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Quantization (physics)

In physics, quantization is a widely observed fact about nature. Many physical properties are quantized, taking discrete values, rather than a continuous range. Notably in atoms and molecules, accounting for the stability of matter. Planck's constant usually plays a crucial role in quantum phenomena.

One also speaks of "field quantization", as in the "quantization of the electromagnetic field", where one refers to photons as field "quanta" (for instance as light quanta).

In a related, but different, sense quantization refers to a theory building procedure. Even though classical physics may fundamentally be seen as a part of quantum physics, the build-up of a quantum theory is sometimes done the other way around, starting from an existing classical theory, and re-casting it to obtain a quantum counterpart. Widely attempted during the first decades of the quantum era (from 1900 to 1925), it produced few lasting successes, as seen from today's viewpoint, because the methods were heuristic, rather than exact. The more exact theories of quantum mechanics now routinely imply quantization.

Second quantization is a special formalism of quantum theory suited to deal with variable numbers of particles. It pertains to quantum field theory and draws its name from a loose understanding of the formalism as quantifying once more an already quantized theory. A quantized theory at first deals with operators and wave functions, while second quantization makes also the wave functions into operators. In this sense second quantization is a full quantization. This is widely used in particle theory, nuclear theory, condensed matter theory, and quantum optics.

Mathematical physicist Edward Nelson once said that "quantization is a mystery, but second quantization is a functor'".

For current theoretical work, more systematic quantization procedures have been developed, as described in the following.

Contents

Some quantization methods

Note that the universe is really inherently quantum and there is not a prior reason why it ought to be describable as the quantization of some classical theory. In fact, since we don't observe classical anticommuting fermion fields, for example, the physical meaning or even relevance of quantization is open to question.

Note also that the fundamental nature of the universe is a subject of debate. To say that the universe is inherently quantum dismisses the possibility of another, more specific and accurate theory and methodology eventually accompanying or replacing quantum mechanics. The current state of science and quantum mechanics is not one of certainty, and quantization also carries that disclaimer. However, regardless of whether or not the inherent nature of the universe is quantum, it most definitely isn't classical!

Canonical quantization

The classical theory is described using a spacelike foliation of spacetime with the state at each slice being described by an element of a symplectic manifold with the time evolution given by the symplectomorphism generated by a Hamiltonian function over the symplectic manifold. The quantum algebra of "operators" is a \hbar-deformation of the algebra of smooth functions over the symplectic space such that the leading term in the Taylor expansion over \hbar of the commutator [A,B] is i\hbar\{A,B\}. (Here, the curly braces denote the Poisson bracket.) In general, this \hbar-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 \hbar deformation of the corresponding Poisson algebra or even Poisson supermanifolds. (The literal classical interpretation of this, of course, does not exist. This is a purely formal procedure.)

Covariant canonical quantization

It turns out there is a way to perform a canonical quantization without having to resort to the noncovariant approach of foliating spacetime and choosing a Hamiltonian. This method is based upon a classical action, but is different from the functional integral approach.

The method does not apply to all possible actions (like for instance actions with a noncausal structure or actions with gauge "flows"). It starts with the classical algebra of all (smooth) functionals over the configuration space. This algebra is quotiented over by the ideal generated by the Euler-Lagrange equations. Then, this quotient algebra is converted into a Poisson algebra by introducing a Poisson bracket derivable from the action, called the Peierls bracket. This Poisson algebra is then \hbar-deformed in the same way as in canonical quantization.

Actually, there is a way to quantize actions with gauge "flows". It involves the Batalin-Vilkovisky formalism.

Path integral quantization

The classical theory is given by an action with the permissible configurations being the ones which are extremal with respect to functional variations of the action. The quantum-mechanical counterpart of this is the path integral formulation.

Geometric quantization

See geometric quantization

Schwinger's variational approach

See quantum action

Last updated: 05-13-2005 07:56:04