The confession of John von Neumann

Between the late 1920s and 1936, von Neumann suggested no less than three
mathematical models for non-relativistic finite quantum systems.

The first one starts with the states given by a Hilbert space, and the
observables given by self-adjoint operators on that space. Generally, these
operators are only densely defined and unbounded, however, they are closed.
This model has an advantage that it allows for the Max Born interpretation
of the states, as given by the respective wave functions, in terms of
probability densities on the configuration space in which the quantum system
is situated.

The second model starts with observables, given by elements in a C*-algebra,
and then the states are defined depending on the given C*-algebra. This
model does, so far, not allow for a Max Born type interpretation of states.

The third model, suggested in 1936 in collaboration with George David
Birkhoff, is only concerned with the logical structure of the observables.
In this model one does even less have a Max Born type interpretation.

After that, till his death in 1957, von Neumann never returned to the issue
of mathematical modelling of quantum systems in his publications.

What is hardly known nowadays is that, in a letter to George David Birkhoff,
von Neumann wrote :

"I WOULD LIKE TO MAKE A CONFESSION WHICH MAY SEEM IMMORAL : I DO NOT BELIEVE
IN HILBERT SPACE ANYMORE."

as quoted in :

G.D. Birkhoff, Proceedings of Symposia in Pure Mathematics, Vol. 2, p. 158,
(Ed. R.P. Dilworth), American Mathematical Society, Rhode Island, 1961.

and according to G.D. Birkhoff, the respective letter of von Neumann was
dated 13 November, 1935.




============================================================
>From :
Emeritus Professor Elemer E. Rosinger
Department of Mathematics
and Applied Mathematics
University of Pretoria
Pretoria
0002 South Africa
e-mail address : [EMAIL PROTECTED]
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