Fields, Observables, and Interpolating Operators

A local field can create or detect particle content because its action on the vacuum can have a nonzero projection onto a stable one-particle sector. If a local operator $O$ satisfies

$$P_{1,r}O(f)\Omega\neq0$$

for some test function $f$, then $O$ is an interpolating operator for particle species $r$. This is a relation between an operator and a state sector, not an identity between them. The same operator can also create multiparticle states; different operators can interpolate the same particle; and a non-Hermitian, charged, or gauge-fixed field need not itself be a physical observable.

Required background. One-Particle States: Mass, Spin, and Relativistic Normalization supplies the one-particle sector and invariant normalization used to define interpolation.

Helpful background. Quantum Fields as Operator-Valued Distributions supplies the smearing and domain language, while Vector Spaces, Duals, Linear Maps, and Bases supplies the linear-dual viewpoint.

Roles that notation often conflates

The words in the title describe different mathematical roles:

Object	Controlled meaning	Particle relation
Particle	A state or stable sector carrying mass, spin, momentum, and internal quantum numbers	It is not an operator or a spacetime function
Local field	A covariant operator-valued distribution $f\mapsto O(f)$	Its action can have projections onto several state sectors
Interpolating operator	A local operator with nonzero vacuum-to-one-particle overlap	“Interpolator” is a role, not a claim of fundamentality
Observable	Physically admissible operator content on the physical state space; measurable quantities use self-adjoint elements or effects	It may have no one-particle overlap at all
Source	An external $c$-number profile paired with an operator	It probes the theory but is neither a state nor an operator
Matrix element	Numerical or distributional data such as $\langle\Psi	O(f)

A real free scalar smeared with a real test function is the simplest case in which a field can furnish observable content and interpolate a particle. The roles separate in a gauge formulation, where a field representative used in a calculation need not define an observable on the physical state space.

Interpolation is a spectral projection

Assume the theory has a Poincaré-invariant vacuum $\Omega$ and an isolated stable one-particle sector of mass $m_r$. With the site’s covariant normalization,

$$\langle\mathbf p',r|\mathbf p,r\rangle = (2\pi)^3\,2E_{\mathbf p,r}\, \delta^{(3)}(\mathbf p'-\mathbf p),$$

the projector onto that scalar sector is

$$\begin{aligned} P_{1,r} &= \int\mathrm d\Pi_{\mathbf p,r}\, |\mathbf p,r\rangle \langle\mathbf p,r|,\\ \mathrm d\Pi_{\mathbf p,r} &= \frac{\mathrm d^3\mathbf p} {(2\pi)^3\,2E_{\mathbf p,r}}. \end{aligned}$$

First subtract a possible vacuum component,

$$\begin{aligned} O_{\mathrm s}(x) &:= O(x)-v_O\mathbf 1,\\ v_O &:= \langle\Omega|O(0)|\Omega\rangle. \end{aligned}$$

For a scalar $O_{\mathrm s}$ and a scalar particle, translation covariance gives

$$\begin{aligned} \langle\mathbf p,r| O_{\mathrm s}(x)|\Omega\rangle &= e^{ip\cdot x}\kappa_{O,r},\\ \kappa_{O,r} &:= \langle\mathbf p,r| O_{\mathrm s}(0)|\Omega\rangle. \end{aligned}$$

Lorentz covariance makes $\kappa_{O,r}$ independent of $\mathbf p$ on the mass shell. If $O_{\mathrm s}$ is Hermitian, the reverse matrix element is

$$\langle\Omega| O_{\mathrm s}(x)|\mathbf p,r\rangle = e^{-ip\cdot x}\kappa_{O,r}^*.$$

Using

$$\widetilde f(p) := \int\mathrm d^4x\, e^{ip\cdot x}f(x),$$

the one-particle projection is therefore

$$\begin{aligned} P_{1,r}O_{\mathrm s}(f)\Omega &= \kappa_{O,r} \int\mathrm d\Pi_{\mathbf p,r}\, \widetilde f(p) |\mathbf p,r\rangle. \end{aligned}$$

Thus $O$ interpolates species $r$ precisely when $\kappa_{O,r}\neq0$. With spin, Lorentz, or internal indices, the constant is replaced by coefficient functions $u_A(p,\sigma)$ constrained by covariance. Compatible quantum numbers are necessary: a symmetry selection rule forces the overlap to vanish when the operator and state transform incompatibly. They are not sufficient, because a dynamically allowed overlap may still happen to be zero.

