124 research outputs found
A Gravitational Theory of the Quantum
The synthesis of quantum and gravitational physics is sought through a
finite, realistic, locally causal theory where gravity plays a vital role not
only during decoherent measurement but also during non-decoherent unitary
evolution. Invariant set theory is built on geometric properties of a compact
fractal-like subset of cosmological state space on which the universe is
assumed to evolve and from which the laws of physics are assumed to derive.
Consistent with the primacy of , a non-Euclidean (and hence non-classical)
state-space metric is defined, related to the -adic metric of number
theory where is a large but finite Pythagorean prime. Uncertain states on
are described using complex Hilbert states, but only if their squared
amplitudes are rational and corresponding complex phase angles are rational
multiples of . Such Hilbert states are necessarily -distant from
states with either irrational squared amplitudes or irrational phase angles.
The gappy fractal nature of accounts for quantum complementarity and is
characterised numerically by a generic number-theoretic incommensurateness
between rational angles and rational cosines of angles. The Bell inequality,
whose violation would be inconsistent with local realism, is shown to be
-distant from all forms of the inequality that are violated in any
finite-precision experiment. The delayed-choice paradox is resolved through the
computational irreducibility of . The Schr\"odinger and Dirac equations
describe evolution on in the singular limit at . By contrast,
an extension of the Einstein field equations on is proposed which reduces
smoothly to general relativity as . Novel proposals for
the dark universe and the elimination of classical space-time singularities are
given and experimental implications outlined
Projective completions of Jordan pairs Part II. Manifold structures and symmetric spaces
We define symmetric spaces in arbitrary dimension and over arbitrary
non-discrete topological fields \K, and we construct manifolds and symmetric
spaces associated to topological continuous quasi-inverse Jordan pairs and
-triple systems. This class of spaces, called smooth generalized projective
geometries, generalizes the well-known (finite or infinite-dimensional) bounded
symmetric domains as well as their ``compact-like'' duals. An interpretation of
such geometries as models of Quantum Mechanics is proposed, and particular
attention is paid to geometries that might be considered as "standard models"
-- they are associated to associative continuous inverse algebras and to Jordan
algebras of hermitian elements in such an algebra
Quantum Algorithms from a Linear Algebra Perspective
The field of quantum computing has gained much attention in recent years due to further advances in the development of quantum computers and the recognition that this new paradigm will greatly endanger many modern encryption practices. This paper gives analysis of some of these algorithms, notably Grover’s database search algorithm, and Shor’s factoring and discrete log algorithms, from the perspective of linear algebra. The consequences these have for modern cryptography are discussed, and a brief overview of the current state of the field is given. Without assuming a physics background, this paper aims to provide a self-contained and mathematically rigorous explanation of quantum algorithms for an undergraduate audience
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