Quantum Yang-Mills Theory, sections 3/6.1/6.2 (Osterwalder-Schrader reconstruction)

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Moments of a Euclidean-invariant, reflection-positive, ergodic, Borel measure on the space of tempered distributions may satisfy the Osterwalder-Schrader axioms.

Referencing norms: OsAxiomsFromChecklist, is_euclidean_invariant, is_reflection_positive, satisfies_os_axioms, satisfies_os_cluster_property, satisfies_os_symmetry

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Two principal approaches -- studying quantum theory on Hilbert space, and studying classical functional integrals -- are related by the Osterwalder-Schrader construction.

Referencing norms: OsterwalderSchraderReconstruction

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One of the achievements of 20th century axiomatic quantum field theory was the discovery of how to convert a Euclidean-invariant field theory on a Euclidean space-time to a Lorentz-invariant field theory on Minkowski space-time, and vice-versa. Wightman used positive energy to establish analytic continuation of expectations of Minkowski field theories to Euclidean space. Kurt Symanzik interpreted the Euclidean expectations as a statistical mechanical ensemble of classical Markov fields [46], with a probability density proportional to exp(-S), where S denotes the Euclidean action functional. E. Nelson reformulated Symanzik's picture and showed that one can also construct a Hilbert space and a quantum-mechanical field from a Markov field [33]. Osterwalder and Schrader then discovered the elementary "reflection-positivity" condition to replace the Markov property. This gave rise to a general theory establishing equivalence between Lorentzian and Euclidean axiom schemes [35]. See also [13].

Referencing norms: OSReconstructionProof, OSReconstructionTheoremClaim, OsterwalderSchraderSource, os-proves-reconstruction-theorem

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[35] K. Osterwalder and R. Schrader, Axioms for Euclidean Green's functions, Comm. Math. Phys. 31 (1973), 83-112, and Comm. Math. Phys. 42 (1975), 281-305.