## Path Integrals in Physics: Volume II Quantum Field Theory, Statistical Physics and other Modern ApplicationsThe path integral approach has proved extremely useful for the understanding of the most complex problems in quantum field theory, cosmology, and condensed matter physics. Path Integrals in Physics: Volume II, Quantum Field Theory, Statistical Physics and other Modern Applications covers the fundamentals of path integrals, both the Wiener and Feynman types, and their many applications in physics. The book deals with systems that have an infinite number of degrees of freedom. It discusses the general physical background and concepts of the path integral approach used, followed by a detailed presentation of the most typical and important applications as well as problems with either their solutions or hints how to solve them. Each chapter is self-contained and can be considered as an independent textbook. It provides a comprehensive, detailed, and systematic account of the subject suitable for both students and experienced researchers. |

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### Contents

Quantum field theory the pathintegral approach | 1 |

311 Systems with an infinite number of degrees of freedom and quantum field theory | 2 |

312 Pathintegral representation for transition amplitudes in quantum field theories | 14 |

quantization via path integrals over Grassmann variables | 21 |

314 Perturbation expansion in quantum field theory in the pathintegral approach | 22 |

315 Generating functionals for Green functions and an introduction to functional methods in quantum field theory | 27 |

316 Problems | 38 |

32 Pathintegral quantization of gaugefield theories | 49 |

431 Permutations and transition amplitudes | 206 |

432 Pathintegral formalism for coupled identical oscillators | 210 |

433 Path integrals and parastatistics | 216 |

434 Problems | 221 |

441 Nonrelativistic field theory at nonzero temperature and the diagram technique | 223 |

442 Euclideantime relativistic field theory at nonzero temperature | 226 |

443 Realtime formulation of field theory at nonzero temperature | 233 |

444 Path integrals in the theory of critical phenomena | 238 |

321 Gaugeinvariant Lagrangians | 50 |

322 Constrained Hamiltonian systems and their pathintegral quantization | 54 |

constrained systems with an infinite number of degrees of freedom | 60 |

324 Pathintegral quantization of YangMills theories | 64 |

325 Covariant generating functional in the YangMills theory | 67 |

326 Covariant perturbation theory for YangMills models | 73 |

327 Higherorder perturbation theory and a sketch of the renormalization procedure for Yang Mills theories | 80 |

328 Spontaneous symmetrybreaking of gauge invariance and a brief look at the standard model of particle interactions | 88 |

329 Problems | 98 |

331 Rearrangements and partial summations of perturbation expansions the I N expansion and separate integration over high and low frequency mo... | 101 |

332 Semiclassical approximation in quantum field theory and extended objects solitons | 110 |

333 Semiclassical approximation and quantum tunneling instantons | 120 |

334 Pathintegral calculation of quantum anomalies | 130 |

335 Pathintegral solution of the polaron problem | 137 |

336 Problems | 144 |

advanced applications of path integrals 341 Pathintegral quantization of a gravitational field in an asymptotically flat spacetime and the correspondin... | 149 |

342 Path integrals in spatially homogeneous cosmological models | 154 |

343 Pathintegral calculation of the topologychange transitions in 2 + 1dimensional gravity | 160 |

344 Hawkings pathintegral derivation of the partition function for black holes | 166 |

345 Path integrals for relativistic point particles and in the string theory | 174 |

346 Quantum field theory on noncommutative spacetimes and path integrals | 185 |

Path integrals in statistical physics | 194 |

41 Basic concepts of statistical physics | 195 |

42 Path integrals in classical statistical mechanics | 200 |

43 Path integrals for indistinguishable particles in quantum mechanics | 205 |

445 Quantum field theory at finite energy | 245 |

446 Problems | 252 |

45 Superfluidity superconductivity nonequilibrium quantum statistics and the pathintegral technique | 257 |

451 Perturbation theory for superfluid Bose systems | 258 |

452 Perturbation theory for superconducting Fermi systems | 261 |

453 Nonequilibrium quantum statistics and the process of condensation of an ideal Bose gas | 263 |

454 Problems | 277 |

46 Nonequilibrium statistical physics in the pathintegral formalism and stochastic quantization | 280 |

calculation of usual integrals by the method of stochastic quantization | 281 |

462 Realtime quantum mechanics within the stochastic quantization scheme | 284 |

463 Stochastic quantization of field theories | 288 |

464 Problems | 293 |

47 Pathintegral formalism and lattice systems | 295 |

471 Ising model as an example of genuine discrete physical systems | 296 |

472 Lattice gauge theory | 302 |

473 Problems | 308 |

I Finitedimensional Gaussian integrals | 311 |

II Table of some exactly solved Wiener path integrals | 313 |

IV Short glossary of selected notions from the theory of Lie groups and algebras | 316 |

V Some basic facts about differential Riemann geometry | 325 |

VI Supersymmetry in quantum mechanics | 329 |

332 | |

337 | |

### Common terms and phrases

action approximation arbitrary Bose gas bosons boundary conditions calculation canonical chiral classical commutation relations configuration consider constraints coordinates corresponding coupling constant covariant defined degrees of freedom denotes density derivative differentiation Dirac divergences eigenstates eigenvalue elements energy equations of motion Euclidean expression external factor fermionic Feynman diagrams finite formula Fourier transform gauge condition gauge fields gauge group gauge invariance gauge theory gravitational Green functions Hamiltonian instanton interaction Lagrangian Langevin equation lattice Lie algebra Lie group Lorentz matrix method metric momenta momentum non-Abelian non-commutative non-zero number of degrees obtain operator oscillator parameter particles partition function path integral path-integral representation permutation perturbation expansion perturbation theory physical potential problem propagator quantum field theory quantum mechanics quarks relativistic renormalization result S-matrix scalar field solution space spacetime spinor statistical stochastic quantization symmetry temperature tensor transition amplitude variables vector Ward-Takahashi identities Yang-Mills theory zero

### Popular passages

Page 336 - Wess J and Bagger J 1983 Supersymmetry and Supergravity (Princeton, NJ: Princeton University Press...