Positivity, Spectrum, Covariance, and Locality Hypotheses

Positivity, spectral support, covariance, locality, and clustering constrain different mathematical carriers. None can be silently substituted for another: a Euclidean kernel may be pointwise positive yet fail reflection positivity, a local covariant field may have an indefinite state space, and positive energy does not by itself imply locality. A generalized free field with continuous Källén–Lehmann weight provides a model in which each condition can be checked separately.

Required background. Theorem-First Claim Records supplies the hypothesis–conclusion grammar; Domains, Signatures, Supports, and Regularity fixes the spaces on which the conditions act; Hilbert Positivity and Unitary Evolution gives the physical positivity condition; and Microcausality and Relativistic Compatibility develops locality. Helpful background. Clustering, Vacuum Assumptions, and Long-Range Correlations explains the additional large-separation hypothesis.

Structural hypotheses and their carriers

Let $\mathfrak F$ be a field $*$-algebra, $\omega$ a state, $W_n$ its Lorentzian vacuum distributions, and $S_n$ candidate Euclidean Schwinger distributions. A useful hypothesis statement identifies the carrier, signature, and quantifiers.

Condition	Carrier and quantified statement	Immediate use	Not implied merely by
Hilbert or state positivity	$\omega(A^*A)\geq0$ for every $A\in\mathfrak F$	GNS quotient and positive Hilbert inner product	covariance, locality, or positive energy
Reflection positivity	$\sum_{i,j}\bar c_i c_j\,S(\Theta F_i\,F_j)\geq0$ for every finite family supported at positive Euclidean time	positive reconstructed inner product	$C(x,y)\geq0$ pointwise or positive Fourier denominator
Positive Hamiltonian	$H\geq0$ in a chosen time evolution	stability for that time translation	the full relativistic spectrum condition
Spectrum condition	$\operatorname{sp}P\subset\overline V_+$ for the joint translation generators	analytic tube properties and causal reconstruction theorems	Hilbert positivity alone
Covariance	$U(a,\Lambda)\phi(f)U(a,\Lambda)^*=D(\Lambda^{-1})\phi(f_{(a,\Lambda)})$ for every group element and test function	frame-independent transformation law	positivity or locality
Locality	$[\phi_a(f),\phi_b(h)]_\mp=0$ whenever the supports are spacelike separated	causal compatibility of local observables	spectral support or covariance alone
Clustering	$\omega(A\alpha_{\lambda a}(B))\to\omega(A)\omega(B)$ for specified spacelike $a$ as $\lambda\to\infty$	vacuum purity/uniqueness consequences and decay of correlations	locality alone

Here $\Theta$ combines Euclidean-time reflection with the order reversal or conjugation appropriate to the field algebra. Reflection positivity is therefore positivity of a nonlocal quadratic form on the positive-time subalgebra. It is not the statement that every numerical value of a covariance is positive.

The Lorentzian spectrum condition concerns the joint spectrum of all translation generators. The weaker statement $H=P^0\geq0$ in one frame allows spacelike joint spectral points and is not Lorentz invariant by itself. Conversely, a positive state can be invariant under a translation representation whose spectrum is not restricted to $\overline V_+$.

Covariance and locality also have separate quantifiers. Covariance compares a field in every transformed test function; locality compares pairs with spacelike-separated supports. A commutator may transform covariantly without vanishing, and a field may be local under a smaller symmetry group without possessing Poincaré covariance.

Reconstruction uses several hypotheses at once

For Euclidean data, Osterwalder–Schrader reconstruction begins with a positive semidefinite form on functionals supported at positive Euclidean time. Quotienting null vectors and completing produces a Hilbert space. Euclidean time translations that preserve the positive half-space generate a contraction semigroup, from which a nonnegative Hamiltonian is obtained. Euclidean invariance and analytic continuation then enter the construction of a relativistic representation, while symmetry, regularity, and clustering serve other parts of the theorem Osterwalder and Schrader 1973, §§ 3–4, pp. 88–103.

The 1975 correction makes clear why this is a theorem chain rather than a single positivity argument: stronger growth and regularity control is needed to pass from Euclidean Green functions to tempered Wightman distributions Osterwalder and Schrader 1975, §§ II–V, pp. 283–297. Reflection positivity supplies the inner product, but it does not independently supply every domain, analyticity, covariance, or temperedness conclusion.

The converse direction has the same discipline. Wightman functions satisfying the full spectral and regularity conditions can often be continued to Euclidean Schwinger functions, but Hilbert positivity alone does not guarantee such a continuation. Each arrow retains the entire source theorem’s hypotheses.

