87 research outputs found
Learning about Quantum Gravity with a Couple of Nodes
Loop Quantum Gravity provides a natural truncation of the infinite degrees of
freedom of gravity, obtained by studying the theory on a given finite graph. We
review this procedure and we present the construction of the canonical theory
on a simple graph, formed by only two nodes. We review the U(N) framework,
which provides a powerful tool for the canonical study of this model, and a
formulation of the system based on spinors. We consider also the covariant
theory, which permits to derive the model from a more complex formulation,
paying special attention to the cosmological interpretation of the theory
Mathematical Methods for 4d N=2 QFTs
In this work we study different aspects of 4d N = 2 superconformal field theories. Not only we
accurately define what we mean by a 4d N = 2 superconformal field theory, but we also invent and
apply new mathematical methods to classify these theories and to study their physical content.
Therefore, although the origin of the subject is physical, our methods and approach are rigorous
mathematical theorems: the physical picture is useful to guide the intuition, but the full mathematical rigor is needed to get deep and precise results. No familiarity with the physical concept
of Supersymmetry (SUSY) is need to understand the content of this thesis: everything will be
explained in due time. The reader shall keep in mind that the driving force of this whole work
are the consequences of SUSY at a mathematical level. Indeed, as it will be detailed in part II, a
mathematician can understand a 4d N = 2 superconformal field theory as a complexified algebraic
integrable system. The geometric properties are very constrained: we deal with special K\ua8ahler
geometries with a few other additional structures (see part II for details). Thanks to the rigidity
of these structures, we can compute explicitly many interesing quantities: in the end, we are able
to give a coarse classification of the space of "action" variables of the integrable system, as well as
a fine classification -- only in the case of rank k = 1 -- of the spaces of "angle" variables.
We were able to classify conical special K\ua8ahler geometries via a number of deep facts of algebraic
number theory, diophantine geometry and class field theory: the perfect overlap between mathematical theorems and physical intuition was astonishing. And we believe we have only scratched
the surface of a much deeper theory: we can probably hope to get much more information than
what we already discovered; of course, a deeper study of the subject -- as well as its generalizations
-- is required.
A 4d N = 2 superconformal field theory can thus be defined by its geometric structure: its scaling
dimensions, its singular fibers, the monodromy around them and so on. But giving a proper and
detailed definition is only the beginning: one may be interested in exploring its physical content. In
particular, we are interested in supersymmetric quantities such as BPS states, framed BPS states
and UV line operators. These quantities, thanks to SUSY, can be computed independently of
many parameters of the theory: this peculiarity makes it possible to use the language of category
theory to analyze the aforementioned aspects. As it will be proven in part V, to each 4d N = 2
superconformal field theory we can associate a web of categories, all connected by functors, that
describe the BPS states, the framed BPS states (IR) and the UV line operators. Hence, following
the old ideas of \u2018t Hooft, it is possible to describe the phase space of gauge theories via categories,
since the vacuum expectation values of such line operators are the order parameters of the confinement/deconfinement phase transitions. Mathematically, the (quantum) cluster algebra of Fomin
and Zelevinski is the structure needed. Moreover, the analysis of BPS objects led us to a deep
understanding of generalized S-dualities. Not only were we able to precisely define -- abstractly and
generally -- what the S-duality group of a 4d N = 2 superconformal field theory should be, but we
were also able to write a computer algorithm to obtain these groups in many examples (with very
high accuracy)
On the foundations of thermodynamics
On the basis of new, concise foundations, this paper establishes the four
laws of thermodynamics, the Maxwell relations, and the stability requirements
for response functions, in a form applicable to global (homogeneous), local
(hydrodynamic) and microlocal (kinetic) equilibrium.
The present, self-contained treatment needs very little formal machinery and
stays very close to the formulas as they are applied by the practicing
physicist, chemist, or engineer. From a few basic assumptions, the full
structure of phenomenological thermodynamics and of classical and quantum
statistical mechanics is recovered.
Care has been taken to keep the foundations free of subjective aspects (which
traditionally creep in through information or probability). One might describe
the paper as a uniform treatment of the nondynamical part of classical and
quantum statistical mechanics ``without statistics'' (i.e., suitable for the
definite descriptions of single objects) and ``without mechanics'' (i.e.,
independent of microscopic assumptions). When enriched by the traditional
examples and applications, this paper may serve as the basis for a course on
thermal physics.Comment: 78 page
Foundations of Quantum Theory: From Classical Concepts to Operator Algebras
Quantum physics; Mathematical physics; Matrix theory; Algebr
Intelligent Systems
This book is dedicated to intelligent systems of broad-spectrum application, such as personal and social biosafety or use of intelligent sensory micro-nanosystems such as "e-nose", "e-tongue" and "e-eye". In addition to that, effective acquiring information, knowledge management and improved knowledge transfer in any media, as well as modeling its information content using meta-and hyper heuristics and semantic reasoning all benefit from the systems covered in this book. Intelligent systems can also be applied in education and generating the intelligent distributed eLearning architecture, as well as in a large number of technical fields, such as industrial design, manufacturing and utilization, e.g., in precision agriculture, cartography, electric power distribution systems, intelligent building management systems, drilling operations etc. Furthermore, decision making using fuzzy logic models, computational recognition of comprehension uncertainty and the joint synthesis of goals and means of intelligent behavior biosystems, as well as diagnostic and human support in the healthcare environment have also been made easier
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