Spectral Decomposition of Two-Point Functions

In a translation-invariant vacuum, insert a complete set of energy–momentum states between two copies of a vacuum-subtracted Hermitian scalar operator. Translation covariance supplies a phase $e^{-ip_X\cdot x}$ for each intermediate state, while Hermiticity and the positive physical inner product turn its coefficient into the nonnegative square $|\langle\Omega|\widehat{\mathcal O}(0)|X\rangle|^2$. A discrete sum at finite regulator or volume therefore becomes a positive momentum-space measure in the infinite-volume limit.

The spectrum condition confines that measure to the closed forward cone, and scalar Poincaré covariance organizes it by invariant mass. This is the spectral decomposition of the vacuum Wightman function. It does not yet produce the time-ordered Källén–Lehmann denominator, an analytic pole-or-cut interpretation, or a thermal spectral sum.

Required background. Operators, Observables, and Matrix Elements supplies the interpretation and normalization of the matrix elements inserted below. The Generating Functional distinguishes Wightman from time-ordered two-point functions and fixes the correlator conventions used in the handoff.

Helpful background. Bounded, Compact, and Integral Operators clarifies how discrete sums become continuum measures, while Spectra, Resolvents, Spectral Measures, and Functional Calculus supplies the abstract spectral-measure language.

Completeness in the vacuum two-point function

Work in four-dimensional Minkowski space with a normalized, translation-invariant vacuum $|\Omega\rangle$ in a positive-metric physical Hilbert space. Let $\mathcal O$ be a Hermitian scalar operator and remove its vacuum expectation:

$$\widehat{\mathcal O} =\mathcal O -\langle\Omega|\mathcal O|\Omega\rangle\mathbf 1, \qquad \langle\Omega|\widehat{\mathcal O}|\Omega\rangle=0.$$

The Wightman two-point distribution is

$$W_{\mathcal O}(x-y) =\langle\Omega| \widehat{\mathcal O}(x)\widehat{\mathcal O}(y) |\Omega\rangle.$$

Point fields are distributional notation. Products with test functions, or a regulator for which the displayed vectors and sums exist, are understood before taking a continuum limit. The subtraction removes the chosen vacuum contribution. If the representation contains other translation-invariant states with nonzero overlap, they require separate treatment rather than being silently removed.

For a discrete regulated spectrum, choose orthonormal joint energy–momentum eigenstates $|X\rangle$ and write

$$\mathbf 1 =|\Omega\rangle\langle\Omega| +\sum_{X\ne\Omega}|X\rangle\langle X|, \qquad P^\mu|X\rangle=p_X^\mu|X\rangle.$$

In infinite volume, the symbol $\sum_X$ below includes the appropriate integrals over generalized states and degeneracy labels. The positive measure formulation will make that replacement precise without squaring momentum delta functions.

The hypotheses have distinct jobs.

Input	What it supplies
Translation-invariant vacuum and translation covariance	Dependence on $x-y$ and the momentum phase of each intermediate state
Completeness in the chosen physical representation	Exhaustion of the state sectors reached by $\widehat{\mathcal O}$
Hermiticity and a positive physical inner product	Nonnegative squared overlaps
Spectrum condition	Support of total energy–momentum in the closed forward cone $\overline V_+$
Scalar Poincaré covariance	Reduction of the Lorentz-invariant measure to the variable $p^2$ with $p^0\ge0$
Distributional regularity	Meaning for smearing, Fourier transformation, and the continuum limit

Local commutativity, clustering, a mass gap, and asymptotic completeness are not used in the basic insertion.

Translation phases and forward spectral support

With the inherited convention

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

vacuum invariance and the eigenvalue equation give

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

Insert the identity at $y=0$. Hermiticity turns the second matrix element into the complex conjugate of the first, so

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

This is the promised positive spectral sum. “Positive” does not mean that the complex-valued distribution $W_{\mathcal O}(x)$ is pointwise nonnegative. It means that $W_{\mathcal O}$ is of positive type. For every test function $f$,

$$\begin{aligned} &\int \mathrm d^4x\,\mathrm d^4y\, f(x)^*W_{\mathcal O}(x-y)f(y) \\ &\qquad= \sum_{X\ne\Omega} \left| \langle\Omega|\widehat{\mathcal O}(0)|X\rangle \right|^2 \left|\widetilde f(p_X)\right|^2 \ge0. \end{aligned}$$

