Euclidean Fields, Reflection Positivity, and OS Reconstruction

Euclidean correlation functions describe a relativistic quantum theory only when they carry enough positivity and analytic control. This chapter follows the construction from a Schwinger hierarchy to a positive Hilbert space and Lorentzian fields.

Enter this chapter

Begin with the hierarchy and reflection-positivity pages, then distinguish the Hilbert-space construction from the full reconstruction theorem. Helpful background. Euclidean correlators and Schwinger functions supplies the physical correlator language. Reflection positivity and OS reconstruction gives the first-pass bridge to Lorentzian theory. Reflection positivity and transfer-matrix criteria gives the regulated lattice analogue.

Topics

1. Euclidean Random Fields and Schwinger Hierarchies — Defines the complete distribution-valued Euclidean correlation hierarchy.
2. Osterwalder–Schrader Axioms and Reflection Positivity — States the structural conditions, including reflection positivity.
3. Reflection Positivity and Hilbert-Space Reconstruction — Constructs the physical inner product by reflection and quotienting.
4. Osterwalder–Schrader Reconstruction — Identifies the hypotheses that license Lorentzian reconstruction.
5. Euclidean Growth, Regularity, and Temperedness Conditions — Explains why fixed-order temperedness does not replace uniform growth.
6. Analytic Continuation between Euclidean and Lorentzian Domains — Tracks analytic domains and their distributional boundary values.
7. Clustering, Vacuum Uniqueness, and Mass-Gap Implications — Separates vacuum uniqueness, decay, gaps, and isolated particle poles.
8. Fermionic, Gauge, and Lattice Reflection Positivity — Tests reflection positivity for fermionic, gauge, and lattice variables.
9. OS–Wightman Comparison Directions and Failure Modes — Identifies which comparison direction fails when a hypothesis is removed.

How the ideas fit together

Reflection positivity makes the quotient inner product nonnegative; it is stronger than ordinary positivity of a covariance. Euclidean time translations then lead toward positive-energy dynamics, while continuation of the complete hierarchy requires additional analytic estimates. The 1973 sufficiency argument was corrected by stronger growth hypotheses in Osterwalder and Schrader 1975, §§ III–IV, pp. 285–289 (PDF). The table keeps those conditions separate from clustering and particle claims.

What each hypothesis licenses

Objects, hypotheses, licensed conclusions, and failure boundaries in Euclidean reconstruction

Euclidean object and domain	OS hypotheses	Reconstruction or result	Excluded converse	Adversarial check
Random distribution or full Schwinger hierarchy on Schwartz test functions	Consistent cylinder laws or a continuous positive-definite characteristic functional; finite, jointly continuous moments when the hierarchy is used	A Euclidean probability law determines compatible moments, while Minlos extension constructs a law from a valid characteristic functional	A symmetric Euclidean-covariant moment list need not exist as, or uniquely determine, a probability law	Test every finite moment matrix; a negative variance or an indeterminate moment problem stops the claimed inference
Positive-time algebra with a fixed reflection across the time-zero plane	Euclidean covariance and symmetry, suitable regularity, and nonnegativity of the reflected form for every positive-time polynomial	Quotienting its null space and completing gives the physical Hilbert space and a contraction semigroup with nonnegative generator	Ordinary covariance positivity does not imply reflection positivity, locality, or a relativistic field theory	Use a positive rational Fourier multiplier with one negative partial-fraction residue and find a negative reflected direction
Complete hierarchy at all orders on Euclidean configuration space	Corrected E0′ or E0″ growth together with Euclidean invariance, reflection positivity, permutation symmetry, and clustering	The corrected reconstruction theorem yields a tempered local positive-energy Wightman hierarchy and its cyclic theory, unique up to unitary equivalence	A valid two-point kernel or fixed-order temperedness does not establish a complete reconstruction	Keep the free two-point function but set the four-point function to zero; polynomial positivity then becomes negative
Ordered imaginary-time regions and their permuted complex tubes	Spectral support, Euclidean symmetry, uniform growth, an unobstructed path inside the named tube, and a specified boundary approach	Analytic functions acquire Wightman, opposite-order, or time-ordered distributional boundary values according to the approach domain	The substitution t = −iτ does not choose an ordering, cross a singularity, or prove numerical stability	Move an energy contour through a pinch and verify that residues or a cut obstruct the proposed continuation
Clustered Euclidean state and connected semigroup correlations	Qualitative clustering on a dense observable set; for a gap, a uniform exponential rate in a sector that detects the full vacuum complement	Qualitative clustering selects a unique vacuum under the stated density assumptions; exponential decay locates a positive spectral threshold	A Hamiltonian gap need not contain an isolated one-particle mass shell, and one channel need not see every low-energy state	Choose a continuous positive spectral measure supported above a positive threshold; decay is exponential but there is no particle pole
Fermionic or gauge lattice algebra on one side of a site- or link-centered reflection	Antilinear graded order reversal, orientation reversal of crossing links, a gauge-invariant positive algebra, and a nonnegative crossing-action expansion	The null quotient gives the positive physical sector and, for the stated Wilson construction, a positive transfer operator at fixed lattice spacing	Gauge-fixed propagator positivity and continuum existence do not follow; an improved action is not automatically reflection positive	Omit fermionic order reversal or replace a reflected link adjoint by the original orientation and watch the square acquire the wrong sign

Structured table data (JSON).

Review the chapter

Does exponential decay of a Euclidean two-point function prove an isolated one-particle pole?

Solution

No. A positive continuous spectral measure supported above a nonzero threshold also produces exponential decay, possibly with a power-law prefactor. An isolated particle requires an atomic spectral contribution; a decay rate alone does not establish one.

Where to go next

Continue to scattering and particle structure to see why an isolated mass shell matters, or constructive QFT to study how suitable Euclidean hierarchies are built.

References

• 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.