On the Choice of Algebra for Quantization

Abstract

In this paper, I examine the relationship between physical quantities and physical states in quantum theories. I argue against the claim made by Arageorgis (1995) that the approach to interpreting quantum theories known as Algebraic Imperialism allows for "too many states". I prove a result establishing that the Algebraic Imperialist has very general resources that she can employ to change her abstract algebra of quantities in order to rule out unphysical states.

Deduction and definability in infinite statistical systems

“…See Ruetsche and Earman (2011) and Ruetsche (2011a) for more on the physical interpretation of certain kinds of states. And see Feintzeig (2017a) for more on the relationship between an algebra and its collection of allowed states.…”

Reductive Explanation and the Construction of Quantum Theories

“…In fact, the results of Feintzeig ([2018a]) show us one way in which a choice of physical states might be employed to aid theory construction. Feintzeig shows that if one is given a physical theory formulated in an algebraic framework, and then one specifies a subspace of the state space corresponding to the physical states, there is a general procedure for constructing a new theory that allows for only physical states.…”

“…The subset of physically reasonable states will generate a subspace V of the space A * of linear functionals. Feintzeig ([2018a]) shows that under very general conditions…”

“…I will assume that there is at least a normative ideal, whether or not it is actually achievable, to construct physical theories that allow only for physical states and rule out unphysical states. Feintzeig ([2018a]) establishes a general procedure for adapting quantum theories to allow for only physical states once a physical state space has been specified, but leaves open how one might go about figuring out which states to deem physical. I will argue that what I call the requirement of reductive explanations provides a way of filling this gap and deciding which states to consider as physical.…”

Why Be regular?, part I

“…This means that whether the weak* topology is physically significant is deeply intertwined with whether the Weyl algebra itself has the physical significance that Halvorson attributes to it, as the natural or correct algebra of quantities for a one particle system. We emphasize that the Weyl algebra is not the only possible choice-and more, mathematical physicists and philosophers have presented a number of arguments that it is the wrong choice (see, e.g., Fannes and Verbeure, 1974;Landsman, 1990a,b;Buchholz and Grundling, 2008;Grundling and Neeb, 2009;Feintzeig, 2018a). Moreover, if one were to make a different choice, one would arrive at a different state space, with a different weak* topology, and a 23 Here we echo a point made forcefully by Fletcher (2016), that different choices of topology encode different senses of approximation-and in any particular case, careful attention must be paid to whether a particular topology captures the salient sense.…”