Equivalently, if $E_P(B)$ is the joint spectral projector of the self-adjoint energy–momentum operators onto a Borel set $B$, then, after the same smearing or regulation,

$$\nu_{\mathcal O}(B) =\langle\Omega| \widehat{\mathcal O}(0)E_P(B)\widehat{\mathcal O}(0) |\Omega\rangle =\left\|E_P(B)\widehat{\mathcal O}(0)|\Omega\rangle\right\|^2 \ge0.$$

Thus $\nu_{\mathcal O}$ is a positive momentum measure and

$$W_{\mathcal O}(x) =\int e^{-ip\cdot x}\,\mathrm d\nu_{\mathcal O}(p), \qquad \operatorname{supp}\nu_{\mathcal O} \subseteq\overline V_+.$$

The inclusion is an application of the spectrum condition, not a consequence of translation covariance alone. The completeness construction and its positive weights are developed in Schwartz 2014, § 24.2.1, pp. 467–469; the original structural construction appears in Lehmann 1954, § 1(a), pp. 344–347.

The positive invariant-mass measure

For a scalar operator and Poincaré-invariant vacuum, the momentum measure is Lorentz invariant. After setting aside any residual zero-momentum atom from an additional translation-invariant state, as qualified above, every relevant orbit in $\overline V_+$ is characterized by

$$\mu^2=p^2\ge0, \qquad p^0\ge0.$$

Define the positive-frequency free-mass distribution

$$\begin{aligned} \Delta_+(x;\mu^2) &=\int\frac{\mathrm d^4p}{(2\pi)^4}\, 2\pi\theta(p^0)\delta(p^2-\mu^2)e^{-ip\cdot x} \\ &=\int\frac{\mathrm d^3\mathbf p} {(2\pi)^3 2E_{\mu,\mathbf p}}\, e^{-ip\cdot x}, \qquad E_{\mu,\mathbf p} =\sqrt{\mathbf p^2+\mu^2}. \end{aligned}$$

Disintegrating $\nu_{\mathcal O}$ over these mass shells gives a unique positive measure $\rho_{\mathcal O}(\mathrm d\mu^2)$ on $[0,\infty)$ such that

$$\boxed{ W_{\mathcal O}(x) =\int_{[0,\infty)} \rho_{\mathcal O}(\mathrm d\mu^2)\, \Delta_+(x;\mu^2) }.$$

With

$$W_{\mathcal O}(x) =\int\frac{\mathrm d^4p}{(2\pi)^4} e^{-ip\cdot x}\widetilde W_{\mathcal O}(p),$$

the same statement is

$$\widetilde W_{\mathcal O}(p) =2\pi\theta(p^0) \int_{[0,\infty)} \rho_{\mathcal O}(\mathrm d\mu^2)\, \delta(p^2-\mu^2).$$

The measure notation is primary. When one writes $\rho_{\mathcal O}(\mu^2)\,\mathrm d\mu^2$, the symbol may include Dirac atoms as well as an ordinary density. A rescaling $\mathcal O\mapsto c\mathcal O$ gives

$$\rho_{\mathcal O} \longmapsto |c|^2\rho_{\mathcal O}.$$

Consequently positivity alone gives neither $\int\rho_{\mathcal O}=1$ nor $Z_{\mathcal O}\le1$ for an arbitrary composite or rescaled operator. Those statements require an additional normalization or sum rule.

Discrete masses and continuum sectors

Suppose the channel contains a stable scalar one-particle state of mass $m$, normalized by

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

Scalar covariance makes the vacuum-to-particle overlap momentum independent up to a phase. Choose that phase so that

$$\langle\Omega|\widehat{\mathcal O}(0)|\mathbf p\rangle =\sqrt{Z_{\mathcal O}}, \qquad Z_{\mathcal O}\ge0.$$

The one-particle part of completeness then gives

$$\begin{aligned} W_{\mathcal O}^{(1)}(x) &=\int\mathrm d\Pi_{\mathbf p}\, Z_{\mathcal O}e^{-ip\cdot x} \\ &=Z_{\mathcal O}\Delta_+(x;m^2), \end{aligned}$$

so the invariant-mass measure contains

$$\rho_{\mathcal O}(\mathrm d\mu^2) \supset Z_{\mathcal O}\, \delta(\mu^2-m^2)\,\mathrm d\mu^2.$$

Other stable states or bound states can add further atoms. In a channel with continuously variable total momentum among several particles or other excitations, the corresponding exact-state overlaps instead produce continuous measure. Schematically, when an $n$-particle description is available,

$$W_{\mathcal O}^{(n)}(x) =\frac{1}{S_n} \int\prod_{r=1}^{n}\mathrm d\Pi_r\, \left|F_n(p_1,\ldots,p_n)\right|^2 e^{-i(p_1+\cdots+p_n)\cdot x},$$

where $F_n=\langle\Omega|\widehat{\mathcal O}(0)|p_1,\ldots,p_n\rangle$ and $S_n$ is the identical-particle symmetry factor. Every integrand is nonnegative, but the support begins only at the lightest invariant mass allowed by the operator’s quantum numbers and the theory’s spectrum.

