353 research outputs found
Elaboration in Dependent Type Theory
To be usable in practice, interactive theorem provers need to provide
convenient and efficient means of writing expressions, definitions, and proofs.
This involves inferring information that is often left implicit in an ordinary
mathematical text, and resolving ambiguities in mathematical expressions. We
refer to the process of passing from a quasi-formal and partially-specified
expression to a completely precise formal one as elaboration. We describe an
elaboration algorithm for dependent type theory that has been implemented in
the Lean theorem prover. Lean's elaborator supports higher-order unification,
type class inference, ad hoc overloading, insertion of coercions, the use of
tactics, and the computational reduction of terms. The interactions between
these components are subtle and complex, and the elaboration algorithm has been
carefully designed to balance efficiency and usability. We describe the central
design goals, and the means by which they are achieved
Mini-Workshop: Product Systems and Independence in Quantum Dynamics
Quantum dynamics, both reversible (i.e., closed quantum systems) and irreversible (i.e., open quantum systems), gives rise to product systems of Hilbert spaces or, more generally, of Hilbert modules. When we consider reversible dynamics that dilates an irreversible dynamics, then the product system of the latter is equal to the product system of the former (or is contained in a unique way). Whenever the dynamics is on a proper subalgebra of the algebra of all bounded operators on a Hilbert space, in particular, when the open system is classical (commutative) it is indispensable that we use Hilbert modules. The product system of a reversible dynamics is intimately related to a filtration of subalgebras that are independent in a state or conditionally independent in a conditional expectation of the reversible system. This has been illustrated in many concrete dilations that have been obtained with the help of quantum stochastic calculus. Here the underlying Fock space or module determines the sort of quantum independence underlying the reversible system. The mini-workshop brought together experts from quantum dynamics, product systems and quantum independence who have contributed to the general theory or who have studied intriguing examples. As the implications of the tight relationship between product systems and independence had so far been largely neglected, we expect from our mini-workshop a strong innovative impulse to this field
A perspective on non-commutative frame theory
This paper extends the fundamental results of frame theory to a
non-commutative setting where the role of locales is taken over by \'etale
localic categories. This involves ideas from quantale theory and from semigroup
theory, specifically Ehresmann semigroups, restriction semigroups and inverse
semigroups. We establish a duality between the category of complete restriction
monoids and the category of \'etale localic categories. The relationship
between monoids and categories is mediated by a class of quantales called
restriction quantal frames. This result builds on the work of Pedro Resende on
the connection between pseudogroups and \'etale localic groupoids but in the
process we both generalize and simplify: for example, we do not require
involutions and, in addition, we render his result functorial. We also project
down to topological spaces and, as a result, extend the classical adjunction
between locales and topological spaces to an adjunction between \'etale localic
categories and \'etale topological categories. In fact, varying morphisms, we
obtain several adjunctions. Just as in the commutative case, we restrict these
adjunctions to spatial-sober and coherent-spectral equivalences. The classical
equivalence between coherent frames and distributive lattices is extended to an
equivalence between coherent complete restriction monoids and distributive
restriction semigroups. Consequently, we deduce several dualities between
distributive restriction semigroups and spectral \'etale topological
categories. We also specialize these dualities for the setting where the
topological categories are cancellative or are groupoids. Our approach thus
links, unifies and extends the approaches taken in the work by Lawson and Lenz
and by Resende.Comment: 69 page
Foundations for Relativistic Quantum Theory I: Feynman's Operator Calculus and the Dyson Conjectures
In this paper, we provide a representation theory for the Feynman operator
calculus. This allows us to solve the general initial-value problem and
construct the Dyson series. We show that the series is asymptotic, thus proving
Dyson's second conjecture for QED. In addition, we show that the expansion may
be considered exact to any finite order by producing the remainder term. This
implies that every nonperturbative solution has a perturbative expansion. Using
a physical analysis of information from experiment versus that implied by our
models, we reformulate our theory as a sum over paths. This allows us to relate
our theory to Feynman's path integral, and to prove Dyson's first conjecture
that the divergences are in part due to a violation of Heisenberg's uncertainly
relations
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