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Perturbation Theory in QFT Is Asymptotic

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Perturbation Theory in QFT Is Asymptotic Perturbative QFT is Asymptotic; is Divergent; is Problematic in Principle Justin Bond1 1Department of Physics, University of California at San Diego, La Jolla, CA 92093 A very brief review is given on what it means for a series to be asymptotic followed by a basic example in non-relativistic quantum mechanics. Finally, Freeman J. Dyson's mostly qualitative argument for why QED should be expected to be divergent in general, is given. In such a case, the theory is either incomplete or new mathematical methods are needed to deal with these divergent expansions to give a more complete understanding of what the theory is saying. The Borel transform is discussed as an effort to define a sum for such a factorially divergent series. INTRODUCTION tion method used for divergent series. But first, we talk about some of the basics about asymptotic expansions Under the current paradigm of quantum field theory, and how they arise in non-relativistic quantum mechan- we often have to do without exact solutions for measur- ics. able quantities. The effective theories that describe inter- actions must contain nonlinear terms in the Lagrangian ASYMPTOTIC EXPANSIONS: A VERY BRIEF in order to couple different Fourier modes and different REMINDER fields[1] (i.e. to get interesting stuff to happen). These terms that make interesting things happen, also make it If one cannot determine an integral in some analytic difficult to solve for measurable quantities. We're often form, another way to tackle the integral is to expand forced to expand an expression for these quantities into it somehow. A common technique is to expand the inte- a power-series. The problem is, however, that the series grand in some power-series for some small parameter and is often asymptotic and thus diverges. Given a partic- compute the integral term by term in the expansion[3]. ular asymptotic expansion in QFT, the precision of the Sometimes the expansion of the integrand works in a par- predicted value of that observable is therefore limited, ticular domain but the integral itself may be over a re- in principle, as opposed to a practical limitation on com- gion in which the domain is not valid. Often this will puting the infinite order term. For practical purposes, we generate an asymptotic expansion. One example for an can ignore this divergence because the answer is right, up asymptotic expansion is the following to some decimal and by the natural ordering of terms in a power-series, we're never at odds with in which order 1 the terms should be added in the expansion. 4 Z 4 I(x) = 4x3ex e−t dt (1) To emphasize, there is no issue with QFT's practical x predictive power, but nonetheless F. J. Dyson reasonably took issue with the in-principle-problem of a divergent for x ! 1. One can expand this to yield series describing physical phenomena. It is not a math- ematically well defined theory. In an interview[2], F. J. 1 4 Z 4 3 3 · 7 Dyson discusses his \ambition of making quantum elec- I(x) = 4x3ex e−t dt = 1 − + + ::: (2) 4x4 (4x4)2 trodynamics into a completely solvable theory...in which x all the renormalizations would be done...and everything 1 This is a power series in the parameter λ = 4x4 . Because could be done, in principle, as accurately as you want." there is a factorial coefficient for each term in the expan- He later realized that these divergences in QED were sion, there will be some x-dependent order after which, likely fundamental to the theory of perturbative QED each successive term of the series will increase and even- itself. Ultimately, he came to the conclusion that either tually diverge. For the above expansion, that order is different mathematical methods were needed or a new 4 roughly nc ≈ x . Thus, as the expansion parameter λ physical theory is needed (which may also include the goes to zero, or as x ! 1, one can take more and more need for new math). terms in the expansion as a good approximation to the In this paper, we discuss the former scenario. In partic- actual function I(x). ular, we discuss the Borel summation which is a summa- AN EXAMPLE FROM NON-RELATIVISTIC QUANTUM MECHANICS [6] It should be noted, however, that there exist quantum field the- ories that are exactly solvable, but from my understanding, they are usually relegated to lower dimensions. You can see section Asymptotic integrals do not just show up in high en- 22.2 of Peskin & Schroeder for more information. ergy QFT, but also ordinary quantum mechanics. Per- 2 quasi-bound states well below the potential maximum. Also, the energies of the perturbed states will take the form of Ea(λ) = Aa(λ) + Sa(λ) where Aa(λ) is a part that is analytic in λ and Sa(λ) is a nonperturbative part that is exponentially small as λ ! 