Categorical Semantics for Schrödinger's Equation

Abstract

Applying ideas from monadic dynamics to the wellestablished framework of categorical quantum mechanics, we provide a novel toolbox for the simulation of finite-dimensional quantum dynamics. We use strongly complementary structures to give a graphical characterisation of quantum clocks, their action on systems and the relevant energy observables, and we proceed to formalise the connection between unitary dynamics and projection-valued spectra. We identify the Weyl canonical commutation relations in the axioms of strong complementarity, and conclude the existence of a dual pair of time/energy observables for finite-dimensional quantum clocks, with the relevant uncertainty principle given by mutual unbias of the corresponding orthonormal bases. We show that Schrödinger's equation can be abstractly formulated as characterising the Fourier transforms of certain Eilenberg-Moore morphisms from a quantum clock to a quantum dynamical system, and we use this to obtain a generalised version of the Feynman's clock construction. We tackle the issue of synchronism of clocks and systems, prove conservation of total energy and give conditions for the existence of an internal time observable for a quantum dynamical system. Finally, we identify our treatment as part of a more general theory of simulated symmetries of quantum systems (of which our clock actions are a special case) and their conservation laws (of which energy is a special case).

Mermin Non-Locality in Abstract Process Theories

“…by and . A more familiar presentation of strongly complementary pairs can be given by observing that they correspond (when both structures have enough classical points to form a basis) to pairs of non-degenerate observables obeying the finitedimensional Weyl form of the Canonical Commutation Relations [16]. Also, we have the following characterisation of strong complementarity in terms of group actions on classical points.…”

Infinite-dimensional Categorical Quantum Mechanics

“…From a countable basis of momentum eigenstates, we define the position eigenstates as Dirac deltas, and construct the position and momentum observables as a unital special commutative †-Frobenius algebras. Furthermore, we show these observables to be strongly complementary: this is the categorical counterpart of the Weyl canonical commutation relations, and opens the way to future applications of the formalism to infinite-dimensional quantum symmetries and dynamics (within the framework of [7]).…”

“…Finally, in the finite-dimensional case it is known [7] that strong complementarity [4,8] corresponds to the Weyl canonical commutation relations, so we expect the classical structures for momentum and position eigenstates to be strongly complementary. This is indeed the case.…”

“…The following theorem shows the relationship between momentum eigenstates and position-space translation in our framework, and relates it to the special case of Equation 4.8. This more general relationship can be extended to cases (such as the finite-dimensional ones of [7]) where momenta cannot be valued in a subset of the reals.…”

Can category-theoretic semantics resolve the problem of the interpretation of the quantum state vector?

“…Let us denote elements of a physical theory as X and elements of the physical reality as Y . Then, using the language of category theory (in a spirit loosely related to the categorical approach to theories of physical systems [2,3] or to the program of categorical quantum mechanics [4,5,6]), the equivalence between the theory and the reality can be expressed as an isomorphism Y ≃ X , i.e., a one-to-one relation (morphism f ) assigning to each element of the reality Y an element of the theory X f : Y → X (1) that can be "undone" in the sense that there is another one-to-one relation (morphism g)…”