Canonical quantization

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.

History

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.

First quantization

Single particle systems

The following exposition is based on Dirac's treatise on quantum mechanics.
In the classical mechanics of a particle, there are dynamic variables which are called coordinates () and momenta (). These specify the ''state'' of a classical system. The canonical structure (also known as the symplectic structure) of classical mechanics consists of Poisson brackets enclosing these variables, such as . All transformations of variables which preserve these brackets are allowed as canonical transformations in classical mechanics. Motion itself is such a canonical transformation.

By contrast, in quantum mechanics, all significant features of a particle are contained in a state $, \psi\rangle$, 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 $\hat$ acting on a state $, \psi_n\rangle$, yielding
$\hat, \psi_n\rangle=E_n, \psi_n\rangle,$
where is the characteristic energy associated to this $, \psi_n\rangle$ eigenstate.

Any state could be represented as a linear combination of eigenstates of energy; for example,
$, \psi\rangle=\sum_^ a_n , \psi_n\rangle ,$where are constant coefficients.

As in classical mechanics, all dynamical operators can be represented by functions of the position and momentum ones, $\hat$ and $\hat$, respectively. The connection between this representation and the more usual wavefunction representation is given by the eigenstate of the position operator $\hat$ representing a particle at position $x$, which is denoted by an element $, x\rangle$ in the Hilbert space, and which satisfies $\hat, x\rangle = x, x\rangle$. Then, $\psi(x)= \langle x, \psi\rangle$.

Likewise, the eigenstates $, p\rangle$ of the momentum operator $\hat$ specify the momentum representation: $\psi(p)= \langle p, \psi\rangle$.

The central relation between these operators is a quantum analog of the above Poisson bracket of classical mechanics, the canonical commutation relation,
hat,\hat= \hat\hat-\hat\hat = i\hbar.

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.

Many-particle systems

When turning to N-particle systems, i.e., systems containing N identical particles (particles characterized by the same quantum numbers such as mass, charge and spin), it is necessary to extend the single-particle state function $\psi(\mathbf)$ to the N-particle state function $\psi(\mathbf_1,\mathbf_2,\dots,\mathbf_N)$. A fundamental difference between classical and quantum mechanics concerns the concept of indistinguishability of identical particles. Only two species of particles are thus possible in quantum physics, the so-called bosons and fermions which obey the following rules for each kind of particle:

* for bosons: $\psi(\mathbf_1,\dots,\mathbf_j,\dots,\mathbf_k,\dots,\mathbf_N)=+\psi(\mathbf_1,\dots,\mathbf_k,\dots,\mathbf_j,\dots,\mathbf_N),$
* for fermions: $\psi(\mathbf_1,\dots,\mathbf_j,\dots,\mathbf_k,\dots,\mathbf_N)=-\psi(\mathbf_1,\dots,\mathbf_k,\dots,\mathbf_j,\dots,\mathbf_N),$

where we have interchanged two coordinates $(\mathbf_j, \mathbf_k)$ 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.

Issues and limitations

Classical and quantum brackets

Dirac's book details his popular rule of supplanting Poisson brackets by commutators:

One might interpret this proposal as saying that we should seek a "quantization map" $Q$ mapping a function $f$ on the classical phase space to an operator $Q_f$ on the quantum Hilbert space such that
Q_ = \frac _f,Q_g/math>
It is now known that there is no reasonable such quantization map satisfying the above identity exactly for all functions $f$ and

Groenewold's theorem

One concrete version of the above impossibility claim is Groenewold's theorem (after Dutch theoretical physicist Hilbrand J. Groenewold), which we describe for a system with one degree of freedom for simplicity. Let us accept the following "ground rules" for the map $Q$. First, $Q$ should send the constant function 1 to the identity operator. Second, $Q$ should take $x$ and $p$ to the usual position and momentum operators $X$ and $P$. Third, $Q$ should take a polynomial in $x$ and $p$ to a "polynomial" in $X$ and $P$, that is, a finite linear combinations of products of $X$ and $P$, which may be taken in any desired order. In its simplest form, Groenewold's theorem says that there is no map satisfying the above ground rules and also the bracket condition
Q_ = \frac _f,Q_g/math>
for all polynomials $f$ and $g$.

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 $f$ and $g$ have degree three. Groenewold's theorem can be stated as follows:

The proof can be outlined as follows. Suppose we first try to find a quantization map on polynomials of degree less than or equal to three satisfying the bracket condition whenever $f$ has degree less than or equal to two and $g$ 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
$x^2 p^2 = \frac \ = \frac \$
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
\frac (x^3),Q(p^3)/math>
does not coincide with
\frac (x^2p),Q(xp^2)
Thus, we have two incompatible requirements for the value of $Q(x^2p^2)$.

Axioms for quantization

If represents the quantization map that acts on functions in classical phase space, then the following properties are usually considered desirable:
#$Q_x \psi = x \psi$ and $Q_p \psi = -i\hbar \partial_x \psi ~~$   (elementary position/momentum operators)
#$f \longmapsto Q_f ~~$   is a linear map
# _f,Q_gi\hbar Q_~~   (Poisson bracket)
#$Q_=g(Q_f)~~$   (von Neumann rule).

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.

Second quantization: field theory

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.

Field operators

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 $, 0\rangle$ 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.

Real scalar field

A scalar field theory provides a good example of the canonical quantization procedure.This treatment is based primarily on Ch. 1 in Classically, a scalar field is a collection of an infinity of oscillator normal modes. It suffices to consider a 1+1-dimensional space-time $\mathbb \times S_1,$ in which the spatial direction is compactified to a circle of circumference 2, rendering the momenta discrete.

The classical Lagrangian density describes an infinity of coupled harmonic oscillators, labelled by which is now a ''label'' (and not the displacement dynamical variable to be quantized), denoted by the classical field ,
$\mathcal(\phi) = \tfrac(\partial_t \phi)^2 - \tfrac(\partial_x \phi)^2 - \tfrac m^2\phi^2 - V(\phi),$
where is a potential term, often taken to be a polynomial or monomial of degree 3 or higher. The action functional is
$S(\phi) = \int \mathcal(\phi) dx dt = \int L(\phi, \partial_t\phi) dt \, .$The canonical momentum obtained via the Legendre transformation using the action is $\pi = \partial_t\phi$, and the classical Hamiltonian is found to be
H(\phi,\pi) = \int dx \left tfrac \pi^2 + \tfrac (\partial_x \phi)^2 + \tfrac m^2 \phi^2 + V(\phi)\right

Canonical quantization treats the variables and as operators with canonical commutation relations at time = 0, given by
$[\phi(x),\phi(y)$= 0, \ \ [\pi(x), \pi(y)">phi(x),\phi(y)">hat x,\hat p ...