2,682 research outputs found
Quantum Picturalism
The quantum mechanical formalism doesn't support our intuition, nor does it
elucidate the key concepts that govern the behaviour of the entities that are
subject to the laws of quantum physics. The arrays of complex numbers are kin
to the arrays of 0s and 1s of the early days of computer programming practice.
In this review we present steps towards a diagrammatic `high-level' alternative
for the Hilbert space formalism, one which appeals to our intuition. It allows
for intuitive reasoning about interacting quantum systems, and trivialises many
otherwise involved and tedious computations. It clearly exposes limitations
such as the no-cloning theorem, and phenomena such as quantum teleportation. As
a logic, it supports `automation'. It allows for a wider variety of underlying
theories, and can be easily modified, having the potential to provide the
required step-stone towards a deeper conceptual understanding of quantum
theory, as well as its unification with other physical theories. Specific
applications discussed here are purely diagrammatic proofs of several quantum
computational schemes, as well as an analysis of the structural origin of
quantum non-locality. The underlying mathematical foundation of this high-level
diagrammatic formalism relies on so-called monoidal categories, a product of a
fairly recent development in mathematics. These monoidal categories do not only
provide a natural foundation for physical theories, but also for proof theory,
logic, programming languages, biology, cooking, ... The challenge is to
discover the necessary additional pieces of structure that allow us to predict
genuine quantum phenomena.Comment: Commissioned paper for Contemporary Physics, 31 pages, 84 pictures,
some colo
A universe of processes and some of its guises
Our starting point is a particular `canvas' aimed to `draw' theories of
physics, which has symmetric monoidal categories as its mathematical backbone.
In this paper we consider the conceptual foundations for this canvas, and how
these can then be converted into mathematical structure. With very little
structural effort (i.e. in very abstract terms) and in a very short time span
the categorical quantum mechanics (CQM) research program has reproduced a
surprisingly large fragment of quantum theory. It also provides new insights
both in quantum foundations and in quantum information, and has even resulted
in automated reasoning software called `quantomatic' which exploits the
deductive power of CQM. In this paper we complement the available material by
not requiring prior knowledge of category theory, and by pointing at
connections to previous and current developments in the foundations of physics.
This research program is also in close synergy with developments elsewhere, for
example in representation theory, quantum algebra, knot theory, topological
quantum field theory and several other areas.Comment: Invited chapter in: "Deep Beauty: Understanding the Quantum World
through Mathematical Innovation", H. Halvorson, ed., Cambridge University
Press, forthcoming. (as usual, many pictures
Higher Structures in M-Theory
The key open problem of string theory remains its non-perturbative completion
to M-theory. A decisive hint to its inner workings comes from numerous
appearances of higher structures in the limits of M-theory that are already
understood, such as higher degree flux fields and their dualities, or the
higher algebraic structures governing closed string field theory. These are all
controlled by the higher homotopy theory of derived categories, generalised
cohomology theories, and -algebras. This is the introductory chapter
to the proceedings of the LMS/EPSRC Durham Symposium on Higher Structures in
M-Theory. We first review higher structures as well as their motivation in
string theory and beyond. Then we list the contributions in this volume,
putting them into context.Comment: 22 pages, Introductory Article to Proceedings of LMS/EPSRC Durham
Symposium Higher Structures in M-Theory, August 2018, references update
Loops and Knots as Topoi of Substance. Spinoza Revisited
The relationship between modern philosophy and physics is discussed. It is
shown that the latter develops some need for a modernized metaphysics which
shows up as an ultima philosophia of considerable heuristic value, rather than
as the prima philosophia in the Aristotelian sense as it had been intended, in
the first place. It is shown then, that it is the philosophy of Spinoza in
fact, that can still serve as a paradigm for such an approach. In particular,
Spinoza's concept of infinite substance is compared with the philosophical
implications of the foundational aspects of modern physical theory. Various
connotations of sub-stance are discussed within pre-geometric theories,
especially with a view to the role of spin networks within quantum gravity. It
is found to be useful to intro-duce a separation into physics then, so as to
differ between foundational and empirical theories, respectively. This leads to
a straightforward connection bet-ween foundational theories and speculative
philosophy on the one hand, and between empirical theories and sceptical
philosophy on the other. This might help in the end, to clarify some recent
problems, such as the absence of time and causality at a fundamental level. It
is implied that recent results relating to topos theory might open the way
towards eventually deriving logic from physics, and also towards a possible
transition from logic to hermeneutic.Comment: 42 page
An alternative Gospel of structure: order, composition, processes
We survey some basic mathematical structures, which arguably are more
primitive than the structures taught at school. These structures are orders,
with or without composition, and (symmetric) monoidal categories. We list
several `real life' incarnations of each of these. This paper also serves as an
introduction to these structures and their current and potentially future uses
in linguistics, physics and knowledge representation.Comment: Introductory chapter to C. Heunen, M. Sadrzadeh, and E. Grefenstette.
Quantum Physics and Linguistics: A Compositional, Diagrammatic Discourse.
Oxford University Press, 201
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