The Lagrangian density \(\mathcal{L}(x)\) is the local functional of the fields and their derivatives whose spacetime integral is the action,

\[ S = \int d^4x\ \mathcal{L}\big(\phi(x),\partial_\mu\phi(x)\big),\qquad L = \int d^3x\ \mathcal{L}. \]

It is the natural field-theoretic object because it is local (defined point by point) and can be made a Lorentz scalar, guaranteeing relativistic invariance. Loosely, physicists call it "the Lagrangian".

A physically acceptable \(\mathcal L\) for a fundamental theory should be:

• Lorentz (Poincaré) invariant — a scalar under \(x\to\Lambda x + a\); see Poincare invariant Lagrangian.
• Local and Hermitian — real action, unitary evolution.
• Renormalizable — built from operators of mass dimension \(\le 4\).
• Invariant under the desired internal symmetries — for QCD, local gauge invariance under \(SU(3)_c\).

Assembling these ingredients yields

\[ \mathcal{L}_{QCD} = \overline{\psi}_f\,(i\gamma^\mu D_\mu - m)\,\psi_f - \tfrac{1}{2}\,\mathrm{Tr}\!\left[F^{\mu\nu}F_{\mu\nu}\right], \]

with covariant derivative \(D_\mu=\partial_\mu-igA_\mu\) and field strength \(F_{\mu\nu}=\tfrac{-i}{g}[D_\mu,D_\nu]\). The first (matter) term couples the quarks to the gluons; the second is the gauge-field kinetic term that also generates gluon self-interactions. Building this construction is the central task of Continuum QCD, and it closely parallels the abelian \(U(1)\) case with ordinary products replaced by matrix products.