SMuK 2021 – scientific programme
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AGPhil: Arbeitsgruppe Philosophie der Physik
AGPhil 2: Quantum Theory 1
AGPhil 2.2: Talk
Tuesday, August 31, 2021, 11:45–12:15, H4
The Representation and Determinable Structure of Quantum Properties — •Samuel C. Fletcher and David E. Taylor — University of Minnesota, Twin Cities
Let us begin with a puzzle. Consider an electron with a two-dimensional Hilbert state space, and the properties of having spin in the x- and y-directions, respectively. On the one hand, it is standard to represent these as the Pauli operators σx and σy, whose eigenvalues represent the values of spin-up and spin-down in their respective directions. And it is well-known that these operators do not commute. On the other hand, it is also commonly acknowledged that projection operators, as self-adjoint operators, can also represent these quantities, whose eigenvalues represent the property obtaining or not. But each of these quantities is only plausibly represented by the identity operator on the Hilbert space, and these operators obviously commute. Operators commute iff the properties they represent are compatible. So the spin-x and spin-y properties are both compatible and not compatible: a contradiction. We propose to resolve this puzzle by denying that self-adjoint operators represent properties simpliciter: rather, they represent a determinable property, whose extension is the domain of the operator, plus a particular level of specification with associated determinates, which are named by the eigenvalues. So the different operators in the puzzle actually reflect different levels of specification of one and the same property. Thus it is not the properties of a quantum system which are incompatible in a non-classical way, but rather the levels of specification.