Osterwalder–Schrader Axioms and Reflection Positivity

The Osterwalder–Schrader input is a hierarchy of Euclidean distributions satisfying compatible regularity, covariance, symmetry, reflection positivity, and—when a unique vacuum is wanted—clustering conditions. Reflection positivity is the condition that converts a chosen Euclidean time reflection into a positive physical inner product; ordinary positivity of a Euclidean covariance is not enough.

Required background. Euclidean random fields and Schwinger hierarchies supplies measures, moments, and cumulants. Positivity, spectrum, covariance, and locality hypotheses distinguishes the positivity notions and their conclusions.

Helpful background. Reflection positivity and OS reconstruction gives the physical overview. Conjugation and reflection positivity in radial quantization compares time reflection with the radial version used in conformal field theory.

The Euclidean conditions

Let $S=(S_0,S_1,\ldots)$ be scalar Schwinger distributions on $\mathbb R^d$, with $S_0=1$. Fix the reflection

$$\theta(\tau,\mathbf x)=(-\tau,\mathbf x)$$

and the open positive half-space $\mathbb R^d_+=\{(\tau,\mathbf x):\tau>0\}$. A useful modern statement separates five logically different requirements.

1. Regularity and growth. Each $S_n$ is a distribution with continuity and uniform-in-$n$ bounds strong enough for the intended reconstruction and boundary-value theorem. “Tempered for every fixed $n$” by itself is not the corrected uniform growth hypothesis.
2. Euclidean covariance. Simultaneous translations and proper Euclidean rotations act covariantly on all arguments and field indices.
3. Permutation symmetry. Bosonic Schwinger functions are symmetric; fermionic components obey the graded version.
4. Reflection positivity. The sesquilinear form made by reflecting one positive-time factor is nonnegative.
5. Clustering. Widely separated groups factorize in a stated translation direction and topology. This condition concerns the vacuum sector, not the existence of the basic quotient.

The original labels E0–E4 and their formulation appear in Osterwalder and Schrader 1973, §3, pp. 87–90. The 1975 paper explicitly corrects the original equivalence claim and supplies stronger growth conditions; one should therefore cite the corrected theorem, not E0 alone, for a full reconstruction statement Osterwalder and Schrader 1975, §§III–IV, pp. 285–289.

The reflected form

For a random-field realization, let $\mathcal A_+$ be the algebra generated by smeared fields $\Phi(f)$ with $\operatorname{supp}f\subset\mathbb R^d_+$. Define the antilinear reflection on scalar bosonic polynomials by

$$\Theta\!\left[c\,\Phi(f_1)\cdots\Phi(f_n)\right] =\overline c\,\Phi(\theta f_n)\cdots\Phi(\theta f_1), \qquad (\theta f)(x)=\overline{f(\theta x)}.$$

For commuting scalar fields the reversal is invisible, but retaining it makes the graded generalization unambiguous. Reflection positivity is

$$(F,F)_{\mathrm{OS}} :=\mathbb E\!\left[(\Theta F)F\right]\geq0, \qquad F\in\mathcal A_+.$$

At hierarchy level, if $F=(f_0,f_1,\ldots,f_N)$ is a finite sequence of test kernels supported at ordered positive times, this expectation is a finite sum of pairings with $S_{m+n}$. Thus reflection positivity can be tested without assuming that an underlying measure has already been constructed. It is stronger than $\mathbb E[\lvert F\rvert^2]\geq0$: the first factor has been reflected across a time-zero hypersurface.

The choice of hypersurface is not an extra violation of Euclidean invariance. Covariance transports the condition to parallel hyperplanes and rotations transport the distinguished direction. What matters is that the chosen reflection and positive-time algebra are stated together.