In finite spatial volume these contributions are generally discrete levels. The continuum emerges only after an infinite-volume limit makes the allowed momenta dense. A threshold is therefore channel dependent; no universal argument sets it to $4m^2$.

Free and interacting scalar checks

For the canonically normalized free real scalar, the previous page established

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

There is one mass-shell contribution in the field’s channel and no interacting continuum. Hence

$$\rho_{\phi}^{(0)}(\mathrm d\mu^2) =\delta(\mu^2-m^2)\,\mathrm d\mu^2,$$

and the decomposition returns

$$W_0(x) =\langle\Omega_0|\phi(x)\phi(0)|\Omega_0\rangle =\Delta_+(x;m^2).$$

Its Fourier transform,

$$\widetilde W_0(p) =2\pi\theta(p^0)\delta(p^2-m^2),$$

checks the phase, the factor of $2\pi$, the positive-energy support, and the covariant state normalization at once.

A continuum does not by itself diagnose interaction. As an independent free-field check, take the centered composite $\mathcal O=:\!\phi^2\!:$, normal ordered in the free vacuum. Wick factorization gives

$$W_{\phi^2}(x) =2\,[\Delta_+(x;m^2)]^2.$$

For total momentum $P\in\overline V_+$ and $s=P^2$, the integrated two-body phase space is

$$\begin{aligned} \Phi_2(s) &=\int\mathrm d\Pi_1\,\mathrm d\Pi_2\, (2\pi)^4\delta^{(4)}(P-p_1-p_2) \\ &=\frac{1}{8\pi} \sqrt{1-\frac{4m^2}{s}}\, \theta(s-4m^2). \end{aligned}$$

Since $\widetilde W_{\phi^2}(P)=2\Phi_2(P^2)$, comparison with $\widetilde W=2\pi\theta(P^0)\rho(P^2)$ yields

$$\rho_{\phi^2}(s) =\frac{1}{8\pi^2} \sqrt{1-\frac{4m^2}{s}}\, \theta(s-4m^2).$$

This positive continuum is a two-particle sector of a free theory, not an unstable one-particle state.

In the simplest interacting channel with one isolated stable scalar and a continuum, the measure has the conditional form

$$\rho_{\mathcal O}(\mathrm d\mu^2) =Z_{\mathcal O}\delta(\mu^2-m_{\mathrm{phys}}^2) \,\mathrm d\mu^2 +\theta(\mu^2-s_0) \rho_{\mathrm{cont}}(\mu^2)\,\mathrm d\mu^2,$$

with $Z_{\mathcal O}>0$ when the overlap is nonzero and $\rho_{\mathrm{cont}}\ge0$. The form is not exhaustive: additional stable bound states add atoms, and a channel without an isolated state has no first term.

For example, consider a one-species theory in which an unbroken $\mathbb Z_2$ symmetry makes $\phi$ and the lightest stable scalar odd, with no additional stable species or odd bound state below $3m_{\mathrm{phys}}$. The operator $\phi$ cannot connect the even vacuum to the even two-particle state made from two such scalars. The lightest allowed multiparticle channel then consists of three copies of the odd scalar, so $s_0=(3m_{\mathrm{phys}})^2$, not $4m_{\mathrm{phys}}^2$. If an even stable species were available, an odd-plus-even channel could instead lower the threshold. This selection-rule check makes the phrase “lightest allowed threshold” operational. The atom-plus-continuum structure and its normalization are also treated in Schwartz 2014, § 24.2.1, pp. 467–469.

