Types of quantum field thories
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 ). 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.
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.
Suppose we’re working with a quantum system that’s nearly a quantum harmonic oscillator, but not quite; that is, the quadratic potential is only a good local approximation to the real potential . Then we can write the Hamiltonian as where is a function of the position and the momentum (or equivalently, of and ) and is small.
Now we solve Schrödinger’s equation perturbatively. We know that
and we assume that
so that it makes sense to solve it perturbatively. Define
After a little work, we find that
and integrating, we get
We feed this equation back into itself recursively to get
So here we have a sum of a bunch of terms; the th term involves 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 , while the dots are interactions with the potential .
As an example, let’s consider and choose so that When acts on a state we get So at each interaction, the system either gains a photon or changes phase and loses a photon.
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 schematically looks like
where the first sum is the perturbation series itself and where the terms with a prefactor of the form are the contributions of the instantons ( is the contribution of the instanton action functional). Since all the derivatives of the function vanish at coupling constant , 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.
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)
A general introduction on divergence of perturbation theory, asymptotic series and non-perturbative effects is for instance on the first pages of
Further discussion is for instance in
Perturbation theory in the context of AQFT is discussed in the following articles.
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
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)
(relation to renormalization)
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