C*-algebarski formalizam kvantne mehanike

master's thesis

• Title (english): C*-algebraic formalism of quantum mechanics
• Author: Matija Livaić
• Mentor: Ilja Gogić (mentor)
• Mentor: Ivica Smolić (komentor)
• Committee member: Ivica Smolić (predsjednik povjerenstva)
• Committee member: Ilja Gogić (član povjerenstva)
• Committee member: Nikola Poljak (član povjerenstva)
• Committee member: Matko Milin (član povjerenstva)
• Committee member: Tihomir Vukelja (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Physics); Zagreb
• Defense date and country: 2021-09-30, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Physics

Abstract

U ovom radu ponudeni su alternativni aksiomi kvantne mahanike za koje se smatra da imaju bolju fizikalnu osnovu. Svakom kvantnom sistemu pridružuje se separabilna unitalna C*-algebra pri čemu su opservable reprezenetirane hermitskim elementima na toj algebri, a stanja su reprezentirana pozitivnim linearnim funkcionalima norme jedan. Prvo su analizirane nužne konceptualne promjene u razvoju kvantne mehanike koje leže u osnovi formalizma. Potom je operacionalnom definicijom opservabli i stanja izgrađen matematički fomalizam pogodan za opis općenitog fizikalnog sistema. Izloženi su osnovni pojmovi i rezultati teorije C*-algebri. Pokazano je da unošenje dodatnih pretpostavki u formalizam - mogućnost simultanog i preciznog odredivanja svih opservabli - vodi na formalizam klasične mehanike. Pokazano je kako se klasični sistemi opisuju komutativnim C*-algebrama. Potom su izloženi glavni rezultati teorije reprezentacije teorije C*-algebri: GNS konstrukcija, Gelfand-Naimarkov teorem te veza izmedu ireducibilnih reprezentacija i čistih stanja. Iz toga su izvedeni osnovni Dirac-von Neumannovi aksiomi. Naposljetku, proučavali smo dinamiku sistema. Pokazano je da se ona može opisati jednoparametarskim grupama *-automorfizama te je pomoću Stoneovog teorema izvedena Schrödingerova jednadžba.

Abstract (english)

In this paper we offer alternative axioms of quantum mechanics which we consider to have a better physical basis. To each quantum system a separable unital C*-algebra is assigned, the observables are represented by hermitian elements of that algebra, while the states are represented as positive linear functionals of norm one. First we analyzed necessary conceptual changes in the development of quantum mechanics that lie in the roots of the formalism. By defining observables and states operationally, we build a formalism suitable for a description of a general physical system. We present the basic concepts and results from the theory of C*-algebras. We show that by inserting additional assumptions - the possibility of a simultaneous and precise determination of all observables - leads to the formalism of classical mechanics. We show that classical systems are represented by commutative C*-algebras. We then present the main results from the theory of representations of C*-algebras: GNS construction, Gelfand-Naimark’s theorem, and the connection between irreducible representations and pure states. From this we deduce the basic Dirac-von Neumann axioms. Finally, we studied the system’s dynamics. It is shown that it can be described by one-parameter groups of *-automorphisms and with the aid of the Stone’s theorem we derive the Schrödinger’s equation.

• Keywords: aksiomi kvantne mehanike; C*-algebre; opservable; stanja; ireducibilne reprezentacije C*-algebri; čista stanja; Gelfand-Naimarkov teorem; jednoparametarske grupe *-automorfizama.
• Keywords (english): axioms of quantum mechanics; C*-algebras; opservables; states; irreducible representations of C*-algebras; pure states; Gelfand-Naimark theorem; oneparameter groups of *-automorphisms
• Language: croatian
• URN:NBN: urn:nbn:hr:217:349241
• Study programme: Title: Physics; specializations in: Research; Course: Research; Study programme type: university; Study level: integrated undergraduate and graduate; Academic / professional title: magistar/magistra fizike (magistar/magistra fizike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2021-11-10 11:44:40