Operators, Observables, and Matrix Elements

A matrix element is not a property of an operator alone. It is data attached to an operator, two states, a Hilbert-space representation, a common domain, and normalization conventions. A diagonal element of a physical observable gives an expectation value; an off-diagonal element controls coherence-sensitive cross terms or a possible transition coupling; and a vacuum-to-state element reveals which state sectors an operator can probe.

That information does not make every field an observable. A field may be an operator-valued distribution, a non-Hermitian charged field, a gauge-fixed variable, or simply a convenient interpolating coordinate. Observable status additionally requires the appropriate adjointness and domain properties and, in a constrained or gauge theory, a well-defined action on the physical state space. This page works in one fixed physical Hilbert-space representation and uses the free real scalar to separate these roles. Measurement protocols, developed scattering form factors, local-observable algebras, and the spectral measure are left to the linked continuations.

Required background. Vacua, States, and Representations supplies the distinction among a state, a vector, and a representation used in expectation values. Fields, Observables, and Interpolating Operators supplies the field–interpolator distinction and the vacuum-to-one-particle overlap used below.

Matrix elements are state–operator data

Let $\mathcal O(x)$ be a local operator-valued distribution. Its honest elementary object is a smearing

$$\mathcal O(f) =\int \mathrm d^4x\,f(x)\mathcal O(x), \qquad f\in C_c^\infty(\mathbb R^{1,3}),$$

defined on a declared dense domain $\mathcal D\subset\mathcal H$. For normalized vectors $|\alpha\rangle,|\beta\rangle\in\mathcal D$,

$$M_{\beta\alpha}[f] =\langle\beta|\mathcal O(f)|\alpha\rangle$$

is a complex number. Sharp-momentum kets and point-field notation are distributional shorthand; their delta functions and normalization factors must remain visible.

Different choices of states ask different physical questions.

Matrix-element class	What it records	What it does not yet supply
$\langle\Omega	\mathcal O	\Omega\rangle$
$\langle\psi	A	\psi\rangle$
$\operatorname{Tr}(\varrho A)$, when the trace exists	The same mean for a density operator $\varrho$ in the chosen representation	A claim that every algebraic state has such a density matrix in this representation
$\langle\beta	\mathcal O	\alpha\rangle$,$\beta\ne\alpha$
$\langle\Omega	\mathcal O	X\rangle$
$\langle\mathbf p’	\mathcal O(0)	\mathbf p\rangle$

The phases of off-diagonal elements depend on state conventions. Under $|\alpha\rangle\mapsto e^{i\theta_\alpha}|\alpha\rangle$ and $|\beta\rangle\mapsto e^{i\theta_\beta}|\beta\rangle$,

$$M_{\beta\alpha} \longmapsto e^{i(\theta_\alpha-\theta_\beta)}M_{\beta\alpha},$$

whereas a diagonal expectation is unchanged. Physical probabilities or cross sections combine amplitudes with the rest of a convention-consistent preparation and detection calculation.

When an operator is an observable

In the traditional sharp-observable description, a real-valued quantity is represented by a self-adjoint operator $A$ and its projection-valued spectral measure $E_A$. If $\Delta\subset\mathbb R$ is a set of outcomes, then

$$\Pr_\psi(A\in\Delta) =\mu_\psi^A(\Delta) \equiv\langle\psi|E_A(\Delta)|\psi\rangle, \qquad \langle A\rangle_\psi =\int_{\mathbb R}a\, \mathrm d\mu_\psi^A(a).$$

When the first moment exists, an expectation value is therefore a moment of a probability distribution, not an individual outcome. More general measurements use effects, positive-operator-valued measures, and instruments; those operational constructions are not developed here.

