## Quantum Field Theory: A Tourist Guide for MathematiciansQuantum field theory has been a great success for physics, but it is difficult for mathematicians to learn because it is mathematically incomplete. Folland, who is a mathematician, has spent considerable time digesting the physical theory and sorting out the mathematical issues in it. Fortunately for mathematicians, Folland is a gifted expositor. The purpose of this book is to present the elements of quantum field theory, with the goal of understanding the behavior of elementary particles rather than building formal mathematical structures, in a form that will be comprehensible to mathematicians. Rigorous definitions and arguments are presented as far as they are available, but the text proceeds on a more informal level when necessary, with due care in identifying the difficulties. The book begins with a review of classical physics and quantum mechanics, then proceeds through the construction of free quantum fields to the perturbation-theoretic development of interacting field theory and renormalization theory, with emphasis on quantum electrodynamics. The final two chapters present the functional integral approach and the elements of gauge field theory, including the Salam-Weinberg model of electromagnetic and weak interactions. |

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### Contents

1 | |

Review of Prequantum Physics | 13 |

Basic Quantum Mechanics | 33 |

Relativistic Quantum Mechanics | 65 |

Free Quantum Fields | 97 |

Quantum Fields with Interactions | 123 |

Renormalization | 191 |

Functional Integrals | 257 |

Gauge Field Theories | 291 |

317 | |

323 | |

### Common terms and phrases

adjoint angular momentum annihilation operators anticommutation antiparticles Bosons calculation canonical charge classical coefﬁcients commutation components consider convergent corresponding counterterms coupling constant creation operators decay deﬁned deﬁnite delta-function denote derivative differential dimensions Dirac equation Dirac matrices divergent Dyson series eigenvalue electromagnetic ﬁeld electron energy external lines factor Fermion Feynman diagrams ﬁeld operators ﬁnd ﬁnite ﬁrst Fock space formula Fourier transform free ﬁelds functional integrals gauge ﬁelds Gaussian Hamiltonian hence Hilbert space inﬁnite inﬁnitesimal inner product interaction invariant Lagrangian linear Lorentz mass shell massless mathematical matrix element momenta neutrinos notation observables obtain outgoing parameters perturbation theory physicists physics position potential quantity quantization quantum ﬁeld theory quarks relativistic renormalization replace result S-matrix satisﬁes scalar ﬁeld self-adjoint signiﬁcance speciﬁc spin spinor subspace symmetry theorem unitary representation vacuum expectation values variables vector ﬁeld vertex vertices Wick rotation Wightman axioms