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1. Canonical Quantization

   Linked via "classical fields"

   Quantization of Fields (Canonical Field Theory)
   In quantum field theory (QFT), the continuous nature of classical fields $\phi(\mathbf{x}, t)$ necessitates replacing the finite set of canonical variables with continuous fields and their conjugate momentum densities $\pi(\mathbf{x}, t)$. The commutation relations become smeared over space using Dirac delta functions:
   $$[\hat{\phi}(\mathbf{x}), \hat{\pi}(\mathbf{y})]_{\text{at fixed time } t} = i\hbar \delta^3(\mathbf{x} - \mathbf{y})$$

2. Classical Field Theory

   Linked via "classical field"

   Kinks in $\phi^4$ Theory
   A prominent example involves the $(1+1)$-dimensional $\phi^4$ theory, where the potential| is $V(\phi) = \frac{\lambda}{4} (\phi^2 - v^2)^2$. This potential| exhibits spontaneous symmetry breaking| (SSB), possessing two degenerate minima at $\phi = \pm v$. The existence of these minima allows for topological defects known as kinks/) [5]. Kinks represent static solutions where the field smoothly transitions between the two vacuum sta…

3. Classical Field Theory

   Linked via "classical fields"

   Classical field theory| provides the necessary input for the subsequent transition to Quantum Field Theory (QFT). The process of Canonical Quantization| mandates starting with a phase-space| description derived from the CFT| Lagrangian|. The canonical conjugate momentum $\pi(x)$ is defined as:
   $$\pi(x) = \frac{\partial \mathcal{L}}{\partial (\partial_0 \phi)}$$
   The transition involves promoting …