0. FREEMAN J. DYSON'S ARGUMENT OF DIVERGENCE IN QED Dyson gives a simple physical argument that indicates the power series expansions in QED will be divergent FIG. 1: A plot of the potential V (x) = x2 and V 0(x) = even after renormalization. Also, the divergence is non- x2 − x4. Because V 0(x = ±∞) = −∞ the states for the trivial, as opposed to the trivial divergences that occur perturbed Hamiltonian are mostly unbound states with the for the scattering amplitude of a free particle where the possibility of only quasi-bound states. particle has a possibility of being captured into a perma- nently bound state. Suppose we have the form of a physical quantity given turbation theory in quantum mechanics is used to deter- as a power series in the charge coupling in QED. The mine how the eigenfunctions change as a result of some power series is obtained by integrating the equations of perturbation of the Hamiltonian. In non-degenerate per- motion of the theory. If the expansion is such that, turbation theory, one can follow the states as the pertur- bation of the form λH is switched on. The assumption 2 2 4 1 F (e ) = a0 + a2e + a4e + ::: < 1 (4) of perturbation theory is that the actual states that are being expanded are analytic functions of the parameter for some nonzero value of e2, then F (e2) is analytic at λ in a circle of finite radius about λ = 0 in the com- e = 0, this must mean that the physical quantity is an plex plane. So there is a requirement that it should be analytic function of e for e = 0. Therefore, if it is an possible to analytically continue to states of differing λ. analytic function, we can allow e2 to extend in the com- Let us now look at a quick example regarding the per- plex plane and it must have a finite radius of convergence turbation to the SHO hamiltonian. Under the perturba- about e2 = 0; so e2 ! −e2 or F (e2) ! F (−e2) should tion, the full hamiltonian will be of the form, also be convergent for sufficiently small values of e2. We can interpret the physical quantity F (−e2) as aris- p2 1 1 ing in a world where like charges attract each other. In H = H + λH = + kx2 + λx4 (3) 0 1 2m 2 4 the classical limit of large particle number and large dis- tances, the potential between charges is just the clas- for positive k. sical Coulomb potential with a sign reversal. One can The form of the unperturbed and perturbed potential imagine a \pathological" state with a large number of is shown in Fig. 1. For λ > 0, both H and H tend 0 electron-positron pairs with the electrons separated from to 1 as q ! 1 and so the states are always bound the positrons in different regions of space such that the states with a discrete spectrum in both cases. For λ < 0, negative Coulomb potential is larger than the rest energy H has only bound states, and H has no true bound 0 and kinetic energy of the particles themselves. states, except for a set of quasi-bound states that are The point is that for any physical state of the system, ultimately delocalized. No matter how small jλj gets, there is a large potential barrier separating that physical 1 4 1 2 q 2k the 4 λx term will always overpower 2 kx for jxj > jλj state with these \pathological" states, in some sense sim- and all the states will have a finite probability of escaping ilar to the quartic potential perturbation of the harmonic the potential due to the tunneling process. oscillator where the quasi-bound states have a finite over- Perturbation theory in quantum mechanics expands lap with states that can escape to regions of arbitrarily the new states in terms of the old ones. How can we get large negative potential. In that case, any quasi-bound spatially extended states from a set of states spatially state is unstable against the spontaneous tunneling be- localized about an origin? We can't! For the harmonic yond the potential maxima. In the case of the alter- oscillator, all the states need to go to 0 at x = 1, there- nate QED where e2 has gone to −e2, any physical state fore all states constructed from these states must have is unstable against the spontaneous generation of many this property as well. So the boundary condition won't particles. Once in such a pathological state, the system let you. The analyticity requirement is not met, and will generate even more particles and the vacuum will therefore the expansion is not entirely valid.
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