A remark on the modal interpretation of algebraic quantum field theory

Rob Clifton

Juan Sebastián Ardenghi, Mario Castagnino, Olímpia Lombardi

Wayne C. Myrvold

Earman and Ruetsche ([2005]) have cast their gaze upon existing no-go theorems for relativistic modal interpretations, and have found them inconclusive. They suggest that it would be more fruitful to investigate modal interpretations proposed for ‘really relativistic theories,’ that is, algebraic relativistic quantum field theories. They investigate the proposal of (Clifton [2000]), and extend Clifton's result that, for a host of states, his proposal yields no definite observables other than multiples of the identity. This leads Earman and Ruetsche to a suspicion that troubles for modal interpretations of such relativistic theories ‘are due less to the Poincaré invariance of relativistic QFT vs. the Galilean invariance of ordinary nonrelativistic QM than to the infinite number of degrees of freedom of former vs. the finite number of degrees of freedom of the latter’ (pp. 577–8). I am skeptical of this suggestion. Though there are troubles for modal interpretations of relativistic quantum field theory that are due to its being a field theory—that is, due to infinitude of the degrees of freedom—they are not the only troubles faced by modal interpretations of quantum theories set in relativistic spacetime; there are also troubles traceable to relativistic causal structure. 1. Introduction 2. Relativity and Causal Structure 3. The Theorems Purified 4. Evolving States in a Relativistic Context

Francisco Calderón

Pieter E. Vermaas

Rob Clifton

Abstract The modal interpretation is actually a family of interpretations sharing the feature that a system’s density operator constrains the possibilities for assigning definite values to its observables (Healey 1989, Dieks & Vermaas 1998). We start with a brief review of the basic idea behind these interpretations from an algebraic point of view (Zimba & Clifton 1998, Halvorson & Clifton 1999). Consider a ‘universe’ comprised of a quantum system U, with finitely many degrees of freedom, represented by a Hilbert space Hu = ®i Hi. At any given time, U will occupy a pure vector state x E Hu that determines a reduced density operator Ds on the Hilbert space Hi of any subsystem S U. Let B(Hs) de­