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(Never) Mind your p's and q's: Von Neumann versus Jordan on the Foundations of Quantum Theory | | Anthony Duncan
; Michel Janssen
; | | Date: |
29 Apr 2012 | | Abstract: | In early 1927, Pascual Jordan (1927b) published his version of what came to
be known as the Dirac-Jordan statistical transformation theory. Later that year
and partly in response to Jordan, John von Neumann (1927a) published the modern
Hilbert space formalism of quantum mechanics. Central to both formalisms are
expressions for conditional probabilities of finding some value for one
quantity given the value of another. Beyond that Jordan and von Neumann had
different views about the appropriate formulation of problems in the new
theory. For Jordan, unable to let go of the analogy to classical mechanics, the
solution of such problems required the identification of sets of canonically
conjugate variables, i.e., p’s and q’s. Jordan (1927e) ran into serious
difficulties when he tried to extend his approach from quantities with fully
continuous spectra to those with wholly or partly discrete spectra. For von
Neumann, not constrained by the analogy to classical physics and aware of the
daunting mathematical difficulties facing the approach of Jordan (and Dirac
(1927)), the solution of a problem in the new quantum mechanics required only
the identification of a maximal set of commuting operators with simultaneous
eigenstates. He had no need for p’s and q’s. Jordan and von Neumann stated the
characteristic new rules for probabilities in quantum mechanics somewhat
differently. Jordan (1927b) was the first to state those rules in full
generality, von Neumann (1927a) rephrased them and then sought to derive them
from more basic considerations (von Neumann, 1927b). In this paper we
reconstruct the central arguments of these 1927 papers by Jordan and von
Neumann and of a paper on Jordan’s approach by Hilbert, von Neumann, and
Nordheim (1928). We highlight those elements that bring out the gradual
loosening of the ties between the new quantum formalism and classical
mechanics. | | Source: | arXiv, 1204.6511 | | Services: | Forum | Review | PDF | Favorites | |
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