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. |
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 | |
Common terms and phrases
adjoint Banach algebra bosons called Chapter classical compact complex connection correlation functions corresponding covariant derivative deduce Definition denote Dirac eigenvalue eigenvector electromagnetic element Euler-Lagrange equations exercise exists fact fermionic Feynman diagrams fiber finite finite-dimensional Fock space formula 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 Proof prove quantization quantum field theory quantum mechanics quark renormalization representation right-hand side satisfy scalar field 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 μ μ μν