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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.
1 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!
1.1 Canonical quantization
The classical theory is described using a spacelike foliation of spacetimeIn special relativity and general relativity, time and three-dimensional space are treated together as a single four-dimensional manifold called spacetime . A point in spacetime may be referred to as an event . Each event has four coordinates t x y z ; or with the state at each slice being described by an element of a symplectic manifoldIn mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. A 2-form ω on a manifold M is called nondegenerate if for every nonzero vector u in the tangent space at a point, there is a vector v such that with the time evolution given by the symplectomorphism generated by a HamiltonianIn physics, Hamiltonian has two distinct but closely related meanings. In classical mechanics, the Hamiltonian is a function describing the state of a mechanical system in terms of position and momentum variables. See Hamiltonian mechanics. In quantum mec 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 commutatorFor an electrical switch that periodically reverses the current see commutator (electric In mathematics the commutator of two elements g and h of a group G is the element g −1 h −1 gh often denoted by [ g h ]. It is equal to the group's identi is . (Here, the curly braces denote the Poisson bracketIn mathematics and classical mechanics, the Poisson bracket is a bilinear map turning two differentiable functions over a symplectic space into a function over that symplectic space. In particular, if we have two functions, A and B then : where ω is.) In general, this -deformation is highly nonunique, which explains the claim that quantization is an art. Now, we look for unitary representationIn mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π g is a unitary operator for every g ∈ G''. The general theory is well-developed in case G is a locally compact (Haus 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.)
