nLab
perturbation theory

Idea

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 correlator?s 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)$$\psi (t)=e^{-\mathrm{itH}}\psi (0),$$\psi(t) = e^{-itH} \psi(0),

and we assume that

(2)$$e^{-\mathrm{itH}}\psi (t)\approx e^{-{\mathrm{itH}}_0}\psi (t)$$e^{-itH}\psi(t) \approx e^{-itH_0} \psi(t)

so that it makes sense to solve it perturbatively. Define

(3)$${\psi }_1(t)=e^{{\mathrm{itH}}_0}{e}^{-\mathrm{itH}}\psi (t)$$\psi_1(t) = e^{itH_0} e^{-itH}\psi(t)

and

(4)$$V_1(t)=e^{{\mathrm{itH}}_0}\lambda V{e}^{-{\mathrm{itH}}_0}.$$V_1(t) = e^{itH_0} \lambda V e^{-itH_0}.

After a little work, we find that

(5)$$\frac{d}{\mathrm{dt}}{\psi }_1(t)=-i{V}_1(t){\psi }_1(t),$$\frac{d}{dt}\psi_1(t) = -i V_1(t) \psi_1(t),

and integrating, we get

(6)$${\psi }_1(t)=-i{\int }_0^t{V}_1(t_0){\psi }_1(t_0){\mathrm{dt}}_0+\psi (0).$$\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)$$\begin{array}{ccc}{\psi }_1(t)& =& -i{\int }_0^t{V}_1(t_0)[-i{\int }_0^{t_0}{V}_1(t_1){\psi }_1(t_1){\mathrm{dt}}_1+\psi (0)]{\mathrm{dt}}_0+\psi (0)\\ & =& [\psi (0)]+[{\int }_0^t{i}^{-1}{V}_1(t_0)\psi (0){\mathrm{dt}}_0]+[{\int }_0^t{\int }_0^{t_0}{i}^{-2}{V}_1(t_0)V_1(t_1){\psi }_1(t_1){\mathrm{dt}}_1{\mathrm{dt}}_0]\\ & =& \sum _{n=0}^{\mathrm{\infty }}{\int }_{t\ge t_0\ge \dots \ge t_{n-1}\ge 0}{i}^{-n}{V}_1(t_0)\cdots V_1(t_{n-1})\psi (0){\mathrm{dt}}_{n-1}\cdots {\mathrm{dt}}_0\\ & =& \sum _{n=0}^{\mathrm{\infty }}(-\lambda i{)}^n{\int }_{t\ge t_0\ge \dots \ge t_{n-1}\ge 0}{e}^{-i(t-t_0)H_0}V{e}^{-i(t_0-t_1)H_0}V\cdots V{e}^{-i(t_{n-1}-0)H_0}\psi (0){\mathrm{dt}}_{n-1}\cdots {\mathrm{dt}}_0.\end{array}$$\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 $V=(z+\frac{d}{\mathrm{dz}})$ and choose $\lambda =\frac{1}{\sqrt{2}}$ so that $\lambda V=p.$ When $V$ acts on a state $\psi =z^n,$ we get $V\psi =z^{n+1}+{\mathrm{nz}}^{n-1}.$ So at each interaction, the system either gains a photon or changes phase and loses a photon.