## Mathematical Aspects of Quantum Field TheoryOver the last century quantum field theory has made a significant impact on the formulation and solution of mathematical problems and inspired powerful advances in pure mathematics. However, most accounts are written by physicists, and mathematicians struggle to find clear definitions and statements of the concepts involved. This graduate-level introduction presents the basic ideas and tools from quantum field theory to a mathematical audience. Topics include classical and quantum mechanics, classical field theory, quantization of classical fields, perturbative quantum field theory, renormalization, and the standard model. The material is also accessible to physicists seeking a better understanding of the mathematical background, providing the necessary tools from differential geometry on such topics as connections and gauge fields, vector and spinor bundles, symmetries and group representations. |

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### Contents

1 | |

2 Quantum mechanics | 14 |

3 Relativity the Lorentz group and Diracs equation | 51 |

4 Fiber bundles connections and representations | 65 |

5 Classical field theory | 93 |

6 Quantization of classical fields | 117 |

7 Perturbative quantum field theory | 153 |

8 Renormalization | 192 |

9 The Standard Model | 204 |

Hilbert spaces and operators | 232 |

C algebras and spectral theory | 258 |

289 | |

293 | |

### Other editions - View all

Mathematical Aspects of Quantum Field Theory Edson de Faria,Welington de Melo No preview available - 2010 |

### Common terms and phrases

adjoint Banach algebra bosons called Chapter classical compact complex connection coordinates correlation functions corresponding covariant derivative deduce define deﬁned Deﬁnition denote dense Dirac eigenvalue eigenvector electromagnetic ﬁeld element Euler–Lagrange equations example exercise exists fact fermionic Feynman diagram ﬁeld ﬁnite ﬁnite-dimensional ﬁrst Fock space formula Fourier transform gauge Gaussian Gelfand-Naimark theorem given Hamiltonian Hence Hilbert space homomorphism inner product interaction invariant isometry isomorphism Klein–Gordon equation Lagrangian Lemma leptons Lie algebra linear functional linear operator Lorentz mathematical matrix measure metric Minkowski momentum norm orthogonal particle perturbative polynomial principal bundle projection-valued measure Proof prove quantization quantum ﬁeld theory quantum mechanics quark reader renormalization representation right-hand side satisﬁes scalar Schwinger Schwinger functions self-adjoint operator shows space H spacetime spectral theorem spectrum spinor subspace symmetry tensor unitary vector bundle vector space vertices Wightman Yang–Mills