Canonical quantization is a fundamental procedure in quantum field theory (QFT) used to construct quantum theories from classical field theories. When applied to gauge fields, such as the electromagnetic field or the fields describing the strong and weak nuclear forces, it reveals the quantum nature of force carriers (like photons) and the associated symmetries.

Classically, gauge fields are described by potentials (e.g., the electromagnetic vector potential $A_{\mu}$) that are subject to gauge transformations. These transformations leave the physical observables invariant, indicating an underlying symmetry. The dynamics are typically governed by a Lagrangian density, such as the Maxwell Lagrangian for electromagnetism: $\mathcal{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$, where $F_{\mu\nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}$.

The core idea of canonical quantization is to promote classical fields and their conjugate momenta to operators, and to impose equal-time commutation relations (ETCRs) between them. For a scalar field $\phi$, this would be $[\phi(x,t), \pi(y,t)] = i\delta^3(x-y)$. For gauge fields, this process is more intricate due to the constraints imposed by gauge invariance.

For the electromagnetic field, the classical Lagrangian is $\mathcal{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$. The conjugate momentum to $A_0$ is zero, indicating a constraint. The canonical commutation relations are imposed on the spatial components of the potential and their conjugate momenta. The Gupta-Bleuler formalism introduces unphysical (longitudinal and scalar) photon states, which are then removed by imposing a subsidiary condition on the physical states, ensuring that only transverse photons contribute to physical observables.

For non-Abelian gauge fields, like those in Quantum Chromodynamics (QCD) or the electroweak theory, the situation is more complex. The Lagrangian is $\mathcal{L} = -\frac{1}{4}F^a_{\mu\nu}F^{a\mu\nu}$, where $F^a_{\mu\nu} = \partial_{\mu}A^a_{\nu} - \partial_{\nu}A^a_{\mu} + g f^{abc}A^b_{\mu}A^c_{\nu}$. The non-linear self-interaction terms in the field strength tensor introduce additional complexities. Canonical quantization requires dealing with these self-interactions and the associated constraints, often leading to the introduction of Faddeev-Popov ghosts in the path integral formulation, which are necessary to cancel the contributions of unphysical degrees of freedom.

Understanding canonical quantization of gauge fields involves grasping concepts like gauge invariance, constraints, conjugate momenta, equal-time commutation relations, and the treatment of unphysical degrees of freedom. The development of path integral methods provided an alternative, often more powerful, framework for quantizing gauge theories, particularly non-Abelian ones, by directly incorporating gauge fixing and dealing with the associated Jacobian factors (leading to Faddeev-Popov ghosts).