Everything about Canonical Quantization totally explained
In
physics,
canonical quantization is one of many procedures for
quantizing a
classical theory. Historically, this was the earliest method to be used to build
quantum mechanics. When applied to a classical
field theory it's also called
second quantization. The word
canonical refers actually to a certain structure of the classical theory (called the
symplectic structure) which is preserved in the quantum theory. This was first emphasized by
Paul Dirac, in his attempt to build
quantum field theory.
History
Commutators were introduced by
Werner Heisenberg;
wavefunctions, by
Erwin Schrödinger. The connection between the two was discovered by
Paul Dirac, who was also the first to apply this technique to the quantization of the
electromagnetic field.
Eugene Wigner and
Pascual Jordan were the first to quantize the electron field, whose quantum mechanics was first investigated by
Dirac. The name
canonical quantization may have been first coined by
Pascual Jordan.
The exposition here leans heavily on Dirac's influential book on quantum mechanics. This route to
quantum mechanics is through the
uncertainty principle. A later development was the
Feynman path integral, a formulation of
quantum theory which emphasizes the role of superposition of quantum amplitudes. The two methods give the same results.
Quantum mechanics
In the
classical mechanics of a particle, one has dynamical variables which are called coordinates (
) and momenta (
). These specify the
state of a classical system. The
canonical structure (also known as the
symplectic structure) of classical mechanics consists of
Poisson brackets between these variables. All transformations which keep these brackets unchanged are allowed as
canonical transformations in classical mechanics.
In quantum mechanics, these dynamical variables become operators acting on a
Hilbert space of
quantum states. The
Poisson brackets (more generally the
Dirac brackets) are replaced by
commutators,
. This readily yields the
uncertainty principle in the form
. (Here, the curly braces denote the
Poisson bracket.) In general, this
-deformation is highly nonunique, which explains the claim that quantization is an art. Now, we look for
unitary representations of this quantum algebra. With respect to such a unitary rep, a symplectomorphism in the classical theory would now correspond to a
unitary transformation. In particular, the time evolution symplectomorphism generated by the classical Hamiltonian is now a unitary transformation generated by the corresponding quantum Hamiltonian.
We could be more general than this. We can work with a
Poisson manifold instead of a symplectic space for the classical theory and perform a
deformation of the corresponding
Poisson algebra or even
Poisson supermanifolds.
Further Information
Get more info on 'Canonical Quantization'.
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