QFT adds two prior questions. First, an unbounded expression must have a suitable domain, and a point field generally has to be smeared before it is an operator at all. Second, the physical observables can form a proper subalgebra of all operators available in an auxiliary description. A gauge-fixed variable, for example, may fail to preserve the physical state space. A charged field may change superselection sector rather than belong to the neutral observable algebra. Fewster and Rejzner 2020, §§ 2.1–2.2, Open PDF pp. 4–6; § 8, pp. 32–33 separates sharp observables, states, representations, field algebras, and gauge-invariant observable subalgebras.

Expression	Correct first classification
$\phi(f)$ for a free real scalar and real test function $f$	A smeared symmetric field operator that can carry observable content and also interpolate a particle, subject to its domain and closure
$\phi(x)$ at one point	An operator-valued distributional symbol, not an ordinary bounded operator
A complex charged field $\Phi(f)$	A generally non-Hermitian, sector-changing field; Hermitian or neutral composites require a separate check
A gauge-fixed potential component $A_\mu(f)$	A useful field variable whose physical-observable status is not established by its matrix elements alone

Hermiticity is therefore not a universal shortcut. For a self-adjoint $A$ and vectors in the relevant domains,

$$\langle\beta|A|\alpha\rangle^* =\langle\alpha|A|\beta\rangle,$$

but that relation alone does not prove self-adjointness, gauge invariance, locality of a measurement, or experimental accessibility.

Vacuum, one-particle, and transition matrix elements

Translation covariance fixes the spacetime phase before dynamics fixes the value at the origin. With

$$\mathcal O(x) =e^{iP\cdot x}\mathcal O(0)e^{-iP\cdot x},$$

momentum eigenstates satisfy

$$\langle\beta|\mathcal O(x)|\alpha\rangle =e^{i(p_\beta-p_\alpha)\cdot x} \langle\beta|\mathcal O(0)|\alpha\rangle.$$

For a translation-invariant vacuum $P^\mu|\Omega\rangle=0$ and a one-particle state $|\mathbf p\rangle$,

$$\langle\Omega|\mathcal O(x)|\mathbf p\rangle =e^{-ip\cdot x} \langle\Omega|\mathcal O(0)|\mathbf p\rangle.$$

This is the kinematic content of a vacuum-to-one-particle overlap. Its being nonzero says that $\mathcal O$ probes that particle sector; it does not say that $\mathcal O$ creates only one particle or is itself an observable. Schwartz 2014, § 24.2, pp. 466–467 gives this overlap and derives its translation phase without requiring the field to be elementary.

Internal symmetries provide a second kinematic filter. If

$$[Q,\mathcal O]=q_{\mathcal O}\mathcal O, \qquad Q|\alpha\rangle=q_\alpha|\alpha\rangle, \qquad Q|\beta\rangle=q_\beta|\beta\rangle,$$

then

$$(q_\beta-q_\alpha-q_{\mathcal O}) \langle\beta|\mathcal O|\alpha\rangle=0.$$

Compatible quantum numbers are necessary for a nonzero element, but they are not sufficient: dynamics may still make the coefficient vanish.

For stable spin-zero external states, define the momentum-transfer orientation

$$q\equiv p'-p, \qquad \langle\mathbf p'|\mathcal O(x)|\mathbf p\rangle =e^{iq\cdot x} \langle\mathbf p'|\mathcal O(0)|\mathbf p\rangle.$$

If $\mathcal O$ is also a Lorentz scalar, covariance restricts the last factor to scalar functions of invariants such as $q^2$ and the external masses. Currents and stress tensors require additional vector or tensor structures. This page fixes the orientation and interpretation only; reduction, crossing, spin projectors, and interacting form factors continue in Scattering.

