Quantum Field Theory (QFT) has proved to be the most useful
strategy for the description of elementary particle interactions
and as such is regarded as a fundamental part of modern theoretical
physics. In most presentations, the emphasis is on the
effectiveness of the theory in producing experimentally testable
predictions, which at present essentially means Perturbative QFT.
However, after more than fifty years of QFT, we still are in the
embarrassing situation of not knowing a single non-trivial (even
non-realistic) model of QFT in 3+1 dimensions, allowing a
non-perturbative control. As a reaction to these consistency
problems one may take the position that they are related to our
ignorance of the physics of small distances and that QFT is only an
effective theory, so that radically new ideas are needed for a
consistent quantum theory of relativistic interactions (in 3+1
dimensions). The book starts by discussing the conflict between
locality or hyperbolicity and positivity of the energy for
relativistic wave equations, which marks the origin of quantum
field theory, and the mathematical problems of the perturbative
expansion (canonical quantization, interaction picture, non-Fock
representation, asymptotic convergence of the series etc.). The
general physical principles of positivity of the energy, Poincare'
covariance and locality provide a substitute for canonical
quantization, qualify the non-perturbative foundation and lead to
very relevant results, like the Spin-statistics theorem, TCP
symmetry, a substitute for canonical quantization, non-canonical
behaviour, 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 Gauss' law constraint. It is responsible
for the conflict between locality of the charged fields and
positivity, it yields the superselection of the (unbroken) gauge
charges, provides a non-perturbative explanation of the Higgs
mechanism in the local gauges, 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 are massive 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.
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