Quantum Field Theory in Categorical Quantum Mechanics

Abstract

We use tools from non-standard analysis to formulate the building blocks of quantum field theory within the framework of categorical quantum mechanics. Building upon previous work, we construct an object of Hilb having quantum fields as states and we show that the usual ladder and field operators can be defined as suitable endomorphisms. We deal with relativistic normalisation and we obtain the Lorentz invariant Heisenberg picture operators. By moving to a coherent perspective-where the classical time and momentum parameters are replaced by wavefunctions over the parameter spaces-we show that ladder operators and field operators can be obtained by applying the same morphism to plane waves and delta functions respectively. Finally, we formulate the commutation relations diagrammatically and we use them to derive the propagator.

Functorial Evolution of Quantum Fields

“…• The categories Hilb, C*alg, W*alg and vNA have some annoying limitations, so in an infinite-dimensional context one can alternatively work with hyperfinite quantum systems [24], which incorporate infinities and infinitesimals to offer additional features-such as duals, traces and unital Frobenius algebras-over plain Hilbert spaces and C*-algebras.…”

“…Our definition of CCA, on the other hand, has no requirement of discreteness. Furthermore, the freedom left in the choice of field category for a CCA allows us to benefit from the full power of the non-standard approach to categorical quantum mechanics [24,32]. As a consequence, we are able to sketch below a formalization within our framework of the continuous limit for the Dirac QCA, following the same lines as the construction of a (1 + d)-dimensional partitioned CCA above.…”

“…Remark 47: The power to express limiting constructions algebraically, without exiting the original framework, is one of the most attractive aspects of non-standard analysis. The dagger compact category + Hilb (and other categories derived from it) can be used to make categorical sense of constructions from quantum field theory [24,32], including other cellular automata with fieldtheoretic continuous limits. The formulation of such limits within our framework is an point of great interest, but is left to future work.…”

Functorial evolution of quantum fields

“…• The categories Hilb, C*alg, W*alg and vNA have some annoying limitations, so in an infinite-dimensional context one can alternatively work with hyperfinite quantum systems [13], which incorporate infinities and infinitesimals to offer additional featuressuch as duals, traces and unital Frobenius algebras-over plain Hilbert spaces and C*-algebras.…”

Causality in Higher Order Process Theories

“…A potential avenue of future research is to generalise the work of [59] to generate interesting examples of HOPTs beyond higher order causal categories, for example by generalising constructions to infinite dimensional process theories [29,30,47,48,49] and time symmetric operational theories [56], alongside modelling sectorial restrictions [86]. Furthermore there is a similarity between the canonical processes of HOPTs and causal inferential theories [76] to be explored alongside the connection to frameworks designed with the direct purpose of modelling string diagrams with open holes [72,73].…”