Free scalar field and stress-tensor matrix elements

Use the site’s covariant one-particle normalization,

$$\langle\mathbf p'|\mathbf p\rangle =(2\pi)^3 2E_{\mathbf p}\, \delta^{(3)}(\mathbf p'-\mathbf p), \qquad \mathrm d\Pi_{\mathbf k} =\frac{\mathrm d^3\mathbf k} {(2\pi)^3 2E_{\mathbf k}}.$$

For the canonically normalized free real scalar,

$$\phi(x) =\int \mathrm d\Pi_{\mathbf k} \left[ a(\mathbf k)e^{-ik\cdot x} +a^\dagger(\mathbf k)e^{ik\cdot x} \right], \qquad |\mathbf p\rangle=a^\dagger(\mathbf p)|\Omega_0\rangle.$$

The matching commutator and vacuum condition are

$$[a(\mathbf k),a^\dagger(\mathbf p)] =(2\pi)^3 2E_{\mathbf k}\, \delta^{(3)}(\mathbf k-\mathbf p), \qquad a(\mathbf k)|\Omega_0\rangle=0.$$

The reciprocal normalization of the commutator and measure gives

$$\langle\Omega_0|\phi(x)|\mathbf p\rangle =e^{-ip\cdot x}, \qquad \langle\Omega_0|\phi(0)|\mathbf p\rangle=1.$$

The free vacuum is even under $\phi\mapsto-\phi$, while a one-particle state is odd. Consequently,

$$\langle\Omega_0|\phi|\Omega_0\rangle=0, \qquad \langle\mathbf p'|\phi(x)|\mathbf p\rangle=0.$$

The field has unit vacuum-to-one-particle overlap but no one-to-one matrix element. This is exactly what an odd interpolating field should do in the centered free theory.

Now choose the canonical free-scalar tensor and normal order it relative to the same vacuum,

$$T_{\mathrm N}^{\mu\nu} \equiv :\!\left[ \partial^\mu\phi\,\partial^\nu\phi -\eta^{\mu\nu} \left( \frac12\partial_\rho\phi\,\partial^\rho\phi -\frac12m^2\phi^2 \right) \right]\!:$$

Normal ordering makes the free-vacuum expectation vanish. The tensor is even under $\phi\mapsto-\phi$, so

$$\langle\Omega_0|T_{\mathrm N}^{\mu\nu}|\Omega_0\rangle=0, \qquad \langle\Omega_0|T_{\mathrm N}^{\mu\nu}(0)|\mathbf p\rangle=0.$$

Between one-particle states, the normal-ordered mode expansion leaves only the $a^\dagger a$ terms. Direct substitution gives

$$\langle\mathbf p'|T_{\mathrm N}^{\mu\nu}(0)|\mathbf p\rangle = p'^\mu p^\nu+p'^\nu p^\mu -\eta^{\mu\nu}(p'\!\cdot p-m^2).$$

This one formula passes three independent checks. First, with $q=p'-p$ and $p^2=p'^2=m^2$,

$$\begin{aligned} q_\mu \langle\mathbf p'|T_{\mathrm N}^{\mu\nu}(0)|\mathbf p\rangle &=(m^2-p'\!\cdot p)p^\nu +(p'\!\cdot p-m^2)p'^\nu \\ &\quad -(p'^\nu-p^\nu)(p'\!\cdot p-m^2) =0. \end{aligned}$$

Second, the forward value is

$$\langle\mathbf p|T_{\mathrm N}^{\mu\nu}(0)|\mathbf p\rangle =2p^\mu p^\nu.$$

Third, the charge $P^\nu=\int\mathrm d^3\mathbf x\,T_{\mathrm N}^{0\nu}(x)$ has

$$\begin{aligned} \langle\mathbf p'|P^\nu|\mathbf p\rangle &=(2\pi)^3 2E_{\mathbf p}p^\nu \delta^{(3)}(\mathbf p'-\mathbf p) \\ &=p^\nu\langle\mathbf p'|\mathbf p\rangle, \end{aligned}$$

as required for the generator of translations. The canonical tensor and its conserved charges are derived in Schwartz 2014, § 3.3.1, pp. 34–36; the quantum matrix element above follows from that tensor and the displayed free mode expansion.

