of ${\mathcal{L}}_{+}^{\uparrow }$) are labelled by a pair $(j_{+},j_{-})$ with $j_{\pm }\in \frac{1}{2}{\mathbb{Z}}_{\ge 0}$, of dimension $(2j_{+}+1)(2j_{-}+1)$. The most-used cases:

The spin of a finite-dimensional representation is $j_{+}+j_{-}$; equivalently, under the rotation subgroup $SO(3)\subset {\mathcal{L}}_{+}^{\uparrow }$, the rep $(j_{+},j_{-})$ decomposes as $j_{+}\otimes j_{-}=|j_{+}-j_{-}|\oplus \cdots \oplus (j_{+}+j_{-})$.

These are the representations carried by classical fields and by field operators in QFT. They are all non-unitary (because $SL(2,\mathbb{C})$ is non-compact), which is why they describe field components, not Hilbert-space states. Unitary representations of the Poincaré group are infinite-dimensional and are labelled differently — see Wigner's Classification below.

Casimir invariants

The Poincaré algebra has two independent Casimir operators (commuting with all generators):

• $P^2\equiv P^{\mu }{P}_{\mu }$ — the mass-squared invariant. On any irreducible unitary representation it acts as a scalar $m^2\ge 0$.
• $W^2\equiv W^{\mu }{W}_{\mu }$, where $W^{\mu }\equiv -\frac{1}{2}{ϵ}^{\mu \nu \rho \sigma }{J}_{\nu \rho }{P}_{\sigma }$ is the Pauli–Lubanski vector. On an irrep with $m>0$ it acts as $-m^{2}s(s+1)$ with $s$ the spin; on a massless irrep its eigenvalues are different (see Wigner classification below).

These two scalars are the only Poincaré-invariant labels of single-particle states, and they are the basis of Wigner's classification.

Little groups

For each non-zero momentum $p^{\mu }$, the little group $G_p\subset {\mathcal{L}}_{+}^{\uparrow }$ is the subgroup of Lorentz transformations leaving $p^{\mu }$ fixed. Wigner's strategy is to classify states first at a standard momentum (a representative of each Lorentz orbit) by their little-group transformation, then propagate to all other momenta by Lorentz boost.

The little group structure is what produces the discrete spin/helicity quantum numbers attached to single-particle states.

Why this matters

Three things in QFT all trace back to the Poincaré group structure laid out above:

1. Wigner classification of single-particle states (next subsection) — uses the Casimirs and little groups.
2. Field representations $(j_{+},j_{-})$ — the table above lists which classical/operator-valued field types are available, used in QFT Postulate 5 and in specifying field content for any specific theory (QED, QCD, ...).
3. Spin–statistics, $CPT$, gauge invariance for massless spin-1 — all derive from the Poincaré structure plus locality / cluster decomposition in the modern story (foundations-modern.md).

Wigner's Classification of Particles

A particle is identified with an irreducible unitary representation of the Poincaré group on a complex separable Hilbert space. Wigner's theorem (1939) classifies these irreps. The result is that every physical irrep is labelled by two Casimirs (mass and spin/helicity); the structure of the classification follows from Mackey's induced-representation theorem applied to the semidirect-product structure ${\mathcal{P}}_{+}^{\uparrow }={\mathbb{R}}^{1,3}⋊SL(2,\mathbb{C})$.

Statement

The irreducible unitary representations of ${\mathcal{P}}_{+}^{\uparrow }$ (or, more precisely, of its universal cover) are classified by:

• Mass-squared $P^2=m^2$ with $m\ge 0$, the eigenvalue of the first Casimir.
• A choice of irrep of the little group of a standard momentum on the corresponding mass shell:
  • $m>0$: little group $SO(3)$, irreps labelled by spin $j\in \{0,\frac{1}{2},1,\frac{3}{2},\dots \}$.
  • $m=0$: little group $ISO(2)$, irreps labelled by helicity $h\in \frac{1}{2}\mathbb{Z}$ (with helicity quantized in integer/half-integer units by $4\pi$ rotation closure of the universal cover; "continuous-spin" reps are mathematically allowed but empirically absent).
  • $m^2<0$ (tachyons): little group $SO(1,2)$ — unphysical, excluded by the spectrum condition (P2).
  • $p^{\mu }=0$: trivial rep — the vacuum.

