Developments in unitary quantum theory

Thesis

This thesis describes advancements in unitary quantum theory. These can be divided into three main topics: Everett’s relative-state formalism, the Page–Wootters construction, and unorthodox quantum theory.

Everett’s relative-state formalism allows one to express the state of one quantum system relative to the state of another with which it is entangled. This formalism has only recently been translated to the Heisenberg picture by myself & Deutsch (2021). Our construction makes explicit the locality of Everettian multiplicity and demonstrates that the relative-states of a quantum system are fully fledged quantum systems in their own right. For instance, the observables of a relative state satisfy an appropriate form of a system’s characteristic algebra, and the relative states can store locally inaccessible information, just like the absolute system.

The Heisenberg-picture relative-state formalism has been used by me (Kuypers (2022a)) to refine a construction due to Page & Wootters (1983). In their construction, it is shown that an external time parameter is unnecessary for explaining change and dynamics in quantum theory because one can instead rely on ‘clock time’ – i.e. the time as represented by a clock. This makes time manifestly physical: clock time is an observable attribute of a clock, whereas the time parameter is not an attribute of anything physical. I have shown (Kuypers (2022a)) that the Page–Wootters construction is a ‘calculus for q-numbers’; that is to say that within that construction, one can define functions of q-numbers (which can be represented by matrices or operators on a Hilbert space) as well as derivatives of such functions.

Both the Page–Wootters construction and Everett’s relative state formalism are here treated in the Heisenberg picture because that picture has the crucial advantage that it is a local description of quantum theory (see §2.5). Therefore, the Heisenberg picture has been used by me (Kuypers (2022b)) to describe quantum systems in chronology-violating regions of spacetime (i.e. regions of spacetime that contain closed timelike curves). The presence of closed timelike curves makes one of the axioms of ‘orthodox’ quantum theory problematic – namely, the axiom that the observables of spacelike-separated systems commute. In unorthodox quantum theory, that axiom is dropped. The foundations of unorthodox quantum theory are presented, and I demonstrate that this theory provides a resolution to the so-called grandfather paradox.

Authors: Kuypers, S

• DOI: 10.5287/ora-xkm5zbr7x
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: heisenberg picture; everett; unitary quantum theory; relative-state formalism; everettian quantum theory; unorthodox quantum theory; chronology-violating regions; deutsch hayden descriptors; closed timelike curves; quantum multiverse
• Subjects: Quantum information theory; Quantum computing; Foundations of quantum theory; Physics