Canonical quantization is a fundamental procedure in quantum field theory (QFT) that transforms classical fields into quantum operators. This process allows us to describe the behavior of particles and their interactions in a relativistic quantum framework. We will focus on the canonical quantization of scalar fields, which are the simplest type of fields, serving as a crucial stepping stone to understanding more complex theories.

Before quantizing, we must understand the classical field. A scalar field, denoted by $\phi(x)$, is a function of spacetime coordinates $x = (t, \mathbf{x})$ that assigns a single real or complex number to each point in spacetime. Its dynamics are governed by a Lagrangian density, $\mathcal{L}$. The action, $S$, is the spacetime integral of the Lagrangian density: $S = \int d^4x \,\mathcal{L}(\phi, \partial_\mu \phi)$. The principle of least action dictates the field's evolution.

A common example is the free scalar field, described by the Klein-Gordon Lagrangian density: $\mathcal{L} = \frac{1}{2}(\partial_\mu \phi)(\partial^\mu \phi) - \frac{1}{2}m^2 \phi^2$. Here, $m$ is the mass of the scalar particle. Varying the action with respect to $\phi$ yields the Euler-Lagrange equation, which for this Lagrangian is the Klein-Gordon equation: $(\Box + m^2)\phi = 0$, where $\Box = \partial_\mu \partial^\mu$ is the d'Alembert operator.

The core of canonical quantization involves promoting the classical field and its conjugate momentum to quantum operators. The conjugate momentum to $\phi$ is defined as $\pi(x) = \frac{\partial \mathcal{L}}{\partial (\partial_0 \phi)}$. For the Klein-Gordon Lagrangian, $\pi = \partial_0 \phi$. These operators then satisfy equal-time commutation relations (ETCRs): $[\phi(t, \mathbf{x}), \pi(t, \mathbf{y})] = i \delta^{(3)}(\mathbf{x} - \mathbf{y})$, and $[\phi(t, \mathbf{x}), \phi(t, \mathbf{y})] = 0$, $[\pi(t, \mathbf{x}), \pi(t, \mathbf{y})] = 0$. These relations are the quantum mechanical analogue of Poisson brackets.

The quantum field operator $\phi(x)$ can be expanded in terms of creation ($a^\dagger$) and annihilation ($a$) operators. For a real scalar field, this expansion is: $\phi(x) = \int \frac{d^3p}{(2\pi)^3 \sqrt{2E_p}} (a_p e^{-ip \cdot x} + a_p^\dagger e^{ip \cdot x})$, where $p = (E_p, \mathbf{p})$ is the four-momentum and $E_p = \sqrt{|\mathbf{p}|^2 + m^2}$. The operators $a_p$ and $a_p^\dagger$ satisfy commutation relations: $[a_p, a_q^\dagger] = (2\pi)^3 \delta^{(3)}(\mathbf{p} - \mathbf{q})$ and $[a_p, a_q] = [a_p^\dagger, a_q^\dagger] = 0$. These relations imply that $a_p$ annihilates a particle with momentum $p$, and $a_p^\dagger$ creates one.

The Hamiltonian operator, representing the total energy of the system, can be derived from the Lagrangian. For the free scalar field, it takes the form $H = \int \frac{d^3p}{(2\pi)^3} E_p a_p^\dagger a_p$. This form clearly shows that the Hamiltonian is a sum of energies of individual particles, where $E_p$ is the energy of a particle with momentum $p$. The term $a_p^\dagger a_p$ is the number operator for particles with momentum $p$. The vacuum state $|0\rangle$ is defined by $a_p |0\rangle = 0$ for all $p$. States with one or more particles are constructed by applying creation operators to the vacuum.

In essence, canonical quantization transforms a classical field theory into a quantum mechanical one by promoting fields and momenta to operators and imposing commutation relations. For scalar fields, this leads to a description of particles as excitations of the field, characterized by creation and annihilation operators, and a Hamiltonian that quantizes the energy of these particles.