Quantum Fields, States, and Observables

A quantum field theory is not a field operator, a Lagrangian, or a Hilbert space taken by itself. It is a compatible structure of observables, states, dynamics, spacetime structure and covariance properties, and locality, together with a controlled prescription for the quantities one wants to calculate. A formulation—canonical operators, functional integrals, Euclidean correlation functions, or local observable algebras—is a way to present some or all of that structure. This chapter separates the objects that every later formulation must keep straight.

The quickest diagnostic is to ask what kind of statement is being made. In continuum QFT, a field is modeled as an operator-valued distribution that may create particle excitations; a particle is a state-sector concept tied to symmetry and spectral structure; an observable is a physically admissible quantity; a state assigns expectation values; and a representation realizes an abstract algebra as operators. These objects are related, but none is automatically identical to another.

The objects and the relations among them

Let $\mathcal A$ denote an abstract unital $*$-algebra of observables. In the algebraic language a state is a normalized positive linear functional,

$$\begin{aligned} \omega&:\mathcal A\longrightarrow\mathbb C,\\ \omega(\mathbf 1)&=1,\\ \omega(A^*A)&\geq 0. \end{aligned}$$

A representation $\pi$ maps $\mathcal A$ to operators on a Hilbert space $\mathcal H_\pi$. When a cyclic vector $\Omega_\omega$ represents the state, expectation values take the familiar form

$$\omega(A) = \langle\Omega_\omega,\pi_\omega(A)\Omega_\omega\rangle.$$

This is a preview of the GNS relation, not a proof of the GNS theorem. Its purpose here is to show why “the state” and “the Hilbert-space representation” answer different questions: $\omega$ specifies expectation values on the algebra, while $\pi_\omega$ gives one concrete operator realization. In a field theory, physically relevant representations need not all be unitarily equivalent. The algebraic definitions, their Hilbert-space realization, and the scope of this inequivalence are developed carefully in Fewster and Rejzner 2020, §§ 2.1–2.3, pp. 4–9.

Object	What it specifies	A test that identifies it	What it must not be confused with
Theory	Observables, allowed states, dynamics, symmetries, locality, and any regulator or limiting prescription needed to define them	Can two formulations be matched on a declared class of observables and states?	A bare action or one formal integral
Formulation	A presentation and calculational language for the theory	Which data are primitive, and which equivalence statement has actually been established?	The physical theory independently of assumptions and limits
Observable algebra	Algebraic relations among physically admissible quantities	Are adjoints, products, commutators, symmetries, and localization specified?	The algebra of every auxiliary or gauge-fixed field
State	Expectation values and probabilities for observables	Does $\omega$ satisfy normalization and positivity?	A vector before a representation has been chosen
Representation	A realization of the algebra by operators on a Hilbert space, with domains where needed	Which algebra is represented, on which space and common domain?	The abstract algebra or the state itself
Vacuum	In Minkowski QFT, ordinarily a translation-invariant state compatible with the positive-energy spectrum condition	What invariance, spectral, purity, and uniqueness assumptions are imposed?	A universally unique empty state
Field	A spacetime-indexed operator-valued distribution, often transforming covariantly	What does the smeared field $\phi(f)$ mean, and how does it transform?	A particle, a pointwise ordinary operator, or necessarily an observable
Particle	Under suitable Minkowski and stability assumptions, a one-particle sector classified by a unitary Poincaré representation	What are the mass, spin or helicity, internal labels, and normalization?	The field used to create or detect that sector
Interpolating operator	An operator with nonzero overlap with a chosen state sector	Is a vacuum–one-particle matrix element nonzero in the chosen charge channel?	A unique physical field or an observable in every description
Locality condition	Compatibility of operations associated with spacelike-separated regions	Which ordinary or graded commutator vanishes, for which fields or observables?	A claim that every gauge-fixed variable is a local observable

The map below makes the direction and qualification of the main relations visible. Inspect especially the routes through algebra, state, and representation, and the two conditional routes out of a field.

