108 research outputs found
Moduli stacks of algebraic structures and deformation theory
We connect the homotopy type of simplicial moduli spaces of algebraic
structures to the cohomology of their deformation complexes. Then we prove that
under several assumptions, mapping spaces of algebras over a monad in an
appropriate diagram category form affine stacks in the sense of Toen-Vezzosi's
homotopical algebraic geometry. This includes simplicial moduli spaces of
algebraic structures over a given object (for instance a cochain complex). When
these algebraic structures are parametrised by properads, the tangent complexes
give the known cohomology theory for such structures and there is an associated
obstruction theory for infinitesimal, higher order and formal deformations. The
methods are general enough to be adapted for more general kinds of algebraic
structures.Comment: several corrections, especially in sections 6 and 7. Final version,
to appear in the J. Noncommut. Geo
Categorical Quantum Dynamics
We use strong complementarity to introduce dynamics and symmetries within the
framework of CQM, which we also extend to infinite-dimensional separable
Hilbert spaces: these were long-missing features, which open the way to a
wealth of new applications. The coherent treatment presented in this work also
provides a variety of novel insights into the dynamics and symmetries of
quantum systems: examples include the extremely simple characterisation of
symmetry-observable duality, the connection of strong complementarity with the
Weyl Canonical Commutation Relations, the generalisations of Feynman's clock
construction, the existence of time observables and the emergence of quantum
clocks.
Furthermore, we show that strong complementarity is a key resource for
quantum algorithms and protocols. We provide the first fully diagrammatic,
theory-independent proof of correctness for the quantum algorithm solving the
Hidden Subgroup Problem, and show that strong complementarity is the feature
providing the quantum advantage. In quantum foundations, we use strong
complementarity to derive the exact conditions relating non-locality to the
structure of phase groups, within the context of Mermin-type non-locality
arguments. Our non-locality results find further application to quantum
cryptography, where we use them to define a quantum-classical secret sharing
scheme with provable device-independent security guarantees.
All in all, we argue that strong complementarity is a truly powerful and
versatile building block for quantum theory and its applications, and one that
should draw a lot more attention in the future.Comment: Thesis submitted for the degree of Doctor of Philosophy, Oxford
University, Michaelmas Term 2016 (273 pages
The Absolute Relativity Theory
This paper is a first presentation of a new approach of physics that we
propose to refer as the Absolute Relativity Theory (ART) since it refutes the
idea of a pre-existing space-time. It includes an algebraic definition of
particles, interactions and Lagrangians. It proposed also a purely algebraic
explanation of the passing of time phenomenon that leads to see usual
Euler-Lagrange equations as the continuous version of the
Knizhnik-Zamolodchikov monodromy. The identification of this monodromy with the
local ones of the Lorentzian manifolds gives the Einstein equation
algebraically explained in a quantized context. A fact that could lead to the
unification of physics. By giving an algebraic classification of particles and
interactions, the ART also proposes a new branch of physics, namely the Mass
Quantification Theory, that provides a general method to calculate the
characteristics of particles and interactions. Some examples are provided. The
MQT also predicts the existence of as of today not yet observed particles that
could be part of the dark matter. By giving a new interpretation of the weak
interaction, it also suggests an interpretation of the so-called dark energy
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