Quantum Fields, States, and Observables
A quantum field theory relates local observables to states, dynamics, and spacetime symmetry. Fields, particles, vacua, and Hilbert-space representations are parts of that description, but they are not interchangeable. This chapter establishes their distinctions before a particular quantization method is chosen.
Enter this chapter
Section titled “Enter this chapter”Begin with What Is a Quantum Field Theory? for the organizing objects. For a particle-based entry, proceed to one-particle states and then to fields as distributions.
Topics
Section titled “Topics”- What Is a Quantum Field Theory? — Separates a theory’s physical specification from the language used to formulate it.
- Why Local Relativistic Quantum Theory Uses Fields — Explains why fields accommodate relativistic locality and variable particle number.
- Vacua, States, and Representations — Distinguishes expectation-value assignments from operator realizations and vacuum choices.
- One-Particle States: Mass, Spin, and Relativistic Normalization — Connects mass and spin labels with the invariant normalization of particle states.
- Multiparticle States, Statistics, and Fock Organization — Organizes identical-particle sectors with the appropriate exchange statistics.
- Quantum Fields as Operator-Valued Distributions — Explains why a continuum field must be smeared with test functions.
- Fields, Observables, and Interpolating Operators — Distinguishes measurable quantities from fields that overlap with particle states.
- Spacelike Compatibility and Local Observables — States the algebraic compatibility of observables in spacelike-separated regions.
- Antiparticles and Charge-Conjugate Excitations — Interprets antiparticles as positive-energy excitations with conjugate charges.
How the ideas fit together
Section titled “How the ideas fit together”An algebra describes how observables combine; a state supplies their expectation values. A representation turns the abstract algebra into Hilbert-space operators, and a vector may represent a state within that realization. These distinctions remain meaningful when no preferred particle description exists. See Fewster and Rejzner 2019, § 2.
Particles enter through appropriate spectral and symmetry properties of states. A field can have an overlap with a one-particle state without being that particle, and gauge-dependent fields need not be observables. Smearing makes the local-field language mathematically usable; it does not itself resolve gauge invariance or asymptotic particle existence.
Review the chapter
Section titled “Review the chapter”Does a nonzero correlation between spacelike-separated observables violate locality?
Solution
No. Locality requires the observables to commute, not their joint expectation value to factorize. A state can correlate commuting observables. Vanishing commutators and absence of correlations are different statements.
Where to go next
Section titled “Where to go next”Continue to Classical Local Fields and Actions to specify a model’s dynamics, or to Canonical Quantization and the Free Scalar for the first complete quantum example.