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 $U(1)\times SU(2)\times SU(3)$ 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, $G_{lc}$ 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.

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