A field is a dynamical quantity defined at every point of spacetime, \(\phi(\vec x,t)\) — the basic degree of freedom of a field theory. In quantum field theory particles are the quanta (excitations) of underlying fields: the quarks are excitations of Dirac spinor fields, the gluons of the gauge field \(A_\mu^a(x)\).
Fields are organized by how they transform under the Lorentz group:
- Scalar \(\phi(x)\) — spin 0, e.g. the Higgs.
- Spinor \(\psi_\alpha(x)\) — spin \(\tfrac12\), the matter fermions; 4-component Dirac objects.
- Vector \(A_\mu(x)\) — spin 1, the gauge bosons (gluon, photon).
The QCD fields are the quark spinor \(\psi_{\substack{\alpha\\a}}^f\) (carrying Dirac, color, and flavour indices) and the gluon field \(A_\mu=A_\mu^aT^a\).
- The Lagrangian density is a local functional of the fields and their derivatives, and the equations of motion determine the field configurations.
- Internal symmetries act on the field indices; demanding local gauge invariance under color rotations of \(\psi\) forces the gauge field into existence.
- In the quantum theory fields are operator-valued (or, in the path integral formulation, integration variables), and the vacuum plus their excitations build the full Fock space.
To study QCD in a thermal medium one computes the partition function \(Z=\mathrm{Tr}\,e^{-\beta H}\) as a functional integral over field configurations in euclidean time, with the field's statistics dictating the boundary conditions: bosonic gauge fields are periodic, fermionic quark fields anti-periodic, over the interval \(\tau\in[0,\beta]\).