Rössler attractor: Difference between revisions
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{{Unreferenced stub|auto=yes|date=December 2009}} | |||
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'''. | |||
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, | |||
:<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> | |||
where * represents [[complex conjugation]]. | |||
The '''Osterwalder–Schrader theorem''' states that Schwinger functions which satisfy these properties can be analytically continued into a [[quantum field theory]]. | |||
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. | |||
Then, | |||
:<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> | |||
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. | |||
==See also== | |||
*[[Wick rotation]] | |||
*[[Konrad Osterwalder]] | |||
{{Statistical mechanics topics}} | |||
{{DEFAULTSORT:Schwinger Function}} | |||
[[Category:Statistical field theories]] | |||
{{Quantum-stub}} | |||
Revision as of 01:58, 30 July 2013
Template:Unreferenced stub 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.
Pick any arbitrary coordinate τ and pick a test function fN with N points as its arguments. Assume fN has its support in the "time-ordered" subset of N points with 0 < τ1 < ... < τN. Choose one such fN for each positive N, with the f's being zero for all N larger than some integer M. Given a point x, let be the reflected point about the τ = 0 hyperplane. Then,
where * represents complex conjugation.
The Osterwalder–Schrader theorem states that Schwinger functions which satisfy these properties can be analytically continued into a quantum field theory.
Euclidean path integrals satisfy reflection positivity formally. Pick any polynomial 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.
Then,
![{\displaystyle \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 _{-}]}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2fd207dbf32725b261dbb75b91374ee1cffc71d)
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, 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.