The Origins of Instantons from Path Integrals

Question

I know that you can come across non-perturbative effects in QFT, particular when the coupling constant lies outside the radius of convergence of the asympototic perturbation series.

From the perspective of canonical quantization I understand how and why such instantons crop up. Specifically, they give (non-trivial) exact solutions of the equations of motion for the full interacting theory. These may be canonically quantized to produce a quantum instanton.

My question relates to the origins of instantons from the path-integral approach. Weinberg (in Volume II of his QFT text) claims that in perturbation theory

we expand the action $S$ around spacetime independent vacuum values of the fields, keeping the leading quadratic term in the exponential $\exp(iS)$

What does he mean by this? I've definitely never done any expansions around vacuum values to derive perturbation theory for the path integral. Doesn't one usually just split e.g. $\exp(iS_{free})\exp(iS_{int})$ and that immediately gives you the series?

He goes on to say that non-perturbative effects

arise because there are stationary points of the action which are spacetime dependent

I understand this in the context of canonical quantization, but it doesn't make much sense to me with the path-integral. Unless of course one introduces the stationary phase approximation, but I don't know its relation to usual pertubation theory. Hopefully this will be clearer once the previous paragraph makes sense to me!

Answer 1

I) This is discussed around eq. (23.7.1) on p. 462 in Ref. 1. The task is to perform the path integral

$$\tag{1} \int_{BC} [d\phi]e^{\frac{i}{\hbar}S[\phi]} ~=~\sum_{\nu}\int\! du \int_{BC_0} [d\phi_q]e^{\frac{i}{\hbar}S[\phi_{cl}+\phi_{\nu,u}+\phi_q]} $$ over fields $\phi$ with some (possible inhomogeneous) boundary conditions $BC$. This is done by splitting the fields

$$\tag{2} \phi~=~\phi_{cl}+\phi_{\nu,u}+\phi_{q}$$

into the following parts.

1. A single distinguished classical solution $\phi_{cl}$ (in the trivial instanton sector). The classical solution $\phi_{cl}$ satisfies the Euler-Lagrange equations with the (possible inhomogeneous) boundary conditions $BC$.

2. A set of instantons $\phi_{\nu,u}$ labelled with discrete topological number $\nu$ and continuous moduli $u$. The instantons $\phi_{\nu,u}$ satisfy the Euler-Lagrange equations with homogeneous boundary conditions $BC_0$. Instantons arise when there isn't a unique solution to the Euler-Lagrange equation with the given boundary conditions $BC$.

3. And quantum fluctuation $\phi_q$ satisfying the homogeneous boundary conditions $BC_0$.

The action

$$\tag{3} S[\phi_{\rm cl}+\phi_{\nu,u}+\phi_q] ~=~S[\phi_{\rm cl}+\phi_{\nu,u}]+S_{2}[\phi_q]+{\cal O}((\phi_q)^3) ~\approx~S[\phi_{\rm cl}+\phi_{\nu,u}]+S_{2}[\phi_q]$$

is then often expanded to quadratic order (denoted $S_2$) in the quantum fluctuations $\phi_q$ leading to a Gaussian path integral. See also eq. (23.7.2) in Ref. 1. Note that the linear term $S_{1}[\phi_q]=0$ in $\phi_q$ vanishes because of Euler-Lagrange equations.

In ordinary perturbation theory without instantons, there is no summation over instanton sectors and integration over moduli.

II) One may wonder if the summation over instanton sectors in eq. (1) constitutes a kind of over-counting of the field configurations in the path integral? E.g. couldn't one reproduce a non-trivial instanton by including sufficiently many (all?) quantum corrections in the trivial sector, etc.?

From an idealized mathematical point of view, the need to sum over instanton sectors may be seen as the mathematical fact that not all $C^{\infty}$ functions are analytic. (Speaking of analycity, it seems relevant to mention the characteristic hallmark of instantons: The (non-trivial) instanton terms in the partition function have non-analytic dependence of the coupling constants of the theory. In short: One cannot reproduce non-perturbative effects by only applying perturbative methods.)

However in practice the path integral over quantum fluctuations is not well-defined (much) beyond the Gaussian approximation. So in practice one may view the decomposition on the rhs. of eq. (1) as a pragmatic definition of the full path integral on the lhs. of eq. (1).

References:

1. S. Weinberg, The Quantum Theory of Fields, Vol. 2, p. 421 and p. 462.