## Quantum Spaces: Poincaré Seminar 2007This book is the seventh in a series of lectures of the S ́ eminaire Poincar ́ e,which is directed towards a large audience of physicists and of mathematicians. The goal of this seminar is to provide up-to-date information about general topics of great interest in physics. Both the theoretical and experimental aspects are covered, with some historical background. Inspired by the Bourbaki seminar in mathematics in its organization, hence nicknamed “Bourbaphi”, the Poincar ́ e Seminar is held twice a year at the Institut Henri Poincar ́ e in Paris, with cont- butions prepared in advance. Particular care is devoted to the pedagogical nature of the presentations so as to ful?ll the goal of being readable by a large audience of scientists. This volume contains the tenth such seminar, held on April 30, 2007. It is devoted to the application of non-commutative geometry and quantum groups to physics. The book starts with a pedagogical introduction to Moyal geometry by V- cent Pasquier, with special emphasis on the quantum Hall e?ect. It is followed by a detailed review of Vincent Rivasseau on non-commutative ?eld theory and the recent advances which lead to its renormalizability and asymptotic safety. The description of the quantum Hall e?ect as a non-commutative ?uid is then treated in detail by Alexios Polychronakos. Integrable spin chains can be studied through quantum groups; their striking agreement with neutron scattering experiments is reviewed by Jean-MichelMaillet. |

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### Contents

LXVIII | 130 |

LXX | 132 |

LXXI | 133 |

LXXII | 134 |

LXXIII | 135 |

LXXIV | 137 |

LXXV | 138 |

LXXVII | 141 |

XI | 16 |

XII | 19 |

XIII | 34 |

XV | 38 |

XVI | 42 |

XVII | 45 |

XVIII | 47 |

XIX | 49 |

XX | 54 |

XXI | 56 |

XXII | 58 |

XXIV | 59 |

XXV | 60 |

XXVII | 62 |

XXVIII | 64 |

XXX | 66 |

XXXI | 68 |

XXXII | 71 |

XXXIII | 76 |

XXXV | 77 |

XXXVI | 78 |

XXXVIII | 80 |

XXXIX | 81 |

XL | 82 |

XLII | 83 |

XLIII | 85 |

XLV | 87 |

XLVI | 88 |

XLIX | 91 |

L | 95 |

LI | 99 |

LII | 100 |

LIII | 102 |

LIV | 108 |

LV | 110 |

LVII | 113 |

LVIII | 114 |

LIX | 115 |

LX | 116 |

LXI | 117 |

LXII | 121 |

LXIII | 122 |

LXV | 124 |

LXVI | 125 |

LXVII | 126 |

LXXVIII | 143 |

LXXIX | 144 |

LXXXI | 146 |

LXXXII | 147 |

LXXXIII | 148 |

LXXXIV | 155 |

LXXXVI | 161 |

LXXXVII | 170 |

LXXXIX | 172 |

XC | 173 |

XCI | 175 |

XCII | 176 |

XCIII | 177 |

XCIV | 178 |

XCVI | 179 |

XCVII | 181 |

XCVIII | 183 |

XCIX | 186 |

C | 187 |

CI | 189 |

CII | 190 |

CIII | 191 |

CIV | 192 |

CV | 194 |

CVI | 195 |

CVII | 196 |

CVIII | 202 |

CIX | 205 |

CX | 206 |

CXI | 208 |

CXII | 209 |

CXIII | 212 |

CXV | 213 |

CXVI | 214 |

CXVII | 215 |

CXVIII | 216 |

CXIX | 218 |

CXX | 220 |

CXXIII | 222 |

CXXIV | 223 |

CXXV | 224 |

CXXVI | 225 |

CXXVII | 226 |

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### Common terms and phrases

amplitude asymptotic Bethe ansatz bosonic chain classical coordinates correlation functions corresponding counterterms covariant defined density dimension dimensional Dirac operator divergent droplet dynamical eigenstates eigenvalues electrons energy equation external factor fermions Feynman finite fluid gauge field gauge theory graph Gross-Neveu model Hamiltonian Heisenberg Hilbert space integral interaction invariant J.M. Maillet JHEP Lagrangian Landau level Lett loops lowest Landau level magnetic field Math matrix model momentum Moyal space NCVQFT non-commutative gauge theory non-commutative geometry non-commutative spaces nontrivial obtained operators oscillators parameters particles perturbative phase space Phys physics planar Poisson brackets Polychronakos polynomials power counting propagator quadratic quantization quantum field theory quantum Hall effect renormalizable representation Rivasseau scalar scale sinh solitons space-time spectral action spin standard model string theory subgraphs symmetry theorem tion topological transformations two-point ultraviolet unitary variables vertex wave functions