Schwinger function

In quantum field theory, the Wightman distributions can be analytically continued to analytic functions in Euclidean space with the domain restricted to the ordered set of points in Euclidean space with no coinciding points. These functions are called the Schwinger functions (named after Julian Schwinger) and they are analytic, symmetric under the permutation of arguments (antisymmetric for fermionic fields), Euclidean covariant and satisfy a property known as reflection positivity.

Details

Pick any arbitrary coordinate τ and pick a test function f_N with N points as its arguments. Assume f_N has its support in the "time-ordered" subset of N points with 0 < τ₁ < ... < τ_N. Choose one such f_N for each positive N, with the f's being zero for all N larger than some integer M. Given a point x, let $\scriptstyle {\bar {x}}$ be the reflected point about the τ = 0 hyperplane. Then,

\sum _{{m,n}}\int d^{d}x_{1}\cdots d^{d}x_{m}\,d^{d}y_{1}\cdots d^{d}y_{n}S_{{m+n}}(x_{1},\dots ,x_{m},y_{1},\dots ,y_{n})f_{m}({\bar {x}}_{1},\dots ,{\bar {x}}_{m})^{*}f_{n}(y_{1},\dots ,y_{n})\geq 0

where * represents complex conjugation.

Osterwalder–Schrader theorem

The Osterwalder–Schrader theorem (named after Konrad Osterwalder and Robert Schrader)^[1] states that Schwinger functions which satisfy these properties can be analytically continued into a quantum field theory.

One way of (formally) constructing Schwinger functions which satisfy the above properties is through the Euclidean path integral. In particular, Euclidean path integrals (formally) satisfy reflection positivity. Let F be any polynomial functional of the field φ which does not depend upon the value of φ(x) for those points x whose τ coordinates are nonpositive. Then

\int {\mathcal {D}}\phi F[\phi (x)]F[\phi ({\bar {x}})]^{*}e^{{-S[\phi ]}}=\int {\mathcal {D}}\phi _{0}\int _{{\phi _{+}(\tau =0)=\phi _{0}}}{\mathcal {D}}\phi _{+}F[\phi _{+}]e^{{-S_{+}[\phi _{+}]}}\int _{{\phi _{-}(\tau =0)=\phi _{0}}}{\mathcal {D}}\phi _{-}F[{\bar {\phi }}_{-}]^{*}e^{{-S_{-}[\phi _{-}]}}.

Since the action S is real and can be split into S+, which only depends on φ on the positive half-space, and S− which only depends upon φ on the negative half-space, and if S also happens to be invariant under the combined action of taking a reflection and complex conjugating all the fields, then the previous quantity has to be nonnegative.

References

↑ Osterwalder, K., and Schrader, R.: "Axioms for Euclidean Green’s functions," Comm. Math. Phys. 31 (1973), 83–112; 42 (1975), 281–305.

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[1] Osterwalder, K., and Schrader, R.: "Axioms for Euclidean Green’s functions," Comm. Math. Phys. 31 (1973), 83–112; 42 (1975), 281–305.

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Schwinger function

Details

Osterwalder–Schrader theorem

See also

References