11 research outputs found
Microscopic description of 2d topological phases, duality and 3d state sums
Doubled topological phases introduced by Kitaev, Levin and Wen supported on
two dimensional lattices are Hamiltonian versions of three dimensional
topological quantum field theories described by the Turaev-Viro state sum
models. We introduce the latter with an emphasis on obtaining them from
theories in the continuum. Equivalence of the previous models in the ground
state are shown in case of the honeycomb lattice and the gauge group being a
finite group by means of the well-known duality transformation between the
group algebra and the spin network basis of lattice gauge theory. An analysis
of the ribbon operators describing excitations in both types of models and the
three dimensional geometrical interpretation are given.Comment: 19 pages, typos corrected, style improved, a final paragraph adde
Quantum Gravity: Motivations and Alternatives
The mutual conceptual incompatibility between GR and QM/QFT is generally seen
as the most essential motivation for the development of a theory of Quantum
Gravity (QG). It leads to the insight that, if gravity is a fundamental
interaction and QM is universally valid, the gravitational field will have to
be quantized, not at least because of the inconsistency of semi-classical
theories of gravity. If this means to quantize GR, its identification of the
gravitational field with the spacetime metric has to be taken into account. And
the resulting quantum theory has to be background-independent. This can not be
achieved by means of quantum field theoretical procedures. More sophisticated
strategies have to be applied. One of the basic requirements for such a
quantization strategy is that the resulting quantum theory has GR as a
classical limit. - However, should gravity not be a fundamental, but an
residual, emergent interaction, it could very well be an intrinsically
classical phenomenon. Should QM be nonetheless universally valid, we had to
assume a quantum substrate from which gravity would result as an emergent
classical phenomenon. And there would be no conflict with the arguments against
semi-classical theories, because there would be no gravity at all on the
substrate level. The gravitational field would not have any quantum properties,
and a quantization of GR would not lead to any fundamental theory. The
objective of a theory of 'QG' would instead be the identification of the
quantum substrate from which gravity results. - The paper tries to give an
overview over the main options for theory construction in the field of QG.
Because of the still unclear status of gravity and spacetime, it pleads for the
necessity of a plurality of conceptually different approaches to QG.Comment: 32 page
Communication protocols and quantum error-correcting codes from the perspective of topological quantum field theory
Topological quantum field theories (TQFTs) provide a general,
minimal-assumption language for describing quantum-state preparation and
measurement. They therefore provide a general language in which to express
multi-agent communication protocols, e.g. local operations, classical
communication (LOCC) protocols. Here we construct LOCC protocols using TQFT,
and show that LOCC protocols induce quantum error-correcting codes (QECCs) on
the agent-environment boundary. Such QECCs can be regarded as implementing, or
inducing the emergence of, spacetimes on such boundaries. We investigate this
connection between inter-agent communication and spacetime using BF and
Chern-Simons theories, and then using topological M-theory.Comment: 52 page
Quantum gravity: motivations and alternatives
The mutual conceptual incompatibility between General Relativity and Quantum Mechanics / Quantum Field Theory is generally seen as the most essential motivation for the development of a theory of Quantum Gravity. It leads to the insight that, if gravity is a fundamental interaction and Quantum Mechanics is universally valid, the gravitational field will have to be quantized, not at least because of the inconsistency of semi-classical theories of gravity. The objective of a theory of Quantum Gravity would then be to identify the quantum properties and the quantum dynamics of the gravitational field. If this means to quantize General Relativity, the general-relativistic identification of the gravitational field with the spacetime metric has to be taken into account. The quantization has to be conceptually adequate, which means in particular that the resulting quantum theory has to be background-independent. This can not be achieved by means of quantum field theoretical procedures. More sophisticated strategies, like those of Loop Quantum Gravity, have to be applied. One of the basic requirements fo
A Panorama Of Physical Mathematics c. 2022
What follows is a broad-brush overview of the recent synergistic interactions
between mathematics and theoretical physics of quantum field theory and string
theory. The discussion is forward-looking, suggesting potentially useful and
fruitful directions and problems, some old, some new, for further development
of the subject. This paper is a much extended version of the Snowmass
whitepaper on physical mathematics [1]
This Week's Finds in Mathematical Physics (1-50)
These are the first 50 issues of This Week's Finds of Mathematical Physics,
from January 19, 1993 to March 12, 1995. These issues focus on quantum gravity,
topological quantum field theory, knot theory, and applications of
-categories to these subjects. However, there are also digressions into Lie
algebras, elliptic curves, linear logic and other subjects. They were typeset
in 2020 by Tim Hosgood. If you see typos or other problems please report them.
(I already know the cover page looks weird).Comment: 242 page
Notes in Pure Mathematics & Mathematical Structures in Physics
These Notes deal with various areas of mathematics, and seek reciprocal
combinations, explore mutual relations, ranging from abstract objects to
problems in physics.Comment: Small improvements and addition