Bogoliubov causality condition explained

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\le y means that x causally precedes y .

Let

S(g) be scattering matrix as a functional of g . The Bogoliubov causality condition in terms of variational derivatives has the form:(\delta)/(\left( \delta g(x)) (\delta S(g))/(\delta g(y)) S ^{\dagger(g)\right)=0} for x\le y.

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. . .