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 , 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.
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.
By contrast, in quantum mechanics, all significant features of a particle are contained in a state , called a quantum state. Observables are represented by operators acting on a Hilbert space of such quantum states.
The eigenvalue of an operator acting on one of its eigenstates represents the value of a measurement on the particle thus represented. For example, the energy is read off by the Hamiltonian operator acting on a state , yielding where is the characteristic energy associated to this eigenstate.
Any state could be represented as a linear combination of eigenstates of energy; for example, where are constant coefficients.
As in classical mechanics, all dynamical operators can be represented by functions of the position and momentum ones, and , respectively. The connection between this representation and the more usual wavefunction representation is given by the eigenstate of the position operator representing a particle at position , which is denoted by an element in the Hilbert space, and which satisfies . Then, .
Likewise, the eigenstates of the momentum operator specify the momentum representation: .
The central relation between these operators is a quantum analog of the above Poisson bracket of classical mechanics, the canonical commutation relation,
This relation encodes (and formally leads to) the uncertainty principle, in the form . This algebraic structure may be thus considered as the quantum analog of the canonical structure of classical mechanics.
where we have interchanged two coordinates of the state function. The usual wave function is obtained using the Slater determinant and the identical particles theory. Using this basis, it is possible to solve various many-particle problems.
One might interpret this proposal as saying that we should seek a "quantization map" mapping a function on the classical phase space to an operator on the quantum Hilbert space such that It is now known that there is no reasonable such quantization map satisfying the above identity exactly for all functions and
Actually, the nonexistence of such a map occurs already by the time we reach polynomials of degree four. Note that the Poisson bracket of two polynomials of degree four has degree six, so it does not exactly make sense to require a map on polynomials of degree four to respect the bracket condition. We can, however, require that the bracket condition holds when and have degree three. Groenewold's theorem Theorem 13.13 can be stated as follows:
The proof can be outlined as follows. Section 13.4 Suppose we first try to find a quantization map on polynomials of degree less than or equal to three satisfying the bracket condition whenever has degree less than or equal to two and has degree less than or equal to two. Then there is precisely one such map, and it is the Weyl quantization. The impossibility result now is obtained by writing the same polynomial of degree four as a Poisson bracket of polynomials of degree three in two different ways. Specifically, we have On the other hand, we have already seen that if there is going to be a quantization map on polynomials of degree three, it must be the Weyl quantization; that is, we have already determined the only possible quantization of all the cubic polynomials above.
The argument is finished by computing by brute force that does not coincide with Thus, we have two incompatible requirements for the value of .
However, not only are these four properties mutually inconsistent, any three of them are also inconsistent! As it turns out, the only pairs of these properties that lead to self-consistent, nontrivial solutions are 2 & 3, and possibly 1 & 3 or 1 & 4. Accepting properties 1 & 2, along with a weaker condition that 3 be true only asymptotically in the limit (see Moyal bracket), leads to deformation quantization, and some extraneous information must be provided, as in the standard theories utilized in most of physics. Accepting properties 1 & 2 & 3 but restricting the space of quantizable observables to exclude terms such as the cubic ones in the above example amounts to geometric quantization.
When the canonical quantization procedure is applied to a field, such as the electromagnetic field, the classical field variables become . 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.
The classical Lagrangian density describes an , labelled by which is now a label (and not the displacement dynamical variable to be quantized), denoted by the classical field , where is a potential term, often taken to be a polynomial or monomial of degree 3 or higher. The action functional is The canonical momentum obtained via the Legendre transformation using the action is , and the classical Hamiltonian is found to be
Canonical quantization treats the variables and as operators with canonical commutation relations at time = 0, given by Operators constructed from and can then formally be defined at other times via the time-evolution generated by the Hamiltonian,
However, since and no longer commute, this expression is ambiguous at the quantum level. The problem is to construct a representation of the relevant operators on a Hilbert space and to construct a positive operator as a quantum operator on this Hilbert space in such a way that it gives this evolution for the operators as given by the preceding equation, and to show that contains a vacuum state on which has zero eigenvalue. In practice, this construction is a difficult problem for interacting field theories, and has been solved completely only in a few simple cases via the methods of constructive quantum field theory. Many of these issues can be sidestepped using the Feynman integral as described for a particular in the article on scalar field theory.
In the case of a free field, with , the quantization procedure is relatively straightforward. It is convenient to Fourier transform the fields, so that The reality of the fields implies that The classical Hamiltonian may be expanded in Fourier modes as where .
This Hamiltonian is thus recognizable as an infinite sum of classical normal mode oscillator excitations , each one of which is quantized in the standard manner, so the free quantum Hamiltonian looks identical. It is the s that have become operators obeying the standard commutation relations, , with all others vanishing. The collective Hilbert space of all these oscillators is thus constructed using creation and annihilation operators constructed from these modes, for which for all , with all other commutators vanishing.
The vacuum is taken to be annihilated by all of the , and is the Hilbert space constructed by applying any combination of the infinite collection of creation operators † to . This Hilbert space is called Fock space. For each , this construction is identical to a quantum harmonic oscillator. The quantum field is an infinite array of quantum oscillators. The quantum Hamiltonian then amounts to where may be interpreted as the number operator giving the number of particles in a state with momentum .
This Hamiltonian differs from the previous expression by the subtraction of the zero-point energy of each harmonic oscillator. This satisfies the condition that must annihilate the vacuum, without affecting the time-evolution of operators via the above exponentiation operation. This subtraction of the zero-point energy may be considered to be a resolution of the quantum operator ordering ambiguity, since it is equivalent to requiring that all creation operators appear to the left of annihilation operators in the expansion of the Hamiltonian. This procedure is known as Wick ordering or normal ordering.
It turns out that commutation relations are useful only for quantizing bosons, for which the occupancy number of any state is unlimited. To quantize fermions, which satisfy the Pauli exclusion principle, anti-commutators are needed. These are defined by .
When quantizing fermions, the fields are expanded in creation and annihilation operators, , , which satisfy
The states are constructed on a vacuum annihilated by the , and the Fock space is built by applying all products of creation operators to . Pauli's exclusion principle is satisfied, because , by virtue of the anti-commutation relations.
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.
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.
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