Typed relations connect ten foundational QFT categories without identifying them. Solid routes mark defining or realization relations, while dashed routes mark a conditional presentation, admissibility test, or state-sector overlap; the map is schematic and not to scale.

Read without the graphic: a theory can admit a formulation and specifies its admissible algebra and states; a positive state on an algebra supports a GNS representation, while a vacuum is a specially qualified state. An observable belongs to the physical algebra, whereas a field becomes observable only when admissibility conditions hold and may interpolate a particle sector only through a nonzero overlap. No reverse implication or identity follows.

The relations are typed; a generic arrow would hide the conditions that make it valid.

Typed relation	Source and target	Qualification
contrasts with	A theory contrasts with any one formulation	Two formulations describe the same physics only after their observable, state, boundary, and limiting data have been matched
represents	$\pi$ represents $\mathcal A$ on $\mathcal H_\pi$	The map, Hilbert space, and operator domains are part of the statement
evaluates	$\omega$ evaluates elements of $\mathcal A$	Positivity and normalization make it a state
is smeared by	A field distribution is smeared by a test function $f$	The test-function class and support determine what localization and convergence claims mean
interpolates	A field interpolates a particle sector	A nonzero matrix element establishes overlap, not identity
is observable when	A field-derived quantity is observable when it meets the theory’s physical admissibility conditions	Gauge invariance, superselection, domains, and localization can matter
requires / continues in	One page requires another, or a bounded discussion continues in a later treatment	Hard prerequisites and later treatments are listed explicitly below

Reversing any one of these relations without extra hypotheses is a common source of false identifications.

Three entry routes

The chapter has no common hard prerequisite. Choose an entry route from the question you are trying to answer; then close the hard prerequisites before entering a dependent leaf.

Question	Start here	Exact hard closure	Observable exit
What data make a QFT more than a formal expression, and why are fields useful?	What Is a Quantum Field Theory?, then Why Local Relativistic Quantum Theory Uses Fields	None; the first page is only recommended before the second	State what a formulation preserves and give a qualified reason for using local fields
How are vacua, particles, multiparticle sectors, and antiparticles organized?	Vacua, States, and Representations, then One-Particle States	Neither entry has a hard prerequisite. Multiparticle States requires one-particle states. Antiparticles requires one-particle states and the interpolating-operator page	Normalize a one-particle wave packet, say when Fock organization applies, and distinguish a positive-energy antiparticle from a negative-frequency term
What does a local quantum field mean?	Quantum Fields as Operator-Valued Distributions	No hard prerequisite for smearing; Fields, Observables, and Interpolating Operators requires one-particle states; Spacelike Compatibility then requires the interpolating-operator page	Specify the test functions, field or observable class, and graded locality statement

Recommended preparation is diagnostic rather than blocking. If the state–representation distinction is opaque, review Banach and Hilbert Spaces, Completion, and Riesz Representation and Unbounded Operators, Domains, Closure, and Adjoints. For particle labels, use Lorentz Field Representations and Poincaré Particle Representations. Multiparticle organization can draw on Direct Sums, Tensor Products, and Index Structure and Graded Algebra, Grassmann Variables, and Berezin Integration. Interpolation uses the dual-space viewpoint in Vector Spaces, Duals, and Linear Maps. For smearing and locality, the repair routes are Test-Function Spaces, Distributions, Support, and Convergence, Convolution, Approximate Identities, and Poisson Summation, Locally Convex, Nuclear, and Rigged Hilbert Spaces, and Hyperbolic Equations and Causal Propagators.

