## Mathematics of Quantization and Quantum FieldsUnifying a range of topics that are currently scattered throughout the literature, this book offers a unique and definitive review of mathematical aspects of quantization and quantum field theory. The authors present both basic and more advanced topics of quantum field theory in a mathematically consistent way, focusing on canonical commutation and anti-commutation relations. They begin with a discussion of the mathematical structures underlying free bosonic or fermionic fields, like tensors, algebras, Fock spaces, and CCR and CAR representations (including their symplectic and orthogonal invariance). Applications of these topics to physical problems are discussed in later chapters. Although most of the book is devoted to free quantum fields, it also contains an exposition of two important aspects of interacting fields: diagrammatics and the Euclidean approach to constructive quantum field theory. With its in-depth coverage, this text is essential reading for graduate students and researchers in departments of mathematics and physics. |

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### Contents

Introduction | 1 |

Operators in Hilbert spaces | 36 |

Tensor algebras | 57 |

Analysis in L2 Rd | 92 |

Measures | 111 |

Algebras | 142 |

Antisymmetric calculus | 159 |

Canonical commutation relations | 173 |

CAR on Fock spaces | 337 |

Orthogonal invariance of CAR algebras | 351 |

Cliﬁord relations | 368 |

Orthogonal invariance of the CAR on Fock spaces | 386 |

Quasifree states | 423 |

Dynamics of quantum ﬁelds | 475 |

Quantum ﬁelds on spacetime | 512 |

Diagrammatics | 555 |

CCR on Fock space | 212 |

Symplectic invariance of CCR | 239 |

Symplectic invariance of the | 266 |

Canonical anticommutation relations | 313 |

Euclidean approach for bosons | 605 |

Interacting bosonic ﬁelds | 641 |

661 | |

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### Common terms and phrases

algebra annihilation operators anti-involution anti-linear anti-symmetric assume Bogoliubov bosonic C*-algebra called CCR algebra CCR representation charged symplectic Clearly commutation conjugation consider covariance decomposition deﬁned Deﬁnition denoted dense describe Dirac equation dual dynamics elements equipped equivalent Euclidean exists a unique fermionic ﬁnite ﬁnite-dimensional ﬁrst ﬁx Fock representation Fock space Gaussian Hamiltonian Hence Hermitian Hilbert space holomorphic identiﬁcation iﬁ implies inﬁnite integral introduce invariant irreducible Kahler space kernel KleinIGordon equation L2 space Lemma Let r G linear measure non-degenerate notation Note obtain orthogonal path space phase space polynomial positive deﬁnite Proof of Thm Prop prove quadratic quantization quantum ﬁeld theory quasi-free real Hilbert space Recall resp satisﬁes scalar product scattering operator Sect self-adjoint operator Subsect subspace symplectic form symplectic space tensor trace-class transformations unitary operator vector space volume form von Neumann algebra Weyl Wick