We discuss deformation quantization in quantum mechanics and quantum field theory. We begin with a discussion of the mathematical question of deforming the commutative algebra of functions on a manifold into a non-commutative algebra by use of an associative product.

We then apply these considerations to the commutative algebra of observables of a classical dynamical system, which may be deformed to the non-commutative algebra of quantum observables. This is the process of deformation quantization, which provides a canonical procedure for finding the measurable quantities of a quantum system. The deformation quantization approach is illustrated, first for the case of a simple harmonic oscillator, then for an oscillator coupled to an external source, and finally for a quantum field theory of scalar bosons, where the well-known formula for the number of quanta emitted by a given external source in terms of the Poisson distribution is reproduced.

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The relation of the star product method to the better-known methods involving the representation of observables as linear operators on a Hilbert space, or the representation of expectation values as functional integrals, is analyzed. In any event one seems to arrive at new revolutions in physics and mathematics every year. This book hopes to convey some of the excitment of this period, but will adopt a relatively pedestrian approach designed to illuminate the relations between quantum and classical.

There will be some discussion of philosophical matters such as measurement, uncertainty, decoherence, etc.

In Chapter 1 connections of QM to deterministic behavior are exhibited in the trajectory representations of Faraggi-Matone. Chapter 1 also includes a review of KP theory and some preliminary remarks on coherent states, density matrices, etc. We develop in Chapter 4 relations between quantization and integrability based on Moyal brackets, discretizations, KP, strings and Hirota formulas, and in Chapter 2 we study the QM of embedded curves and surfaces illustrating some QM effects of geometry. Chapter 3 is on quantum integrable systems, quantum groups, and modern deformation quantization.

Chapter 5 involves the Whitham equations in various roles mediating between QM and classical behavior.

Thus in Chapter 5 we will try to give some conceptual background for susy, gauge theories, renormalization, etc. In Chapter 6 we continue the deformation quantization then by exhibiting material based on and related to noncommutative geometry and gauge theory. We are always looking for ways to improve customer experience on Elsevier. We would like to ask you for a moment of your time to fill in a short questionnaire, at the end of your visit.

If you decide to participate, a new browser tab will open so you can complete the survey after you have completed your visit to this website. Note that w is a unitary representation. Whereas compact matrix pseudogroups are typically versions of Drinfeld-Jimbo quantum groups in a dual function algebra formulation, with additional structure, the bicrossproduct ones are a distinct second family of quantum groups of increasing importance as deformations of solvable rather than semisimple Lie groups.

This quantum group was linked to a toy model of Planck scale physics implementing Born reciprocity when viewed as a deformation of the Heisenberg algebra of quantum mechanics.

## Algebraic constructive quantum field theory: Integrable models and deformation techniques

Also, starting with any compact real form of a semisimple Lie algebra g its complexification as a real Lie algebra of twice the dimension splits into g and a certain solvable Lie algebra the Iwasawa decomposition , and this provides a canonical bicrossproduct quantum group associated to g. For su 2 one obtains a quantum group deformation of the Euclidean group E 3 of motions in 3 dimensions.

From Wikipedia, the free encyclopedia. Algebraic construct of interest in theoretical physics.

## Thematic Programs

Basic notions. Subgroup Normal subgroup Quotient group Semi- direct product Group homomorphisms kernel image direct sum wreath product simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable List of group theory topics.

- Table of Contents.
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Finite groups. Discrete groups Lattices.

## Integrability, Duality and Deformed Symmetry | ANU Mathematical Sciences Institute

Topological and Lie groups. Algebraic groups. Linear algebraic group Reductive group Abelian variety Elliptic curve. Main article: Crystal base.

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