14 Mathematical Description of Infinitely Extended Quantum
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
14.1 Local Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
14.2 Asymptotic Abelianess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
15 Physically Relevant Representations . . . . . . . . . . . . . . . . . . . . . . . . 101
16 Cluster Property and Pure Phases . . . . . . . . . . . . . . . . . . . . . . . . . 105
17 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
17.1 Spin Systems with Short-Range Interactions . . . . . . . . . . . . . . . 111
17.2 Free Bose Gas. Bose–Einstein Condensation . . . . . . . . . . . . . . 113
17.3 * Appendix: The Infinite Volume Dynamics
for Short-Range Spin Interactions . . . . . . . . . . . . . . . . . . . . . . 116
18 Symmetry Breaking in Quantum Systems . . . . . . . . . . . . . . . . . . . . 121
18.1 Wigner Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
18.2 Spontaneous Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . 124
18.3 Symmetry Breaking Order Parameter . . . . . . . . . . . . . . . . . . . . 127
19 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
19.1 Heisenberg Ferromagnet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
19.2 Bose–Einstein Condensation . . . . . . . . . . . . . . . . . . . . . . . . . . 130
19.3 Massless Field in d 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
20 Constructive Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . 133
21 Symmetry Breaking in the Ising Model . . . . . . . . . . . . . . . . . . . . . 139
22 * Thermal States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
22.1 Gibbs States and KMS Condition . . . . . . . . . . . . . . . . . . . . . . 148
22.2 GNS Representation Defined by a Gibbs State . . . . . . . . . . . . . 151
22.3 KMS States in the Thermodynamical Limit . . . . . . . . . . . . . . . 154
22.4 Pure Phases. Extremal and Primary KMS States . . . . . . . . . . . . 156
23 Fermi and Bose Gases at Non-zero Temperature . . . . . . . . . . . . . . 159
23.1 Free Fermi and Bose Gases in Finite Volume . . . . . . . . . . . . . . 159
23.2 Free Fermi Gas in the Thermodynamical Limit . . . . . . . . . . . . . 160
23.3 Free Bose Gas in the Thermodynamical Limit . . . . . . . . . . . . . 163
23.4 Bose–Einstein Condensation and Symmetry Breaking . . . . . . . . 164
24 Quantum Fields at Non-zero Temperature . . . . . . . . . . . . . . . . . . . 167
25 Breaking of Continuous Symmetries: Goldstone’s Theorem . . . . . . 169
25.1 The Goldstone Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
25.2 A Critical Look at the Hypotheses of Goldstone Theorem . . . . . 173
25.3 The Goldstone Theorem with Mathematical Flavour . . . . . . . . . 182
xii
Contents