First QFT application: the massive free covariance

For $m>0$, write $\omega_{\mathbf p}=\sqrt{\mathbf p^2+m^2}$. This is the exact free-field test case behind reflection positivity and OS reconstruction. Performing the Euclidean-energy integral gives

$$C_m(\tau,\mathbf x) =\int\frac{d^{d-1}\mathbf p}{(2\pi)^{d-1}} \frac{e^{i\mathbf p\cdot\mathbf x}e^{-\omega_{\mathbf p}\lvert\tau\rvert}} {2\omega_{\mathbf p}}.$$

For a test function $f$ supported at positive time, the reflected two-point form is

$$\begin{aligned} C_m(\theta f,f) &=\int\frac{d^{d-1}\mathbf p}{(2\pi)^{d-1}} \frac{1}{2\omega_{\mathbf p}} \left\lvert \int_0^\infty d\tau\int d^{d-1}\mathbf x\, e^{-\omega_{\mathbf p}\tau-i\mathbf p\cdot\mathbf x}f(\tau,\mathbf x) \right\rvert^2\\ &\geq0. \end{aligned}$$

This factorization is the central independent check: the spatial Fourier transform exposes the positive semigroup kernel $e^{-\tau\omega_{\mathbf p}}$. For the Gaussian measure, positivity of the covariance form lifts to polynomial reflection positivity through Wick’s rule (equivalently, bosonic Fock-space second quantization). Euclidean covariance, bosonic symmetry, and the required massive clustering can also be checked directly. The construction of the resulting Hilbert space is carried out on the next page.

Ordinary covariance positivity is not reflection positivity

Consider the rotationally invariant multiplier

$$C(p^2)=\frac{1}{(p^2+m_1^2)(p^2+m_2^2)}, \qquad 0<m_1<m_2.$$

It is positive for real Euclidean momentum, so $\int \overline{\widetilde f(p)}C(p^2)\widetilde f(p)\,d^dp\geq0$. Yet

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

has a negative spectral residue. The reflected form can select that contribution and become negative. More generally, for real rational covariances without poles on the positive real axis, reflection positivity holds exactly when all poles are simple, lie on the negative real axis, and have nonnegative residues Arici et al. 2018, Theorem 3.7 and §§2–3. This is the promised adversarial test: Euclidean invariance and ordinary covariance positivity survive, but the Hilbert-space reconstruction fails at its norm.

Scope of the remaining axioms

Reflection positivity alone does not produce a relativistic local theory. Euclidean covariance is used to build spacetime generators; symmetry and analyticity enter locality after continuation; growth estimates control distributional boundary values; clustering selects the vacuum behavior. Conversely, clustering is not required merely to form the reflection-positive quotient, and reflection positivity does not imply that a continuum measure exists.

For gauge or fermion fields, neither the naive scalar algebra nor the naive reflection is correct. One must specify graded order reversal, link reflection, and the gauge-invariant positive algebra. Those variants are treated in Fermionic, gauge, and lattice reflection positivity.

Exercise

Let $f(\tau,\mathbf x)=h(\tau)g(\mathbf x)$ with $h$ supported in $\tau>0$. Derive the free reflected norm above and identify its null condition.

Solution

Spatial Fourier transformation and the kernel formula give

$$C_m(\theta f,f)=\int\frac{d^{d-1}\mathbf p}{(2\pi)^{d-1}} \frac{\lvert\widetilde g(\mathbf p)\rvert^2}{2\omega_{\mathbf p}} \left\lvert\int_0^\infty e^{-\omega_{\mathbf p}\tau}h(\tau)\,d\tau\right\rvert^2.$$

It vanishes precisely when the product of $\widetilde g(\mathbf p)$ and the Laplace transform of $h$ evaluated at $\omega_{\mathbf p}$ vanishes for almost every $\mathbf p$ with respect to $d^{d-1}\mathbf p/(2\omega_{\mathbf p})$. A nonzero positive-time test function can therefore represent a null vector.

References

• Arici, Francesca, Daniel Becker, Chris Ripken, Frank Saueressig, and Walter D. van Suijlekom. “Reflection Positivity in Higher Derivative Scalar Theories.” Journal of Mathematical Physics 59 (2018): 082302. doi:10.1063/1.5027231. Open preprint.
• Osterwalder, Konrad, and Robert Schrader. “Axioms for Euclidean Green’s Functions.” Communications in Mathematical Physics 31 (1973): 83–112. doi:10.1007/BF01645738. Open PDF.
• Osterwalder, Konrad, and Robert Schrader. “Axioms for Euclidean Green’s Functions II.” Communications in Mathematical Physics 42 (1975): 281–305. doi:10.1007/BF01608978. Open PDF.