Physics:Bogoliubov causality condition

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 [math]\displaystyle{ g: M\to [0,1] }[/math] defined on the Minkowski space [math]\displaystyle{ M }[/math]. This function characterizes the intensity of the interaction in different space-time regions: the value [math]\displaystyle{ g(x)=0 }[/math] at a point [math]\displaystyle{ x }[/math] corresponds to the absence of interaction in [math]\displaystyle{ x }[/math], [math]\displaystyle{ g(x)=1 }[/math] corresponds to the most intense interaction, and values between 0 and 1 correspond to incomplete interaction at [math]\displaystyle{ x }[/math]. For two points [math]\displaystyle{ x,y\in M }[/math], the notation [math]\displaystyle{ x\le y }[/math] means that [math]\displaystyle{ x }[/math] causally precedes [math]\displaystyle{ y }[/math].

Let [math]\displaystyle{ S(g) }[/math] be scattering matrix as a functional of [math]\displaystyle{ g }[/math]. The Bogoliubov causality condition in terms of variational derivatives has the form:

[math]\displaystyle{ \frac{\delta}{\delta g(x)}\left(\frac{\delta S(g)}{\delta g(y)} S^\dagger(g)\right)=0 \mbox{ for } x\le y. }[/math]

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.