Preparation diagnostic

Try this without looking ahead	Ready	Unsure	Repair
Given an algebra element $A$, explain why $\omega(A^*A)\geq0$ and an operator domain answer different questions	You distinguish a positive functional from the domain of its represented operator	You describe both as “normalizing the Hilbert space”	Review the Hilbert-space and unbounded-operator pages linked above
For $p^2=m^2$ and $p^0>0$, name the invariant measure and the massive one-particle labels	You give $\mathrm d^3\mathbf p/(2E_{\mathbf p})$ up to a declared $2\pi$ convention, plus mass and spin	You use $\mathrm d^3\mathbf p$ with no compensating normalization or call a Lorentz-field index the particle spin	Review the Lorentz and Poincaré representation page linked above
Build a two-identical-particle state and exchange its factors	You can state when the symmetric or antisymmetric projection is used	Tensor order, particle labels, and statistics are being conflated	Review the tensor-product and graded-algebra pages linked above
Replace $\phi(x)$ by a smeared expression and state what fixes its localization	You name the test-function class and support	The integral is treated as an ordinary pointwise operator average with no topology or support	Review the distributions, convolution, and nuclear-space pages linked above
Say which free-scalar kernel has causal support	You name the Pauli–Jordan commutator or the retarded or advanced kernel	You name the Feynman two-point function	Review the causal-propagator page linked above

Convention bridge

This chapter inherits the site conventions rather than creating a second convention system.

Datum	Convention used here	Invariant or consistency check
Spacetime	$\eta_{\mu\nu}=\operatorname{diag}(+1,-1,-1,-1)$ and $p\cdot x=p^0t-\mathbf p\cdot\mathbf x$	Positive-energy mass shell $p^2=m^2$, $p^0>0$
Units and modes	$\hbar=c=1$; $e^{-ip\cdot x}$ is the positive-frequency plane wave when $p^0>0$	The Hamiltonian assigns positive energy to creation-operator excitations
Particle states	The relativistic ket normalization and measure displayed below	A wave packet has a finite positive norm
State choice	Particle language presumes a declared positive-energy vacuum representation in Minkowski spacetime	Vacuum invariance and the spectrum condition are stated rather than inferred from notation
Fields	The test-function space is declared; distributional equality means equality after smearing	Support and convergence are checked on test functions
Brackets	Ordinary commutators for bosonic observables; graded commutators when fermionic fields are included	Spacelike compatibility names the field or observable class

A normalization test for the particle route

With the site convention $\eta_{\mu\nu}=\operatorname{diag}(+1,-1,-1,-1)$, a stable massive one-particle sector has $p^2=m^2$ and $p^0>0$. Its massive irreducible Poincaré labels can be written

$$P^2=m^2, \qquad W^2=-m^2s(s+1),$$

where $W^\mu$ is the Pauli–Lubanski vector. Massless sectors require helicity or a more general little-group analysis rather than this massive-spin formula. A common relativistic normalization is

$$\begin{aligned} \langle\mathbf p',\sigma'|\mathbf p,\sigma\rangle &=2E_{\mathbf p}(2\pi)^3\\ &\quad\times \delta^{(3)}(\mathbf p'-\mathbf p) \delta_{\sigma'\sigma}. \end{aligned}$$ $$\mathrm d\Pi_p = \frac{\mathrm d^3\mathbf p} {(2\pi)^3\,2E_{\mathbf p}}.$$

The delta-normalized kets are generalized states. A normalizable wave packet

$$|\psi\rangle = \sum_\sigma\int\mathrm d\Pi_p\, \psi_\sigma(\mathbf p)|\mathbf p,\sigma\rangle$$

has $\langle\psi|\psi\rangle=\sum_\sigma\int\mathrm d\Pi_p\,|\psi_\sigma(\mathbf p)|^2$ in this convention. This one calculation checks the measure, the state normalization, and every factor of $2E_{\mathbf p}$ at once. The Poincaré classification and normalization are derived in Weinberg 1995, § 2.5, pp. 62–73.