Sketch of derivation (Mackey induction)

The classification proceeds in five steps. Each is mechanical given the Lie-algebra and topology data already in § Lorentz and Poincaré Groups; we sketch the logic and defer technical proofs to references.

Step 1 — Diagonalize translations. ${\mathbb{R}}^{1,3}\subset {\mathcal{P}}_{+}^{\uparrow }$ is abelian and normal; on any unitary representation it can be simultaneously diagonalized, with eigenvalues labelled by a four-momentum $p^{\mu }$ (the spectrum of $P^{\mu }$). So the Hilbert space decomposes as a direct integral

$$\mathcal{H}={\int }^{\oplus }d\mu (p){\mathcal{H}}_p,P^{\mu }|p,\sigma ⟩=p^{\mu }|p,\sigma ⟩,$$

with $\sigma$ labelling extra degrees of freedom at each $p$.

Step 2 — Lorentz orbits. Lorentz transformations move $p$ around. Irreducibility forces the support of $\mu$ to be a single Lorentz orbit $\mathcal{O}\subset {\mathbb{R}}^{1,3}$ (otherwise the representation would split into pieces supported on different orbits). The orbits are:

(Backward-pointing orbits with $p^0<0$ are excluded by the spectrum condition.)

Step 3 — Pick a standard momentum and identify the little group. For each orbit $\mathcal{O}$, pick a representative $\bar{p}$ and define the little group

$$G_{\bar{p}}=\{\Lambda \in SL(2,\mathbb{C}):\Lambda \bar{p}=\bar{p}\}$$

— the Lorentz transformations that fix $\bar{p}$. The little groups for each orbit (computed by direct algebra):

The relevant little group is the universal cover (since we want projective reps of ${\mathcal{P}}_{+}^{\uparrow }$, ordinary reps of its universal cover; see math/group-theory.md § Connectedness and discrete components).

Step 4 — Mackey induction: irreps of ${\mathcal{P}}_{+}^{\uparrow }$ ↔ irreps of $G_{\bar{p}}$. Mackey's induced-representation theorem (a general result for semidirect products $N⋊G$ with $N$ abelian) states:

The irreducible unitary representations of ${\mathcal{P}}_{+}^{\uparrow }$ supported on a Lorentz orbit $\mathcal{O}$ are in bijective correspondence with the irreducible unitary representations of the little group $G_{\bar{p}}$ of any standard momentum $\bar{p}\in \mathcal{O}$.

Concretely, given an irrep $\sigma$ of $G_{\bar{p}}$, the induced rep ${\mathrm{Ind}}_{G_{\bar{p}}}^{{\mathcal{P}}_{+}^{\uparrow }}(\sigma )$ is built on the Hilbert space $L^2(\mathcal{O})\otimes V_{\sigma }$. A boost $\Lambda$ acts by $\Lambda :(p,v)\mapsto (\Lambda p,W(\Lambda ,p)v)$, where the Wigner rotation $W(\Lambda ,p)\in G_{\bar{p}}$ is the little-group-valued rotation that compensates for the $p$-dependent boost (see Weinberg Vol. 1 §2.5 for the explicit construction). Irreducibility on the ${\mathcal{P}}_{+}^{\uparrow }$ side ⇔ irreducibility of $\sigma$ on the $G_{\bar{p}}$ side.

Step 5 — Classify irreps of each little group.

• $G_{\bar{p}}=SU(2)$ (massive): irreps are the spin-$j$ reps with $j\in \{0,\frac{1}{2},1,\frac{3}{2},\dots \}$ of dimension $2j+1$. (Standard $SU(2)$ representation theory; see math/group-theory.md § Worked examples: $SO(2)$ and $SU(2)$.)
• Universal cover of $ISO(2)$ (massless): the abelian subgroup ${\mathbb{R}}^2$ has unitary characters labelled by a vector $(a,b)\in {\mathbb{R}}^2$. Two cases:
  • $(a,b)=0$: trivial action, leaving the rotation generator to label irreps by helicity $h$. For the cover, $h$ is quantized in $\frac{1}{2}\mathbb{Z}$.
  • $(a,b)\ne 0$: the continuous-spin representations, parameterized by $\rho =\sqrt{a^2+b^2}>0$. Allowed mathematically, not observed in nature (their absence is empirical, sometimes posed as an extra "Wigner condition").
• $SU(1,1)$ (tachyonic): irreps exist but tachyonic states violate causality — excluded by the spectrum condition (P2).