A generalized free scalar with continuous spectrum

Fix $m_0>0$ and a positive Borel measure $\rho$ on $[m_0^2,\infty)$ with sufficiently mild growth, for example

$$d\rho(s)=\frac{1}{\Lambda^2} e^{-(s-m_0^2)/\Lambda^2} \mathbf1_{[m_0^2,\infty)}(s)\,ds.$$

For Schwartz test functions on Minkowski space, define

$$W_2(f,g) = \int_{m_0^2}^{\infty}d\rho(s) \int\frac{d^4p}{(2\pi)^3} \theta(p^0)\delta(p^2-s) \widetilde f(-p)\widetilde g(p).$$

More generally, it is enough that $\int(1+s)^{-N}d\rho(s)<\infty$ for some $N$ so that the expression defines a tempered distribution. Set all truncated $n$-point functions to zero except $W_2^T=W_2$; Wick’s rule then defines a generalized free field. This is the class introduced and analyzed as a local field theory by Greenberg Greenberg 1961, §§ II–IV, pp. 160–170.

Hilbert positivity

For every test function $f$,

$$W_2(\bar f,f) = \int d\rho(s) \int\frac{d^4p}{(2\pi)^3} \theta(p^0)\delta(p^2-s) \lvert\widetilde f(p)\rvert^2 \geq0.$$

The bosonic Fock construction over the resulting one-particle space makes the Wick moments a positive state. Positivity depends on $d\rho\geq0$. A signed spectral weight can retain Lorentz covariance and locality of the commutator while destroying the Hilbert-space norm.

Spectral support and covariance

Each mass-shell measure is invariant under the proper orthochronous Lorentz group and supported in $\overline V_+$. Their positive mixture is therefore Poincaré covariant, and the one-particle spectral support is

$$\{p\in\overline V_+:p^2\in\operatorname{supp}\rho\}.$$

The full Fock spectrum consists of finite sums of these momenta and remains in $\overline V_+$ because the forward cone is convex. This implements the Källén–Lehmann idea that a positive spectral measure decomposes the two-point function into free mass components Källén 1952, pp. 426–433 and Lehmann 1954, §§ 2–3, pp. 345–352.

Locality

The commutator is the spectral integral

$$[\phi(x),\phi(y)] = i\int d\rho(s)\,\Delta_s(x-y)\mathbf1,$$

where every Pauli–Jordan distribution $\Delta_s$ vanishes at spacelike separation. Pairing with spacelike-separated test functions may be interchanged with the finite positive spectral integral in the displayed example, so the commutator vanishes. Locality is proved componentwise; it is not inferred from positivity.

Clustering

Because the spectral support is bounded away from zero, the two-point function tends to zero under large spacelike translations of separated test functions. Wick’s rule then shows that every cross-contraction between the two clusters vanishes, leaving precisely the product of their vacuum expectations. Thus the field clusters. If the measure reaches $s=0$, the rate and even the relevant smeared limit require a new infrared analysis; locality alone does not settle it.

No isolated particle shell

The chosen $\rho$ is purely absolutely continuous. It has no atom $Z\delta(s-m^2)$, so the two-point function contains no isolated one-particle mass hyperboloid with residue $Z>0$. It has a continuum beginning at $m_0^2$, even though the theory is positive, covariant, local, gapped in energy, and clustering. An isolated stable particle requires an atomic contribution separated appropriately from the remaining spectral support. The detailed physical interpretation belongs to the Källén–Lehmann Representation.

This example also separates “free” from “generalized free.” Its higher correlations obey Wick’s rule, yet its two-point function is not the propagator of a single Klein–Gordon mass and the field satisfies no finite-order Klein–Gordon equation when the spectral support is continuous.

Euclidean reflection positivity of the spectral mixture

The Euclidean covariance associated with the same measure is

$$C_E(p_E) = \int_{m_0^2}^{\infty} \frac{d\rho(s)}{p_E^2+s}.$$

Each free massive covariance is reflection positive. For every positive-time test function $f$ its reflection form $Q_s(f)$ is nonnegative; hence Tonelli’s theorem gives

$$Q_\rho(f)=\int d\rho(s)\,Q_s(f)\geq0.$$

Positive superposition therefore preserves reflection positivity. This argument uses positivity of the spectral measure and the reflected quadratic form, not just positivity of the denominator.

Failure test: pointwise positivity is not reflection positivity

Let $0<m_1<m_2$ and consider the rotationally invariant Euclidean covariance

$$C(p_E^2) = \frac{1}{(p_E^2+m_1^2)(p_E^2+m_2^2)}.$$

It is strictly positive for real Euclidean momentum. Nevertheless,

$$C(p_E^2) = \frac{1}{m_2^2-m_1^2} \left( \frac{1}{p_E^2+m_1^2} - \frac{1}{p_E^2+m_2^2} \right),$$

so its two simple negative-axis poles have residues of opposite sign. For a real rational covariance with no pole on the nonnegative real axis, reflection positivity holds exactly when all poles lie on the negative real axis, are simple, and have nonnegative residues Arici et al. 2018, Definition 1.1 and Theorem 3.7, pp. 1–8. This covariance therefore fails reflection positivity. An explicit positive-time test function can isolate the negative-residue component and make the reflected quadratic form negative.

The strongest surviving statements are Euclidean invariance, ordinary pointwise positivity of the momentum-space function, and ultraviolet decay. A positive reconstructed Hilbert space does not follow. Nor does multiplying by an overall positive constant repair the negative residue.

