Quantization (physics)

In physics, quantization is the process of transition from a classical understanding of physical phenomena to a newer understanding known as quantum mechanics. (It is a procedure for constructing a quantum field theory starting from a classical field theory.) This is a generalization of the procedure for building quantum mechanics from classical mechanics. Also related is field quantization, as in the 'quantization of the electromagnetic field', referring to photons as field 'quanta' (for instance as light quanta). This procedure is basic to theories of particle physics, nuclear physics, condensed matter physics, and quantum optics. In physics, quantization is the process of transition from a classical understanding of physical phenomena to a newer understanding known as quantum mechanics. (It is a procedure for constructing a quantum field theory starting from a classical field theory.) This is a generalization of the procedure for building quantum mechanics from classical mechanics. Also related is field quantization, as in the 'quantization of the electromagnetic field', referring to photons as field 'quanta' (for instance as light quanta). This procedure is basic to theories of particle physics, nuclear physics, condensed matter physics, and quantum optics. Quantization converts classical fields into operators acting on quantum states of the field theory. The lowest energy state is called the vacuum state. The reason for quantizing a theory is to deduce properties of materials, objects or particles through the computation of quantum amplitudes, which may be very complicated. Such computations have to deal with certain subtleties called renormalization, which, if neglected, can often lead to nonsense results, such as the appearance of infinities in various amplitudes. The full specification of a quantization procedure requires methods of performing renormalization. The first method to be developed for quantization of field theories was canonical quantization. While this is extremely easy to implement on sufficiently simple theories, there are many situations where other methods of quantization yield more efficient procedures for computing quantum amplitudes. However, the use of canonical quantization has left its mark on the language and interpretation of quantum field theory. Canonical quantization of a field theory is analogous to the construction of quantum mechanics from classical mechanics. The classical field is treated as a dynamical variable called the canonical coordinate, and its time-derivative is the canonical momentum. One introduces a commutation relation between these which is exactly the same as the commutation relation between a particle's position and momentum in quantum mechanics. Technically, one converts the field to an operator, through combinations of creation and annihilation operators. The field operator acts on quantum states of the theory. The lowest energy state is called the vacuum state. The procedure is also called second quantization. This procedure can be applied to the quantization of any field theory: whether of fermions or bosons, and with any internal symmetry. However, it leads to a fairly simple picture of the vacuum state and is not easily amenable to use in some quantum field theories, such as quantum chromodynamics which is known to have a complicated vacuum characterized by many different condensates. Even within the setting of canonical quantization, there is difficulty associated to quantizing arbitrary observables on the classical phase space. This is the ordering ambiguity: Classically the position and momentum variables x and p commute, but their quantum mechanical counterparts do not. Various quantization schemes have been proposed to resolve this ambiguity, of which the most popular is the Weyl quantization scheme. Nevertheless, the Groenewold–van Hove theorem says that no perfect quantization scheme exists. Specifically, if the quantizations of x and p 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. See Groenewold's theorem for one version of this result. There is a way to perform a canonical quantization without having to resort to the non covariant 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 (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 ℏ {displaystyle hbar } -deformed in the same way as in canonical quantization. There is also a way to quantize actions with gauge 'flows'. It involves the Batalin–Vilkovisky formalism, an extension of the BRST formalism.