Perturbation Theory

Prerequisites: Canonical Quantum Mechanics. Full path (17 pages)

Intuitive
Concrete
Abstract

Intuitive

Perturbation theory is a large collection of iterative methods for obtaining approximate solutions to problems involving a small parameter $\epsilon$ . These methods are so powerful that sometimes it is actually advisable to introduce a parameter $\epsilon$ temporarily into a difficult problem having no small parameter, and then finally to set $\epsilon =1$ to recover the original problem. This apparently artificial conversion to a perturbation problem may be the only way to make progress.

The thematic approach of perturbation theory is to decompose a tough problem into an infinite number of relatively easy ones. Hence, perturbation theory is most useful when the first few steps reveal the important features of the solution and the remaining ones give small corrections.

Advanced mathematical methods for scientists and engineers by Bender and Orszag

Concrete

We split the Hamiltonian into a part we can solve exactly and a small remainder, $H = H_0 + \lambda V$ , and expand the quantity we want in powers of $\lambda$ . We set $\hbar = c = 1$ .

Quantum mechanics. For a non-degenerate eigenstate $|n\rangle$ of $H_0$ with energy $E_n$ , Rayleigh-Schrödinger perturbation theory gives

$E_n(\lambda) = E_n + \lambda \langle n|V|n\rangle + \lambda^2 \sum_{m\neq n} \frac{|\langle m|V|n\rangle|^2}{E_n - E_m} + \mathcal{O}(\lambda^3).$

The expansion is useful when the matrix elements of $\lambda V$ are small compared with the level spacings $E_n - E_m$ .

Time evolution. In the interaction picture the operators evolve with $H_0$ , and the perturbation becomes $V_I(t) = e^{iH_0 t}\, V\, e^{-iH_0 t}$ . The time evolution operator between $t_0$ and $t$ is the Dyson series

$U_I(t,t_0) = T \exp\left(-i\lambda\int_{t_0}^{t} dt'\, V_I(t')\right) = \sum_{k=0}^{\infty} \frac{(-i\lambda)^k}{k!} \int_{t_0}^{t} dt_1 \cdots \int_{t_0}^{t} dt_k\; T\big[V_I(t_1)\cdots V_I(t_k)\big],$

where $T$ orders the operators with later times to the left.

Quantum field theory. For a real scalar field with $\mathcal{L} = \frac12 \partial_\mu\phi\,\partial^\mu\phi - \frac12 m^2\phi^2 - \frac{\lambda}{4!}\phi^4$ (signature $(+,-,-,-)$ ), the interaction Hamiltonian is $H_I(t) = \int d^3x\, \frac{\lambda}{4!}\phi_I^4$ . Correlation functions in the interacting vacuum $|\Omega\rangle$ are given by the Gell-Mann-Low formula

$\langle\Omega| T\,\phi(x_1)\cdots\phi(x_n)|\Omega\rangle = \frac{\langle 0| T\,\phi_I(x_1)\cdots\phi_I(x_n)\exp\left(-i\int d^4z\, \frac{\lambda}{4!}\phi_I(z)^4\right)|0\rangle}{\langle 0| T\exp\left(-i\int d^4z\, \frac{\lambda}{4!}\phi_I(z)^4\right)|0\rangle},$

where the time integrals run over $(-\infty,\infty)$ with a small rotation $t \to t(1-i\epsilon)$ that projects the free vacuum $|0\rangle$ onto $|\Omega\rangle$ . Expanding the exponential and applying Wick's theorem writes every term as a product of Feynman propagators

$D_F(x-y) = \int \frac{d^4p}{(2\pi)^4}\, \frac{i\, e^{-ip\cdot(x-y)}}{p^2 - m^2 + i\epsilon},$

and each term is drawn as a Feynman diagram. The denominator removes the vacuum bubbles.