Normal ordering here is a free-vacuum prescription, not a general definition of an interacting renormalized stress tensor. Even in the free theory, an improvement

$$T_\xi^{\mu\nu} =T_{\mathrm N}^{\mu\nu} +\xi(\eta^{\mu\nu}\Box-\partial^\mu\partial^\nu) :\!\phi^2\!:$$

shifts the off-forward one-particle matrix element by

$$2\xi(q^\mu q^\nu-\eta^{\mu\nu}q^2),$$

while leaving the forward normalization unchanged. The integrated momentum is also unchanged provided the improvement surface term vanishes, as it does for suitable wave packets or decay conditions. The conserved charge is robust; a local off-forward representative contains additional convention data.

What matrix elements do not establish

Tempting inference	Missing step
“$\langle\Omega	\mathcal O
“$\mathcal O=\mathcal O^\dagger$, so it is measurable.”	A point-field symbol, domain, self-adjoint realization, physical-state condition, and measurement prescription remain to be checked
“$	\langle\beta
“A one-to-one matrix element is an $S$-matrix amplitude.”	A form factor contains a local insertion; asymptotic reduction and connected scattering normalization are separate constructions
“This matrix element vanishes, so the operator vanishes.”	A symmetry selection rule, state mismatch, or kinematic zero can kill one element while others remain nonzero
“The displayed $T^{\mu\nu}$ is unique.”	Improvement terms and interacting renormalization can change local off-forward matrix elements while preserving the conserved momentum

Check your understanding

Prompt	A successful check
Rephase $	\alpha\rangle$and$
Compare $\phi(x)$, $\phi(f)$, and a self-adjoint observable	State the distribution, domain, and physical-state qualifications instead of using “Hermitian” as a shortcut
Apply the charge selection rule	Derive $q_\beta-q_\alpha=q_{\mathcal O}$ as a necessary condition for a nonzero matrix element
Check the free-scalar zeros	Use $\mathbb Z_2$ parity to explain why $\langle p’
Test the stress tensor	Verify conservation, the $2p^\mu p^\nu$ forward limit, and the normalization of $P^\nu$
Route the next question	Send completeness and positivity to spectral decomposition, a readout protocol to Quantum Information, and interacting form factors to Scattering

If the state, representation, and operator distinctions are not yet stable, use Quantum states and operators repair before continuing.

Where the matrix elements lead

For a vacuum-subtracted operator, the two-point function gathers all vacuum-to-state overlaps. Schematically, with discrete sums and continuum phase-space integrals both understood,

$$\langle\Omega| \widehat{\mathcal O}(x) \widehat{\mathcal O}^{\dagger}(0) |\Omega\rangle = \sum_X e^{-ip_X\cdot x} \left| \langle\Omega|\widehat{\mathcal O}(0)|X\rangle \right|^2.$$

The squared overlaps suggest positivity, but this page stops before converting the sum into an invariant-mass measure or proving its support. That is the task of Spectral Decomposition of Two-Point Functions.

Different questions take different exits:

• Local Measurement Instruments in QFT adds effects, outcome probabilities, state updates, localization, and operational validity.

• Form Factors and Local Operator Insertions develops asymptotic reduction, connected normalization, tensor decomposition, and crossing.

• Haag–Kastler Nets and Locality develops local observable algebras without identifying them with a chosen field coordinatization.

• Local and Composite Operator Insertions treats coincident products, sources, and the need for an interacting definition of composite insertions.

References

• Fewster, Christopher J., and Katarzyna Rejzner. “Algebraic Quantum Field Theory—an Introduction.” In Felix Finster, Domenico Giulini, Johannes Kleiner, and Jürgen Tolksdorf, eds., Progress and Visions in Quantum Theory in View of Gravity: Bridging Foundations of Physics and Mathematics, 1–61. Cham: Birkhäuser, 2020. DOI. arXiv:1904.04051v2. Open PDF.

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. First ed. Cambridge: Cambridge University Press, 2014. DOI.