Independent checks

Spectral-measure check. Test $W_2(\bar f,f)$ with wave packets localized near each part of the mass spectrum. A negative component of $\rho$ yields a negative norm and exposes failure of state positivity.

Causal-support check. Write the commutator as a mass integral of $\Delta_s$ and verify its pairing with spacelike-separated smearings vanishes before drawing conclusions about locality.

Particle check. Decompose $d\rho=Z\delta(s-m^2)+d\rho_c$. Only an atom contributes an isolated mass shell. A threshold singularity in $d\rho_c$ is not a delta function.

Reflection check. Fourier transform only the Euclidean time variable. For a free mass component, the positive-time kernel is proportional to $e^{-\omega_s(\mathbf p)(t+t')}/(2\omega_s)$, a rank-one positive kernel at each $\mathbf p$. A negative spectral residue reverses its sign for a suitable test packet.

Independence check. Changing the sign of one spectral component preserves Poincaré covariance and spacelike support of the commutator but violates Hilbert positivity. This supplies a direct counterexample to any implication from covariance plus locality to positivity.

Common pitfalls

Conflating positive energy with positive norm. The support of a two-point distribution can lie in the forward cone while its spectral density has a negative coefficient. Spectral location and Hilbert positivity are separate tests.

Reading a threshold as a particle pole. A continuous density beginning at $m_0^2$ gives a branch cut or continuum, not a normalizable one-particle state of fixed mass.

Using the free massive scalar as a universal template. Its single mass shell, exponential Euclidean decay, and simple reflection-positive kernel are special features. Generalized-free, massless, thermal, boundary, and indefinite-metric models alter different rows of the hypothesis table.

Exercises

1. Positivity under spectral mixing. Let $d\rho$ be a positive measure satisfying the stated growth bound. Prove that the mixed two-point function is of positive type.

Solution

For each $s$, the mass-shell distribution gives

$$W_{2,s}(\bar f,f) = \int\frac{d^4p}{(2\pi)^3} \theta(p^0)\delta(p^2-s)\lvert\widetilde f(p)\rvert^2\geq0.$$

Integrating this nonnegative quantity against $d\rho(s)\geq0$ preserves the inequality. The growth hypothesis ensures that the integral is finite as a distributional pairing. Wick’s rule then gives the positive bosonic Gaussian state on the polynomial field algebra.

2. Locality of the mixture. Explain why a continuous mass spectrum does not enlarge the causal support of the commutator.

Solution

Every $\Delta_s$ has support in the closed light cone, independent of $s$. If $f$ and $h$ have spacelike-separated supports, $\Delta_s(f,h)=0$ for every $s$. Therefore

$$\int d\rho(s)\,\Delta_s(f,h)=0.$$

The union or integral over masses changes timelike behavior but cannot create support at a spacelike-separated pair.

3. Detect an isolated mass. For $d\rho(s)=Z\delta(s-m^2)ds+r(s)ds$ with $Z>0$ and $r$ supported above $(m+\epsilon)^2$, identify the isolated one-particle contribution.

Solution

The delta term gives

$$Z\,\theta(p^0)\delta(p^2-m^2),$$

an isolated positive mass hyperboloid with one-particle norm scaled by $Z$. The continuous term begins a finite distance above it and describes a continuum. If $Z=0$, no isolated shell remains even if $r(s)$ diverges at its threshold.

4. Locate the reflection-positivity failure. Compute the residues of $[(z+m_1^2)(z+m_2^2)]^{-1}$ and apply the rational criterion.

Solution

At $z=-m_1^2$ the residue is $(m_2^2-m_1^2)^{-1}>0$; at $z=-m_2^2$ it is $(m_1^2-m_2^2)^{-1}<0$. The poles are simple and on the negative real axis, but the second residue violates the nonnegative-residue condition. Reflection positivity fails despite $C(z)>0$ for $z\geq0$.

References

• Arici, Francesca, Daniel Becker, Chris J. Fewster, Rainer Verch, and Miles Visser. “Reflection Positivity in Higher Derivative Scalar Theories.” Journal of Mathematical Physics 59 (2018): 082301. DOI. Open PDF.
• Greenberg, Oscar W. “Generalized Free Fields and Models of Local Field Theory.” Annals of Physics 16 (1961): 158–176. DOI.
• Källén, Gunnar. “On the Definition of the Renormalization Constants in Quantum Electrodynamics.” Helvetica Physica Acta 25 (1952): 417–434. INSPIRE record.
• Lehmann, Harry. “Über Eigenschaften von Ausbreitungsfunktionen und Renormierungskonstanten quantisierter Felder.” Il Nuovo Cimento 11 (1954): 342–357. DOI.
• Osterwalder, Konrad, and Robert Schrader. “Axioms for Euclidean Green’s Functions.” Communications in Mathematical Physics 31 (1973): 83–112. DOI. Open PDF.
• Osterwalder, Konrad, and Robert Schrader. “Axioms for Euclidean Green’s Functions II.” Communications in Mathematical Physics 42 (1975): 281–305. DOI. Open PDF.