Canonical quantization

Quantum field theory
History of...

In physics, canonical quantization is a procedure for quantizing a classical theory while attempting to preserve the formal structure of the classical theory, to the extent possible. Historically, this was Werner Heisenberg's route to obtaining 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 that is preserved to the extent possible in canonical quantization. This method was used in the context of quantum field theory by Paul Dirac, in his construction of quantum electrodynamics. In the field theory context, it is also called second quantization, in contrast to the semi-classical first quantization.

Contents

History

Commutators were introduced by Werner Heisenberg, wavefunctions by Erwin Schrödinger. The connection between the two was discovered by Paul Dirac, who was also the first to apply this technique to the quantization of the electromagnetic field. Eugene Wigner and Pascual Jordan were the first to quantize the electron field, whose quantum mechanics was first investigated by Dirac. The name canonical quantization may have been first coined by Pascual Jordan.

Quantum mechanics

The following exposition is based largely on Dirac's classic book on quantum mechanics.[1] In the classical mechanics of a particle, there are dynamic variables which are called coordinates (x) and momenta (p). These specify the state of a classical system. The canonical structure (also known as the symplectic structure) of classical mechanics consists of Poisson brackets between these variables. All transformations which keep these brackets unchanged are allowed as canonical transformations in classical mechanics.

In quantum mechanics, observables are represented by operators acting on a Hilbert space of quantum states. The value of an operator on one of its eigenstates represents the value of a measurement. In particular, position and momentum are observables for a point particle, and are represented by quantum operators. An eigenvector of the position operator X representing a particle at position x may be denoted by an element |x\rangle of the Hilbert space, which satisfies X|x\rangle = x|x\rangle. More general states may be constructed by superposition, e.g. |\psi\rangle = \int dx \psi(x)|x\rangle. The function \psi(x) is the wave function corresponding to this state in the Hilbert space, and may also be expressed as \psi(x) = \langle x|\psi\rangle. The Poisson brackets of classical mechanics are replaced by commutators,

[X,P] = XP-PX = i\hbar.

This construction leads to the uncertainty principle in the form\Delta x \Delta p \geq \frac{\hbar}{2}. This algebraic structure may be considered a quantum analog of the canonical structure of classical mechanics.

Second quantization: field theory

Quantum mechanics was successful at describing non-relativistic systems with fixed numbers of particles, but a new framework is 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 quantum field theory, the classical field variable becomes a quantum operator. The amplitude of the field becomes quantized, and the quanta are identified with individual particles.

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 to the equation 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 term second quantization in the early literature, which is still used to describe field quantization, even though the modern interpretation 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 hidden, but 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, this is not an issue.

Field operators

Quantum mechanically, fields 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> 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.[2] For simplicity, the quantization can be carried in a 1+1 dimensional space-time \mathbb{R}\times S_1, in which the spatial direction is compactified to a circle of circumference 2π, rendering the momenta discrete. The classical Lagrangian density is then

\mathcal{L}(\phi) = \frac{1}{2}(\partial_t \phi)^2 - \frac{1}{2}(\partial_x \phi)^2 - \frac{1}{2} m^2\phi^2 - V(\phi),

where V(\phi) 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{L}(\phi) dx dt = \int L(\phi, \partial_t\phi) dt.

The canonical momentum obtained via the Legendre transform using the action L is \pi = \partial_t\phi, and the classical Hamiltonian is found to be

H(\phi,\pi) = \int dx \left[\frac{1}{2} \pi^2 %2B \frac{1}{2} (\partial_x \phi)^2 %2B \frac{1}{2} m^2 \phi^2 %2B V(\phi)\right].

Canonical quantization treats the variables \phi(x) and \pi(x) as operators with canonical commutation relations at time t = 0 given by

[\phi(x),\phi(y)] = 0, \ \ [\pi(x), \pi(y)] = 0, \ \ [\phi(x),\pi(y)] = i\hbar \delta(x-y).

Operators constructed from \phi and \pi can then formally be defined at other times via the time-evolution generated by the Hamiltonian:

\mathcal{O}(t) = e^{itH} \mathcal{O} e^{-itH}.

However, since \phi and \pi do not commute, this expression is ambiguous at the quantum level. The problem is to construct a representation of the relevant operators \mathcal{O} on a Hilbert space \mathcal{H} and to construct a positive operator H as a quantum operator on this Hilbert space in such a way that it gives this evolution for the operators \mathcal{O} as given by the preceding equation, and to show that \mathcal{H} contains a vacuum state |0> on which H 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 V(\phi) in the article on scalar field theory.