For free fields and for stable asymptotic sectors, identical-particle states are organized by symmetric or antisymmetric tensor powers,

$$\mathcal F_\pm(\mathcal H_1) = \bigoplus_{n=0}^{\infty} \mathcal S_\pm\mathcal H_1^{\otimes n}.$$

This does not assert that the exact Hilbert space of every interacting QFT is one universal free Fock space. It is the correct organization for the free models in this volume and, when scattering assumptions hold, for asymptotic particle states. Coleman’s progression from relativistic one-particle states through occupation numbers to the free scalar makes this scope visible in Coleman 2019, §§ 1.2–3.4, pp. 6–45.

Smearing, interpolation, and locality

The central page of this chapter is the operator-valued-distribution page. The point symbol $\phi(x)$ is shorthand for a distributional object; the controlled operator is obtained by smearing with a test function,

$$\begin{aligned} \phi(f)&=\int\mathrm d^4x\,f(x)\phi(x),\\ f&\in C_c^\infty(\mathbb R^{1,3}). \end{aligned}$$

or with the test-function class declared by the formulation. For unbounded fields, the smeared operators must act on a suitable common dense domain. Smearing controls localization and distributional limits; it does not by itself define products at coincident points. The continuum field, algebra, state, and locality data are set out in Hollands and Wald 2015, § 1, pp. 4–5, and § 2.1, pp. 9–12.

For a chosen vacuum $|\Omega\rangle$, particle species and charge channel, the schematic condition

$$\langle\mathbf p,\sigma,q|\Phi_i(0)|\Omega\rangle \neq 0$$

says that $\Phi_i$ interpolates that one-particle sector. For a canonically normalized scalar channel, a corresponding matrix element is often denoted $Z^{1/2}$ after its phase and state normalization have been fixed. It does not say that $\Phi_i$ is the particle. Different fields can have overlap with the same sector, field redefinitions can change the overlap, and a charged or gauge-fixed field need not itself be a physical observable. Vacuum–one-particle overlap and its normalization are developed in Coleman 2019, § 13.5, pp. 279–281, while the role of field redefinitions is bounded in Weinberg 1995, § 7.7, pp. 331–332.

Locality must name the objects being compared. For homogeneous fields define the graded commutator

$$[F,G]_{\mathrm{gr}} = FG-(-1)^{|F||G|}GF.$$

A local field system satisfies $[F(f),G(g)]_{\mathrm{gr}}=0$ when the supports of $f$ and $g$ are spacelike separated, subject to its field-content and domain hypotheses. Under the usual fermion-parity superselection rule, physical observables are even, so spacelike-separated observable algebras commute in the ordinary sense. In a gauge-fixed description, locality of auxiliary fields and locality of gauge-invariant observables are separate claims.

The free real scalar gives the first concrete check:

$$[\phi(f),\phi(g)] = i\Delta(f,g)\mathbf 1,$$

where the Pauli–Jordan distribution $\Delta$ has causal support, so the commutator vanishes for spacelike-separated test-function supports. Commutativity is compatibility, not statistical independence: it does not imply $\omega(AB)=\omega(A)\omega(B)$, and vacuum Wightman correlations can remain nonzero at spacelike separation. This is also not a statement about the Feynman two-point function, which is generally nonzero there. The local-observable formulation and its causality requirement are stated in Fewster and Rejzner 2020, § 4.1, pp. 13–15; the free-field construction from particles and causal commutators is developed in Weinberg 1995, §§ 5.1–5.2, pp. 191–206.

Two threads to carry through the chapter

Free real scalar. Start with an abstract canonical algebra, choose its positive-energy vacuum representation, and obtain a smeared Hermitian field. The field has nonzero vacuum-to-one-particle matrix elements; the one-particle states carry the massive spin-zero representation; symmetric tensor powers organize the free multiparticle sectors; and the Pauli–Jordan commutator supplies the first spacelike-compatibility test. Each clause names a different object or relation.