The criterion does not require a particle to have an elementary field in a Lagrangian. A composite local operator can interpolate a stable bound state. Weinberg proves the corresponding isolated-pole statement for arbitrary local operators in Weinberg 1995, § 10.2, pp. 430–431 and 434–436. His convention has signature $(-+++)$, so his pole at $q^2=-m^2$ is the site’s $p^2=m^2$; zero versus nonzero overlap is unchanged.

First application: the free real scalar

Use the conventions already fixed in this chapter:

$$\begin{aligned} \mathrm d\mu_{\mathbf k} &:= \frac{\mathrm d^3\mathbf k} {(2\pi)^3\sqrt{2E_{\mathbf k}}},\\ \phi(x) &= \int\mathrm d\mu_{\mathbf k} \Bigl[ a(\mathbf k)e^{-ik\cdot x}\\ &\qquad+ a^\dagger(\mathbf k)e^{ik\cdot x} \Bigr],\\ [a(\mathbf k),a^\dagger(\mathbf p)] &= (2\pi)^3\delta^{(3)}(\mathbf k-\mathbf p),\\ |\mathbf p\rangle &= \sqrt{2E_{\mathbf p}}\, a^\dagger(\mathbf p)\Omega_0. \end{aligned}$$

The only term that contributes to $\langle\Omega_0|\phi(x)|\mathbf p\rangle$ is the annihilation part. Indeed,

$$\langle\Omega_0| a(\mathbf k)|\mathbf p\rangle = \sqrt{2E_{\mathbf p}}\, (2\pi)^3\delta^{(3)}(\mathbf k-\mathbf p).$$

Substitution gives the normalization check

$$\begin{aligned} \langle\Omega_0| \phi(x)|\mathbf p\rangle &= \int \frac{\mathrm d^3\mathbf k} {(2\pi)^3} \sqrt{\frac{E_{\mathbf p}}{E_{\mathbf k}}}\\ &\quad\times e^{-ik\cdot x} (2\pi)^3\delta^{(3)} (\mathbf k-\mathbf p)\\ &= e^{-ip\cdot x}. \end{aligned}$$

In particular,

$$\langle\Omega_0|\phi(0)|\mathbf p\rangle = 1.$$

The factors of $(2\pi)^3$ and $2E_{\mathbf p}$ have canceled, so the free scalar has unit vacuum-to-one-particle overlap in the site convention. Equivalently,

$$\begin{aligned} \langle\mathbf p|\phi(f)|\Omega_0\rangle &= \widetilde f(p),\\ \langle\Omega_0|\phi(f)|\mathbf p\rangle &= \widetilde f(-p),\\ P_1\phi(f)\Omega_0 &= \int\mathrm d\Pi_{\mathbf p}\, \widetilde f(p)|\mathbf p\rangle. \end{aligned}$$

Coleman 2019, §§ 3.3–3.4, pp. 37–46 constructs this free scalar from creation and annihilation operators. Coleman’s free creator obeys a bare delta normalization; the conversion

$$a(\mathbf p) = (2\pi)^{3/2}a_{\mathrm C}(\mathbf p)$$

produces the site commutator and the same unit overlap.

