# Contents

## Idea

### General

Perturbation theory is a general method of finding (or even defining) the solution of equations of mathematical physics by expanding them with respect to a small parameter in the vicinity of known, defined or well-understood solution (for which the small parameter is $0$). It is used in the study of PDEs involving operators depending on small parameter, in classical and celestical mechanics, in quantum mechanics, and in the statistical and quantum field theory.

One of the varieties of perturbation theory provides a method to make sense of and handle the path integral involved in the quantization of classical field theory to quantum field theory.

It is based on the observation that the quantization of free classical field theories, whose action functional contains only the kinetic term, is well understood; therefore, the quantization of a functional consisting of a kinetic term and polynomial interaction terms may be expanded like a Taylor series in the interaction terms, thus yielding what looks like a series of correlators in a free field theory. If the coupling constant – the parameter in front of the interaction terms – is small enough, one says one is in the weakly coupled regime of the theory and expects this perturbation series to approximate the desired answer. Usually, even for that to work the action functional first has to be subjected to renormalization.

### More details

Suppose we’re working with a quantum system that’s nearly a quantum harmonic oscillator, but not quite; that is, the quadratic potential ${V}_{0}=\frac{1}{2}k{x}^{2}-\frac{1}{2}$ is only a good local approximation to the real potential ${V}_{0}+\lambda V$. Then we can write the Hamiltonian as $H={H}_{0}+\lambda V,$ where $V$ is a function of the position $x$ and the momentum $p$ (or equivalently, of $z=\frac{p+ix}{\sqrt{2}}$ and $\frac{d}{\mathrm{dz}}$) and $\lambda$ is small.

Now we solve Schrödinger’s equation perturbatively. We know that

(1)$\psi \left(t\right)={e}^{-\mathrm{itH}}\psi \left(0\right),$\psi(t) = e^{-itH} \psi(0),

and we assume that

(2)${e}^{-\mathrm{itH}}\psi \left(t\right)\approx {e}^{-{\mathrm{itH}}_{0}}\psi \left(t\right)$e^{-itH}\psi(t) \approx e^{-itH_0} \psi(t)

so that it makes sense to solve it perturbatively. Define

(3)${\psi }_{1}\left(t\right)={e}^{{\mathrm{itH}}_{0}}{e}^{-\mathrm{itH}}\psi \left(t\right)$\psi_1(t) = e^{itH_0} e^{-itH}\psi(t)

and

(4)${V}_{1}\left(t\right)={e}^{{\mathrm{itH}}_{0}}\lambda V{e}^{-{\mathrm{itH}}_{0}}.$V_1(t) = e^{itH_0} \lambda V e^{-itH_0}.

After a little work, we find that

(5)$\frac{d}{\mathrm{dt}}{\psi }_{1}\left(t\right)=-i{V}_{1}\left(t\right){\psi }_{1}\left(t\right),$\frac{d}{dt}\psi_1(t) = -i V_1(t) \psi_1(t),

and integrating, we get

(6)${\psi }_{1}\left(t\right)=-i{\int }_{0}^{t}{V}_{1}\left({t}_{0}\right){\psi }_{1}\left({t}_{0}\right){\mathrm{dt}}_{0}+\psi \left(0\right).$\psi_1(t) = -i\int_0^t V_1(t_0) \psi_1(t_0) dt_0 + \psi(0).

We feed this equation back into itself recursively to get

(7)$\begin{array}{ccc}{\psi }_{1}\left(t\right)& =& -i{\int }_{0}^{t}{V}_{1}\left({t}_{0}\right)\left[-i{\int }_{0}^{{t}_{0}}{V}_{1}\left({t}_{1}\right){\psi }_{1}\left({t}_{1}\right){\mathrm{dt}}_{1}+\psi \left(0\right)\right]{\mathrm{dt}}_{0}+\psi \left(0\right)\\ & =& \left[\psi \left(0\right)\right]+\left[{\int }_{0}^{t}{i}^{-1}{V}_{1}\left({t}_{0}\right)\psi \left(0\right){\mathrm{dt}}_{0}\right]+\left[{\int }_{0}^{t}{\int }_{0}^{{t}_{0}}{i}^{-2}{V}_{1}\left({t}_{0}\right){V}_{1}\left({t}_{1}\right){\psi }_{1}\left({t}_{1}\right){\mathrm{dt}}_{1}{\mathrm{dt}}_{0}\right]\\ & =& \sum _{n=0}^{\infty }{\int }_{t\ge {t}_{0}\ge \dots \ge {t}_{n-1}\ge 0}{i}^{-n}{V}_{1}\left({t}_{0}\right)\cdots {V}_{1}\left({t}_{n-1}\right)\psi \left(0\right){\mathrm{dt}}_{n-1}\cdots {\mathrm{dt}}_{0}\\ & =& \sum _{n=0}^{\infty }\left(-\lambda i{\right)}^{n}{\int }_{t\ge {t}_{0}\ge \dots \ge {t}_{n-1}\ge 0}{e}^{-i\left(t-{t}_{0}\right){H}_{0}}V{e}^{-i\left({t}_{0}-{t}_{1}\right){H}_{0}}V\cdots V{e}^{-i\left({t}_{n-1}-0\right){H}_{0}}\psi \left(0\right){\mathrm{dt}}_{n-1}\cdots {\mathrm{dt}}_{0}.\end{array}$\array{ \psi_1(t) & = & -i \int_0^t V_1(t_0) \left[-i\int_0^{t_0} V_1(t_1) \psi_1(t_1) dt_1 + \psi(0) \right] dt_0 + \psi(0) \\ & = & \left[\psi(0)\right] + \left[\int_0^t i^{-1} V_1(t_0)\psi(0) dt_0\right] + \left[\int_0^t\int_0^{t_0} i^{-2} V_1(t_0)V_1(t_1) \psi_1(t_1) dt_1 dt_0\right] \\ & = & \sum_{n=0}^{\infty} \int_{t \ge t_0 \ge \ldots \ge t_{n-1} \ge 0} i^{-n} V_1(t_0)\cdots V_1(t_{n-1}) \psi(0) dt_{n-1}\cdots dt_0 \\ & = & \sum_{n=0}^{\infty} (-\lambda i)^n \int_{t \ge t_0 \ge \ldots \ge t_{n-1} \ge 0} e^{-i(t-t_0)H_0} V e^{-i(t_0-t_1)H_0} V \cdots V e^{-i(t_{n-1}-0)H_0} \psi(0) dt_{n-1}\cdots dt_0. }

