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| | Nice to meet you, I am Marvella Shryock. To gather cash is a factor that I'm completely addicted to. For years he's been living in North Dakota and his family members loves it. Hiring is my occupation.<br><br>Feel free to surf to my site - [http://www.hamcass.org/index.php?document_srl=268708&mid=gido std testing at home] |
| In [[quantum field theory]], the [[Wightman distribution]]s can be [[Analytic continuation|analytically continued]] to analytic functions in [[Euclidean space]] with the [[Domain (mathematics)|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 field]]s), Euclidean covariant and satisfy a property known as '''reflection positivity'''.
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| Pick any arbitrary coordinate τ and pick a [[test function]] ''f''<sub>''N''</sub> with ''N'' points as its arguments. Assume ''f''<sub>''N''</sub> has its [[Support (mathematics)|support]] in the "time-ordered" subset of ''N'' points with 0 < τ<sub>1</sub> < ... < τ<sub>''N''</sub>. Choose one such ''f''<sub>''N''</sub> for each positive ''N'', with the f's being zero for all ''N'' larger than some integer ''M''. Given a point ''x'', let <math>\scriptstyle \bar{x}</math> be the reflected point about the τ = 0 [[hyperplane]]. Then,
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| :<math>\sum_{m,n}\int d^dx_1 \cdots d^dx_m\, d^dy_1 \cdots d^dy_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</math> | |
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| where * represents [[complex conjugation]].
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| The '''Osterwalder–Schrader theorem''' states that Schwinger functions which satisfy these properties can be analytically continued into a [[quantum field theory]].
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| Euclidean [[functional integration|path integral]]s satisfy reflection positivity formally. Pick any polynomial [[functional (mathematics)|functional]] ''F'' of the field φ which doesn't depend upon the value of φ(''x'') for those points ''x'' whose τ coordinates are nonpositive.
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| Then,
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| :<math>\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_-]}.</math>
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| Since the action ''S'' is real and can be split into ''S''<sub>+</sub> which only depends on φ on the positive half-space and ''S''<sub>−</sub> which only depends upon φ on the negative half-space, 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.
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| ==See also==
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| *[[Wick rotation]]
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| *[[Konrad Osterwalder]]
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| {{Statistical mechanics topics}}
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| {{DEFAULTSORT:Schwinger Function}}
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| [[Category:Statistical field theories]]
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| {{Quantum-stub}}
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