Quantum fields, states, and observables

The classical action from the previous page names fields and dynamics, but it does not by itself define a quantum theory. QFT also needs a quantum algebra or construction, states, observables, consistency conditions, and any regulator or limiting prescription needed to give those objects meaning. This page separates the central concepts before later calculations compress them into familiar notation.

Required background. The quantum-mechanics diagnostic checks states, operators, spectra, and pictures. The classical fields and relativity diagnostic checks Lorentz representations and particle labels.

What must be specified beyond an action?

Writing a Lagrangian density is the beginning of a construction, not its end. For a definite claim, identify at least:

• the spacetime, field content, parameters, symmetries, and action or other dynamical data;
• the algebra of quantities being represented and the meaning of products at coincident points;
• a state or rule for selecting states, including boundary or asymptotic data;
• the physical observables and how their expectation values are obtained;
• the regulator, renormalization, continuum limit, or approximation when the formal expressions are not already defined; and
• the regime in which the claim applies.

Different formulations can encode the same physical content while using different intermediate objects. A canonical construction emphasizes a Hilbert space and operator algebra; a functional integral emphasizes correlation functions and sources; an algebraic formulation begins from local observable algebras. Equivalence is a statement about matched physical quantities under stated assumptions, not about formulas looking identical. What Is a Quantum Field Theory? develops this comparison in detail.

Algebra, representation, and state

An abstract $*$-algebra $\mathcal A$ says which sums, products, adjoints, and commutators are meaningful. A representation $\pi$ realizes those elements as operators on a vector or Hilbert space. A state is a positive normalized linear functional,

$$\omega(A^*A)\ge0, \qquad \omega(1)=1,$$

which assigns expectation values. In a chosen Hilbert-space representation, $\omega(A)$ may be written as $\langle\psi|\pi(A)|\psi\rangle$ or $\operatorname{Tr}[\rho\,\pi(A)]$. The state is not the observable, and the Hilbert space is not the vacuum. A vacuum is a state with additional symmetry and spectral properties.

This distinction becomes essential in systems with infinitely many degrees of freedom, where inequivalent representations can occur. The free-field pages ahead choose a Fock representation suited to the Minkowski vacuum; that useful choice is not a theorem that one universal Fock space describes every state, background, or interacting theory.

Fields are not particles

A quantum field is a local operator-valued distribution. The meaningful object is normally a smeared field

$$\phi(f)=\int\mathrm d^d x\,f(x)\phi(x)$$

for an allowed test function $f$ and a stated operator domain. The symbol $\phi(x)$ is convenient distributional shorthand, not generally an everywhere-defined operator that can be multiplied at the same point without care.

A particle, when that language applies, is a feature of a state sector. A stable one-particle state is labeled by a mass shell, momentum, and spin or helicity representation. With covariant normalization,

$$\langle\mathbf p',\sigma'|\mathbf p,\sigma\rangle =2E_{\mathbf p}(2\pi)^{d-1} \delta^{(d-1)}(\mathbf p'-\mathbf p)\delta_{\sigma'\sigma},$$

and the corresponding completeness measure contains $\mathrm d^{d-1}\mathbf p/[(2\pi)^{d-1}2E_{\mathbf p}]$. The normalization and measure must be changed together.

A field can interpolate a particle when $\langle0|\phi(0)|p,\sigma\rangle\ne0$, but this overlap does not identify the field with the particle. Two different operators can overlap the same one-particle state, and one operator can couple to both one- and multiparticle sectors. Resonances and infraparticles further limit the sharp-particle picture.

Fields are not automatically observables

An observable belongs to the physical content on which the theory assigns measurable expectation values or probabilities. Some fields are observables after suitable smearing and renormalization; others are useful coordinates in a larger description.

