Take the tour ×  Definition 1:  The notion of independent
systems has a precise meaning in probabilities. It states that
the (joint) probability or finding the system ($S_1S_2$) in the
configuration ($C_1C_2$) is equal to the probability of finding
the system ($S_1$) in the configuration ($C_1$) times the
probability of finding the system ($S_2$) in the configuration
($C_2$).  Definition 2:  However, in we consider fields systems,
the practical tool is Lagrangians. So I should say that 2
systems are independent if :  $$ Lagrangian (S_1S_2) =
Lagrangian (S_1) + Lagrangian (S_2)$$  The question:  Now, what
is the relationship between these 2 definitions? They could be
only compatible, or they could be equivalent in the field
domain. Is there a way to "demonstrate" the latter from the
former ?  share|improve this question You might be interested in
this question and its answer by David Bar Moshe. (It doesn't
answer your question exactly, but seems related.) –  Nathaniel
May 4 at 9:59 Yes, It is very interesting,