Conclusion

Combining Steps 1–5: every irreducible unitary representation of ${\mathcal{P}}_{+}^{\uparrow }$ supported on a forward orbit (massive or massless, $h$ discrete) is labelled by

• mass $m\ge 0$, and
• spin $j\in \frac{1}{2}{\mathbb{Z}}_{\ge 0}$ (massive) or helicity $h\in \frac{1}{2}\mathbb{Z}$ (massless).

This is Wigner's classification. The labels are exactly the two Casimirs $P^2=m^2$ and (for $m>0$) $W^2=-m^{2}j(j+1)$ where $W^{\mu }$ is the Pauli–Lubanski vector. The full proof — including the technical analysis of induced representations, projective representations, and continuous-spin exclusion — is laid out in:

• Weinberg, The Quantum Theory of Fields, Vol. 1, Ch. 2 (the standard physics treatment).
• Streater & Wightman, PCT, Spin and Statistics, and All That, Ch. 1 (axiomatic version).
• Tung, Group Theory in Physics, Chs. 9–10 (representation-theoretic emphasis).
• Bargmann–Wigner (1948), the original induced-representation construction.

Classical Fields and Lagrangians

A classical field is a function $\varphi :{\mathbb{M}}^4\to V$ taking values in some target space $V$ carrying a representation of the Lorentz group:

• Scalar field $\varphi (x)$: trivial representation, e.g. the Higgs field.
• Spinor field ${\psi }_{\alpha }(x)$: spin-$\frac{1}{2}$ representation of $SL(2,\mathbb{C})$, e.g. the Dirac field.
• Vector field $A_{\mu }(x)$: the four-vector representation, e.g. the electromagnetic potential.
• Tensor / spinor-tensor fields: higher-spin generalizations.

Dynamics are encoded in a Lagrangian density $\mathcal{L}(\varphi ,{\partial }_{\mu }\varphi )$, a Lorentz scalar, with action $S[\varphi ]=\int d^{4}x\mathcal{L}$. The classical equations of motion follow from the Euler–Lagrange equations

$$\frac{\partial \mathcal{L}}{\partial \varphi }-{\partial }_{\mu }\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }\varphi )}=0.$$

Worked example: free real scalar field and the Klein–Gordon equation

The simplest non-trivial Lagrangian field theory is a single real scalar $\varphi (x)$ with the free scalar Lagrangian

$${\mathcal{L}}_{\varphi }=\frac{1}{2}{\partial }_{\mu }\varphi {\partial }^{\mu }\varphi -\frac{1}{2}{m}^2{\varphi }^2.$$

This is essentially uniquely fixed by demanding: Lorentz invariance, polynomial in $\varphi$ and $\partial \varphi$, at most two derivatives (for second-order field equations), reflection symmetry $\varphi \to -\varphi$, and a kinetic term with conventional sign and normalization. Applying the Euler–Lagrange equation gives

$${\partial }_{\mu }\frac{\partial {\mathcal{L}}_{\varphi }}{\partial ({\partial }_{\mu }\varphi )}-\frac{\partial {\mathcal{L}}_{\varphi }}{\partial \varphi }={\partial }_{\mu }{\partial }^{\mu }\varphi +m^2\varphi =0,$$

i.e. the Klein–Gordon equation

$$\boxed{(\square +m^2)\varphi (x)=0,\square \equiv {\partial }^{\mu }{\partial }_{\mu }.}$$

In this Lagrangian route the Klein–Gordon equation is derived, not postulated; the postulate has moved from "the equation" to "the Lagrangian ${\mathcal{L}}_{\varphi }$".

