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 $g: M\to [0,1]$ 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:

$\frac{\delta}{\delta g(x)}\left(\frac{\delta S(g)}{\delta g(y)} S^\dagger(g)\right)=0 \mbox{ for } x\le y.$

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. ISBN 079230540X. ISBN 978-0792305408.