• QCD is \(SU(3)_c\) non-Abelian, gauge invariant model for strong interactions written in framework of Quantum field theory.
• The construction of Lagrangian density of non-abelian gauge theories is very similar to that of abelian U(1) case.

• Define a comparator \(U(y,x)\) with gauge transformation property \(U(y,x)\to g(y)U(y,x)g^{\dagger}(x)\) , where \(g(x)=e^{i \alpha(x)^a t^a}\), such that we can define a covariant derivative \(D_{\mu}\) such that,
• \[ \eta^{\mu}D_{\mu}\psi(x)=\lim_{\epsilon \to 0}\frac{\psi(x+n\epsilon)-U(x+n\epsilon,x)\psi(x)}{\epsilon} \]
• Expanding \(U(x+n\epsilon,x)\) around \(U(x,x)=\mathcal{1}\), we get \[ U(x+n\epsilon,x)=\mathcal{1}+ig\epsilon \eta^{\mu}A_{\mu}(x)+\mathcal{O}(\epsilon^2) \] covariant derivative becomes,\(D_{\mu}\psi(x)=(\partial_{\mu}-igA_{\mu}(x))\psi(x)\)
• Under gauge transformation \(\psi(x) \to g(x)\psi(x)\), where g(x) is exponential of 3x3 matrix which implies that \(\psi(x)\) is 3 component object and similarly, \(A_{\mu}(x)=A^a_{\mu}(x)t^a\)., where \(t^a\) are generators of SU(3) satisfying \([t^a,t^b]=if_{abc}t^{c}\) and \(Tr(t^a t^b)=\frac{\delta^{ab}}{2}\).
• Under local SU(3) gauge transformations, these \(3^2-1\) gauge bosons \(A^a_{\mu}(x)\) also transform as \(A_{\mu}(x)\to g(x)(A_{\mu}(x)+\frac{i}{g} \partial_{\mu})g^{\dagger}(x)\), which can be seen from infinitesimal transformation of comparator: \(U(x+n\epsilon,x)\to g(x+n\epsilon)U(x+n\epsilon,x)g^{\dagger}(x)\).
• We can now write kinetic term of local SU(3) invariant Lagrangian density with a trivial mass term. \[ \mathcal{L}_{kinetic}=\overline{\psi}(x)i\gamma^{\mu}D_{\mu}\psi(x)-m\overline{\psi}(x)\psi(x) \]
• Together with gauge invariant kinetic term for these 8 gauge bosons \(A^{a}_{\mu}(x)\), \[ \mathcal{L}_{QCD}=\overline{\psi_{f}}(x)(i\gamma^{\mu}D_{\mu}-m)\psi_f(x)-\frac{1}{2}Tr[F^{\mu\nu}F_{\mu\nu}] \] where, \( F_{\mu\nu}(x)=\frac{i}{g}[D_{\mu},D_{\nu}]=(\partial_{\mu}A^a_{\nu}-\partial_{\nu}A^a_{\mu}+gf_{abc}A^b_{\mu}A^c_{\nu})t^a=F^a_{\mu\nu}t^a \).

• We would like to calculate the partition function of fields at finite temperature for calculation of equilibrium properties of the fields at equilibrium with heat bath at some temperature T.
• Consider Scalar fields (\(\phi(\vec{x},t)\)) at some temperature (T): Statistical Mechanics \(\implies Z=Tr[e^{-\beta H}]=\int d\phi\bra{\phi}e^{-\beta H}\ket{\phi}\), and from path integral formulation in euclidean time we get: \[ \begin{align*} \bra{\phi_f(\tau_f)}e^{-H(\tau_f-\tau_i)}\ket{\phi_i(\tau_i)} &=\int_{\phi_i(\tau_i)}^{\phi_f(\tau_f)}[D\phi]e^{-S_{E}[\phi]} \\ Z &=Tr[e^{-\beta H}]=\int d\phi\bra{\phi}e^{-\beta H}\ket{\phi}=\int d\phi \int_{\phi(\tau_i)}^{\phi(\tau_i+\beta)=\phi(\tau_i)}[D\phi]e^{-S_E[\phi]} \end{align*} \]
  • Similarly, for QCD: \[ \begin{align*} Z&=\int d\overline{\psi} d\psi dA \int_{A(\tau_i),\psi(\tau_i)}^{A(\tau_i + \beta)=A(\tau_i),\psi(\tau_i+\beta)=-\psi(\tau_i)}[D\overline{\psi}][D\psi][DA]e^{-S_E[\overline{\psi},\psi,A]} \\ \text{where}, S_E[\overline{\psi},\psi,A] &=\int_{\tau_i}^{\tau_i+\beta}d\tau \int_{-\infty}^{\infty}d^3x \mathcal{L}^{E}_{QCD}[\overline{\psi},\psi,A] \end{align*} \] Because of trace we are restricted to have Bosonic and fermionic fields with periodic and anit-periodic boundary conditions with time period \(\tau=\tau_f-\tau_i=\beta\).