Published online by Cambridge University Press: 24 October 2008
The probability relations which can occur between two separated physical systems are discussed, on the assumption that their state is known by a representative in common. The two families of observables, relating to the first and to the second system respectively, are linked by at least one match between two definite members, one of either family. The word match is short for stating that the values of the two observables in question determine each other uniquely and therefore (since the actual labelling is irrelevant) can be taken to be equal. In general there is but one match, but there can be more. If, in addition to the first match, there is a second one between canonical conjugates of the first mates, then there are infinitely many matches, every function of the first canonical pair matching with the same function of the second canonical pair. Thus there is a complete one-to-one correspondence between those two branches (of the two families of observables) which relate to the two degrees of freedom in question. If there are no others, the one-to-one correspondence persists as time advances, but the observables of the first system (say) change their mates in the way that the latter, i.e. the observables of the second system, undergo a certain continuous contact-transformation.
† The whole mathematical treatment is familiar to mathematicians in dealing with an “unsymmetrical kernel” Ψ (x, y). See Courant-Hilbert, , Methoden der mathematischen Physik, 2nd edition, p. 134.Google Scholar
* In order to adapt this proof to the case when the biorthogonal development is not unique, just replace the biorthogonal development by a particular one, on which you fix your attention.
* To make the earlier text conform to the present simplified wording, replace x 2 + x′ by P and p′ − p 2 by X. Then X and P are canonical conjugates. The mating (x − with P and p − with X) has to be cross-wise, though.
† In fact it persists anyhow, but as a rule in a very much more complicated form.