An interacting field is not a pure creator

For a general local operator, even after subtracting its vacuum expectation value, let $P_0:=|\Omega\rangle\langle\Omega|$ denote the vacuum projector. Then

$$\begin{aligned} O_{\mathrm s}(f)\Omega &= P_{1,r}O_{\mathrm s}(f)\Omega\\ &\quad+ (1-P_0-P_{1,r}) O_{\mathrm s}(f)\Omega. \end{aligned}$$

The second term can contain other one-particle species and multiparticle continuum states. A nonzero $\kappa_{O,r}$ therefore says that $O$ creates particle $r$ from the vacuum. The adjoint $O^\dagger$ supplies the reverse annihilating matrix element; a Hermitian $O$ performs both roles. None of these statements says that $O(f)\Omega$ is purely one particle.

For an interacting stable scalar, one often writes

$$\langle\mathbf p|\phi_{\mathrm s}(0)|\Omega\rangle = \sqrt{Z_\phi}$$

and rescales the subtracted field so that this overlap is one. Here $Z_\phi$ is a field-normalization constant tied to the chosen operator. Unit overlap is a convention, not proof that the field obeys a free equation or creates no continuum. Coleman states these assumptions, excludes an unstable meson from the one-particle construction, and builds the normalized wave-packet creator in Coleman 2019, § 13.5, pp. 279–281.

The isolated mass shell is an explicit hypothesis. If the translation spectrum contains no isolated one-particle subspace for species $r$, then the projector $P_{1,r}$ used above is unavailable and this criterion cannot be applied. Resonance and infraparticle examples of that failure are deferred to the later particle-language page.

One particle, many interpolating operators

Rescaling an operator changes its overlap without changing the particle:

$$\begin{aligned} O'(x)&=c\,O(x),\\ \kappa_{O',r}&=c\,\kappa_{O,r}. \end{aligned}$$

More generally, if $R$ has the same allowed quantum numbers but

$$\langle\mathbf p,r|R(0)|\Omega\rangle=0,$$

then $O'=cO+R$ interpolates the same species whenever $c\kappa_{O,r}\neq0$, while its other matrix elements and multiparticle content can differ.

There is an especially transparent on-shell check. Let a local scalar $O$ interpolate a mass-$m$ particle and define

$$O_\ell := O+\ell^2(\Box+m^2)O.$$

Since

$$(\Box+m^2)e^{ip\cdot x} = (-p^2+m^2)e^{ip\cdot x} = 0$$

on that mass shell,

$$\langle\mathbf p|O_\ell(x)|\Omega\rangle = \langle\mathbf p|O(x)|\Omega\rangle.$$

The two local operators can still differ in off-shell correlators, contact terms, and other spectral sectors. This elementary observation is not the equivalence theorem.

The source-exact statement is enough here. Under an infinitesimal continuous local redefinition of the fields, the induced change of the action is proportional, up to integrations by parts, to the field equations. A coupling variation of this form is redundant: masses and S-matrix elements are unchanged even though matrix elements of the field variables can change. Weinberg 1995, § 7.7, pp. 331–332 gives this continuous local-redefinition argument and a scalar rescaling example. It does not establish a general equivalence theorem for arbitrary finite, nonlocal, singular, or regulator-dependent transformations; full asymptotic reduction belongs downstream.

Sources probe fields; observables require more

An external source is a classical profile paired with an operator,

$$S_J = S+ \int\mathrm d^4x\, J^A(x)O_A(x).$$

Under an invertible linear change of operator basis, the source components transform dually so that the pairing $J^AO_A$ is unchanged. A nonlinear field redefinition does not preserve this single linear pairing by itself: one must transform the full source functional and, in general, include sources for the induced composite operators. In either description, a source can select spacetime, Lorentz, and internal quantum numbers, but it is not a particle, state, or observable. The developed source-derivative machinery belongs to the generating-functional page.

Observability is a separate question. A physical quantity needs appropriate adjointness and domains on the physical state space; the algebraic distinction between observables, states, and representations is developed in Fewster and Rejzner 2019, arXiv:1904.04051v2, §§ 2.1–2.2, PDF pp. 4–6; § 8, PDF pp. 32–33. Hermiticity alone does not settle those questions.

