Quantization (physics) Guide, Meaning , Facts, Information and Description

In physics, quantization is the formulation of a classical theory in the formalism of quantum physics. Even though classical physics stems from quantum theory, the build up of a quantum theory is often made the other way around, starting from existing classical physics to derive the more fundamental quantum counterpart. For instance one can speak of the quantization of the electromagnetic field.

Quantization in quantum theory is the taking of discrete rather than continuous values for some physical quantities (e.g. the total energy of a black body). When a quantity can only take on integer multiples of some base value, the smallest possible intervals between the discrete values are quanta. The size of the quanta typically varies from system to system, but the Planck constant usually playes a crucial role in it. Using wave functions and applying boundary conditions on them often does 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. According to the formal explanation a quantized theory deals with operators and normal wave functions, while second quantization makes also the wave functions operators, that makes the number of particles also quantized, i.e. half an electron is not possible. In this sense second quantization is a full quantization.

It has been said that quantization is a mystery, but second quantization is a functor.

Some quantization methods

Note that the universe is really inherently quantum and there is no a priori 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 -deformation of the algebra of smooth functions over the symplectic space such that the leading term in the Taylor expansion over of the commutator is . (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. (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. This 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 -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.

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