Three routes to Klein–Gordon

For completeness, the Klein–Gordon equation can be reached three ways, with the genuine input shifting in each:

All three give the same equation. The physical input differs: a substitution rule (A), a choice of Lagrangian (B), or the Casimir-eigenvalue definition of a massive spin-0 species (C). Routes B and C are the modern viewpoint; Route A survives only as a heuristic motivating Dirac's first-order ansatz in QED-historical §1.1.

As a field operator

After quantization (whether canonical or in the Wigner-construction sense), $\varphi (x)$ is promoted to an operator-valued distribution $\hat{\varphi }(x)$ on Fock space, still satisfying $(\square +m^2)\hat{\varphi }=0$ as an operator equation. Its mode expansion and ladder structure are given below in § Fock Space; its propagator $⟨0|T\hat{\varphi }(x)\hat{\varphi }(y)|0⟩$ enters Feynman-rule calculations; the Klein–Gordon operator $(\square +m^2)$ reappears as the external-leg amputation operator in § LSZ Reduction Formula.

The free Dirac field is the spin-$\frac{1}{2}$ analogue (postulating ${\mathcal{L}}_{\psi }=\bar{\psi }(i{\gamma }^{\mu }{\partial }_{\mu }-m)\psi$ instead, derived in QED/historical.md § 1.4); the free photon field is the massless spin-1 analogue, from ${\mathcal{L}}_{\text{EM}}$ in QED/historical.md § 0.2. Each is built by the same Route-B recipe: write down the simplest Lorentz-scalar Lagrangian in the appropriate field, derive the EOM, quantize.

Canonical Structure

The conjugate momentum to a field $\varphi (x)$ is

$$\pi (x)=\frac{\partial \mathcal{L}}{\partial ({\partial }_0\varphi (x))}.$$

The classical Hamiltonian density is $\mathcal{H}=\pi {\partial }_0\varphi -\mathcal{L}$.

Operator-Valued Distributions

In QFT, fields cannot be ordinary operator-valued functions of $x$ — products like $\varphi (x{)}^2$ are too singular. Instead, $\varphi (x)$ is an operator-valued tempered distribution: it is well-defined only after smearing against a Schwartz test function $f\in \mathcal{S}({\mathbb{M}}^4)$,

$$\varphi (f)=\int d^{4}xf(x)\varphi (x),$$

yielding an (unbounded) operator on a dense domain $\mathcal{D}\subset \mathcal{H}$.

Fock Space

For a free field of mass $m$ and spin $s$, the Hilbert space is the Fock space

$$\mathcal{F}({\mathcal{H}}_1)=\underset{n=0}{\overset{\infty }{⨁}}{\mathcal{H}}_1^{{\otimes }_{s}n}\text{(bosonic, symmetrized)},$$

or with antisymmetrization ${\otimes }_a$ for fermions. Here ${\mathcal{H}}_1$ is the one-particle Hilbert space (an irreducible Wigner representation). Fock space is built from a vacuum $|0⟩$ via creation ($a^{†}$) and annihilation ($a$) operators satisfying canonical (anti)commutation relations:

$$[a_{\mathbf{p}},a_{\mathbf{q}}^{†}]=(2\pi {)}^3{\delta }^3(\mathbf{p}-\mathbf{q})\text{(bosons)},$$ $$\{a_{\mathbf{p}},a_{\mathbf{q}}^{†}\}=(2\pi {)}^3{\delta }^3(\mathbf{p}-\mathbf{q})\text{(fermions)}.$$

Terminology — ladder operators. The pair $(a,a^{†})$ is collectively called ladder operators (or raising/lowering operators): $a^{†}$ raises the particle number by one (${\mathcal{H}}_n\to {\mathcal{H}}_{n+1}$), $a$ lowers it (${\mathcal{H}}_n\to {\mathcal{H}}_{n-1}$, with $a|0⟩=0$). The name comes from the harmonic oscillator in QM (see QM/heisenberg-picture.md), where the same $\hat{a},{\hat{a}}^{†}$ algebra moves between energy eigenstates $|n⟩$ on the "ladder" of equally spaced levels. The QFT usage is the same algebra applied per Fourier mode: each momentum mode $\mathbf{p}$ of a free field is an independent harmonic oscillator, and $a_{\mathbf{p}}^{†},a_{\mathbf{p}}$ are its ladder operators. Particle number is the count of excitations across all modes. In QFT-specific contexts "creation/annihilation operators" is more common than "ladder operators", but the terms are interchangeable.

