Bogoliubov causality condition

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

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

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

• 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.