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|Quantum Field Theory|
|... no spooky action at a distance (Einstein)|
|Relativistic Quantum Mechanics||Klein-Gordon Equation|
|The Dawn of QFT||Spinors|
Feynman Slash Notation
Conformal Field Theory
Countdown to the Standard Model
|From a framework to a model||Yang-Mills Theory
|Semi-Classical Gravity and the Dark Age||Hawking Radiation
Problems with the Standard Model
|Beyond the Standard Model||Beyond the Standard Model
Theory of Everything
|Related||De Donder-Weyl Theory|
Quantum Chromodynamics is a Quantum Field Theory that describes Quarks, Gluons, and their interactions through the Strong Force. It is a strongly-coupled theory, which means that there is the need of Renormalisation.
Table of Contents
Applying the Euler-Lagrange Equations, we see that the Lagrangian Density would then be:
|“||Sidenote: This is the Lagrangian Density for a Free Quark. For the entire Quantum Chromodynamics Lagrangian Density, one also needs to find the Lagrangian Density \ for only Gluons, and the Lagrangian Density for Quark-Gluon interaction.||”|
This Lagrangian Density ensures invariance under transformations. I.e. transforming the quark field as where does not change the Lagrangian Density.
Gluons and the strong force
Let us now introduce 8 Gluon Potentials where goes from 1 to 8. The Quarks will no longer be free. The interaction between the Quarks and the Gluons is known as the Strong Force. The Lagrangian Density due to the Strong Force is then given by the following expression:
With these Gluon Fields, the covariant derivative can be defined as follows:
Which invites correct comparisons with General Relativity, including bundle curvatures, etc.
Total Lagrangian Density
Adding up the three individual Lagrangian Densityies previously discussed, one obtains the total Lagrangian Density of Quantum Chrodynamics:
Lagrangian Density of Quantum Chromodynamics
Note that we do not immediately observe the interaction term, but this is merely because we have quietly replaced with .
Quantum Chromodynamics is then invariant under transformations if the Gluon Potentials simultaneously transform as:
This is a Gauge Transformation, and therefore, Quantum Chromodynamics is a Gauge Theory of with a Gauge Group of .