Example. At first order in $\lambda$ the two-point function receives the term $\frac{-i\lambda}{4!}\int d^4z\,\langle 0|T\phi_I(x)\phi_I(y)\phi_I(z)^4|0\rangle$ . Contracting $\phi(x)$ with one of the four fields at $z$ (4 ways) and $\phi(y)$ with one of the remaining three (3 ways) leaves one self-contraction, so the connected result is

$\frac{-i\lambda}{2}\int d^4z\, D_F(x-z)\, D_F(0)\, D_F(z-y).$

The factor $\frac12 = \frac{4\cdot 3}{4!}$ is the symmetry factor of the diagram. $D_F(0)$ is a divergent loop integral, and it is the first place where renormalization is needed. For $2\to 2$ scattering the lowest order amplitude is simply $i\mathcal{M} = -i\lambda$ .

In practice each order in the coupling corresponds to diagrams with more vertices and more loops. In QED the expansion parameter is effectively $\alpha/\pi \approx 2\times 10^{-3}$ , so a few orders already give very precise predictions. In QCD at low energies the coupling is of order one and the expansion fails there.

No-Nonsense Quantum Mechanics by Jakob Schwichtenberg, section "Perturbation Theory"
No-Nonsense Quantum Field Theory by Jakob Schwichtenberg, chapter "Interacting Quantum Fields"

Abstract

Perturbation theory produces a formal power series $\sum_n a_n g^n$ in the coupling. In interacting quantum field theories, and already in quantum mechanics, this series is in general not convergent but asymptotic: for fixed $N$ ,

$\Big| f(g) - \sum_{n=0}^{N} a_n g^n \Big| = \mathcal{O}(g^{N+1}) \quad (g\to 0),$

while for fixed $g \neq 0$ the partial sums diverge as $N \to \infty$ . Two arguments make this plausible. First, the number of Feynman diagrams at order $n$ grows factorially, and typically $a_n \sim n!\,A^{-n}$ up to powers of $n$ . Second, Dyson's argument for QED: for $e^2 < 0$ like charges attract, the vacuum is unstable against the creation of many pairs, so $f$ cannot be analytic in a disc around $e^2 = 0$ .

For coefficients $a_n \sim n!\,A^{-n}$ the terms $a_n g^n$ decrease until $n \approx A/g$ and grow afterwards. Truncating at the smallest term leaves an error of order $e^{-A/g}$ . Terms of this size are what non-perturbative effects such as instantons contribute, since a classical Euclidean solution with action $S_0/g$ gives a factor $e^{-S_0/g}$ , which has a vanishing Taylor series at $g = 0$ .

A standard way to give meaning to the series is Borel summation. One defines the Borel transform $B(t) = \sum_n a_n t^n / n!$ , which has a finite radius of convergence when $a_n$ grows like $n!$ , continues it analytically, and sets

$f(g) = \int_0^\infty dt\; e^{-t}\, B(g t).$

Term by term the integral reproduces $\sum_n a_n g^n$ , because $\int_0^\infty dt\, e^{-t} t^n = n!$ . The ground state energy of the quartic anharmonic oscillator is Borel summable in this sense. When $B(t)$ has singularities on the positive real axis, the integral must be deformed above or below them, and the two choices differ by terms of order $e^{-A/g}$ . In the double-well potential this ambiguity is related to the instanton-anti-instanton contribution. In four-dimensional gauge theories there are additional singularities, called renormalons, which come from the integration over loop momenta in particular classes of diagrams and are not associated with classical solutions.

The modern view is that the complete answer is a trans-series, a combination of the perturbative series and of sectors weighted by $e^{-kA/g}$ , each with its own power series, and that the large-order behaviour of the perturbative coefficients and the non-perturbative sectors constrain each other (resurgence). For the structure of the perturbative expansion itself, the combinatorics of renormalization has a formulation in terms of a Hopf algebra of Feynman graphs (Connes and Kreimer).

Amplitudes for Astrophysicists I: Known Knowns by Daniel J. Burger et. al.

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