34 research outputs found
Constructive pointfree topology eliminates non-constructive representation theorems from Riesz space theory
In Riesz space theory it is good practice to avoid representation theorems
which depend on the axiom of choice. Here we present a general methodology to
do this using pointfree topology. To illustrate the technique we show that
almost f-algebras are commutative. The proof is obtained relatively
straightforward from the proof by Buskes and van Rooij by using the pointfree
Stone-Yosida representation theorem by Coquand and Spitters
Quantum theory realizes all joint measurability graphs
Joint measurability of sharp quantum observables is determined pairwise, and
so can be captured in a graph. We prove the converse: any graph, whose vertices
represent sharp observables, and whose edges represent joint measurability, is
realised by quantum theory. This leads us to show that it is not always
possible to use Neumark dilation to turn unsharp observables into sharp ones
with the same joint measurability relations, highlighting a caveat in the
church of the larger Hilbert space".Comment: 5 page
Topos quantum theory with short posets
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.Comment: 14 page
Tarski monoids: Matui's spatial realization theorem
We introduce a class of inverse monoids, called Tarski monoids, that can be
regarded as non-commutative generalizations of the unique countable, atomless
Boolean algebra. These inverse monoids are related to a class of etale
topological groupoids under a non-commutative generalization of classical Stone
duality and, significantly, they arise naturally in the theory of dynamical
systems as developed by Matui. We are thereby able to reinterpret a theorem of
Matui on a class of \'etale groupoids as an equivalent theorem about a class of
Tarski monoids: two simple Tarski monoids are isomorphic if and only if their
groups of units are isomorphic. The inverse monoids in question may also be
viewed as countably infinite generalizations of finite symmetric inverse
monoids. Their groups of units therefore generalize the finite symmetric groups
and include amongst their number the classical Thompson groups.Comment: arXiv admin note: text overlap with arXiv:1407.147
Bohrification of operator algebras and quantum logic
Following Birkhoff and von Neumann, quantum logic has traditionally been
based on the lattice of closed linear subspaces of some Hilbert space, or, more
generally, on the lattice of projections in a von Neumann algebra A.
Unfortunately, the logical interpretation of these lattices is impaired by
their nondistributivity and by various other problems. We show that a possible
resolution of these difficulties, suggested by the ideas of Bohr, emerges if
instead of single projections one considers elementary propositions to be
families of projections indexed by a partially ordered set C(A) of appropriate
commutative subalgebras of A. In fact, to achieve both maximal generality and
ease of use within topos theory, we assume that A is a so-called Rickart
C*-algebra and that C(A) consists of all unital commutative Rickart
C*-subalgebras of A. Such families of projections form a Heyting algebra in a
natural way, so that the associated propositional logic is intuitionistic:
distributivity is recovered at the expense of the law of the excluded middle.
Subsequently, generalizing an earlier computation for n-by-n matrices, we
prove that the Heyting algebra thus associated to A arises as a basis for the
internal Gelfand spectrum (in the sense of Banaschewski-Mulvey) of the
"Bohrification" of A, which is a commutative Rickart C*-algebra in the topos of
functors from C(A) to the category of sets. We explain the relationship of this
construction to partial Boolean algebras and Bruns-Lakser completions. Finally,
we establish a connection between probability measure on the lattice of
projections on a Hilbert space H and probability valuations on the internal
Gelfand spectrum of A for A = B(H).Comment: 31 page
A topos for algebraic quantum theory
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 Bohr's 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 Scott's 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).Comment: 52 pages, final version, to appear in Communications in Mathematical
Physic
Intuitionistic quantum logic of an n-level system
A decade ago, Isham and Butterfield proposed a topos-theoretic approach to
quantum mechanics, which meanwhile has been extended by Doering and Isham so as
to provide a new mathematical foundation for all of physics. Last year, three
of the present authors redeveloped and refined these ideas by combining the
C*-algebraic approach to quantum theory with the so-called internal language of
topos theory (see arXiv:0709.4364). The goal of the present paper is to
illustrate our abstract setup through the concrete example of the C*-algebra of
complex n by n matrices. This leads to an explicit expression for the pointfree
quantum phase space and the associated logical structure and Gelfand transform
of an n-level system. We also determine the pertinent non-probabilisitic
state-proposition pairing (or valuation) and give a very natural
topos-theoretic reformulation of the Kochen--Specker Theorem. The essential
point is that the logical structure of a quantum n-level system turns out to be
intuitionistic, which means that it is distributive but fails to satisfy the
law of the excluded middle (both in opposition to the usual quantum logic).Comment: 26 page
Piecewise Boolean Algebras and Their Domains
We characterise piecewise Boolean domains, that is, those domains that arise
as Boolean subalgebras of a piecewise Boolean algebra. This leads to equivalent
descriptions of the category of piecewise Boolean algebras: either as piecewise
Boolean domains equipped with an orientation, or as full structure sheaves on
piecewise Boolean domains.Comment: 11 page