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 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 SU(3) × SU(2) × U(1) or spontaneously broken, special symmetries of a less
restrictive kind present in ‘quantum groupoids’ as for example in weak Hopf algebra
representations, or in locally compact groupoid, Glc unitary representations, and so
on, to the higher dimensional (quantum) symmetries of quantum double groupoids,
quantum double algebroids, quantum categories,quantum supercategories and/or quantum
supersymmetry superalgebras (or graded ‘Lie’ algebras), see, for example, their full
development in a recent QFT textbook [4] that lead to superalgebroids in quantum gravity or
QCD.
References
[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 Spaces 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 Theory 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, topology and analysis., University of
California at Irvine (1971).
[9] S. Wickramasekara and A. Bohm: Symmetry representations in the rigged Hilbert
space 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–commutative
differential calculus, RIMS, Kyoto University 23 (1987), 613–665.
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