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Four and a Half Axioms for Finite Dimensional Quantum Mechanics | | Alexander Wilce
; | | Date: |
30 Dec 2009 | | Abstract: | I discuss a set of strong, but probabilistically intelligible, axioms from
which one can {em almost} derive the appratus of finite dimensional quantum
theory. Stated informally, these require that systems appear completely
classical as restricted to a single measurement, that different measurements,
and likewise different pure states, be equivalent (up to the action of a
compact group of symmetries), and that every state be the marginal of a
bipartite non-signaling state perfectly correlating two measurements.
This much yields a mathematical representation of measurements and states
that is already very suggestive of quantum mechanics. In particular, in any
theory satisfying these axioms, measurements can be represented by orthonormal
subsets of, and states, by vectors in, an ordered real Hilbert space -- in the
quantum case, the space of Hermitian operators, with its usual tracial inner
product. One final postulate (a simple minimization principle, still in need of
a clear interpretation) forces the positive cone of this space to be
homogeneous and self-dual and hence, to be the the state space of a formally
real Jordan algebra. From here, the route to the standard framework of
finite-dimensional quantum mechanics is quite short. | | Source: | arXiv, 0912.5530 | | Services: | Forum | Review | PDF | Favorites | |
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