Why positivity matters

A Euclidean probability measure must satisfy a special positivity condition if it is to reconstruct a physical Hilbert space in the usual framework. Positivity of the probability density alone does not supply that condition.

Prerequisite: Euclidean correlators. For the algebraic background, see Gaussian measures and semigroups.

By the end:

• Distinguish ordinary covariance positivity from reflected positivity.
• Explain why a nonnegative Hamiltonian need not have a positive gap.

Reflect one half of Euclidean time

Write a Euclidean point as $(t,\mathbf x)$ and reflect time by $\theta(t,\mathbf x)=(-t,\mathbf x)$. Let $F$ be a functional of fields supported at positive times. The reflection operation $\Theta$ reflects its arguments and includes complex conjugation.

Reflection positivity requires

$$\langle(\Theta F)F\rangle_E\geq0$$

for the appropriate class of such functionals. More generally,

$$(F,G)_{\mathrm{OS}}=\langle(\Theta F)G\rangle_E$$

should be a positive semidefinite sesquilinear form. This differs from the ordinary probabilistic inequality $\langle|F|^2\rangle_E\geq0$: the first factor has been reflected to the opposite side of a time slice.

That reflection is what makes the expression suitable for interpreting one half of Euclidean space as a state preparation and the other as its adjoint.

A small calculation that exposes the structure

Consider a centered Gaussian field in one Euclidean dimension with covariance

$$C(t-s)=\frac{1}{2m}e^{-m|t-s|}, \qquad m>0.$$

For $F=\sum_j c_j\phi(t_j)$ with every $t_j>0$,

$$\begin{aligned} \langle(\Theta F)F\rangle_E &=\sum_{i,j}\overline c_i c_j\,C(t_i+t_j)\\ &=\frac{1}{2m} \left|\sum_j c_j e^{-mt_j}\right|^2\geq0. \end{aligned}$$

The time arguments add because one insertion is reflected. The resulting matrix is positive semidefinite. For a Gaussian field, the covariance criterion extends to the appropriate polynomial functionals through its Gaussian structure.

This elementary example is closely related to a quantum harmonic oscillator. It shows how decay in Euclidean time can coexist with positivity of the reconstructed inner product.

A positive measure that fails the test

Let $X$ and $Y$ be independent centered real Gaussian variables of variance one, and define

$$\phi(t)=X\cos(\omega t)+Y\sin(\omega t), \qquad \omega>0.$$

This is an ordinary, well-defined Gaussian random field. Its covariance is

$$\mathbb E[\phi(t)\phi(s)]=\cos\bigl(\omega(t-s)\bigr).$$

It is positive definite as a covariance: for any coefficients, the variance of the corresponding linear combination is nonnegative.

But take $F=\phi(t_*)$ at $t_*=\pi/(2\omega)>0$. Then

$$\mathbb E[\phi(-t_*)\phi(t_*)] =\cos(2\omega t_*)=-1.$$

Reflection positivity fails. An ordinary positive Gaussian probability measure has produced a negative value for the would-be reflected norm. The example demonstrates that the two positivity requirements are genuinely different.

From a semidefinite form to states

In the reconstruction picture, identify functionals of zero reflected norm and quotient them out. Complete the remaining inner-product space. Under the further required hypotheses, this yields a Hilbert space.

Positive Euclidean time translations then act as a semigroup with the form

$$T(t)=e^{-tH}, \qquad H\geq0.$$

The Osterwalder–Schrader framework connects suitable Euclidean correlation functions with a relativistic quantum theory. Reflection positivity is one crucial ingredient; regularity, symmetry, and the other required conditions cannot simply be omitted.

Notice what has not yet been proved: $H\geq0$ allows spectral values arbitrarily close to zero. Establishing a mass gap requires an additional bound separating the vacuum from the rest of the spectrum.

Why gauge theories need care

Gauge-fixed fields may be described using auxiliary spaces or measures that do not have the positivity properties of the physical observables. A mass-gap argument must identify the physical space on which its positive Hamiltonian acts.

Gauge-invariant lattice formulations are helpful because positivity can be studied before taking a continuum limit, with the action and reflection chosen appropriately. The property must then survive in the limiting objects used for reconstruction.

Likewise, a positivity inequality used in a numerical bootstrap must be valid for the stated dimension, observable set, and regulator. The SU(3) bootstrap study illustrates how positivity conditions become constraints on loop expectation values. Their individual assumptions remain part of every bound.

Exercises

1. Find a null vector

In the exponential covariance example with $t_1\ne t_2$, choose nonzero $c_1,c_2$ so that the reflected norm vanishes.

Hint. Set $c_1e^{-mt_1}+c_2e^{-mt_2}=0$.

Solution. For example, take $c_1=e^{-mt_2}$ and $c_2=-e^{-mt_1}$. This explains why reconstruction starts with a semidefinite form and removes null vectors.

2. Positivity versus a gap

Can a positive self-adjoint Hamiltonian have no positive spectral gap?

Hint. Consider multiplication by $E$ on functions supported on $E>0$, with a vacuum added separately.

Solution. Yes. Its spectrum can contain every positive energy, however small, while the vacuum has energy zero. Positivity is necessary but does not establish a positive threshold.

Continue at your own depth

Which positivity are you testing? develops this lesson without adding a prerequisite to the next lesson.

The reconstruction references must be read as a pair: Osterwalder–Schrader I (1973) and II (1975). The latter extends and corrects the earlier formulation. The finite Gaussian calculations here illustrate the positive form; they do not replace the corrected regularity, covariance and reconstruction hypotheses. See the reconstruction guide for the scope of that statement.

Read further

• Osterwalder and Schrader, Axioms for Euclidean Green’s functions.
• Guo and collaborators, Bootstrapping SU(3) Lattice Yang–Mills Theory.

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