Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA topos for algebraic quantum theory
131
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohrs idea that the empirical content of quantum physics is accessible only through classical physics, we show how a C*-algebra of observables A induces a topos T(A) in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*-algebra. According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum S(A) in T(A), which in our approach plays the role of a quantum phase space of the system. Thus we associate a locale (which is the topos-theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a C*-algebra (which is the noncommutative notion of a space). In this setting, states on A become probability measures (more precisely, valuations) on S(A), and self-adjoint elements of A define continuous functions (more precisely, locale maps) from S(A) to Scotts interval domain. Noting that open subsets of S(A) correspond to propositions about the system, the pairing map that assigns a (generalized) truth value to a state and a proposition assumes an extremely simple categorical form. Formulated in this way, the quantum theory defined by A is essentially turned into a classical theory, internal to the topos T(A).
rate research
Read More
Topos quantum mechanics, developed by Isham et. al., creates a topos of presheaves over the poset V(N) of abelian von Neumann subalgebras of the von Neumann algebra N of bounded operators associated to a physical system, and established several results, including: (a) a connection between the Kochen-Specker theorem and the non-existence of a global section of the spectral presheaf; (b) a version of the spectral theorem for self-adjoint operators; (c) a connection between states of N and measures on the spectral presheaf; and (d) a model of dynamics in terms of V(N). We consider a modification to this approach using not the whole of the poset V(N), but only its elements of height at most two. This produces a different topos with different internal logic. However, the core results (a)--(d) established using the full poset V(N) are also established for the topos over the smaller poset, and some aspects simplify considerably. Additionally, this smaller poset has appealing aspects reminiscent of projective geometry.
We provide an algebraic formulation of C.Rovellis relational quantum theory that is based on suitable notions of non-commutative higher operator categories, originally developed in the study of categorical non-commutative geometry. As a way to implement C.Rovellis original intuition on the relational origin of space-time, in the context of our proposed algebraic approach to quantum gravity via Tomita-Takesaki modular theory, we tentatively suggest to use this categorical formalism in order to spectrally reconstruct non-commutative relational space-time geometries from categories of correlation bimodules between operator algebras of observables. Parts of this work are joint collaborations with: Dr.Roberto Conti (Sapienza Universita di Roma), Assoc.Prof.Wicharn Lewkeeratiyutkul (Chulalongkorn University, Bangkok), Dr.Rachel Dawe Martins (Istituto Superior Tecnico, Lisboa), Dr.Matti Raasakka (Paris 13 University), Dr.Noppakhun Suthichitranont.
We interpret ontological models for finite-dimensional quantum theory as functors from the category of finite-dimensional Hilbert spaces and bounded linear maps to the category of measurable spaces and Markov kernels. This uniformises several earlier results, that we analyse more closely: Pusey, Barrett, and Rudolphs result rules out monoidal functors; Leifer and Maroneys result rules out functors that preserve a duality between states and measurement; Aaronson et als result rules out functors that adhere to the Schrodinger equation. We also prove that it is possible to have epistemic functors that take values in signed Markov kernels.
In previous work, we used an $infty$-categorical version of ultraproducts to show that, for a fixed height $n$, the symmetric monoidal $infty$-categories of $E_{n,p}$-local spectra are asymptotically algebraic in the prime $p$. In this paper, we prove the analogous result for the symmetric monoidal $infty$-categories of $K_{p}(n)$-local spectra, where $K_{p}(n)$ is Morava $K$-theory at height $n$ and the prime $p$. This requires $infty$-categorical tools suitable for working with compactly generated symmetric monoidal $infty$-categories with non-compact unit. The equivalences that we produce here are compatible with the equivalences for the $E_{n,p}$-local $infty$-categories.
We derive a uniqueness result for non-Cartesian composition of systems in a large class of process theories, with important implications for quantum theory and linguistics. Specifically, we consider theories of wavefunctions valued in commutative involutive semirings -- as modelled by categories of free finite-dimensional modules -- and we prove that the only bilinear compact-closed symmetric monoidal structure is the canonical one (up to linear monoidal equivalence). Our results apply to conventional quantum theory and other toy theories of interest in the literature, such as real quantum theory, relational quantum theory, hyperbolic quantum theory and modal quantum theory. In computational linguistics they imply that linear models for categorical compositional distributional semantics (DisCoCat) -- such as vector spaces, sets and relations, and sets and histograms -- admit an (essentially) unique compatible pregroup grammar.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
