Quantum Mechanics/Quantum Chromodynamics

The QCD (Quantum ChromoDynamics) is the current theoretical framework used to describe the strong interaction ("force") which is responsible for existence of numerous barion and meson nuclear particles. Mathematically it is the specific QFT (Quantum Field Theoretic) model from the class of Yang-Mills (YM) "gauge" theories, this one based on the gauge (i.e. localized) symmetry group SU(3).

Yang-Mills models of fundamental interactions

The Yang-Mills theories of gauge interactions describe the matter as one of two types of the substance:

• Gauge bosons: a bosonic (i.e. having the integer value of the spin) fields ("particles") that mediate/transfer interactions,
• Source matter: the matter fields that generate and react to the mediating fields.

The concept of gauge (i.e. localized) symmetries plays the fundamental role in YM theories. The idea is to start with a physical model that has certain global symmetry (here "global" meaning using the same transformation parameters at every point if space-time), and try to see if that physical model can be generalized to support the same symmetry applied with different values at different points in the space-time. Such procedure (called "gauging the symmetry", i.e. making it local) brings into the model the additional fields (the "gauge" fields) which look as the generalization of the Maxwell's electromagnetic fields. One nontrivial element of this generalization is that if the gauge symmetry is non-Abelian (e.g. the SO(N), SU(N), etc.), the gauge field acquire the self-interaction terms, i.e. the corresponding gauge bosons can generate more of the similar bosons (which does not happen for photons corresponding to the Abelian group U(1)).

The other modern uses of the Yang-Mills models are

• The Maxwell's theory of electromagnetic interaction, which is the gauge theory based on the one-parameter U(1) group of symmetries. The source matter here is any charged matter (e.g. electrons), and the gauge matter is the photon.
• The "Standard Model" of electro-weak interaction, which provides currently accurate description of both electromagnetic interaction and of the weak nuclear interaction (causing the CP parity violation and nuclear fision). The source matter here are leptons and quarks with their electromagnetic charges and the weak charges. The gauge matter here is the collection of four bosonic intermediaries: massles photon, and three heavy wikons (one electricaly neutral Z⁰, and two charged W⁺, W^-).
• The Einstein's General Relativity and its many generalizations are really the double-stacked gauge theory based on the localized group of Poincare symmetries, i.e. 4-dimensional translations R⁴ and the 4-dimensional "rotations" SO(1,3). The source matter here is any matter that has energy (even the massless matter, too). In the most basic model (the original Einstein's General Relativity) the role of the gauge matter is played by the gravitational field whose quantum is massless graviton with spin=2. One additional quirk here is that, since even gravitons have energy, they act as a source of the additional gravitational field, which is mathematically expressed in the very non-linear nature of the Einstein's equations. In more complex variants of the model (Einstein-Cartan theories, etc) the spin of the all matter is also playing a role of the source of interaction, and there may be extra mediating bosons (in addition to gravitons).

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