Complex scalar and charge. A free complex scalar has the schematic expansion

$$\Phi(x) = \int\mathrm d\Pi_p \left[ a(\mathbf p)e^{-ip\cdot x} +b^\dagger(\mathbf p)e^{+ip\cdot x} \right].$$

Both $a^\dagger$ and $b^\dagger$ create positive-energy excitations; their charge labels are opposite. The negative-frequency term multiplying $b^\dagger$ does not create a negative-energy physical state. Moreover, the charged field $\Phi$ transforms nontrivially under the global symmetry, while invariant combinations and the suitably defined free Noether current provide different candidates for observables. In an interacting theory, that coincident composite current requires its own renormalized definition. This thread becomes the chapter’s first test of the distinction among field, charge sector, particle, antiparticle, and observable. The particle–antiparticle content of causal scalar fields is developed in Weinberg 1995, §§ 5.1–5.2, pp. 191–206.

The nine pages in order

The order below is the chapter order. “Requires” names only a hard prerequisite inside this chapter.

Page	Purpose and first check	Requires	Stopping boundary and next treatment
1. What Is a Quantum Field Theory?	Separate physical structure from canonical, functional, Euclidean, Wightman, and local-observable presentations	None	Theorem-level comparisons go to Mathematical QFT; continuum construction goes to the volume that supplies the required method
2. Why Local Relativistic Quantum Theory Uses Fields	Explain why locality, many degrees of freedom, and particle creation make fields natural without claiming a unique logical derivation	None; page 1 is recommended	Rigorous localization and no-go theorems go to Mathematical QFT
3. Vacua, States, and Representations	Distinguish an algebra, a state, a vacuum, and a representation before invoking Fock space	None	GNS proofs, folia, and superselection sectors go to Mathematical QFT
4. One-Particle States: Mass, Spin, and Relativistic Normalization	Classify a stable one-particle sector and normalize a wave packet	None	General representation theory goes to Mathematical Methods; unstable states and LSZ go to Perturbative QFT and Scattering
5. Multiparticle States, Statistics, and Fock Organization	Build symmetric or antisymmetric sectors and state the scope of the Fock description	One-particle states	Interacting asymptotic completeness and scattering construction go to Perturbative QFT and Scattering
6. Quantum Fields as Operator-Valued Distributions	Replace the point symbol by $\phi(f)$ and track support, domains, and distributional limits	None	Common invariant-domain theorems and axiom-first field theory go to Mathematical QFT
7. Fields, Observables, and Interpolating Operators	Use a vacuum-to-one-particle matrix element to distinguish interpolation from identity	One-particle states	LSZ reduction goes to Perturbative QFT and Scattering; gauge-invariant construction goes to Symmetry and Gauge Structure
8. Spacelike Compatibility and Local Observables	State the graded field relation and the ordinary observable-algebra relation, then test the free scalar commutator	Fields, observables, and interpolating operators	Structural microcausality returns in Chapter 11; local-net theorems go to Mathematical QFT
9. Antiparticles and Charge-Conjugate Excitations	Separate positive-energy antiparticles from negative-frequency mode terms	One-particle states; fields, observables, and interpolating operators	Spinor charge conjugation appears in the fermion chapter; general discrete symmetries return in Chapter 11

Typed continuations

The overview stops where a neighboring page supplies the developed treatment.

Question leaving this chapter	Continue with
General Poincaré representation theory	Lorentz Field Representations and Poincaré Particle Representations
States, GNS representations, and folia	States, GNS Representations, and Folia
Operator-valued fields and controlled invariant domains	Wightman Fields, Domains, and Axioms
Passing from pointlike fields to local algebras	From Pointlike Fields to Nets and Affiliated Operators
Local observable nets and locality	Haag–Kastler Nets and Locality
Superselection sectors	Superselection Sectors and DHR Reconstruction
Asymptotic reduction	LSZ Reduction: Poles, Residues, and Stable External States
Gauge-invariant observable construction	Gauge Fields, Redundancy, and Observable Content
Spinor reality and antiparticle conventions	Majorana Fields and Reality Conditions and CPT: Hypotheses, Content, and Limits

