# An Introduction to Non-Perturbative Foundations of Quantum Field Theory

## Franco Strocchi

### Abstract

The book begins by discussing i) the conflict between locality or hyperbolicity and positivity of the energy for relativistic wave equations, which marks the origin of quantum field theory, and ii) the mathematical problems of the perturbative expansion (canonical quantization, interaction picture, non-Fock representation, asymptotic convergence of the series, and so on). The general physical principles of positivity of the energy, Poincaré covariance, and locality provide a substitute for canonical quantization, qualify the non-perturbative foundations, and lead to very relevant results, such ... More

The book begins by discussing i) the conflict between locality or hyperbolicity and positivity of the energy for relativistic wave equations, which marks the origin of quantum field theory, and ii) the mathematical problems of the perturbative expansion (canonical quantization, interaction picture, non-Fock representation, asymptotic convergence of the series, and so on). The general physical principles of positivity of the energy, Poincaré covariance, and locality provide a substitute for canonical quantization, qualify the non-perturbative foundations, and lead to very relevant results, such as the spin–statistics theorem, TCP symmetry, a substitute for canonical quantization, non-canonical behavior, the Euclidean formulation at the basis of the functional integral approach, the non-perturbative definition of the S-matrix (LSZ, Haag–Ruelle–Buchholz theory). A characteristic feature of gauge field theories is the Gauss law constraint; it is responsible for the conflict between locality of the charged fields and positivity, which yields the superselection of the (unbroken) gauge charges, provides a non-perturbative explanation of the Higgs mechanism in the local gauges, and implies the infraparticle structure of the charged particles in QED and the breaking of the Lorentz group in the charged sectors. A non-perturbative proof of the Higgs mechanism is discussed in the Coulomb gauge: the vector bosons corresponding to the broken generators cannot be massless, and their two-point function dominates the Goldstone spectrum, thus excluding the occurrence of massless Goldstone bosons. The solution of the U(1) problem in QCD, the theta vacuum structure, and the inevitable breaking of the chiral symmetry in each theta sector are derived solely from the topology of the gauge group, without relying on the semiclassical instanton approximation.

*Keywords: *
non-perturbative foundations of QFT and of gauge QFT,
non-perturbative treatment of Higgs mechanism and of chiral breaking in QCD

### Bibliographic Information

Print publication date: 2013 |
Print ISBN-13: 9780199671571 |

Published to Oxford Scholarship Online: May 2013 |
DOI:10.1093/acprof:oso/9780199671571.001.0001 |

### Authors

#### Affiliations are at time of print publication.

Franco Strocchi, *author*

Senior Research Fellow, National Institute for Nuclear Research, Pisa, Italy

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