In the case of a free field, with V(\phi) = 0, the procedure is relatively straightforward. It is convenient to Fourier transform the fields, so that

\phi_k = \int \phi(x) e^{-ikx} dx, \ \ \pi_k = \int \pi(x) e^{-ikx} dx.

The reality of the fields imply that \phi_{-k} = \phi_k^\dagger, \pi_{-k} = \pi_k^\dagger,, and the commutation relations become [\phi_k,\pi_k^\dagger] = [\phi_k^\dagger,\pi_k] = i, with all others vanishing. The Hamiltonian may be expanded in Fourier modes as

H=\frac{1}{2}\sum_{k=-\infty}^{\infty}\left[\pi_k \pi_k^\dagger %2B \omega_k^2\phi_k\phi_k^\dagger\right],

where \omega_k = \sqrt{k^2%2Bm^2}. The Hilbert space is constructed using creation and annihilation operators constructed from these modes,

a_k = \frac{1}{\sqrt{2\hbar\omega_k}}\left(\omega_k\phi_k %2B i\pi_k\right), \ \ a_k^\dagger = \frac{1}{\sqrt{2\hbar\omega_k}}\left(\omega_k\phi_k^\dagger - i\pi_k^\dagger\right),

for which [a_k,a_k^\dagger] = 1 for all k, with all other commutators vanishing. The vacuum |0> is taken to be annihilated by all of the a_k, and \mathcal{H} is the Hilbert space constructed by applying any combination of the creation operators a_k^\dagger to |0>. This Hilbert space is called Fock space. For each k, this construction is identical to the quantum harmonic oscillator. The quantum Hamiltonian can be defined to be

H = \sum_{k=-\infty}^{\infty} \hbar\omega_k a_k^\dagger a_k = \sum_{k=-\infty}^{\infty} \hbar\omega_k N_k,

where N_k may be interpreted as the number operator giving the number of particles in a state with momentum k. This Hamiltonian differs from the previous expression by the subtraction of the zero-point energy \hbar\omega_k/2 of each harmonic oscillator. This satisfies the condition that H 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.

Other fields

All other fields can be quantized by a generalization of this procedure. Vector or tensor fields simply have more components, and independent creation and destruction operators must be introduced for each independent component. If a field has any internal symmetry, then creation and destruction operators must be introduced for each component of the field related to this symmetry as well. If there is a gauge symmetry, then the number of independent components of the field must be carefully analyzed to avoid over-counting equivalent configurations, and gauge-fixing may be applied if needed.

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 \{A,B\} = AB %2B BA. When quantizing fermions, the fields are expanded in creation and annihilation operators \theta_k^\dagger, \ \theta_k which satisfy

\{\theta_k,\theta_l^\dagger\} = \delta_{kl}, \ \ \{\theta_k, \theta_l\} = 0, \ \ \{\theta_k^\dagger, \theta_l^\dagger\} = 0.

The states are constructed on a vacuum |0> annihilated by the \theta_k, and the Fock space is built by applying all products of creation operators \theta_k^\dagger to |0>. Pauli's exclusion principle is satisfied because (\theta_k^\dagger)^2|0\rangle = 0 due to the anti-commutation relations.

Condensates

The construction of the scalar field states above assumed that the potential was minimized at \phi = 0, so that the vacuum minimizing the Hamiltonian satisifes \langle 0|\phi|0\rangle=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 \phi = v. This occurs for example, if V(\phi) = g\phi^4 and m^2<0, for which the minimum energy is found at v = \pm m/\sqrt{g}. The value of v in one of these vacua may be considered a condensate of the field \phi. Canonical quantization then can be carried out for the shifted field \phi(x,t)-v, and particle states with respect to the shifted vacuum are defined by quantizing the shifted field. This construction is used in the Higgs mechanism in the standard model of particle physics.

Mathematical quantization

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] 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.

See also

References

  1. ^ Dirac, P. A. M. (1982). Principles of Quantum Mechanics. USA: Oxford University Press. ISBN 0198520115. 
  2. ^ This treatment is based primarily on Ch. 1 in Connes, Alain; Marcolli, Matilde (2008). Noncommutative Geometry, Quantum Fields, and Motives. American Mathematical Society. ISBN 0821842102. http://www.alainconnes.org/docs/bookwebfinal.pdf. 

Historical References

General Technical References

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