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Topological Concepts in Gauge Theories
In these lecture notes, an introduction to topological concepts and methods in studies of gauge field theories is presented. The three paradigms of...
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Method of Invariant Grids
The method of invariant grids is developed for a grid-based computation of invariant manifolds.
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Mathematical Notation and Some Terminology
– The operator L from space W to space E: L : W → E
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Invariance Equation in Differential Form
Definition of invariance in terms of motions and trajectories assumes, at least, existence and uniqueness theorems for solutions of the original...
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Entropy, Quasiequilibrium, and Projectors Field
Projection operators Py contribute both to the invariance equation (3.2), and to the film extension of the dynamics (4.5). Limiting results, exact...
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Chiral Anomalies and Topology
When a field theory has a symmetry, global or local like in gauge theories, in the tree or classical approximation formal manipulations lead to...
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Nonlinear superposition formulae of integrable partial differential equations by the singular manifold method
We study by the singular manifold method a few 1+1-dimensional partial differential equations which possess N-soliton solutions for arbitrary N, i.e....
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Sato Theory and Transformation Groups. A Unified Approach to Integrable Systems
More than 20 years ago, it was discovered that the solutions of the Kadomtsev-Petviashvili (KP) hierarchy constitute an infinite-dimensional...
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Discrete Painlevé Equations: A Review
We present a review of what is current knowledge about discrete Painlevé equations. We start with a historical introduction which explains how the...
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Nonlinear Waves, Solitons, and IST
These lectures are written for a wide audience with diverse backgrounds. The subject is approached from a general perspective and overly detailed...
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Integrability – and How to Detect It
We present a physicist’s approach to integrability and its detection. Starting from specific examples we present a working definition of what is...
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Bilinear Formalism in Soliton Theory
A brief survey of the bilinear formalism discovered by Hirota is given. First, the procedure to obtain soliton solutions of nonlinear evolution...
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4 Abelian Gauge Fixing
The formation of monopoles and their condensation in the QCD ground state is a feature which is related to abelian gauge fixing, discussed in this...
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Introduction
The state of a physical system is the mathematical description of our knowledge of it, and provides information on its future and past. A state...
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2 Quantum Tomographic Methods
The state of a physical system is the mathematical object that provides a complete information on the system. The knowledge of the state is...
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9 Quantum Operations on Qubitsand Their Characterization
Information encoded in quantum system has to obey rules of quantum physics which impose strict bounds on state estimation and on possible...
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5 Galilei Invariant Elementary Particles
In this chapter we apply the general theory of symmetry actions as developed in Chaps. 2 and 3 to the Galilei groups of Chap. 4 splitting the...
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4 The Galilei Groups
In this chapter we describe the Galilei group and its universal central extension both in 3 + 1 and in 2 + 1 dimensions.
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A Appendix
This dictionary gives the definitions and the basic properties of most of the mathematical concepts that are freely used in the book. No references...