David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
International Journal of Theoretical Physics 38:2441-2484 (1999)
The centerpiece of Jeffrey Bub's book Interpreting the Quantum World is a theorem (Bub and Clifton 1996) which correlates each member of a large class of no-collapse interpretations with some 'privileged observable'. In particular, the Bub-Clifton theorem determines the unique maximal sublattice L(R,e) of propositions such that (a) elements of L(R,e) can be simultaneously determinate in state e, (b) L(R,e) contains the spectral projections of the privileged observable R, and (c) L(R,e) is picked out by R and e alone. In this paper, we explore the issue of maximal determinate sets of observables using the tools provided by the algebraic approach to quantum theory; and we call the resulting algebras of determinate observables, "maximal *beable* subalgebras". The capstone of our exploration is a generalized version of Bub and Clifton's theorem that applies to arbitrary (i.e., both mixed and pure) quantum states, to Hilbert spaces of arbitrary (i.e., both finite and infinite) dimension, and to arbitrary observables (including those with a continuous spectrum). Moreover, in the special case covered by the original Bub-Clifton theorem, our theorem reproduces their result under strictly weaker assumptions. This added level of generality permits us to treat several topics that were beyond the reach of the original Bub-Clifton result. In particular: (a) We show explicitly that a (non-dynamical) version of the Bohm theory can be obtained by granting privileged status to the position observable. (b) We show that Clifton's (1995) characterization of the Kochen-Dieks modal interpretation is a corollary of our theorem in the special case when the density operator is taken as the privileged observable. (c) We show that the 'uniqueness' demonstrated by Bub and Clifton is only guaranteed in certain special cases -- viz., when the quantum state is pure, or if the privileged observable is compatible with the density operator. We also use our results to articulate a solid mathematical foundation for certain tenets of the orthodox Copenhagen interpretation of quantum theory. For example, the uncertainty principle asserts that there are strict limits on the precision with which we can know, simultaneously, the position and momentum of a quantum-mechanical particle. However, the Copenhagen interpretation of this fact is not simply that a precision momentum measurement necessarily and uncontrollably disturbs the value of position, and vice-versa; but that position and momentum can never in reality be thought of as simultaneously determinate. We provide warrant for this stronger 'indeterminacy principle' by showing that there is no quantum state that assigns a sharp value to both position and momentum; and, a fortiori, that it is mathematically impossible to construct a beable algebra that contains both the position observable and the momentum observable. We also prove a generalized version of the Bub-Clifton theorem that applies to "singular" states (i.e., states that arise from non-countably-additive probability measures, such as Dirac delta functions). This result allows us to provide a mathematically rigorous reconstruction of Bohr's response to the original EPR argument -- which makes use of a singular state. In particular, we show that if the position of the first particle is privileged (e.g., as Bohr would do in a position measuring context), the position of the second particle acquires a definite value by virtue of lying in the corresponding maximal beable subalgebra. But then (by the indeterminacy principle) the momentum of the second particle is not a beable; and EPR's argument for the simultaneous reality of both position and momentum is undercut.
|Keywords||No keywords specified (fix it)|
|Categories||categorize this paper)|
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
|Through your library|
References found in this work BETA
No references found.
Citations of this work BETA
Laura Ruetsche (2011). Why Be Normal? Studies in History and Philosophy of Science Part B 42 (2):107-115.
Similar books and articles
Hans Halvorson (2001). On the Nature of Continuous Physical Quantities in Classical and Quantum Mechanics. Journal of Philosophical Logic 30 (1):27-50.
Paul Busch & Pekka J. Lahti (1985). A Note on Quantum Theory, Complementarity, and Uncertainty. Philosophy of Science 52 (1):64-77.
Thomas Schürmann, About Conditional Probabilities of Events Regarding the Quantum Mechanical Measurement Process.
Amit Hagar & Meir Hemmo (2006). Explaining the Unobserved: Why Quantum Theory Ain't Only About Information. Foundations of Physics 36 (9):1295-1234.
Jeffrey Bub (1991). The Problem of Properties in Quantum Mechanics. Topoi 10 (1):27-34.
J. Bub, R. Clifton & S. Goldstein (2000). Revised Proof of the Uniqueness Theorem for 'No Collapse' Interpretations of Quantum Mechanics. Studies in History and Philosophy of Science Part B 31 (1):95-98.
Jeffrey Bub (1995). Interference, Noncommutativity, and Determinateness in Quantum Mechanics. Topoi 14 (1):39-43.
Yuichiro Kitajima (2004). A Remark on the Modal Interpretation of Algebraic Quantum Field Theory. Physics Letters A 331 (3-4):181-186.
Yuichiro Kitajima (2005). Interpretations of Quantum Mechanics in Terms of Beable Algebras. International Journal of Theoretical Physics 44 (8):1141-1156.
J. Bub & R. Clifton (1996). A Uniqueness Theorem for 'No Collapse' Interpretations of Quantum Mechanics. Studies in History and Philosophy of Science Part B 27 (2):181-219.
Added to index2009-01-28
Total downloads12 ( #150,141 of 1,692,596 )
Recent downloads (6 months)3 ( #75,733 of 1,692,596 )
How can I increase my downloads?