ix
16.3.2 Quantum field theory . . . . . . . . . . . . . . . . . . 167
16.3.3 Connection with statistical mechanics . . . . . . . . . 171
16.4 Renormalization, and the renormalization group, on the lattice 172
16.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 172
16.4.2 Two one-dimensional examples . . . . . . . . . . . . . 174
16.4.3 Connections with particle physics . . . . . . . . . . . . 177
16.5 Lattice QCD . . . . . . . . . . . . . . . . . . . . . . . . . . 182
16.5.1 Introduction, and the continuum limit . . . . . . . . . 182
16.5.2 Th
at c qq pote tial . . . . . . . . . . . . . . . . . . 184
e st i ¯
n
16.5.3 Calculation of α(M
2 ) . . . . . . . . . . . . . . . . . . 187
Z
16.5.4 Hadron masses . . . . . . . . . . . . . . . . . . . . . . 189
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
VII Spontaneously Broken Symmetry
193
17 Spontaneously Broken Global Symmetry
195
17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
17.2 The Fabri–Picasso theorem . . . . . . . . . . . . . . . . . . . 197
17.3 Spontaneously broken symmetry in condensed matter physics 199
17.3.1 The ferromagnet . . . . . . . . . . . . . . . . . . . . . 199
17.3.2 The Bogoliubov superfluid . . . . . . . . . . . . . . . . 202
17.4 Goldstone’s theorem . . . . . . . . . . . . . . . . . . . . . . . 209
17.5 Spontaneously broken global U(1) symmetry: the Goldstone
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
17.6 Spontaneously broken global non-Abelian symmetry . . . . . 216
17.7 The BCS superconducting ground state . . . . . . . . . . . . 219
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
18 Chiral Symmetry Breaking
227
18.1 The Nambu analogy . . . . . . . . . . . . . . . . . . . . . . . 228
18.1.1 Two flavour QCD and SU(2) f L ×SU(2) f R . . . . . . . 231
18.2 Pion decay and the Goldberger–Treiman relation . . . . . . . 235
18.3 Effective Lagrangians . . . . . . . . . . . . . . . . . . . . . . 239
18.3.1 The linear and non-linear σ-models . . . . . . . . . . . 239
18.3.2 Inclusion of explicit symmetry breaking: masses for pions and quarks . . . . . . . . . . . . . . . . . . . . . . 245
18.3.3 Extension to SU(3) f L ×SU(3) f R . . . . . . . . . . . . . 247
18.4 Chiral anomalies . . . . . . . . . . . . . . . . . . . . . . . . . 249
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
19 Spontaneously Broken Local Symmetry
255
19.1 Massive and massless vector particles . . . . . . . . . . . . . 255
19.2 The generation of ‘photon mass’ in a superconductor: Ginzburg–
Landau theory and the Meissner effect . . . . . . . . . . . . 260
19.3 Spontaneously broken local U(1) symmetry: the Abelian Higgs
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
16.3.2 Quantum field theory . . . . . . . . . . . . . . . . . . 167
16.3.3 Connection with statistical mechanics . . . . . . . . . 171
16.4 Renormalization, and the renormalization group, on the lattice 172
16.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 172
16.4.2 Two one-dimensional examples . . . . . . . . . . . . . 174
16.4.3 Connections with particle physics . . . . . . . . . . . . 177
16.5 Lattice QCD . . . . . . . . . . . . . . . . . . . . . . . . . . 182
16.5.1 Introduction, and the continuum limit . . . . . . . . . 182
16.5.2 Th
at c qq pote tial . . . . . . . . . . . . . . . . . . 184
e st i ¯
n
16.5.3 Calculation of α(M
2 ) . . . . . . . . . . . . . . . . . . 187
Z
16.5.4 Hadron masses . . . . . . . . . . . . . . . . . . . . . . 189
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
VII Spontaneously Broken Symmetry
193
17 Spontaneously Broken Global Symmetry
195
17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
17.2 The Fabri–Picasso theorem . . . . . . . . . . . . . . . . . . . 197
17.3 Spontaneously broken symmetry in condensed matter physics 199
17.3.1 The ferromagnet . . . . . . . . . . . . . . . . . . . . . 199
17.3.2 The Bogoliubov superfluid . . . . . . . . . . . . . . . . 202
17.4 Goldstone’s theorem . . . . . . . . . . . . . . . . . . . . . . . 209
17.5 Spontaneously broken global U(1) symmetry: the Goldstone
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
17.6 Spontaneously broken global non-Abelian symmetry . . . . . 216
17.7 The BCS superconducting ground state . . . . . . . . . . . . 219
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
18 Chiral Symmetry Breaking
227
18.1 The Nambu analogy . . . . . . . . . . . . . . . . . . . . . . . 228
18.1.1 Two flavour QCD and SU(2) f L ×SU(2) f R . . . . . . . 231
18.2 Pion decay and the Goldberger–Treiman relation . . . . . . . 235
18.3 Effective Lagrangians . . . . . . . . . . . . . . . . . . . . . . 239
18.3.1 The linear and non-linear σ-models . . . . . . . . . . . 239
18.3.2 Inclusion of explicit symmetry breaking: masses for pions and quarks . . . . . . . . . . . . . . . . . . . . . . 245
18.3.3 Extension to SU(3) f L ×SU(3) f R . . . . . . . . . . . . . 247
18.4 Chiral anomalies . . . . . . . . . . . . . . . . . . . . . . . . . 249
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
19 Spontaneously Broken Local Symmetry
255
19.1 Massive and massless vector particles . . . . . . . . . . . . . 255
19.2 The generation of ‘photon mass’ in a superconductor: Ginzburg–
Landau theory and the Meissner effect . . . . . . . . . . . . 260
19.3 Spontaneously broken local U(1) symmetry: the Abelian Higgs
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264