Gauge theories add a further distinction. A nonzero overlap computed for a field variable in a gauge-fixed description does not by itself show that the variable defines an observable on the physical state space. The separation between physical photon polarizations, nonphysical components of the covariant potential, and gauge fixing is presented in Schwartz 2014, § 8.4.2 and §§ 8.5–8.6, pp. 126–132. This page uses only that negative conclusion; the construction of gauge-invariant, dressed, charged-sector, or cohomological operators is deferred to the dedicated gauge-observables page. The qualification does not diminish the calculational role of fields. It prevents that role from being mistaken for a measurement claim.

Limits and handoffs

Interpolation can fail or require revision for several distinct reasons:

• incompatible Lorentz or internal quantum numbers force $\kappa_{O,r}=0$;
• a compatible operator can still have zero overlap dynamically;
• nonzero overlap does not isolate a particle from the rest of $O(f)\Omega$;
• if no isolated one-particle spectral projection exists, the $\kappa$-based criterion is unavailable;
• gauge-variant overlap does not by itself define a physical observable; and
• a one-particle overlap and pole are not, by themselves, the hypotheses of LSZ scattering theory.

The next chapter-local step is Spacelike Compatibility and Local Observables, which uses the field-versus-observable distinction. Antiparticles and Charge-Conjugate Excitations later separates field frequency components from positive-energy particle states.

For the broader interpretation of matrix elements, continue to Operators, Observables, and Matrix Elements. Developed source differentiation belongs to Generating Functional. Stable asymptotic projection and reduction belong to LSZ Reduction: Poles, Residues, and Stable External States. Gauge-theory construction belongs to Gauge-Invariant and Dressed Observables, while local observable-algebra criteria belong to Haag–Kastler Nets and Locality. The breakdown of sharp particle language is treated in Resonances, Infraparticles, and Limits of Particle Language.

Common pitfalls

“The field is the particle.” A field is an operator-valued distribution; a particle is a state or sector. Their relation is the matrix element $\kappa_{O,r}$.

“A nonzero overlap means the field creates only one particle.” It fixes the one-particle projection. An interacting field acting on the vacuum generally also has multiparticle components.

“A Hermitian field is automatically observable.” Hermiticity addresses adjointness, not gauge invariance, preservation of the physical state space, operator domains, or local-algebra membership.

“Zero overlap means the particle does not exist.” It means only that this operator does not interpolate that particle. Another operator with the correct quantum numbers may have nonzero overlap.

“A field redefinition proves physical equivalence by inspection.” Physical invariance requires the hypotheses of the relevant equivalence theorem and consistent transformation of every dependent object.

Check your understanding

1. Derive $\langle\Omega_0|\phi(x)|\mathbf p\rangle=e^{-ip\cdot x}$ from the displayed free-field normalization.

   Answer

   The creation part gives zero against the vacuum bra. The annihilation part contracts with $\sqrt{2E_{\mathbf p}}a^\dagger(\mathbf p)\Omega_0$, producing $(2\pi)^3\sqrt{2E_{\mathbf p}}\,\delta^{(3)}(\mathbf k-\mathbf p)$. The field measure contributes $[(2\pi)^3\sqrt{2E_{\mathbf k}}]^{-1}$. The delta function sets $\mathbf k=\mathbf p$, cancels every normalization factor, and leaves $e^{-ip\cdot x}$.

2. Suppose $O_{\mathrm s}(f)\Omega$ has a nonzero one-particle projection and a nonzero continuum projection. Which claims are justified?

   Answer

   The nonzero one-particle projection makes $O$ an interpolator for that stable species. It does not make $O_{\mathrm s}(f)\Omega$ a pure one-particle state, make $O$ unique or fundamental, or make it a physical observable. Those are separate questions about projection, coordinatization, and physical admissibility.

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.
• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014. Publisher.
• Weinberg, Steven. The Quantum Theory of Fields. Volume I: Foundations. Cambridge: Cambridge University Press, 1995. DOI.