66 research outputs found
Supersymmetric Field-Theoretic Models on a Supermanifold
We propose the extension of some structural aspects that have successfully
been applied in the development of the theory of quantum fields propagating on
a general spacetime manifold so as to include superfield models on a
supermanifold. We only deal with the limited class of supermanifolds which
admit the existence of a smooth body manifold structure. Our considerations are
based on the Catenacci-Reina-Teofillatto-Bryant approach to supermanifolds. In
particular, we show that the class of supermanifolds constructed by
Bonora-Pasti-Tonin satisfies the criterions which guarantee that a
supermanifold admits a Hausdorff body manifold. This construction is the
closest to the physicist's intuitive view of superspace as a manifold with some
anticommuting coordinates, where the odd sector is topologically trivial. The
paper also contains a new construction of superdistributions and useful results
on the wavefront set of such objects. Moreover, a generalization of the
spectral condition is formulated using the notion of the wavefront set of
superdistributions, which is equivalent to the requirement that all of the
component fields satisfy, on the body manifold, a microlocal spectral condition
proposed by Brunetti-Fredenhagen-K\"ohler.Comment: Final version to appear in J.Math.Phy
Fault-tolerance in two-dimensional topological systems
This thesis is a collection of ideas with the general goal of building, at least in the abstract, a local fault-tolerant quantum computer. The connection between quantum information and topology has proven to be an active area of research in several fields. The introduction of the toric code by Alexei Kitaev demonstrated the usefulness of topology for quantum memory and quantum computation. Many quantum codes used for quantum memory are modeled by spin systems on a lattice, with operators that extract syndrome information placed on vertices or faces of the lattice. It is natural to wonder whether the useful codes in such systems can be classified. This thesis presents work that leverages ideas from topology and graph theory to explore the space of such codes. Homological stabilizer codes are introduced and it is shown that, under a set of reasonable assumptions, any qubit homological stabilizer code is equivalent to either a toric code or a color code. Additionally, the toric code and the color code correspond to distinct classes of graphs. Many systems have been proposed as candidate quantum computers. It is very desirable to design quantum computing architectures with two-dimensional layouts and low complexity in parity-checking circuitry. Kitaev\u27s surface codes provided the first example of codes satisfying this property. They provided a new route to fault tolerance with more modest overheads and thresholds approaching 1%. The recently discovered color codes share many properties with the surface codes, such as the ability to perform syndrome extraction locally in two dimensions. Some families of color codes admit a transversal implementation of the entire Clifford group. This work investigates color codes on the 4.8.8 lattice known as triangular codes. I develop a fault-tolerant error-correction strategy for these codes in which repeated syndrome measurements on this lattice generate a three-dimensional space-time combinatorial structure. I then develop an integer program that analyzes this structure and determines the most likely set of errors consistent with the observed syndrome values. I implement this integer program to find the threshold for depolarizing noise on small versions of these triangular codes. Because the threshold for magic-state distillation is likely to be higher than this value and because logical CNOT gates can be performed by code deformation in a single block instead of between pairs of blocks, the threshold for fault-tolerant quantum memory for these codes is also the threshold for fault-tolerant quantum computation with them. Since the advent of a threshold theorem for quantum computers much has been improved upon. Thresholds have increased, architectures have become more local, and gate sets have been simplified. The overhead for magic-state distillation has been studied, but not nearly to the extent of the aforementioned topics. A method for greatly reducing this overhead, known as reusable magic states, is studied here. While examples of reusable magic states exist for Clifford gates, I give strong reasons to believe they do not exist for non-Clifford gates
Anti-Newtonian Expansions, Hadamard States, and the Spatial Functional Renormalization Group
The validity of general relativity almost up to the Big Bang entails that the Einstein equations themselves can be used to study the detailed structure of spacetime in the vicinity of the singularity. Within the cosmological paradigm of a Friedmann-Lema\^{i}tre spacetime, where a description of physics in terms of quantum field theories (QFTs) on such a curved background is deemed to be valid, this mandates the existence of a pre-inflationary epoch following the Big Bang. Accepting this physically well-motivated scenario of a pre-inflationary phase as valid, this thesis aims to develop a customized theoretical framework for interacting scalar QFTs on generic Friedmann-Lema\^{i}tre backgrounds. Importantly, such a framework cannot be in Euclidean signature (due to ill-definedness of a Wick rotation), nor should it be tailored towards de Sitter spacetime. Motivated by the subdominance of spatial gradients in the approach to the singularity, the major themes of this thesis are variants of spatial averaging and spatial gradient expansions in relation to the dependence on the underlying vacuum-like state.
