Algebras of observables of nearly canonical physical theories. II

G. K. Tolokonnikov

Vasily Evgenevich Tarasov

Bob Coecke, Ross Duncan

This paper has two tightly intertwined aims: (i) to introduce an intuitive and universal graphical calculus for multi-qubit systems, the ZX -calculus, which greatly simplifies derivations in the area of quantum computation and information. (ii) To axiomatize complementarity of quantum observables within a general framework for physical theories in terms of dagger symmetric monoidal categories. We also axiomatize phase shifts within this framework. Using the well-studied canonical correspondence between graphical calculi and dagger symmetric monoidal categories, our results provide a purely graphical formalisation of complementarity for quantum observables. Each individual observable, represented by a commutative special dagger Frobenius algebra, gives rise to an Abelian group of phase shifts, which we call the phase group. We also identify a strong form of complementarity, satisfied by the Z - and X -spin observables, which yields a scaled variant of a bialgebra.

Tracy Lupher

This dissertation gives a general account of the properties of unitarily inequivalent representations (UIRs) in both canonical quantum field theory and algebraic quantum field theory. A simple model is constructed and then used to show how to build a broad spectrum of UIRs including a version of Haag’s theorem. Haag and Kastler,P, two of the founding fathers of algebraic quantum field theory, argue that the problems posed by UIRs are solved by adopting a notion of equivalence that is weaker than unitary equivalence, which they refer to as physical equivalence. In the dissertation, it is shown that their notion does not provide a suitable classificatory schema. Some of the most important physical representations fail to satisfy the mathematical conditions of their notion. However, Haag and Kastler's notion has an unexpected connection with classical observables. A theorem is proven in which two representations make the same predictions with respect to all classical observables if and only if they satisfy their notion of physical equivalence. Following Haag and Kastler's lead, it was claimed by most proponents of algebraic quantum field theory that all physical content resides in a specific class of observables. It is shown in the dissertation that such claims are exaggerated and misleading. UIRs are used to elucidate the nature of quantum field theory by showing that UIRs have different expectation values for some classical observables of the system, such as temperature and chemical potential, which are not in Haag and Kastler’s specific class. It is shown how UIRs may be used to construct classical observables. To capture the physical content of quantum field theory it is shown that a much larger algebra than that of Haag and Kastler is necessary. Finally, the arguments that UIRs are incommensurable theories are shown to be flawed. The lesson of UIRs is that the mathematical structures in both canonical quantum field theory and Haag and Kastler’s version of algebraic quantum field theory are not sufficient to capture all of the physical content that UIRs represent. A suitable algebraic structure for quantum field theory is provided in the dissertation.

Peter Bongaarts

Robert Olkiewicz