Contents

# 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 perturbative 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 $z = \frac{p+i x}{\sqrt{2}}$ and $\frac{d}{dz}$) and $\lambda$ is small.

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

$\psi(t) = e^{-itH} \psi(0),$

and we assume that

$e^{-itH}\psi(t) \approx e^{-itH_0} \psi(t)$

so that it makes sense to solve it perturbatively. Define

$\psi_1(t) = e^{itH_0} e^{-itH}\psi(t)$

and

$V_1(t) = e^{itH_0} \lambda V e^{-itH_0}.$

After a little work, we find that

$\frac{d}{dt}\psi_1(t) = -i V_1(t) \psi_1(t),$

and integrating, we get

$\psi_1(t) = -i\int_0^t V_1(t_0) \psi_1(t_0) dt_0 + \psi(0).$

We feed this equation back into itself recursively to get

$\array{ \psi_1(t) & = & -i \int_0^t V_1(t_0) \left[-i\int_0^{t_0} V_1(t_1) \psi_1(t_1) dt_1 + \psi(0) \right] dt_0 + \psi(0) \\ & = & \left[\psi(0)\right] + \left[\int_0^t i^{-1} V_1(t_0)\psi(0) dt_0\right] + \left[\int_0^t\int_0^{t_0} i^{-2} V_1(t_0)V_1(t_1) \psi_1(t_1) dt_1 dt_0\right] \\ & = & \sum_{n=0}^{\infty} \int_{t \ge t_0 \ge \ldots \ge t_{n-1} \ge 0} i^{-n} V_1(t_0)\cdots V_1(t_{n-1}) \psi(0) dt_{n-1}\cdots dt_0 \\ & = & \sum_{n=0}^{\infty} (-\lambda i)^n \int_{t \ge t_0 \ge \ldots \ge t_{n-1} \ge 0} e^{-i(t-t_0)H_0} V e^{-i(t_0-t_1)H_0} V \cdots V e^{-i(t_{n-1}-0)H_0} \psi(0) dt_{n-1}\cdots dt_0. }$

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 $V = (z + \frac{d}{dz})$ and choose $\lambda = \frac{1}{\sqrt{2}}$ so that $\lambda V = p.$ When $V$ acts on a state $\psi = z^n,$ we get $V \psi = z^{n+1} + nz^{n-1}.$ 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

$Z(g) = \sum_n a_n g^n + e^{-A/g} \sum_n a_n^{(1)} g^n + \mathcal{O}(e^{-2A/g}) \,,$

where the first sum is the perturbation series itself and where the terms with a prefactor of the form $\exp(-A/g)$ are the contributions of the instantons ($A$ is the contribution of the instanton action functional). Since all the derivatives of the function $g \mapsto e^{-1/g}$ 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. The mathematics behind this is called resurgence theory.

### 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

The argument that the perturbation series of realistic quantum field theories such as QED necessarily diverges goes back to

• Freeman Dyson, Divergence of perturbation theory in quantum electrodynamics, Phys. Rev. 85, 631, 1952 (spire)

Abstract: An argument is presented which leads tentatively to the conclusion that all the power-series expansions currently in use in quantum electrodynamics are divergent after the renormalization of mass and charge. The divergence in no way restricts the accuracy of practical calculations that can be made with the theory, but raises important questions of principle concerning the nature of the physical concepts upon which the theory is built.

and is made more precise in

• Lev Lipatov, Divergence of the Perturbation Theory Series and the Quasiclassical Theory, Sov.Phys.JETP 45 (1977) 216–223 (pdf)

recalled for instance in

• Igor Suslov, section 1 of Divergent perturbation series, Zh.Eksp.Teor.Fiz. 127 (2005) 1350; J.Exp.Theor.Phys. 100 (2005) 1188 (arXiv:hep-ph/0510142)

• Justin Bond, last section of Perturbative QFT is Asymptotic; is Divergent; is Problematic in Principle (pdf)

• Alexander P. Bakulev, Dmitry Shirkov, section 1.1 of Inevitability and Importance of Non-Perturbative Elements in Quantum Field Theory, Proceedings of the 6th Mathematical Physics Meeting, Sept. 14–23, 2010, Belgrade, Serbia (ISBN 978-86-82441-30-4), pp. 27–54 (arXiv:1102.2380)

• Stefan Hollands, Robert Wald, section 4.1 of Quantum fields in curved spacetime, Physics Reports Volume 574, 16 April 2015, Pages 1-35 (arXiv:1401.2026)

• Marco Serone, from 2:46 on in A look at $\phi^4_2$ using perturbation theory (recording)

For the example of phi^4 theory this non-convergence of the perturbation series is discussed in

• Robert Helling, p. 4 of Solving classical field equations (pdf)

For the example of phi^4 theory this non-convergence of the perturbation series is discussed in

• Robert Helling, p. 4 of Solving classical field equations (pdf)

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

• Carl M. Bender, Carlo Heissenberg, Convergent and Divergent Series in Physics (arXiv:1703.05164)

Further discussion is for instance in

### In AQFT

Perturbation theory in the spirit of AQFT, namely in locally covariant perturbative quantum field theory 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)

Reviews includes

and a textbook acount is in

• Katarzyna Rejzner, Perturbative Algebraic Quantum Field Theory, Mathematical Physics Studies, Springer 2016 (pdf)

Further developments in perturbation theory in AQFT on curved spacetimes icludes

### 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