Quantum Spaces (eBook)

Poincaré Seminar 2007

Vincent Rivasseau (Herausgeber)

eBook Download: PDF

2007 | 2007
XII, 228 Seiten
Springer Basel (Verlag)
978-3-7643-8522-4 (ISBN)

Infos

This book confirms noncommutative geometry as an increasingly useful tool for the description of intricate condensed matter phenomena. It describes the striking progress recently made in gathering all the interactions and fields of the standard model into a non-commutative geometry on a simple internal space. Coverage also details the very recent technique of renormalization of quantum field theories on non-commutative space-time.

Inhalt

Contents 6
Foreword 11
Quantum Hall Effect and Non-commutative Geometry 12
1. Introduction 12
2. Lowest Landau level physics 14
3. Interactions 18
4. Skyrmion and nonlinear s-model 23
References 27
Non-commutative Fluids 118
1. Introduction 118
2. Review of non-commutative spaces 119
3. Non-commutative gauge theory 123
4. Connection with .uid mechanics 131
5. The non-commutative description of quantum Hall states 139
6. The quantum matrix Chern-Simons model 147
7. The non-commutative Euler picture and bosonization 153
8. t ´a p´a.ta ... . . . (and it all keeps .owing. . . ) 164
References 164
Heisenberg Spin Chains: From Quantum Groups to Neutron Scattering Experiments 169
1. Introduction 169
2. Heisenberg spin chain and algebraic Bethe ansatz 178
3. Correlation functions: Finite chain 186
4. Correlation functions: Infinite chain 194
5. Exact and asymptotic results 198
6. Conclusion and perspectives 203
References 204
Non-commutative Geometry and the Spectral Model of Space-time 210
1. Background 210
2. Why non-commutative spaces 212
3. What is a non-commutative geometry? 213
4. Inner fluctuations of a spectral geometry 215
5. The spectral action principle 216
6. The finite non-commutative geometry 219
7. The spectral action for M × F and the standard model 222
8. Detailed form of the bosonic action 223
9. Detailed form of the spectral action without gravity 225
10. Predictions 227
11. Final remarks 232
References 233

Textauszug

Quantum Hall Effect and Non-commutative Geometry (p. 1-2)

Vincent Pasquier

1. Introduction

Our aim is to introduce the ideas of non-commutative geometry through the example of the Quantum Hall Effect (QHE). We present a few concrete situations where the concepts of non-commutative geometry find physical applications. The Quantum Hall Effect [1–4] is a remarkable example of a purely experimental discovery which "could" have been predicted because the tools required are not extremely sophisticated and were known at the time of the discovery. What was missing was a good understanding of topological rigidity produced by quantum mechanics whose consequences can be tested at a macroscopic level: The quantized integers of the conductivity are completely analogous to the topological numbers one encounters in the study of fiber-bundles.

One can give a schematic description of the Quantum Hall Effect as follows. It deals with electrons constrained to move in a two-dimensional semiconductor sample in a presence of an applied magnetic field perpendicular to the sample. Due to the magnetic field, the Hilbert space of an electron is stratified into Landau levels separated by an energy gap (called the cyclotron frequency and proportional to the applied field). Each Landau level has a macroscopic degeneracy given by the area of the sample divided by a quantum of area (inversely proportional to the field) equal to 2?l2 where the length l is the so-called magnetic length. It is useful to think of the magnetic length as a Plank constant l2 ¡« . The limit of a strong magnetic field is very analogous to a classical limit.

The electrons behave much like incompressible objects occupying a quantum of area. Thus, when their number times 2?l2 is exactly equal a multiple of the area, it costs the energy gap to add one more electron. This discontinuity in the energy needed to add one more electron is at the origin of the incompressibility of the electron fluid. The number of electrons occupying each unit cell is called the filling factor, and the transverse conductivity is quantized each time the filling factor is exactly an integer.

We shall stick to this simple explanation, although this cannot be the end of the story. Indeed, if it was correct, the filling factor being linear in the magnetic field, the quantization of the conductance should be observed only at specific values of the magnetic field. In fact, it is observed on regions of finite width called plateaux, and it is necessary to invoke the impurities and localized states to account for these plateaus. Roughly speaking, some of the states are localized and do not participate to the conductance. These states are populated when the magnetic field is in a plateau. We refer the reader to a previous Poincar´e seminar for an introduction to these effects [4]. What is important for us here is that (although counter-intuitive) it is possible to realize experimentally situations where the filling factor is exactly an integer (or a fraction as we see next).

It came as a surprise (rewarded by the Nobel prize1) when the Quantum Hall Effect was observed at non-integer filling factors which turn out to always be simple fractions. To explain these fractions, it was necessary to introduce some very specific wave functions and to take into account the interactions between the electrons. The proposed wave functions are in some sense variational, although they carry no adjustable free parameters.

Produkt-Details

• Erscheint lt. Verlag: 22.12.2007
• Reihe/Serie: Progress in Mathematical Physics
• Reihe/Serie: Progress in Mathematical Physics
• Zusatzinfo: XII, 228 p.
• Verlagsort: Basel
• Sprache: englisch
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Theoretische Physik
• Themenwelt: Technik
• Schlagworte: Hall Effect • noncommutative fluids • Noncommutative Geometry • Quantum Hall Effect • quantum space
• ISBN-10: 3-7643-8522-7 / 3764385227
• ISBN-13: 978-3-7643-8522-4 / 9783764385224