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