A free scalar field admits the mode expansion

$$\varphi (x)=\int \frac{d^{3}p}{(2\pi {)}^3}\frac{1}{\sqrt{2{\omega }_{\mathbf{p}}}}(a_{\mathbf{p}}{e}^{-ip\cdot x}+a_{\mathbf{p}}^{†}{e}^{+ip\cdot x}),{\omega }_{\mathbf{p}}=\sqrt{{\mathbf{p}}^2+m^2}.$$

In an interacting theory, no such Fock space exists for the full interacting field (this is the content of Haag's theorem — see Remarks), but Fock spaces remain the appropriate description for asymptotic in/out states.

Vacuum

The vacuum $|0⟩$ is the lowest-energy, Poincaré-invariant state. For a free theory it is annihilated by all $a_{\mathbf{p}}$. In an interacting theory the physical vacuum differs from the free (Fock) vacuum and is in general unitarily inequivalent to it.

States vs. Fields: Why QFT Looks "Operator-Heavy"

A reader coming from non-relativistic QM may notice that QFT seems to focus almost entirely on operators (the field operators $\varphi (x)$, $\psi (x)$, $A_{\mu }(x)$ and their correlators) while saying very little about states. This is real — and worth being explicit about.

The state is still primary in principle, but in practice it is usually fixed implicitly to be one of:

• the vacuum $|0⟩$ — for vacuum correlation functions and most perturbative computations,
• an asymptotic Fock state $|p_1,s_1;p_2,s_2;\dots ⟩$ — for S-matrix calculations,
• a coherent state of a bosonic field — for connecting to classical fields and for IR problems,
• a bound-state wavefunction (positronium, hydrogen) — handled non-perturbatively (Bethe–Salpeter), and rarely written down explicitly,
• a density matrix — for thermal QFT, decoherence, and open systems.

Several reasons drive the operator-centric emphasis:

1. Heisenberg picture is manifestly Lorentz-covariant. Putting all spacetime dependence into operators $\varphi (x)=\varphi (\mathbf{x},t)$ avoids singling out the time slice that the Schrödinger picture requires.
2. The state space is Fock space, not $L^2({\mathbb{R}}^3)$. A multi-particle state is a function over arbitrary numbers of particles with arbitrary momenta; there is no useful single "wavefunction in position space" to write down.
3. Most observables of interest are scattering amplitudes. Prepare an asymptotic in-state, evolve, take the overlap with an asymptotic out-state — the details of the state during the interaction never appear; only $⟨\text{out}|\hat{S}|\text{in}⟩$ does, computed from operator correlators via LSZ.
4. Haag's theorem (see Remarks) says the interacting vacuum and Fock states are unitarily inequivalent to the free ones — there is no concrete Hilbert space on which to write down "the interacting state." So one works with operator correlators and asymptotic states only.

Warning: the Field Operator $\psi (x)$ Is Not a Wavefunction

This is a major terminological collision. In some derivations of QED (notably the historical / Dirac-equation route), $\psi (x)$ starts out as a single-particle relativistic wavefunction — a state. After second quantization, the symbol $\psi (x)$ is reused to denote a field operator on Fock space. From that point on:

• $\psi (x)$ is an operator, not a state.
• $\psi (x)|0⟩$ is the (improper) state of one particle localized at $x$.
• The matrix element $⟨0|\psi (x)|p,s⟩=u(p,s)e^{-ip\cdot x}$ is a Dirac wavefunction, but it's the matrix element of the field operator between two particular states — not the state itself.

Conflating these two meanings of $\psi (x)$ is one of the most common sources of confusion when crossing from QM to QFT.