The electromagnetic potential $A_\mu$ is the standard warning. Gauge-related potentials describe the same physical configuration, so a gauge-fixed component is not automatically a physical observable. The field strength $F_{\mu\nu}$ is gauge invariant, while charged observables may require dressing or nonlocal structure. Gauge fixing can still make propagators and perturbation theory calculable; “not directly observable” does not mean “useless.”

Locality also depends on what is being compared. Bosonic observables at spacelike separation are expected to commute in an appropriate local theory; fermionic fields obey graded relations; gauge-fixed fields require further qualification. Spacelike Compatibility and Local Observables states those distinctions precisely.

One free scalar, several different objects

In the Minkowski-vacuum Fock representation of a free real scalar, compare:

Expression	Meaning	Not automatically
$	0\rangle$	the chosen vacuum vector
$a^\dagger(\mathbf p)	0\rangle$	a generalized one-particle momentum state
$\phi(f)$	a smeared field operator	a particle
$\langle0	\mathrm T\phi(x)\phi(y)	0\rangle$
a pole at $p^2=m^2$	evidence for stable one-particle spectral content under the needed assumptions	a definition of the whole theory
$\int\mathcal D\phi\,e^{iS[\phi]}$	formal Lorentzian functional notation until its contour, regulator, and boundary data are supplied	a complete nonperturbative definition merely by being written

The same abstract theory can expose these objects in different formulations. What matters is the map among their physical predictions, not treating any single notation as self-explanatory.

Checks and limitations

• A correlator depends on a state and ordering prescription; identical denominators do not make Feynman, retarded, advanced, and Wightman functions the same distribution.
• A pole may indicate a stable particle only with spectral, positivity, and isolation assumptions. Unstable resonances are not normalizable asymptotic states.
• Pointwise products of fields typically require a definition such as renormalized composite operators; smearing one field does not solve every coincident-product problem.
• The Fock-space picture is powerful for free fields and perturbations around a suitable vacuum, but it is not the only QFT framework.

Exercises

1. Same particle, different fields. Suppose local operators $\mathcal O_1$ and $\mathcal O_2$ have the same conserved quantum numbers and both have nonzero vacuum-to-one-particle matrix elements for $|p\rangle$. What follows, and what does not?

Solution

Both operators can interpolate the same one-particle sector and can produce a pole at the same physical mass in suitable two-point functions. It does not follow that $\mathcal O_1=\mathcal O_2$, that their residues agree, or that either operator couples only to that state. Their multiparticle spectral weights, normalization, short-distance behavior, and gauge properties may differ. The particle is identified by the state-sector data, not by choosing one unique field name.

2. Classify the data. For a free electromagnetic field, classify $A_\mu(x)$, $F_{\mu\nu}(f)$, the vacuum, a one-photon state, and $\langle0|\mathrm T A_\mu(x)A_\nu(y)|0\rangle$ as field, observable, state, particle state, or correlator. Identify the extra choice in the last item.

Solution

$A_\mu$ is a gauge potential in a redundant field description. $F_{\mu\nu}(f)$ is a smeared gauge-invariant field and a candidate local observable. The vacuum is a state; a one-photon vector is a state in the physical one-particle sector; and the final expression is a time-ordered, state-dependent two-point distribution of gauge potentials. To define its inverse kinetic operator one must choose a gauge-fixing prescription (as well as the Feynman boundary condition already indicated by $\mathrm T$). Its gauge dependence does not make gauge-invariant predictions gauge dependent.

Continue to Canonical quantization and the free scalar, where these distinctions become an explicit algebra, Fock state, field operator, and propagator. Return to Classical fields, actions, and local dynamics if the difference between action data and quantum data is still unclear.

References

• Rudolf Haag, Local Quantum Physics: Fields, Particles, Algebras, second edition, Springer, 1996, doi:10.1007/978-3-642-61458-3.
• Robert M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics, University of Chicago Press, 1994, publisher page.
• Steven Weinberg, The Quantum Theory of Fields, Volume I: Foundations, Cambridge University Press, 1995, doi:10.1017/CBO9781139644167.