Four and a Half Axioms for Finite-Dimensional Quantum Probability

In Yemima Ben-Menahem & Meir Hemmo (eds.), Probability in Physics. Springer. pp. 281--298 (2012)

Abstract

It is an old idea, lately out of fashion but now experiencing a revival, that quantum mechanics may best be understood, not as a physical theory with a problematic probabilistic interpretation, but as something closer to a probability calculus per se. However, from this angle, the rather special C *-algebraic apparatus of quantum probability theory stands in need of further motivation. One would like to find additional principles, having clear physical and/or probabilistic content, on the basis of which this apparatus can be reconstructed. In this paper, I explore one route to such a derivation of finite-dimensional quantum mechanics, by means of a set of strong, but probabilistically intelligible, axioms. Stated very 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 (basic, discrete) measurements as 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 state space of a formally real Jordan algebra. From here, the route to the standard framework of finite-dimensional quantum mechanics is quite short.

Download options

PhilArchive



    Upload a copy of this work     Papers currently archived: 72,694

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Analytics

Added to PP
2014-01-30

Downloads
41 (#280,680)

6 months
1 (#388,311)

Historical graph of downloads
How can I increase my downloads?

References found in this work

No references found.

Add more references

Similar books and articles

Quantum Theory From Four of Hardy's Axioms.Rüdiger Schack - 2003 - Foundations of Physics 33 (10):1461-1468.
Quantum Mechanics on Finite Groups.Stan Gudder - 2006 - Foundations of Physics 36 (8):1160-1192.
Survey of a Quark Model.Stanley P. Gudder - 1982 - Foundations of Physics 12 (11):1041-1055.