Time-Ordering Operator

The time-ordering operator $T$ is the instruction to permute a product of operators so that those with larger time arguments stand to the left:

$$T[A_1(t_1)A_2(t_2)\cdots A_n(t_n)]=A_{\sigma (1)}(t_{\sigma (1)})A_{\sigma (2)}(t_{\sigma (2)})\cdots A_{\sigma (n)}(t_{\sigma (n)}),$$

where $\sigma$ is the permutation that gives $t_{\sigma (1)}>t_{\sigma (2)}>\cdots >t_{\sigma (n)}$. For the two-operator case:

$$T[A(t_1)B(t_2)]=\theta (t_1-t_2)A(t_1)B(t_2)+\theta (t_2-t_1)B(t_2)A(t_1).$$

For fermionic operators an extra sign $(-1{)}^{\sigma }$ from the permutation is included (so $T[\psi (t_1)\bar{\psi }(t_2)]=-T[\bar{\psi }(t_2)\psi (t_1)]$).

$T$ is not an operator on Fock space — it is a notational rule for handling non-commuting operators at different times. It plays a central role in:

• The Dyson series for the S-matrix, $S=T\exp (-i\int H_{\text{int}}dt)$ (see QED/historical.md §0.6).
• Time-ordered correlators / Green's functions (next subsection), which are the inputs to the LSZ reduction formula.
• Path-integral derivations of operator correlators (see QM/path-integral.md §1.6 for the full discussion in the simpler QM setting).

Notation collision warning. The same symbol $T$ is used in QFT for the transition operator $T=-i(S-1)$ — a genuine operator on Fock space — that appears in the splitting $S=1+iT$. The two meanings are entirely unrelated; context disambiguates: time-ordering $T$ always sits in front of a product of operators ($T[\cdots ]$ or $T{\varphi }_1\cdots {\varphi }_n$); transition $T$ sits between a bra and a ket as $⟨f|T|i⟩$. See QED/historical.md §0.6 for the corresponding callout in context.

Correlation Functions

The fundamental observables of QFT are vacuum expectation values of products of fields:

• Wightman functions: $W_n(x_1,\dots ,x_n)=⟨0|\varphi (x_1)\cdots \varphi (x_n)|0⟩$.
• Time-ordered (Green's) functions: $G_n(x_1,\dots ,x_n)=⟨0|T\varphi (x_1)\cdots \varphi (x_n)|0⟩$, where $T$ orders fields by decreasing $x^0$ (with a sign for fermion exchanges).

Time-ordered correlators are what enter the LSZ reduction formula (next subsection) to compute $S$-matrix elements.

LSZ Reduction Formula

The Lehmann–Symanzik–Zimmermann (LSZ) reduction formula is the bridge between time-ordered correlators of local fields (what perturbation theory and Feynman rules naturally compute) and S-matrix elements between asymptotic states (what observables require).

Setup. Pick any local field $\varphi (x)$ with non-zero matrix element to a one-particle state of the species of interest:

$$⟨0|\varphi (x)|p⟩\ne 0.$$

Such a $\varphi$ is called an interpolating field for that species. Different choices of $\varphi$ give the same on-shell $S$-matrix elements (this is the LSZ equivalence of field redefinitions).

The formula for $n+m$ external particles (in the simplest scalar case; spin and species labels suppressed):

$$⟨p_1,\dots ,p_n;\text{out}|q_1,\dots ,q_m;\text{in}⟩=\prod _{i=1}^n\left[i\int d^4{x}_i{e}^{i{p}_i\cdot x_i}({\square }_{x_i}+m^2)\right]\prod _{j=1}^m\left[i\int d^4{y}_j{e}^{-i{q}_j\cdot y_j}({\square }_{y_j}+m^2)\right]⟨0|T\varphi (x_1)\cdots \varphi (x_n)\varphi (y_1)\cdots \varphi (y_m)|0⟩.$$

In words: each external leg contributes a Klein–Gordon operator $(\square +m^2)$ (which on the mass shell extracts the residue of the single-particle pole in momentum space) plus a Fourier factor; everything else is the time-ordered $(n+m)$-point correlator.

What it actually says. When the external momenta are on-shell ($p_i^2=m^2$, $p_i^0>0$), the time-ordered correlator $⟨0|T\varphi \cdots \varphi |0⟩$ has poles in each external momentum at $p^2=m^2$ from the propagation of single-particle intermediate states (the Källén–Lehmann spectral representation). The LSZ recipe extracts the residue at all those poles simultaneously and identifies it with the on-shell $S$-matrix element.

