  In this paper the notion of an EPR state for the composite S of two quantum
systems S1, S2, relative to S2 and a set O of bounded observables of S2, is
introduced in the spirit of classical examples of Einstein-Podolsky-Rosen and
Bohm. We restrict ourselves mostly to EPR states of finite norm. The main
results are contained in Theorem 3,4,5,6 in section III and imply that if the
EPR states of finite norm relative to (S2, O) exist, then the elements of O
have discrete probability distributions and the Von Neuman algebra generated by
O is essentially inbeddable inside S1 by an antiunitary map. The EPR states
then correspond to the different imbeddings and certain additional parameters,
and are explicitely given by formulae which generalize the famous example of
Bohm. If O generates all bounded observables, S2 must be of finite dimension
and can be imbedded inside S1 by an antiunitary map, and the EPR states
relative to S2 are then in canonical bijection with the different imbeddings of
S2 inside S1; moreover they are given by formulae which are exactly those of
the generalized Bohm states. The notion of EPR states of infinite norm is also
explored and it is shown that the original state of Einstein-Podolsky-Rosen can
be realized as a renormalized limit of EPR states of finite quantum systems
considered by Weyl, Schwinger and many others. Finally, a family of states of
infinite norm generalizing the Einstein-Podolsky-Rosen example is explicitly
given.
