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

Related concepts

• perturbation theory

• renormalization

• Feynman diagram

• non-perturbative quantum field theory

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)

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

• Romeo Brunetti, Klaus Fredenhagen, Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds Commun. Math. Phys. 208:623-661 (2000) (arXiv:math-ph/9903028)

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

Further developments along these lines are then

• Michael Dütsch, Klaus Fredenhagen, Algebraic quantum field theory, perturbation theory, and the loop expansion, Commun. Math. Phys. 219:5-30 (2001) (arXiv:hep-th/0001129)

• Michael Dütsch, Klaus Fredenhagen, Perturbative algebraic quantum field theory and deformation quantization, Proceedings of the Conference on Mathematical Physics in Mathematics and Physics, Siena June 20-25 (2000) (arXiv:hep-th/0101079)

(relation to deformation quantization)

• Romeo Brunetti, Michael Dütsch, Klaus Fredenhagen, Perturbative Algebraic Quantum Field Theory and the Renormalization Groups, Adv. Theor. Math. Physics 13 (2009), 1541-1599 (arXiv:0901.2038)

(relation to renormalization)

• Michael Dütsch, Klaus Fredenhagen, A local (perturbative) construction of observables in gauge theores: the example of qed, Commun. Math. Phys. 203 (1999), no.1, 71-105, (arXiv:hep-th/9807078).

(relation to gauge theory and QED)

In BV-BRST formalism

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

• Kevin Costello, Owen Gwilliam, Factorization algebras in quantum field theory (pdf)

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

• Owen Gwilliam, Theo Johnson-Freyd, How to derive Feynman diagrams for finite-dimensional integrals directly from the BV formalism (2011) (arXiv:1202.1554)