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Quantum field theory

theoretical framework for constructing quantum mechanical models of systems classically represented by an infinite number of degrees of freedom

In theoretical physics, quantum field theory (QFT) is the theoretical framework for constructing quantum mechanical models of subatomic particles in particle physics and quasiparticles in condensed matter physics. QFT treats particles as excited states of the underlying physical field, so these are called field quanta.


  • Among the many significant ideas and developments that connect Mathematics with contemporary Physics one of the most intriguing is the role that Quantum Field Theory (QFT) plays in Geometry and Topology. We can argue back and forth on the relevance of such a role, but the perspective QFT offers is often surprising and far reaching. Examples abound, and a fine selection is provided by the revealing insights offered by Yang–Mills theory into the topology of 4-manifolds, by the relation between Knot Theory and topological QFT, and most recently by the interaction between Strings, Riemann moduli space, and enumerative geometry. Doubtless many of the most striking connections suggested by physicists failed to pass the censorship of the Department of Mathematics, and so do not appear in the above official list.
  • The disillusionment with QFT as a basis for the theory of elementary particles was also premature. What was missing was many ingredients, including the identification of the underlying gauge symmetry of the weak interactions, the concept of spontaneous symmetry breaking that could explain how this symmetry was hidden, the identification of the fundamental constituents of the nucleons as colored quarks, the discovery of asymptotic freedom which explained how the elementary colored constituents of hadrons could be seen at short distances yet evade detection through confinement, and the identification of the underlying gauge symmetry of the strong interactions. Once these were discovered, it was but a short step to the construction of the standard model, a gauge theory modeled on QED, which opened the door to the understanding of mesons and nucleons.
  • In parallel with the changes it brought in our attitude toward symmetries, the birth of the Standard Model marked changes also in our attitude toward quantum field theory. After 't Hooft's breakthrough in 1971, it became clear that the old problem of infinities in the weak interactions had been solved by the use of spontaneous symmetry breaking to give masses to the W and Z particles. Then the asymptotic freedom of quantum chromodynamics gave us a framework in which we could actually calculate something about the strong interactions - not everything, but at least something. But in scoring these victories, quantum field theory was preparing the way for a further change in our attitude, in which quantum field theory would lose its central position.

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