Table Of ContentProgress in Mathematical Physics
Volume53
Editors-in-Chief
Anne Boutet de Monvel, Université Paris VII Denis Diderot
Gerald Kaiser, The Virginia Center for Signals and Waves,AustinTexas, USA
Editorial Board
Sir M. Berry,University of Bristol, UK
C. Berenstein, University of Maryland, College Park, USA
P. Blanchard, University of Bielefeld, Germany
A.S. Fokas, University of Cambridge, UK
D. Sternheimer,Université de Bourgogne, Dijon, France
C.Tracy,University of California, Davis, USA
Quantum Spaces
Poincaré Seminar 2007
Bertrand Duplantier
Vincent Rivasseau
Editors
Birkhäuser Verlag
Basel Boston Berlin
(cid:3) (cid:3) (cid:3)
Editors:
Bertrand Duplantier Vincent Rivasseau
Service de Physique Théorique Laboratoire de Physique Théorique
Orme des Merisiers Université Paris XI
CEA - Saclay 91405 Orsay Cedex
91191 Gif-sur-Yvette Cedex France
France e-mail: [email protected]
e-mail: [email protected]
2000 Mathematics Subject Classification: 81V70, 81R50, 81R60, 81T75, 81T17, 82B23
Library of Congress Control Number: 2007933916
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Contents
Foreword.................................................................. xi
V. Pasquier
Quantum Hall Effect and Non-commutative Geometry
1 Introduction.......................................................... 1
2 Lowest Landau level physics.......................................... 3
2.1. Single particle in a magnetic field............................... 3
2.2. Non-commutative product...................................... 5
3 Interactions.......................................................... 7
3.1. Spring in a magnetic field....................................... 7
3.2. Structure of the bound state.................................... 8
3.3. Bosons at ν =1................................................ 9
4 Skyrmion and nonlinear σ-model..................................... 12
4.1. Non-commutative Belavin–Polyakovsoliton..................... 13
References............................................................ 16
V. Rivasseau
Non-commutative Renormalization
1 Introduction.......................................................... 19
2 Commutative renormalization,a blitz review......................... 34
2.1. Functional integral............................................. 34
2.2. Feynman rules.................................................. 38
2.3. Scale analysis and renormalization.............................. 42
2.4. The BPHZ theorem............................................ 45
2.5. The Landau ghost and asymptotic freedom..................... 47
3 Non-commutative field theory........................................ 49
3.1. Field theory on Moyal space.................................... 49
3.1.1. The Moyal space RD.......................................... 49
θ
3.1.2. The Φ(cid:2)4 theory on R4 Moyal space........................... 52
θ
3.1.3. UV/IR mixing................................................ 53
3.2. The Grosse-Wulkenhaar breakthrough.......................... 54
3.3. The non-commutative Gross-Neveu model...................... 56
4 Multi-scale analysis in the matrix basis............................... 58
vi Contents
4.1. A dynamical matrix model..................................... 58
4.1.1. From the direct space to the matrix basis..................... 58
4.1.2. Topology of ribbon graphs.................................... 59
4.2. Multi-scale analysis............................................. 60
4.2.1. Bounds on the propagator.................................... 60
4.2.2. Power counting............................................... 62
5 Hunting the Landau ghost............................................ 64
5.1. One loop....................................................... 64
5.2. Two and three loops............................................ 66
5.3. The general Ward identity...................................... 68
5.3.1. Proof of Theorem 5.1......................................... 71
5.3.2. Bare identity................................................. 76
5.4. The RG flow................................................... 76
6 Propagatorson non-commutative space............................... 77
6.1. Bosonic kernel.................................................. 78
6.2. Fermionic kernel................................................ 78
6.3. Bounds......................................................... 80
6.4. Propagatorsand renormalizability.............................. 81
7 Direct space.......................................................... 82
7.1. Short and long variables........................................ 82
7.2. Routing, Filk moves............................................ 83
7.2.1. Oriented graphs.............................................. 83
7.2.2. Position routing.............................................. 83
7.2.3. Filk moves and rosettes....................................... 85
7.2.4. Rosette factor................................................ 85
7.3. Renormalization................................................ 87
7.3.1. Four-point function........................................... 87
7.3.2. Two-point function........................................... 88
7.3.3. The Langmann-Szabo-Zarembomodel........................ 88
7.3.4. Covariantmodels............................................. 88
8 Parametric representation............................................ 91
8.1. Ordinary Symanzik polynomials................................ 91
8.2. Non-commutative hyperbolic polynomials,
the non-covariantcase.......................................... 95
8.3. Non-commutative hyperbolic polynomials,
the covariantcase.............................................. 99
9 Conclusion........................................................... 100
References............................................................ 102
Contents vii
A.P. Polychronakos
Non-commutative Fluids
1 Introduction.......................................................... 109
