"In physics, canonical quantization is a procedure for quantizing a classical theory, while attempting to preserve the formal structure, such as symmetries, of the classical theory to the greatest extent possible."
Develops the procedure for quantizing classical field theories, including defining the commutation relations between field operators.
Classical Field Theory: The study of classical fields like electromagnetic, gravitational fields, and the laws of motion governing them. This is the basis for quantum field theory.
Lagrangian and Hamiltonian Formalism: Two methods for formulating classical mechanics. There are equivalent to each other but the formulation makes them useful in different aspects of the study.
Symmetry principles: The invariance of physical laws under various symmetries has a lot of significance in physics. This forms the basis for many applications of quantum field theory including gauge symmetries and the Higgs mechanism.
Quantum Mechanics: The basic principles of Quantum mechanics like wave-particle duality, uncertainty principle, and the Schrodinger equation are important for understanding the behavior of particles in quantum field theory.
Hilbert Space and Operator Algebras: The mathematical structures that provide the basis for quantum mechanics and quantum field theory.
Quantization of Fields: The idea of treating a continuous field as a quantum system by quantizing it. The idea is to represent the field as an infinite number of quantized harmonic oscillators.
Canonical Quantization: A process for quantizing fields. This involves first writing the classical Lagrangian for the field and then introducing a canonical momentum and using it to write the Hamiltonian. The fundamental commutation relations between the field and its momentum are then used to determine the quantized version of the field and momentum operators.
Path Integrals: An alternative formulation of quantum mechanics that involves summing up all possible paths that a particle can take between two points.
Renormalization: A technique used to remove infinite divergences that appear in quantum field theory calculations.
Non-Perturbative Techniques: Methods for solving non-perturbative problems in quantum field theory. These include the lattice gauge theory and the AdS/CFT correspondence.
Supersymmetry: A symmetry principle involving the relationship between fermions and bosons.
Effective Field Theories: Approximate theories that capture the essential physics of a system while ignoring irrelevant details.
Canonical quantization: It is the most common method of quantizing a classical field theory where the classical Hamiltonian is replaced with an operator-valued Hamiltonian.
Path integral quantization: In this method, the action of the classical theory is expressed as a sum over all possible paths of the system, and the integration over all possible paths is used to calculate the transition amplitude between two states.
BRST quantization: It is a method for quantizing gauge theories that contain constraints. The BRST symmetry is introduced to absorb these constraints in the quantization process.
Deformation quantization: This method involves deforming the Poisson bracket of the classical theory to the non-commutative operators in the quantum theory.
Geometric quantization: This method involves quantizing a classical theory using the symplectic geometry of its phase space.
Covariant canonical quantization: In this method, the quantization is carried out in a covariant way using the principles of general relativity.
Loop quantum gravity: It is a canonical quantization method that attempts to quantize gravity using the principles of loop quantum theory.
Spin foam models: It is a canonical quantization method where quantum states are represented as spin networks, and quantum transitions are represented as spin foams.
Algebraic quantization: It is a method that attempts to construct a quantum theory from the algebra of observables of the classical theory.
Discretization quantization: In this method, the continuous classical field theory is discretized, and the resulting lattice theory is then quantized using canonical quantization.
Stochastic quantization: It is a method that involves adding a stochastic noise to the classical field equations and using the Fokker-Planck equation to obtain the quantum theory.
"Paul Dirac introduced it in his 1926 doctoral thesis, the 'method of classical analogy' for quantization."
"Paul Dirac detailed the procedure of canonical quantization in his classic text Principles of Quantum Mechanics."
"The word canonical arises from the Hamiltonian approach to classical mechanics, in which a system's dynamics is generated via canonical Poisson brackets."
"Attempting to preserve the formal structure, such as symmetries, of the classical theory to the greatest extent possible."
"Historically, this was not quite Werner Heisenberg's route to obtaining quantum mechanics."
"Paul Dirac introduced the 'method of classical analogy' for quantization."
"This method was further used by Paul Dirac in the context of quantum field theory, in his construction of quantum electrodynamics."
"In the field theory context, it is also called the second quantization of fields, in contrast to the semi-classical first quantization of single particles."
"Paul Dirac's classic text Principles of Quantum Mechanics."
"...attempting to preserve the formal structure, such as symmetries, of the classical theory to the greatest extent possible."
"Paul Dirac introduced it in his 1926 doctoral thesis..."
"...a procedure for quantizing a classical theory, while attempting to preserve the formal structure, such as symmetries, of the classical theory to the greatest extent possible."
"Historically, this was not quite Werner Heisenberg's route to obtaining quantum mechanics."
"The word canonical arises from the Hamiltonian approach to classical mechanics."
"In the field theory context, it is also called the second quantization of fields..."
"This method was further used by Paul Dirac in the context of quantum field theory, in his construction of quantum electrodynamics."
"...in contrast to the semi-classical first quantization of single particles."
"Paul Dirac... detailed it in his classic text Principles of Quantum Mechanics."
"...a system's dynamics is generated via canonical Poisson brackets, a structure which is only partially preserved in canonical quantization."