Positivity and correlation inequalities

The word “positive” can refer to several different statements. Each can be useful; confusing them can turn a valid calculation into an invalid claim about a physical Hilbert space or a mass gap.

Prerequisites: probability measures and lesson 10.
Outcomes: test a small positive matrix; identify the reflection in a reflected form; explain why finitely many successful tests do not establish all reconstruction requirements.

Ordinary expectations give Gram matrices

Let $F_1,\ldots,F_k$ be square-integrable complex random variables for a positive probability measure $\mu$. Their Gram matrix is $M_{ij}=\mathbb E[\overline{F_i}F_j]$. For any coefficients $c_i$,

$$c^\dagger Mc=\mathbb E\left[\left|\sum_i c_iF_i\right|^2\right]\geq0.$$

Thus $M$ is positive semidefinite. Taking two centered variables gives a covariance matrix

$$M=\begin{pmatrix}a&z\\\overline z&b\end{pmatrix}, \qquad a,b\geq0,\qquad |z|^2\leq ab.$$

This is Cauchy–Schwarz in matrix form. With $a=b=1$, the off-diagonal correlation cannot have magnitude greater than one. Positivity constrains allowed data without determining those data uniquely.

The logic is one-way for a finite test: a measured matrix that violates positivity cannot come from the assumed positive measure and observables, but passing one matrix test does not construct a measure with every required correlation.

Reflection changes the form

Reflect Euclidean time by $\theta(t,\mathbf x)=(-t,\mathbf x)$. For positive-time observables $F$, the reflected expression is

$$\mathbb E\bigl[\overline{F(\theta\phi)}F(\phi)\bigr].$$

This is not an ordinary squared modulus on each configuration. Its nonnegativity requires an additional property of the measure and reflection. For a translation-invariant Gaussian process with covariance $K(t-s)$, linear positive-time probes test the matrix $R_{ij}=K(t_i+t_j)$, with $t_i>0$.

A concrete counterexample prevents the two notions from being conflated. Set

$$X(t)=A\cos(\omega t)+B\sin(\omega t),$$

where $A,B$ are independent centered real Gaussians of variance one and $\omega>0$. Its covariance is $K(t-s)=\cos[\omega(t-s)]$. Ordinary covariance positivity holds because the process is explicitly built from two Gaussian variables. Equivalently, its covariance matrices are Gram matrices of the vectors $(\cos\omega t,\sin\omega t)$.

For one reflected probe at $t=\pi/(2\omega)$,

$$K(2t)=\cos\pi=-1.$$

The reflected form fails. The positive probability measure on $(A,B)$ did not guarantee reflection positivity in the time label $t$.

An exponential passes this linear reflected test

For $K_m(t)=e^{-m|t|}/(2m)$, $m>0$, and positive times,

$$\sum_{ij}\overline c_i c_jK_m(t_i+t_j) =\frac1{2m}\left|\sum_j c_je^{-mt_j}\right|^2\geq0.$$

The factor $1/(2m)$ fixes the usual one-dimensional massive covariance normalization. This calculation establishes the displayed quadratic form for linear probes. In a Gaussian theory higher correlations are determined by the covariance, but a general interacting reconstruction needs its full set of hypotheses and observables.

The original framework is Osterwalder–Schrader, Axioms for Euclidean Green’s functions (1973). Its sequel, II (1975), abstract and revised reconstruction framework, explicitly extends and corrects the first paper. A mnemonic list of positivity and symmetry properties is not a substitute for the complete regularity and reconstruction assumptions. The construction page explains the null quotient and physical-time interpretation.

Where correlation inequalities enter research

A positive-semidefinite matrix constraint can bound combinations of Wilson-loop expectations when combined with identities satisfied by those expectations. Adding more observables may strengthen a bound. It does not follow merely from adding rows that the hierarchy converges to the unique infinite-volume continuum theory.

Similarly, a covariance bound has to specify distance, connected subtraction, constants, and uniformity before it can imply decay. An inequality concerning an auxiliary Markov generator does not become a physical spectral-gap bound by calling both generators “positive.” See the semigroup bridge and the bootstrap guide.

Exercise: diagnose two reports

Report A states that $M=\begin{pmatrix}1&1.2\\1.2&1\end{pmatrix}$ is an exact covariance matrix of two real unit-variance observables. Report B tests ten reflected matrices, finds them nonnegative, and declares all Osterwalder–Schrader requirements proved. What fails?

Solution. In A, the vector $(1,-1)$ gives $v^\mathsf{T}Mv=-0.4$, or equivalently $\det M=-0.44$. The exact covariance claim is inconsistent; numerical errors would need a separate uncertainty analysis. B has checked a finite subset of an infinite family and has not checked the other reconstruction assumptions. Finite successful tests are evidence about those tests only.

Try the positivity lab and return to the Gaussian bridge. Keep the kind of positivity visible whenever you use the word.