Bogoliubov causality condition

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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.


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.


  • 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.