Why this is significant.

• No Hamiltonian required. LSZ takes correlators as input. Correlators can be computed from a Lagrangian via path integrals, but they could equally come from lattice simulations, the conformal bootstrap, integrability, or any other source. So LSZ provides an $H$-free route to $S$-matrix elements (compare the Møller-operator construction in foundations-modern.md §2.0.1, which does require $H=H_0+H_{\text{int}}$).
• Field-redefinition equivalence. Replacing $\varphi \to \varphi +g[\varphi ]$ for any local function $g$ leaves on-shell $S$-matrix elements invariant. This is why effective field theories with different operator bases can describe identical physics, and why the "fundamental field" choice is largely conventional.
• Bridges Feynman rules to observables. Step 6 of any QFT calculation (in QED, QED/historical.md §5.2, and elsewhere) implicitly uses LSZ to convert amputated time-ordered diagrams into $S$-matrix elements: the $(\square +m^2)$ operators on the external legs cancel against the external propagators of an amputated diagram, leaving exactly $i\mathcal{M}$ (up to wavefunction renormalization factors $\sqrt{Z}$).

For the full derivation see Peskin–Schroeder Ch. 7.2 or Weinberg Vol. 1 §10.3. For its role in pinning down asymptotic states without invoking $H$, see foundations-modern.md §2.0.1.

S-Matrix and Cross Sections

The S-matrix $S$ maps asymptotic in-states (free particles in the far past) to asymptotic out-states (free particles in the far future):

$$⟨f,\text{out}|i,\text{in}⟩=⟨f|S|i⟩.$$

Writing $S=1+iT$ with $⟨f|T|i⟩=(2\pi {)}^4{\delta }^4(p_f-p_i){\mathcal{M}}_{fi}$, the invariant amplitude $\mathcal{M}$ determines physical observables: differential cross sections $d\sigma$, decay rates $d\Gamma$, etc. The full master formulas connecting $\mathcal{M}$ to measurable rates and the necessary kinematic ingredients (flux factor, Lorentz-invariant phase space, units, the optical theorem) are collected separately in Cross Sections and Decay Rates; the broader observable inventory is in Observables.

Symmetries and Noether Currents

A continuous symmetry of the action implies, by Noether's theorem, the existence of a conserved current $j^{\mu }(x)$ with ${\partial }_{\mu }{j}^{\mu }=0$ and a conserved charge $Q=\int d^{3}x{j}^0$. In the quantum theory, $Q$ generates the symmetry on operators via $[Q,\varphi ]=i\delta \varphi$.

Symmetries are classified as:

• Global (parameter independent of $x$) vs. local / gauge (parameter depends on $x$).
• Internal (acting on field indices) vs. spacetime (Poincaré, conformal, ...).
• Continuous (Lie group) vs. discrete ($C$, $P$, $T$).

Gauge Fields

A gauge theory has a local internal symmetry $G$. To make the Lagrangian invariant one introduces a gauge field $A_{\mu }^a$ valued in the Lie algebra of $G$, and a covariant derivative $D_{\mu }={\partial }_{\mu }-ig{A}_{\mu }^a{T}^a$, where $T^a$ are generators of $G$ in the relevant representation. The field strength

$$F_{\mu \nu }^a={\partial }_{\mu }{A}_{\nu }^a-{\partial }_{\nu }{A}_{\mu }^a+g{f}^{abc}{A}_{\mu }^b{A}_{\nu }^c$$

(with structure constants $f^{abc}$) generalizes the electromagnetic $F_{\mu \nu }$. Quantization requires gauge fixing to remove redundant degrees of freedom.

Regularization and Renormalization

Naive computations in interacting QFT yield divergent loop integrals. Regularization (cutoff, dimensional, Pauli–Villars, lattice) parametrizes the divergences; renormalization absorbs them into a redefinition of a finite number of parameters (masses, couplings, field normalizations). A theory is renormalizable if this can be done with finitely many counterterms; otherwise it is an effective field theory valid only below some energy scale.

The renormalization group describes how renormalized parameters depend on the chosen energy scale $\mu$, governed by beta functions $\beta (g)=\mu \partial g/\partial \mu$.