String Theory

Sunday 9 June 2013

Correspondence principle

In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics (or by the old quantum theory) reproduces classical physics in the limit of large quantum numbers. In other words, it says that for large orbits and for large energies, quantum calculations must agree with classical calculations.

The principle was formulated by Niels Bohr in 1920, though he had previously made use of it as early as 1913 in developing his model of the atom.

The term is also used more generally, to represent the idea that a new theory should reproduce the results of older well-established theories in those domains where the old theories work.

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] 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.

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 $\{A,B\} = AB + BA$ . 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.

Real scalar field

A scalar field theory provides a good example of the canonical quantization procedure. Classically, a scalar field is a collection of an infinity of oscillator normal modes. 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

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

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 quantization 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 $\phi_{-k} = \phi_k^\dagger, \pi_{-k} = \pi_k^\dagger,$ .

The classical Hamiltonian may be expanded in Fourier modes as

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

where $\omega_k = \sqrt{k^2+m^2}$ .

This Hamiltonian is thus recognizable as an infinite sum of classical normal mode oscillator excitations $\phi_k$ , each one of which is quantized in the standard manner, so the free quantum Hamiltonian looks identical. It is the $\phi_k$ s that have become operators obeying the standard commutation relations, $[\phi_k,\pi_k^\dagger] = [\phi_k^\dagger,\pi_k] = i\hbar$ , with all others vanishing. The collective Hilbert space of all these oscillators is thus constructed using creation and annihilation operators constructed from these modes,

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

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 infinite collection of creation operators $a_k^\dagger$ to $|0\rangle$ . This Hilbert space is called Fock space. For each $k$ , 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

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

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.

Issues and Limitations of First qantizations

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

$\{A,B\} \longmapsto \tfrac{1}{i \hbar} [\hat{A},\hat{B}] ~.$

This rule is not as simple or well-defined as it appears. It is ambiguous when products of classical observables are involved which correspond to noncommuting products of the analog operators, and fails in polynomials of sufficiently high order.

For example, the reader is encouraged to check the following pair of equalities invented by Groenewold,assuming only the commutation relation $[\hat{x},\hat{p}]=i\hbar$ :

$\begin{align} \{x^3,p^3\}+\tfrac{1}{12}\{\{p^2,x^3\},\{x^2,p^3\}\}&=0 \\ \tfrac{1}{i\hbar}[\hat{x}^3,\hat{p}^3]+\tfrac{1}{12i\hbar}\left[\tfrac{1}{i\hbar}[\hat{p}^2,\hat{x}^3],\tfrac{1}{i\hbar}[\hat{x}^2,\hat{p}^3]\right]&=-3\hbar^2~.\end{align}$

The right-hand-side "anomaly" term −3ħ² is not predicted by application of the above naive quantization rule. In order to make this procedure more rigorous, one might hope to take an axiomatic approach to the problem. If Q represents the quantization map that acts on functions f 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

$[Q_f,Q_g]=i\hbar Q_{\{f,g\}}$ (Poisson bracket)

$Q_{g \circ f}=g(Q_f)$ (von Neumann rule)

However, not only are these four properties mutually inconsistent, any three of them is 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 ħ→0 (see Moyal bracket) is 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.