So here we have a sum of a bunch of terms; the $n$th term involves $n$ interactions with the potential interspersed with evolving freely between the interactions, and we integrate over all possible times at which those interactions could occur.

Here’s an example Feynman diagram for this simple system, representing the fourth term in the sum above:

The lines represent evolving under the free Hamiltonian ${H}_{0}$, while the dots are interactions with the potential $V$.

As an example, let’s consider $V=\left(z+\frac{d}{\mathrm{dz}}\right)$ and choose $\lambda =\frac{1}{\sqrt{2}}$ so that $\lambda V=p.$ When $V$ acts on a state $\psi ={z}^{n},$ we get $V\psi ={z}^{n+1}+{\mathrm{nz}}^{n-1}.$ So at each interaction, the system either gains a photon or changes phase and loses a photon.

## Properties

### Divergence/Convergence

Despite what one might naively expect, the perturbation series of natural quantum field theories have a vanishing radius of convergence, they are asymptotic series.

Roughly this can be understood as follows: since the pertrubation is in the coupling constant about vanishing coupling, a non-zero radius of convergence would imply that the theory is finite also for negative coupling (where “things fly apart”), which will not happen in realistic theories.

More in detail, theories with non-perturbative effects such as instantons field configurations (such as Yang-Mills theory, hence QCD, QED), branes (such as string theory), etc., are expected to have a path integral which as a function of the coupling constant $g$ schematically looks like

$Z\left(g\right)=\sum _{n}{a}_{n}{g}^{n}+{e}^{-A/g}\sum _{n}{a}_{n}^{\left(1\right)}{g}^{n}+𝒪\left({e}^{-2A/g}\right)\phantom{\rule{thinmathspace}{0ex}},$Z(g) = \sum_n a_n g^n + e^{-A/g} \sum_n a_n^{(1)} g^n + \mathcal{O}(e^{-2A/g}) \,,

where the first sum is the perturbation series itself and where the terms with a prefactor of the form $\mathrm{exp}\left(-A/g\right)$ are the contributions of the instantons ($A$ is the contribution of the instanton action functional). Since all the derivatives of the function $g↦{e}^{-1/g}$ vanish at coupling constant $g=0$, the Taylor series of this part of the path integral does not appear in perturbation series, even though it is present. Therefore this is called a non-perturbative effect.

See the references below for details.

## References

### General

A solid mathematical formulation of perturbation theory has been given in

• K. Hepp.: Théorie de la Renormalisation Lect. Notes in Phys. Springer (1969)

• O. Steinmann, Perturbation expansion in axiomatic field theory Lect. Notes in Phys. 11, Springer (1971)

### On Divergence/Convergence and Non-perturbative effects

A general introduction on divergence of perturbation theory, asymptotic series and non-perturbative effects is for instance on the first pages of

• Marcos Marino, Lectures on non-perturbative effects in large N gauge theories, matrix models and strings (arXiv:1206.6272)

Further discussion is for instance in

### In AQFT

Perturbation theory in the context of AQFT is discussed in the following articles.

The observation that in perturbation theory the Stückelberg-Bogoliubov-Epstein-Glaser local S-matrices yield a local net of observables was first made in

• V. Il’in, D. Slavnov, Observable algebras in the S-matrix approach Theor. Math. Phys. 36 , 32 (1978) 578-585

which was however mostly ignored and forgotten. It is taken up again in

(a quick survey is in section 8, details are in section 2).

Further developments along these lines are then

(relation to deformation quantization)

(relation to renormalization)

(relation to gauge theory and QED)

### In BV-BRST formalism

Perturbative quantization in BV-BRST formalism is nicely systematically discussed in section 5 of

in the broad context of factorization algebras (see there for further references). In particular the relation to Feynman diagrams is discussed in

Revised on July 14, 2013 23:04:35 by Urs Schreiber (82.113.106.95)