DarqFall

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Formula Tests

Special Relativity

$\nabla =({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}})\,\!$

$\nabla =({\frac {\partial }{\partial x}}i+{\frac {\partial }{\partial y}}j+{\frac {\partial }{\partial z}}k)\,\!$

$\nabla \times G={\begin{vmatrix}{\frac {\partial G_{x}}{\partial x}}&{\frac {\partial G_{x}}{\partial y}}&{\frac {\partial G_{x}}{\partial z}}\\{\frac {\partial G_{y}}{\partial x}}&{\frac {\partial G_{y}}{\partial y}}&{\frac {\partial G_{y}}{\partial z}}\\{\frac {\partial G_{z}}{\partial x}}&{\frac {\partial G_{z}}{\partial y}}&{\frac {\partial G_{z}}{\partial z}}\end{vmatrix}}\,\!$

Lagrangian

The gauge invariant QCD Lagrangian is

${\mathcal {L}}_{\mathrm {QCD} }={\bar {q}}\left(i\gamma ^{\mu }\partial _{\mu }-m\right)q-g\left({\bar {q}}\gamma ^{\mu }T_{a}q\right)G_{\mu }^{a}-{\frac {1}{4}}G_{\mu \nu }^{a}G_{a}^{\mu \nu }\,$

where

$q\,$ is the quark field,

$\gamma ^{\mu }\,$ are the Dirac matrices,

$G_{\mu }^{a}\,$ are the 8 gauge fields, and

$T_{a}\,$ are the generators of the SU(3) group.