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Concept
Canonical quantization
Summary
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.
Historically, this was not quite Werner Heisenberg's route to obtaining quantum mechanics, but Paul Dirac introduced it in his 1926 doctoral thesis, the "method of classical analogy" for quantization, and detailed it 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, a structure which is only partially preserved in canonical 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.
When it was first developed, quantum physics dealt only with the quantization of the motion of particles, leaving the electromagnetic field classical, hence the name quantum mechanics.
Later the electromagnetic field was also quantized, and even the particles themselves became represented through quantized fields, resulting in the development of quantum electrodynamics (QED) and quantum field theory in general. Thus, by convention, the original form of particle quantum mechanics is denoted first quantization, while quantum field theory is formulated in the language of second quantization.
First quantization
The following exposition is based on Dirac's treatise on quantum mechanics.
In the classical mechanics of a particle, there are dynamic variables which are called coordinates (x) and momenta (p). These specify the state of a classical system. The canonical structure (also known as the symplectic structure) of classical mechanics consists of Poisson brackets enclosing these variables, such as {x, p} = 1.
Official source
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