The first of these themes is the Anti-Newtonian expansion in a spatially discretized setting. In this framework, the solution of a QFT decouples into two sub-problems: (i) the solution of the cosmological quantum mechanics; and (ii) the solution of the combinatorial problem that allows one to analytically control the terms of a “spatial hopping” expansion. The second theme is a novel manifestly Lorentzian formulation of the Functional Renormalization Group. The key differences of this formulation compared to the standard Euclidean setting are (i) it necessitates the incorporation of state-dependent aspects directly into the flow equation formalism; and (ii) the purely {\it spatial} mode modulation leads to an additional contribution to the renormalization of the Newton constant. In a full quantum gravity computation, the latter would quantitatively affect the interplay between the matter and gravity sectors. Within the asymptotic safety scenario, this interplay is believed to resolve the triviality of scalar field theories. This has found phenomenological applications, which are however yet provisionary, because the infrared regime of the flow equation is neither well-posed nor controlled. This thesis prepares the tools to address this situation
Integral Transformation, Operational Calculus and Their Applications
The importance and usefulness of subjects and topics involving integral transformations and operational calculus are becoming widely recognized, not only in the mathematical sciences but also in the physical, biological, engineering and statistical sciences. This book contains invited reviews and expository and original research articles dealing with and presenting state-of-the-art accounts of the recent advances in these important and potentially useful subjects
New Developments in Geometric Function Theory
The book contains papers published in a Special Issue of Axioms, entitled "New Developments in Geometric Function Theory". An Editorial describes the 14 papers devoted to the study of complex-valued functions which present new outcomes related to special classes of univalent and bi-univalent functions, new operators and special functions associated with differential subordination and superordination theories, fractional calculus, and certain applications in geometric function theory
S-functions, spectral functions of hyperbolic geometry, and vertex operators with applications to structure for Weyl and orthogonal group invariants
In this paper we analyze the quantum homological invariants (the Poincare polynomials of the sI(N) link homology). In the case when the dimensions of homologies of appropriate topological spaces are precisely known, the procedure of the calculation of the Kovanov-Rozansky type homology, based on the Euler-Poincare formula can be appreciably simplified. We express the formal character of the irreducible tensor representation of the classical groups in terms of the symmetric and spectral functions of hyperbolic geometry. On the basis of Labastida-Marino-Ooguri-Vafa conjecture, we derive a representation of the Chern-Simons partition function in the form of an infinite product in terms of the Ruelle spectral functions (the cases of a knot, unknot, and links have been considered). We also derive an infinite-product formula for the orthogonal Chem-Simons partition functions and analyze the singularities and the symmetry properties of the infinite-product structures. (C) 2016 The Authors. Published by Elsevier B.V.Peer reviewe
Background Independent Quantum Gravity: A Status Report
The goal of this article is to present an introduction to loop quantum
gravity -a background independent, non-perturbative approach to the problem of
unification of general relativity and quantum physics, based on a quantum
theory of geometry. Our presentation is pedagogical. Thus, in addition to
providing a bird's eye view of the present status of the subject, the article
should also serve as a vehicle to enter the field and explore it in detail. To
aid non-experts, very little is assumed beyond elements of general relativity,
gauge theories and quantum field theory. While the article is essentially
self-contained, the emphasis is on communicating the underlying ideas and the
significance of results rather than on presenting systematic derivations and
detailed proofs. (These can be found in the listed references.) The subject can
be approached in different ways. We have chosen one which is deeply rooted in
well established physics and also has sufficient mathematical precision to
ensure that there are no hidden infinities. In order to keep the article to a
reasonable size, and to avoid overwhelming non-experts, we have had to leave
out several interesting topics, results and viewpoints; this is meant to be an
introduction to the subject rather than an exhaustive review of it.Comment: 125 pages, 5 figures (eps format), the final version published in CQ
Geometrical Theory of Analytic Functions
The book contains papers published in the Mathematics Special Issue, entitled "Geometrical Theory of Analytic Functions". Fifteen papers devoted to the study concerning complex-valued functions of one variable present new outcomes related to special classes of univalent functions, differential equations in view of geometric function theory, quantum calculus and its applications in geometric function theory, operators and special functions associated with differential subordination and superordination theories and starlikeness, and convexity criteria
- …