Structure constants: Difference between revisions

Revision as of 19:03, 15 November 2013

Bogoliubov causality condition is a causality condition for scattering matrix (S-matrix) in axiomatic quantum field theory. The condition was introduced in axiomatic quantum field theory by Nikolay Bogolyubov in 1955.

Formulation

In axiomatic quantum theory, S-matrix is considered as a functional of a function $g : M \to [0, 1]$ defined on the Minkowski space $M$ . This function characterizes the intensity of the interaction in different space-time regions: the value $g (x) = 0$ at a point $x$ corresponds to the absence of interaction in $x$ , $g (x) = 1$ corresponds to the most intense interaction, and values between 0 and 1 correspond to incomplete interaction at $x$ . For two points $x, y \in M$ , the notation $x \leq y$ means that $x$ causally precedes $y$ .

Template:FrameboxLet $S (g)$ be scattering matrix as a functional of $g$ . The Bogoliubov causality condition in terms of variational derivatives has the form:

\frac{δ}{δ g (x)} (\frac{δ S (g)}{δ g (y)} S^{†} (g)) = 0 for x \leq y .

Template:Frame-footer

References

N. N. Bogoliubov, A. A. Logunov, I. T. Todorov (1975): Introduction to Axiomatic Quantum Field Theory. Reading, Mass.: W. A. Benjamin, Advanced Book Program.
N. N. Bogoliubov, A. A. Logunov, A. I. Oksak, I. T. Todorov (1990): General Principles of Quantum Field Theory. Kluwer Academic Publishers, Dordrecht [Holland]; Boston. ISBN 0-7923-0540-X. ISBN 978-0-7923-0540-8.

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+'''Bogoliubov causality condition''' is a [[causality conditions|causality condition]] for [[S-matrix|scattering matrix]] (''S''-matrix) in [[axiomatic quantum field theory]]. The condition was introduced in axiomatic quantum field theory by [[Nikolay Bogolyubov]] in 1955.
+==Formulation==
+In axiomatic quantum theory, ''S''-matrix is considered as a [[functional (mathematics)|functional]] of a function <math>g: M\to [0,1]</math> defined on the [[Minkowski space]] <math>M</math>. This function characterizes the intensity of the interaction in different space-time regions: the value <math>g(x)=0</math> at a point <math>x</math> corresponds to the absence of interaction in <math>x</math>, <math>g(x)=1</math> corresponds to the most intense interaction, and values between 0 and 1 correspond to incomplete interaction at <math>x</math>. For two points <math>x,y\in M</math>, the notation <math>x\le y</math> means that <math>x</math> causally precedes <math>y</math>.
+{{framebox|gray|color=#CCFFFF}}Let <math>S(g)</math> be scattering matrix as a functional of <math>g</math>. The Bogoliubov causality condition in terms of [[variational derivative]]s has the form:
+:::<math>\frac{\delta}{\delta g(x)}\left(\frac{\delta S(g)}{\delta g(y)} S^\dagger(g)\right)=0 \mbox{ for } x\le y. </math>{{frame-footer}}
+== References ==
+*N. N. Bogoliubov, A. A. Logunov, I. T. Todorov (1975): ''Introduction to Axiomatic Quantum Field Theory''. Reading, Mass.: W. A. Benjamin, Advanced Book Program.
+*N. N. Bogoliubov, A. A. Logunov, A. I. Oksak, I. T. Todorov (1990): ''General Principles of Quantum Field Theory''. Kluwer Academic Publishers, Dordrecht [Holland]; Boston. ISBN 0-7923-0540-X. ISBN 978-0-7923-0540-8.
+[[Category:Quantum field theory]]

Structure constants: Difference between revisions

Revision as of 19:03, 15 November 2013

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