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The notion of quantization refers to the process that is carried out to develop the construction of a quantum model, starting from a description of classical physics. In this way, it is considered a classical theory and transformed into quantum theory. Therefore, to understand what quantization is, you must first know how to differentiate between classical physics and quantum physics.

Classical physics:

 

  1. The states are all the values ​​of the observables that we can measure. Positions, moments, energies, and any other physical magnitude.
  2. One does not de facto distinguish between observables and states. One can measure all observables without disturbing the state. That is if the system is at position x = 3 at the time of measurement, the state does not change. And we can simultaneously measure any observable.
Quantum physics:

 

  1. The states are all the values ​​of the observables that we can measure. Positions, moments, energies, and any other physical magnitude.
  2. One does not de facto distinguish between observables and states. One can measure all observables without disturbing the state. That is if the system is at position x = 3 at the time of measurement, the state does not change. And we can simultaneously measure any observable.

Quantization Process

Geometric quantization infographic
You have a classical theory with its states and its observables. The states are represented by the values ​​of the functions position, moment, energies, angular momentum, etc.

Quantize is to find a procedure to find the states and the corresponding observables from classical theory. That is, we have to find the appropriate representation of the observables as operators.

We have to comment that the quantization process is not unique. There are several mathematically well-defined ways to find a quantum theory starting from a known classical one. Unfortunately, there is no rule to apply one quantization method or another, although there are cases where clearly some are easier to carry out than others. It depends a bit on the intuition of the physicist to apply one scheme or another. Obviously, if a physicist can quantize a classical theory in various ways, the resulting theories must be consistent and equivalent.

Different Types

Canonical, algebraic, geometric, Weyl, and covariant are some of the quantization procedures that can be performed. We must bear in mind that the theories that result from the quantization of the same classical theory have to be equivalent and consistent, regardless of the method used.

 

  • Canonical: In this branch of physics, canonical quantization is a process by which a theory based on the rules of classical physics can be quantized while preserving, as far as possible, the essential structure taking into account the symmetry of the classical theory.
  • Geometric: Geometric quantization is a mathematical procedure to build a quantum theory corresponding to a specific classical theory from the symplectic formalism. Geometric quantization is a procedure for which there is no general algorithm or exact recipe. The idea of ​​geometric quantization is to make certain analogies manifest between classical theory and quantum theory. 
  • Weyl: In the branch of physics that comprises quantum mechanics, taking into account Schrödinger’s image, there is the inverse and reversible process of transformation or mapping from functions formulated in quantum space and Hilbert space operators to phase space in classical physics, this is the Wigner-Weyl transform or the Weyl-Wigner transform. Very often, the mapping or transformation of systems from phase space to operators is called Weyl quantization. On the other hand, the reverse transformation process, from operators to functions in phase space, is known as Wigner transformation or mapping. It is also important to note the two processes are opposite. This transformation or mapping was first developed by Hermann Weyl in 1927. At that time, the scientist was attempting to map symmetric phase space functions to operators, which ultimately produced what we now call Weyl quantization. It is now known that Weyl quantization does not fulfill all the expected and necessary properties needed for quantization, so it is common that it often produces unphysical results. On the other hand, because Weyl quantization has many interesting and positive qualities that scientists find appealing, when one is researching and searching for the best, simple and consistent quantization method, Weyl quantization performs the process with excellent results.
  • Covariant: There are basically two methods: the oldest (which is the one we will follow here), in which the description is given through the coordinates of the Xµ chord and the Virasoro constrictions impose restrictions on the Fock space; The second and more modern form of quantization has geometric bases, in which Faddeev-Popov ghosts are introduced, and the BRST symmetries and the currents of these symmetries are identified. One can distinguish between first and second quantization procedures. In the first case, models of a particle are built, while in the second, systems of multiple equal particles are analyzed.
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