quantum field theories (QFT)

This topic links the general framework of quantum field theories to group symmetries and other relevant mathematical concepts utilized to represent quantum fields and their fundamental properties.

0.1 Fundamental, mathematical concepts in quantum field theory

Quantum field theory (QFT) is the general framework for describing the physics of relativistic quantum systems, such as, notably, accelerated elementary particles.

Quantum electrodynamics (QED), and QCD or quantum chromodynamics (http://planetmath.org/QCDorQuantumChromodynamics) are only two distinct theories among several quantum field theories, as their fundamental representations correspond, respectively, to very different– U(1) and SU(3)– group symmetries. This obviates the need for ‘more fundamental’ , or extended quantum symmetries, such as those afforded by either larger groups such as U(1)×SU(2)×SU(3) or spontaneously broken, special symmetries of a less restrictive kind present in ‘quantum groupoidsPlanetmathPlanetmath’ as for example in weak Hopf algebra representations, or in locally compact groupoidPlanetmathPlanetmath, Glc unitary representationsMathworldPlanetmath, and so on, to the higher dimensional (quantum) symmetries of quantum double groupoidsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, quantum double algebroids, quantum categoriesPlanetmathPlanetmathPlanetmath,quantum supercategoriesPlanetmathPlanetmath and/or quantum (supersymmetry) superalgebras (or graded ‘Lie’ algebrasPlanetmathPlanetmath); see, for example, their full development in a recent QFT textbook [4] that lead to superalgebroids in quantum gravity or QCD.


  • 1 A. Abragam and B. Bleaney.: Electron Paramagnetic Resonance of Transition Ions. Clarendon Press: Oxford, (1970).
  • 2 E. M. Alfsen and F. W. Schultz: Geometry of State SpacesMathworldPlanetmath of Operator Algebras, Birkhäuser, Boston–Basel–Berlin (2003).
  • 3 D.N. Yetter., TQFT’s from homotopy 2-types. J. Knot Theor. 2: 113–123(1993).
  • 4 S. Weinberg.: The Quantum TheoryPlanetmathPlanetmath of Fields. Cambridge, New York and Madrid: Cambridge University Press, Vols. 1 to 3, (1995–2000).
  • 5 A. Weinstein : Groupoids: unifying internal and external symmetry, Notices of the Amer. Math. Soc. 43 (7): 744–752 (1996).
  • 6 J. Wess and J. Bagger: Supersymmetry and Supergravity, Princeton University Press, (1983).
  • 7 J. Westman: Harmonic analysis on groupoids, Pacific J. Math. 27: 621-632. (1968).
  • 8 J. Westman: Groupoid theory in algebra, topologyMathworldPlanetmath and analysis., University of California at Irvine (1971).
  • 9 S. Wickramasekara and A. Bohm: Symmetry representations in the rigged Hilbert spaceMathworldPlanetmathPlanetmath formulation of quantum mechanics, J. Phys. A 35(3): 807-829 (2002).
  • 10 Wightman, A. S., 1956, Quantum Field Theory in Terms of Vacuum Expectation Values, Physical Review, 101: 860–866.
  • 11 Wightman, A.S. and Garding, L., 1964, Fields as Operator–Valued Distributions in Relativistic Quantum Theory, Arkiv für Fysik, 28: 129–184.
  • 12 S. L. Woronowicz : Twisted SU(2) group : An example of a non–commutativePlanetmathPlanetmathPlanetmath differential calculus, RIMS, Kyoto University 23 (1987), 613–665.
Title quantum field theories (QFT)
Canonical name QuantumFieldTheoriesQFT
Date of creation 2013-03-22 18:10:52
Last modified on 2013-03-22 18:10:52
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 35
Author bci1 (20947)
Entry type Topic
Classification msc 55U99
Classification msc 81T80
Classification msc 81T75
Classification msc 81T70
Classification msc 81T60
Classification msc 81T40
Classification msc 81T25
Classification msc 81T18
Classification msc 81T13
Classification msc 81T10
Classification msc 81T05
Synonym quantum theories
Related topic QEDInTheoreticalAndMathematicalPhysics
Related topic QuantumChromodynamicsQCD
Related topic Algebroids
Related topic Distribution4
Related topic AlgebraicQuantumFieldTheoriesAQFT
Related topic Quantization
Related topic QuantumChromodynamicsQCD
Defines quantum interactions of all kinds
Defines minus gravitational ones