What the decomposition does not establish

Tempting conclusion	Missing or failed step
“$W_{\mathcal O}(x)$ is pointwise nonnegative.”	Positivity is the smeared quadratic form, or equivalently positivity of the measure, not a pointwise order on a complex distribution
“Translation covariance forces positive energy.”	Translation covariance gives momentum labels and phases; forward support is a separate spectrum assumption
“Every two-point function has a nonnegative scalar density.”	The result uses a diagonal Hermitian correlator in a positive physical Hilbert space; non-Hermitian pairs, off-diagonal correlators, ghosts, and gauge-fixed indefinite spaces need different statements
“A continuum proves the theory is interacting.”	The free composite $:\!\phi^2\!:$ already has two-particle continuum support
“An atom or continuum already determines a pole or cut.”	Analytic structure follows only after constructing the time-ordered representation and studying its boundary value
“The spectral measure determines all dynamics.”	A two-point measure does not fix higher correlators, operator products, interactions, the vacuum representation, or scattering data
“Locality was proved or used.”	The basic two-point completeness argument uses no commutator at spacelike separation; stronger QFT consequences need additional hypotheses
“The same formula is a thermal Lehmann sum.”	A thermal state introduces Boltzmann weights, transitions between excited states, and positive- and negative-frequency structures

For a non-Hermitian operator, the diagonal pairing $\langle\Omega|\mathcal O(x)\mathcal O^\dagger(0)|\Omega\rangle$ can recover a positive measure under the same physical-Hilbert-space assumptions. For an operator multiplet, the spectral measure is positive semidefinite as a matrix-valued measure; individual off-diagonal entries need not be nonnegative.

The generating functional supplies the time-ordered correlator

$$D_{F,\mathcal O}(x) =\theta(x^0)W_{\mathcal O}(x) +\theta(-x^0)W_{\mathcal O}(-x)$$

for the declared bosonic vacuum ordering, subject to the usual distributional equal-time qualification. General composite time-ordered products can also require local contact or subtraction terms. Neither issue changes the positive Wightman measure derived above.

Check your understanding

Locate each hypothesis. Starting from the completeness sum, identify the exact line at which translation covariance, Hermiticity, Hilbert-space positivity, and the spectrum condition enter.

Check

Translation covariance produces $e^{-ip_X\cdot x}$. Hermiticity converts the second matrix element into the conjugate of the first. The positive inner product makes its product a nonnegative norm square. The spectrum condition is used only afterward, to place $p_X$ and the measure in $\overline V_+$.

Recover the free measure. Insert $\rho_{\phi}^{(0)}(\mathrm d\mu^2)=\delta(\mu^2-m^2)\mathrm d\mu^2$ into the invariant-mass decomposition.

Check

The $\mu^2$ integral evaluates the integrand at $m^2$, giving $W_0(x)=\Delta_+(x;m^2)$. Fourier transformation then gives $2\pi\theta(p^0)\delta(p^2-m^2)$.

Test operator normalization. Replace $\mathcal O$ by $c\mathcal O$ and track the matrix elements and measure.

Check

Every vacuum-to-state matrix element gains $c$, so every squared overlap gains $|c|^2$. Therefore $\rho\mapsto|c|^2\rho$; positivity is unchanged, but no normalization-independent bound on its total weight follows.

Diagnose a threshold. In the one-species unbroken-$\mathbb Z_2$ setup above, decide whether the odd operator $\phi$ can reach a two-particle state made from two odd scalars.

Check

No. The two-particle state is even, so its matrix element with an odd operator and an even vacuum vanishes. Under the stated absence of any additional stable species or allowed odd bound state below $3m_{\mathrm{phys}}$, the first multiparticle continuum in that channel is the three-particle sector.

From spectral support to Källén–Lehmann

The result can now be summarized without analytic language:

$$\begin{aligned} \text{complete physical states} &\Longrightarrow \text{squared overlaps} \\ &\Longrightarrow \text{positive forward-cone measure} \\ &\Longrightarrow \rho_{\mathcal O}(\mathrm d\mu^2). \end{aligned}$$

The Källén–Lehmann Representation next converts this Wightman measure into a time-ordered two-point representation with the common $+i0$ boundary prescription and the necessary regularity and local-term qualifications. Pole, threshold, and cut interpretations remain one further step away.

Wightman Functions and Spectral Support develops the theorem-oriented distributional treatment. Thermal Propagators and Spectral Representations replaces the vacuum insertion by the thermal one. A runnable reconstruction from supplied model measures would be a separate computational continuation requiring declared discretization, convergence, and error controls; no such numerical result is used as evidence for the continuum statement here.

References

• Lehmann, Harry. “On the Properties of Propagation Functions and Renormalization Constants of Quantized Fields.” Il Nuovo Cimento 11 (1954): 342–357. DOI.

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