What the chapter establishes—and what it does not

At the chapter exit, you should be able to:

• distinguish a theory from a formulation and state which observable and state a claimed equivalence concerns;
• distinguish an abstract algebra, a state, a vacuum, and a Hilbert-space representation;
• normalize massive one-particle states and wave packets in the declared relativistic convention;
• state when symmetric or antisymmetric Fock organization is justified;
• replace a point field by a smeared operator-valued distribution;
• explain why a field may interpolate a particle without being that particle or necessarily being observable; and
• state a spacelike-compatibility test that names the field or observable class and the ordinary or graded commutator.

The chapter does not prove the GNS theorem, classify all representations, construct superselection sectors, derive LSZ reduction, or build gauge-invariant observables in an interacting gauge theory. Those are explicit continuations, not missing steps. Canonical construction of the free scalar begins in Chapter 3, while the theorem-oriented map of covariance, spectrum, positivity, locality, spin–statistics, and CPT appears in Chapter 11.

Review the chapter

Prompt	A satisfactory answer includes	Repair route
A reader wants an S-matrix element from a gauge-fixed charged field but cannot normalize $	\mathbf p\rangle$or explain$\Phi(f)$. Choose an entry route, diagnose the missing preparation, correct the claim “the field is the observable particle,” and name the downstream treatment.	Enter through the particle-state route and close its one-particle normalization before interpolation; use the smearing route to repair the distributional gap. The particle is a state-sector concept, the field may interpolate it, the chosen state supplies expectations, and the gauge-fixed field is not automatically observable. Complete reduction continues in the LSZ page linked under typed continuations
In the free scalar theory, classify $\mathcal A$, $\omega_0$, $(\mathcal H_0,\pi_0)$, $\phi(f)$, and $	\mathbf p\rangle$.	Respectively: algebra, vacuum state, vacuum representation, smeared field, and generalized one-particle state; no pair is declared identical
Starting from the displayed relativistic ket normalization, show that the wave-packet norm uses $\mathrm d\Pi_p$.	The delta function, $2E_{\mathbf p}$, and $(2\pi)^3$ cancel consistently, leaving a positive integral	Re-read One-Particle States and the Lorentz-representation preparation linked above
State the free-scalar spacelike-locality test without mentioning the Feynman propagator.	Smeared fields, spacelike-separated supports, the Pauli–Jordan commutator, and its causal support are all named	Re-read Quantum Fields as Operator-Valued Distributions and Spacelike Compatibility
Explain the $b^\dagger e^{+ip\cdot x}$ term in the complex scalar.	$b^\dagger$ creates a positive-energy excitation with the conjugate charge; frequency sign is not physical-energy sign	Re-read Antiparticles and Charge-Conjugate Excitations

References

• Coleman, Sidney. Lectures of Sidney Coleman on Quantum Field Theory. Edited by Bryan Gin-ge Chen, David Derbes, David Griffiths, Brian Hill, Richard Sohn, and Yuan-Sen Ting. Singapore: World Scientific, 2019. DOI.
• Fewster, Christopher J., and Kasia Rejzner. “Algebraic Quantum Field Theory—an Introduction.” In Progress and Visions in Quantum Theory in View of Gravity, 1–61. Cham: Birkhäuser, 2020. DOI. Open PDF, arXiv:1904.04051v2.
• Hollands, Stefan, and Robert M. Wald. “Quantum Fields in Curved Spacetime.” Physics Reports 574 (2015): 1–35. DOI. Open PDF, arXiv:1401.2026v2.
• Weinberg, Steven. The Quantum Theory of Fields. Volume I: Foundations. Cambridge: Cambridge University Press, 1995. DOI.