String Theory

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.

String Theory

Sunday, 9 June 2013

Real scalar field

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