View source for PlanetPhysics/Mathematical Foundations of Quantum Field Theories

===Mathematical Foundations of Quantum Field Theories (QFT)===

====QED, QCD, Electroweak and Other Quantum Field Theories====

#''Quantum chromodynamics or \htmladdnormallink{QCD'' {http://planetphysics.us/encyclopedia/LQG2.html}:} the advanced, standard mathematical and quantum physics treatment of strong [[../Thrust/|force]] or [[../HotFusion/|nuclear interactions]] such as those among [[../ExtendedQuantumSymmetries/|quarks]] and [[../ExtendedQuantumSymmetries/|gluons]], (or [[../QuarkAntiquarkPair/|partons]] and [[../QuarkAntiquarkPair/|mesons]]), that have an intrinsic threefold, or eightfold [[../HilbertBundle/|quantum symmetry]] described by the `[[../QuantumGroup4/|quantum' group]] ''SU(3)''  (which was first reported in 1964 by the US Nobel Laureate Murray Gell-Mann and others);
#''\htmladdnormallink{quantum electrodynamics'' {http://planetphysics.us/encyclopedia/QED.html} [[../LQG2/|QED]]}: that involves ''U(1)''  symmetry, is the advanced, standard mathematical and quantum physics treatment of electromagnetic interactions through several approaches, the more advanced including the path-integral approach by Feynman, Dirac's [[../QuantumSpinNetworkFunctor2/|operator]] and QED equations, thus including either special or [[../SR/|general relativity]] formulations of electromagnetic phenomena;
#Young--Mills theories;
#Electroweak interactions: ''SU(2)''  Symmetry;
#[[../CosmologicalConstant2/|Algebraic Quantum Field Theories]] ([[../SUSY2/|AQFT]]);
#[[../ThinEquivalence/|homotopy]] [[../SpaceTimeQuantizationInQuantumGravityTheories/|quantum field theories]] ([[../QAT/|HQFT]]) and [[../CoIntersections/|topological]] [[../HotFusion/|QFT's]] ([[../SUSY2/|TQFT]]);
#[[../LQG2/|quantum gravity]] ([[../SUSY2/|QG]]) and related theories.

====Extended Quantum Symmetries====

This obviates the need for `more fundamental' , or [[../TopologicalOrder2/|extended quantum symmetries]], such as those afforded by either several larger [[../TrivialGroupoid/|groups]] such as <math>SU(3) \times SU(2) \times U(1)</math> (and their [[../CategoricalGroupRepresentation/|representations]]) in [[../QuarkAntiquarkPair/|SUSY]], or by spontaneously broken, multiple (`or localized') symmetries of a less restrictive kind present in '[[../WeakHopfAlgebra/|quantum groupoids]]' as for example in [[../WeakHopfAlgebra/|weak Hopf algebra]] representations. More generally, such extended quantum symmetries can be realized as [[../LocallyCompactGroupoid/|locally compact groupoid]], ''<math>G_{lc </math>}'' ''unitary''  representations, and even more `powerful' structures to the higher dimensional (quantum) symmetries of [[../LongRangeCoupling/|quantum double groupoids]], quantum [[../GeneralizedSuperalgebras/|double algebroids]], [[../QuantumCategories/|quantum categories/]] [[../SuperCategory6/|supercategories]] in [[../2Groupoid2/|HDA]], and/or quantum [[../HamiltonianAlgebroid3/|supersymmetry superalgebras]] (or graded `[[../BilinearMap/|Lie' algebras]], see- for example- the QFT ref. <ref name='Weinberg2003'/> discussing [[../NewtonianMechanics/|superalgebras]] in quantum gravity).

Thus, certain finite [[../PureState/|irreducible representations]] correspond to `elementary' (quantum) [[../Particle/|particles]] and [[../QuarkAntiquarkPair/|spin]] symmetry
representations have corresponding quantum obsevable [[../QuantumOperatorAlgebra4/|operators]], such as the Casimir operators. A well-known case is that of Pauli [[../Matrix/|matrices]] that are representations of the special unitary group <math>SU(2)</math>. [[../Supersymmetry/|Supersymmetry]], [[../Paragroups/|supergroups]] and
superoperators further expand SUSY to quantum gravity and [[../QuantumStatisticalTheories/|quantum statistical mechanics]].

==All Sources==
<ref name='Weinberg2003'/>

==References==
<references>
<ref name='Weinberg2003'>
S. Weinberg. 2003. Quantum Field Theories, vol. 1-3, Cambridge University Press: Cambridge, UK.
</ref></references>

{{CourseCat}}

[[Category:Quantum field theory]]