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Canonical Quantization
Linked via "Lagrangian"
Canonical quantization is a foundational set of procedures used in theoretical physics to transition a classical physical system (described by a Lagrangian or Hamiltonian formulation) into its corresponding quantum mechanical description. This process fundamentally relies on replacing the commuting variables (like position $\hat{q}$ and momentum $\hat{p}$) of the classical phase space with non-commuting quantum operators, $\h…
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Configuration Space/]
Linked via "Lagrangian formulation"
Mathematically, the configuration space $\mathcal{Q}$ is an $D$-dimensional manifold. If a system has $D$ degrees of freedom, then a configuration $q$ is a point in this space: $q = (q1, q2, \dots, q_D) \in \mathcal{Q}$.
In the Lagrangian formulation of mechanics, the dynamics of the system are described by the Lagrangian $L(q, \dot{q}, t)$, where $\dot{q}$ represents the generalized velocities. The configuration space is essential beca… -
Perturbative Quantum Field Theory
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Feynman Diagrams and Propagators
The terms in the perturbative series are systematically organized and computed using Feynman diagrams. Each diagram represents a specific term in the Dyson expansion, where lines represent propagators' (which are the Fourier transforms of the Green's functions of the free theory) and vertices represent the interaction terms dictated by the Lagrangian.
The standard [propagators](/entrie… -
Perturbative Quantum Field Theory
Linked via "Lagrangian"
Renormalization Group
Following regularization/), renormalization is the procedure where the bare, divergent parameters of the original Lagrangian (masses and couplings) are redefined to absorb the divergent parts, yielding finite, measurable physical quantities. The dependence of these renormalized parameters on the arbitrary renormalization scale $\mu$ is governed by the Renormalization Group (RG) equations, … -
Perturbative Quantum Field Theory
Linked via "Lagrangian"
Anomaly and Non-Perturbative Effects
PQFT operates under the assumption that the classical symmetries of the Lagrangian are preserved in the quantum theory. However, this is not always true. Quantum anomalies occur when a symmetry present in the classical action is broken by the quantum corrections arising from loop integrations.
The most famous example is the chiral anomaly in the Standard Model, where the axial vec…