TY - JOUR
T1 - The Space of Measurement Outcomes as a Spectral Invariant for Non-Commutative Algebras
AU - Spitters, Bas
PY - 2012/7/1
Y1 - 2012/7/1
N2 - The recently developed technique of Bohrification associates to a (unital) C*-algebra A 1. the Kripke model, a presheaf topos, of its classical contexts; 2. in this Kripke model a commutative C*-algebra, called the Bohrification of A; 3. the spectrum of the Bohrification as a locale internal in the Kripke model. We propose this locale, the 'state space', as a (n intuitionistic) logic of the physical system whose observable algebra is A. We compute a site which externally captures this locale and find that externally its points may be identified with partial measurement outcomes. This prompts us to compare Scott-continuity on the poset of contexts and continuity with respect to the C*-algebra as two ways to mathematically identify measurement outcomes with the same physical interpretation. Finally, we consider the not-not-sheafification of the Kripke model on classical contexts and obtain a space of measurement outcomes which for commutative C*-algebras coincides with the spectrum. The construction is functorial on the category of C*-algebras with commutativity reflecting maps.
AB - The recently developed technique of Bohrification associates to a (unital) C*-algebra A 1. the Kripke model, a presheaf topos, of its classical contexts; 2. in this Kripke model a commutative C*-algebra, called the Bohrification of A; 3. the spectrum of the Bohrification as a locale internal in the Kripke model. We propose this locale, the 'state space', as a (n intuitionistic) logic of the physical system whose observable algebra is A. We compute a site which externally captures this locale and find that externally its points may be identified with partial measurement outcomes. This prompts us to compare Scott-continuity on the poset of contexts and continuity with respect to the C*-algebra as two ways to mathematically identify measurement outcomes with the same physical interpretation. Finally, we consider the not-not-sheafification of the Kripke model on classical contexts and obtain a space of measurement outcomes which for commutative C*-algebras coincides with the spectrum. The construction is functorial on the category of C*-algebras with commutativity reflecting maps.
KW - Bohrification
KW - Boolean valued models
KW - Measurement
KW - Sheaves
UR - http://www.scopus.com/inward/record.url?scp=84861581302&partnerID=8YFLogxK
U2 - 10.1007/s10701-011-9619-3
DO - 10.1007/s10701-011-9619-3
M3 - Journal article
AN - SCOPUS:84861581302
SN - 0015-9018
VL - 42
SP - 896
EP - 908
JO - Foundations of Physics
JF - Foundations of Physics
IS - 7
ER -