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Quantum field theory is the quantum theory of force fields mediating elementary particle interactions. It is a successful fundamental theory of physics as it describes through the Standard Model all known elementary particle interactions except gravity. Although ever since its inception in 1967, the Standard Model continues to be verified in accelerator experiments, most recently, with the discovery of what appears to be the Higgs particle at CERN, quantum field theory still presents considerable physical and mathematical challenges.
On the physics side, while the perturbative regime, namely the regime in which the approximation that the interactions are weak is valid, is very well understood and leads to spectacular predictions, important nonperturbative physical phenomena, such as the permanent confinement of quarks, are out of reach so far.
On the mathematical side, a mathematically precise formulation of the quantum field theory of the Standard Model beyond formal perturbation theory is still lacking.
Gaining a better understanding of quantum field theory is particularly important as its field of applications exceeds by far the realm of particle physics. Indeed, quantum field theory appears as an essential tool and basic ingredient in condensed matter physics, statistical mechanics, string theory, but also in pure mathematics, such as parts of differential geometry, topology, noncommutative algebra, probability theory and combinatorics.