The Osterwalder-Schrader axioms are conditions on the Schwinger functions of a Euclidean quantum field theory that
allow a relativistic quantum field theory
on Minkowski space to be reconstructed (Osterwalder
and Schrader 1973, 1975). In a common scalar-field formulation, 
and the 
are tempered distributions
on 
satisfying the following conditions.
1. Regularity: the distributions obey growth bounds sufficient for the reconstruction.
2. Euclidean invariance: they are invariant under the Euclidean group.
3. Reflection positivity: if 
reverses the Euclidean time coordinate and 
is supported in the positive-time half-space, then 
.
4. Symmetry: they are invariant under permutations of their arguments.
5. Cluster property: they factor in the limit when two groups of arguments are separated by increasingly large translations.
Reflection positivity is the central positivity condition. It defines a positive semidefinite sesquilinear form on positive-time observables. Dividing out its null vectors and completing gives a Hilbert space, while positive Euclidean-time translations act as a contraction semigroup. An analytic continuation then produces the relativistic theory and its Wightman functions. The 1975 paper extended and corrected the original reconstruction argument (Osterwalder and Schrader 1975).
