Quantization (in British Englishquantisation) is the systematic transition procedure from a classical understanding of physical phenomena to a newer understanding known asquantum mechanics. It is a procedure for constructingquantum mechanics fromclassical mechanics. A generalization involving infinite degrees of freedom isfield quantization, as in the "quantization of theelectromagnetic field", referring tophotons as field "quanta" (for instance aslight quanta). This procedure is basic to theories ofatomic physics, chemistry,particle physics,nuclear physics,condensed matter physics, andquantum optics.
In 1901, whenMax Planck was developing thedistribution function ofstatistical mechanics to solve theultraviolet catastrophe problem, he realized that the properties ofblackbody radiation can be explained by the assumption that the amount of energy must be in countable fundamental units, i.e. amount of energy is not continuous butdiscrete. That is, a minimum unit of energy exists and the following relationship holdsfor the frequency. Here, is called thePlanck constant, which represents the amount of the quantum mechanical effect. It means a fundamental change of mathematical model of physical quantities.
In 1905,Albert Einstein published a paper, "On a heuristic viewpoint concerning the emission and transformation of light", which explained thephotoelectric effect on quantizedelectromagnetic waves.[1] Theenergy quantum referred to in this paper was later called "photon". In July 1913,Niels Bohr used quantization to describe the spectrum of a hydrogen atom in his paper "On the constitution of atoms and molecules".
The preceding theories have been successful, but they are very phenomenological theories. However, the French mathematicianHenri Poincaré first gave a systematic and rigorous definition of what quantization is in his 1912 paper "Sur la théorie des quanta".[2][3]
The term "quantum physics" was first used in Johnston'sPlanck's Universe in Light of Modern Physics. (1931).
Canonical quantization developsquantum mechanics fromclassical mechanics. One introduces acommutation relation amongcanonical coordinates. Technically, one converts coordinates to operators, through combinations ofcreation and annihilation operators. The operators act onquantum states of the theory. The lowest energy state is called thevacuum state.
Even within the setting of canonical quantization, there is difficulty associated to quantizing arbitrary observables on the classical phase space. This is theordering ambiguity: classically, the position and momentum variablesx andp commute, but their quantum mechanical operator counterparts do not. Variousquantization schemes have been proposed to resolve this ambiguity,[4] of which the most popular is theWeyl quantization scheme. Nevertheless,Groenewold's theorem dictates that no perfect quantization scheme exists. Specifically, if the quantizations ofx andp are taken to be the usual position and momentum operators, then no quantization scheme can perfectly reproduce the Poisson bracket relations among the classical observables.[5]
There is a way to perform a canonical quantization without having to resort to the non covariant approach offoliating spacetime and choosing aHamiltonian. 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 (for instance, actions with a noncausal structure or actions withgauge "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 theEuler–Lagrange equations. Then, this quotient algebra is converted into a Poisson algebra by introducing a Poisson bracket derivable from the action, called thePeierls bracket. This Poisson algebra is then ℏ -deformed in the same way as in canonical quantization.
Inquantum field theory, there is also a way to quantize actions withgauge "flows". It involves theBatalin–Vilkovisky formalism, an extension of theBRST formalism.
One of the earliest attempts at a natural quantization was Weyl quantization, proposed byHermann Weyl in 1927.[6] Here, an attempt is made to associate a quantum-mechanical observable (a self-adjoint operator on a Hilbert space) with a real-valued function on classical phase space. The position and momentum in this phase space are mapped to the generators of the Heisenberg group, and the Hilbert space appears as a group representation of the Heisenberg group. In 1946, H. J. Groenewold[7] considered the product of a pair of such observables and asked what the corresponding function would be on the classical phase space. This led him to discover the phase-space star-product of a pair of functions.More generally, this technique leads to deformation quantization, where the ★-product is taken to be a deformation of the algebra of functions on a symplectic manifold or Poisson manifold. However, as a natural quantization scheme (afunctor), Weyl's map is not satisfactory.
For example, the Weyl map of the classical angular-momentum-squared is not just the quantum angular momentum squared operator, but it further contains a constant term3ħ2/2. (This extra term offset is pedagogically significant, since it accounts for the nonvanishing angular momentum of the ground-state Bohr orbit in the hydrogen atom, even though the standard QM ground state of the atom has vanishingl.)[8]
As a mererepresentation change, however, Weyl's map is useful and important, as it underlies the alternateequivalentphase space formulation of conventional quantum mechanics.
In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. It attempts to carry out quantization, for which there is in general no exact recipe, in such a way that certain analogies between the classical theory and the quantum theory remain manifest. For example, the similarity between the Heisenberg equation in the Heisenberg picture of quantum mechanics and the Hamilton equation in classical physics should be built in.
A more geometric approach to quantization, in which the classical phase space can be a general symplectic manifold, was developed in the 1970s byBertram Kostant andJean-Marie Souriau. The method proceeds in two stages.[9] First, once constructs a "prequantum Hilbert space" consisting of square-integrable functions (or, more properly, sections of a line bundle) over the phase space. Here one can construct operators satisfying commutation relations corresponding exactly to the classical Poisson-bracket relations. On the other hand, this prequantum Hilbert space is too big to be physically meaningful. One then restricts to functions (or sections) depending on half the variables on the phase space, yielding the quantum Hilbert space.
A classical mechanical theory is given by anaction with the permissible configurations being the ones which are extremal with respect to functionalvariations of the action. A quantum-mechanical description of the classical system can also be constructed from the action of the system by means of thepath integral formulation.