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Reflection Positivity

Reflection positivity is a positivity condition on the correlation functions of a Euclidean quantum field theory. Let theta denote reflection of the Euclidean time coordinate. In a functional-integral formulation, the condition requires

<(thetaF)F>_E>=0

for suitable functionals F supported at positive Euclidean times, with the corresponding sesquilinear version for linear combinations of functionals.

Reflection positivity is the positivity axiom in the Osterwalder-Schrader axioms. It permits the reconstructed state space to carry a positive semidefinite inner product, which is essential for obtaining a quantum theory on Minkowski space from its Schwinger functions.

See also

Euclidean Quantum Field Theory, Inner Product, Minkowski Space, Osterwalder-Schrader Axioms, Schwinger Function

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References

Osterwalder, K. and Schrader, R. "Axioms for Euclidean Green's Functions." Comm. Math. Phys. 31, 83-112, 1973. https://doi.org/10.1007/BF01645738.Osterwalder, K. and Schrader, R. "Axioms for Euclidean Green's Functions. II." Comm. Math. Phys. 42, 281-305, 1975. https://doi.org/10.1007/BF01608978.

Cite this as:

Weisstein, Eric W. "Reflection Positivity." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ReflectionPositivity.html

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