2 Review of non-commutative spaces................................... 110
2.1. The operator formulation....................................... 110
2.2. Weyl maps, Wigner functions and ∗-products................... 113
3 Non-commutative gauge theory....................................... 114
3.1. Background-independent formulation........................... 115
3.2. Superselection of the non-commutative vacuum................. 116
3.3. Non-commutative Chern-Simons action......................... 117
3.4. Level quantization for the non-commutative
Chern-Simons action........................................... 121
4 Connection with fluid mechanics...................................... 122
4.1. Lagrange and Euler descriptions of fluids....................... 122
4.2. Reparametrization symmetry and its non-commutative
avatar.......................................................... 124
4.3. Gauging the symmetry......................................... 125
4.4. Non-commutative fluids and the Seiberg-Witten map........... 126
5 The non-commutative description of quantum Hall states............. 130
5.1. Non-commutative Chern-Simons description of
the quantum Hall fluid......................................... 130
5.2. Quasiparticle and quasihole classical states ..................... 132
5.3. Finite number of electrons: the Chern-Simons matrix model..... 133
5.4. Quantum Hall ‘droplet’ vacuum................................ 134
5.5. Excited states of the model..................................... 135
5.6. Equivalence to the Calogero model............................. 137
6 The quantum matrix Chern-Simons model............................ 138
6.1. Quantization of the filling fraction.............................. 138
6.2. Quantum states................................................ 141
6.3. Final remarks on the matrix model............................. 143
7 The non-commutative Euler picture and bosonization................. 144
7.1. Density description of fermionic many-body systems............ 144
7.2. The correspondence to a non-commutative fluid................. 146
7.3. Quantization and the full many-body correspondence........... 147
7.4. Higher-dimensional non-commutative bosonization.............. 148
8 τα´ πα´ντα ρ(cid:7)˜ι ... (and it all keeps flowing...)........................ 155
References............................................................ 155
viii Contents
J.-M. Maillet
Heisenberg Spin Chains: From Quantum Groups to
Neutron Scattering Experiments
1 Introduction.......................................................... 161
2 Heisenberg spin chain and algebraic Bethe ansatz..................... 170
2.1. Algebraic Bethe ansatz......................................... 170
2.2. Description of the spectrum.................................... 172
2.3. Drinfel’d twist and F-basis...................................... 173
2.4. Solution of the quantum inverse problem....................... 175
2.5. Scalar products................................................. 176
2.6. Action of operatorsA, B, C, D on a general state.............. 177
3 Correlation functions: Finite chain.................................... 178
3.1. Matrix elements of local operators.............................. 178
3.2. Elementary blocks of correlationfunctions...................... 179
3.3. Two-point functions............................................ 181
3.4. Towards the comparisonwith neutron scattering
experiments.................................................... 183
4 Correlation functions: Infinite chain.................................. 186
4.1. The thermodynamic limit....................................... 187
4.2. Elementary blocks.............................................. 189
5 Exact and asymptotic results......................................... 190
5.1. Exact results at ∆=1/2....................................... 190
5.1.1. The emptiness formation probability.......................... 190
5.1.2. The two-point function of σz.................................. 191
5.2. Asymptotic results............................................. 192
5.3. Asymptotic behavior of the two-point functions................. 194
6 Conclusion and perspectives.......................................... 195
Acknowledgments.................................................... 196
References............................................................ 196
A. Connes
Non-commutative Geometry and the Spectral Model of Space-time
1 Background.......................................................... 203
2 Why non-commutative spaces........................................ 205
3 What is a non-commutative geometry?............................... 206
4 Inner fluctuations of a spectral geometry............................. 208
5 The spectral action principle......................................... 209
6 The finite non-commutative geometry F.............................. 212
6.1. The representation of A in H ................................ 212
F F
6.2. The unimodular unitary group SU(A )......................... 213
F
6.3. The classification of Dirac operators............................ 214
Contents ix
7 The spectral action for M ×F and the standard model............... 215
8 Detailed form of the bosonic action................................... 216
9 Detailed form of the spectral action without gravity.................. 218
10 Predictions........................................................... 220
10.1. Unification of couplings........................................ 220
10.2. See-saw mechanism for neutrino masses........................ 220
10.3. Mass relation Y (S)=4g2..................................... 222
2
10.4. The Higgs scattering parameter............................... 223
10.5. Naturalness................................................... 223
10.6. Gravitationalterms........................................... 224
11 Final remarks........................................................ 225
References............................................................ 226
