Классикалық теорияны кванттық механикаға келтіру процесі

Process of converting a classical physical theory into one compatible with quantum mechanics

In physics, canonical quantization is a procedure for quantizing a classical theory, while attempting to preserve the formal structure, such as symmetries, of the classical theory to the greatest extent possible. Historically, this was not quite Werner Heisenberg's route to obtaining quantum mechanics, but Paul Dirac introduced it in his 1926 doctoral thesis, the "method of classical analogy" for quantization, and detailed it in his classic text Principles of Quantum Mechanics. The word canonical arises from the Hamiltonian approach to classical mechanics, in which a system's dynamics is generated via canonical Poisson brackets, a structure which is only partially preserved in canonical quantization. This method was further used by Paul Dirac in the context of quantum field theory, in his construction of quantum electrodynamics. In the field theory context, it is also called the second quantization of fields, in contrast to the semi classical first quantization of single particles.

When it was first developed, quantum physics dealt only with the quantization of the motion of particles, leaving the electromagnetic field classical, hence the name quantum mechanics. Later the electromagnetic field was also quantized, and even the particles themselves became represented through quantized fields, resulting in the development of quantum electrodynamics (QED) and quantum field theory in general. Thus, by convention, the original form of particle quantum mechanics is denoted first quantization, while quantum field theory is formulated in the language of second quantization.

Quantum mechanics was successful at describing non relativistic systems with fixed numbers of particles, but a new framework was needed to describe systems in which particles can be created or destroyed, for example, the electromagnetic field, considered as a collection of photons. It was eventually realized that special relativity was inconsistent with single particle quantum mechanics, so that all particles are now described relativistically by quantum fields. When the canonical quantization procedure is applied to a field, such as the electromagnetic field, the classical field variables become quantum operators. Thus, the normal modes comprising the amplitude of the field are simple oscillators, each of which is quantized in standard first quantization, above, without ambiguity. The resulting quanta are identified with individual particles or excitations. For example, the quanta of the electromagnetic field are identified with photons. Unlike first quantization, conventional second quantization is completely unambiguous, in effect a functor, since the constituent set of its oscillators are quantized unambiguously. Historically, quantizing the classical theory of a single particle gave rise to a wavefunction. The classical equations of motion of a field are typically identical in form to the (quantum) equations for the wave function of one of its quanta. For example, the Klein–Gordon equation is the classical equation of motion for a free scalar field, but also the quantum equation for a scalar particle wave function. This meant that quantizing a field appeared to be similar to quantizing a theory that was already quantized, leading to the fanciful term second quantization in the early literature, which is still used to describe field quantization, even though the modern interpretation detailed is different. One drawback to canonical quantization for a relativistic field is that by relying on the Hamiltonian to determine time dependence, relativistic invariance is no longer manifest. Thus it is necessary to check that relativistic invariance is not lost. Alternatively, the Feynman integral approach is available for quantizing relativistic fields, and is manifestly invariant. For non relativistic field theories, such as those used in condensed matter physics, Lorentz invariance is not an issue.

Quantum mechanically, the variables of a field (such as the field's amplitude at a given point) are represented by operators on a Hilbert space. In general, all observables are constructed as operators on the Hilbert space, and the time evolution of the operators is governed by the Hamiltonian, which must be a positive operator. A state annihilated by the Hamiltonian must be identified as the vacuum state, which is the basis for building all other states. In a non interacting (free) field theory, the vacuum is normally identified as a state containing zero particles. In a theory with interacting particles, identifying the vacuum is more subtle, due to vacuum polarization, which implies that the physical vacuum in quantum field theory is never really empty. For further elaboration, see the articles on the quantum mechanical vacuum and the vacuum of quantum chromodynamics. The details of the canonical quantization depend on the field being quantized, and whether it is free or interacting.

The construction of the scalar field states above assumed that the potential was minimized at φ = 0, so that the vacuum minimizing the Hamiltonian satisfies 1=⟨φ⟩ = 0, indicating that the vacuum expectation value (VEV) of the field is zero. In cases involving spontaneous symmetry breaking, it is possible to have a non zero VEV, because the potential is minimized for a value φ = v This occurs for example, if with g > 0 and m^(2) > 0, for which the minimum energy is found at The value of v in one of these vacua may be considered as condensate of the field φ. Canonical quantization then can be carried out for the shifted field φ(x,t) − v, and particle states with respect to the shifted vacuum are defined by quantizing the shifted field. This construction is utilized in the Higgs mechanism in the standard model of particle physics.

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 an ħ deformation of the algebra of smooth functions over the symplectic space such that the leading term in the Taylor expansion over ħ of the commutator [A, B] expressed in the phase space formulation is iħ{A, B} (Here, the curly braces denote the Poisson bracket. The subleading terms are all encoded in the Moyal bracket, the suitable quantum deformation of the Poisson bracket.) In general, for the quantities (observables) involved,
and providing the arguments of such brackets, ħ deformations are highly nonunique—quantization is an "art", and is specified by the physical context. (Two different quantum systems may represent two different, inequivalent, deformations of the same classical limit, ħ → 0.) Now, one looks for unitary representations of this quantum algebra. With respect to such a unitary representation, a symplectomorphism in the classical theory would now deform to a (metaplectic) unitary transformation. In particular, the time evolution symplectomorphism generated by the classical Hamiltonian deforms to a unitary transformation generated by the corresponding quantum Hamiltonian. A further generalization is to consider a Poisson manifold instead of a symplectic space for the classical theory and perform an ħ deformation of the corresponding Poisson algebra or even Poisson supermanifolds.

In contrast to the theory of deformation quantization described above, geometric quantization seeks to construct an actual Hilbert space and operators on it. Starting with a symplectic manifold , one first constructs a prequantum Hilbert space consisting of the space of square integrable sections of an appropriate line bundle over On this space, one can map all classical observables to operators on the prequantum Hilbert space, with the commutator corresponding exactly to the Poisson bracket. The prequantum Hilbert space, however, is clearly too big to describe the quantization of
One then proceeds by choosing a polarization, that is (roughly), a choice of variables on the dimensional phase space. The quantum Hilbert space is then the space of sections that depend only on the chosen variables, in the sense that they are covariantly constant in the other directions. If the chosen variables are real, we get something like the traditional Schrödinger Hilbert space. If the chosen variables are complex, we get something like the Segal–Bargmann space.

Silvan S. Schweber: QED and the men who made it, Princeton Univ. Press, 1994,