How to Renormalize the Schrödinger Equation
Lectures at the VIII Jorge André Swieca Summer School
(Brazil, Feb. 1997)
\abstracts
These lectures illustrate the key ideas of modern renormalization theory and effective field theories in the context of simple nonrelativistic quantum mechanics and the Schrödinger equation. They also discuss problems in QED, QCD and nuclear physics for which rigorous potential models can be derived using renormalization techniques. They end with an analysis of nucleonnucleon scattering based effective theory.
1 Renormalization Revisited
These lectures are about effective field theories — lowenergy approximations to arbitrary highenergy physics — and therefore they are about modern renormalization theory.[1]
Despite the complexity of most textbook accounts, renormalization is based upon a very familiar and simple idea: a probe of wavelength is insensitive to details of structure at distances much smaller than . This means that we can mimic the real shortdistance structure of the target and probe by simple shortdistance structure. For example, a complicated current source of size that generates radiation with wavelengths is accurately mimicked by a sum of pointlike multipole currents (, , etc). In thinking about the longwavelength radiation it is generally much easier to treat the source as a sum of multipoles than to deal with the true current directly. This is particularly true since usually only one or two multipoles are needed for sufficient accuracy. The multipole expansion is a simple example of a renormalization analysis.
In a quantum field theory, QED for example, the quantum fluctuations probe arbitrarily short distances. This is evident when one computes radiative corrections in perturbation theory. Ultraviolet divergences, coming from loop momenta (or wavelengths ), result in infinite contributions — radiative corrections seem infinitely sensitive to short distance behavior. Even ignoring the infinities, this poses a serious conceptual problem since we don’t really know what happens as . For example, there might be new supersymmetric interactions, or superstring properties might become important, or electrons and muons might have internal structure. The situation is saved by renormalization theory which tells us that we don’t really need to know what happens at very large momenta in order to understand lowmomentum experiments. As in the multipole expansion, we can mimic the complex highmomentum, shortdistance structure of the real theory, whatever it is, by a generic set of simple pointlike interactions.
The transformation from the real theory to a simpler effective theory, valid for lowmomentum processes, is achieved in two steps. First we introduce a momentum cutoff that is of order the momentum at which new as yet unknown physics becomes important. Only momenta are retained when calculating radiative corrections.^{1}^{1}1For clarity’s sake we adopt a simple cutoff as our regulator here. In actual calculations one generally tailors the regulator to optimize the calculation. This means that our radiative corrections include only physics that we understand, and that there are no longer infinities. Of course we don’t really know the scale at which new physics will be discovered, but, as we shall see, results are almost independent of provided it is much larger than the momenta in the range being probed experimentally.
The second step is to add local interactions to the lagrangian (or hamiltonian). These mimic the effects of the true shortdistance physics. Any radiative correction that involves momenta above the cutoff is necessarily highly virtual, and, by the uncertainty principle, must occur over distances of order or less. Such corrections will appear to be local to lowmomentum probes whose wavelengths are large compared with . Thus the correct lagrangian for cutoff QED consists of the normal lagrangian together with a series of correction terms:
(1)  
where couplings , , and are dimensionless. The correction terms are nonrenormalizable, but that does not lead to problems because we keep the cutoff finite. These new interactions are far simpler to work with than the supersymmetric/superstring/…interactions that they simulate.
The correction terms in cutoff QED modify the predictions of the theory. The modifications, however, are small if is large. Contributions from the term, for example, are suppressed by , where is the typical momentum in the process under study. The next two terms are suppressed by , and so on. In principle there are infinitely many correction terms in , forming a series in ; but, when working to a given precision (ie, to a given order in ), only a finite number of these terms is important. Indeed none of these correction terms seems important in any highprecision test of QED. This indicates that the scale for new physics, , is quite large — probably of order a few TeV or larger.
Expansion (1) provides a useful parameterization for the effects of new physics on lowmomentum processes. The form of the cutoff lagrangian is independent of the new physics. It is only the numerical values of the couplings , …that contain information about the new physics. (The couplings are analogous to the multipole moments of a current in our example above.) Thus the implications of a highprecision test of QED can be expressed in a modelindependent way as limits on or values for these couplings.
In these lectures I illustrate the powerful techniques of modern renormalization theory in a series of fully workedout examples. These examples are all based upon the standard Schrödinger equation; they require nothing more than elementary quantum mechanics and some simple numerical analysis. And yet, as I discuss, several important problems in QED, QCD and nuclear physics can be rigorously formulated as potential models using renormalization techniques.
I begin, in Section 2, with an illustration of both nonperturbative and perturbative renormalization in the context of the Schrödinger equation. We will explore many aspects of renormalization familiar from applications in quantum field theory. In particular we will see in detail how to design an effective theory to model a particular set of lowenergy data. In Section 3 I discuss the physical conditions that lead to potential models, and, briefly, how they are used in QED and QCD. Finally, in Section 4, I describe how to use our effective potential theory in a systematic analysis of lowenergy nucleonnucleon interactions.
2 Renormalizing the Schrödinger Equation
In this section I illustrate the construction and use of an effective theory by designing one that reproduces a given collection of lowenergy data. I begin by discussing the data we will use. I then describe an obvious, wellknown procedure for modelling such data, and its limitations. Next I construct a somewhat less obvious effective theory that overcomes these limitations. This modern approach allows us to model the data with arbitrary precision at low energies. My goal throughout is to demonstrate how to use effective theories; I am less interested in proving the formalism mathematically correct. Consequently most examples are solved numerically. Nevertheless the insight gained from analytic calculations is ultimately indispensable. Thus I end this section with a very simple analytic calculation, using oneloop perturbation theory, that illustrates several of the key ideas.
2.1 Synthetic Data
To begin we need a collection of lowenergy data that describe a particular physical system. While we could use real experimental data from a physically interesting system, it is better here to avoid the complexities associated with experimental error by generating “synthetic data.” I illustrate the application to real problems later in the lectures, once we have worked through the formalism.
I generated synthetic data for use in these lectures by inventing a simple physical system, and then solving the Schrödinger equation that describes it. I obtained binding energies, lowenergy phase shifts, and matrix elements. For the system, I chose the familiar oneparticle Coulombic atom, but with a shortrange potential in addition to the Coulomb potential:
(2) 
where
(3) 
I arbitrarily set and .
For our purposes, the shortrange potential may be anything; I made one up. The whole point of effective theories is that we can systematically design them directly from lowenergy data, with no knowledge of the shortdistance dynamics. Thus the form of is irrelevant to our analysis. To underscore this point I will not reveal the functional form of the I used. In what follows we need only know that it has finite range.
Such shortrange interactions are common in real Coulombic atoms. The effect of the proton’s finite size on the hydrogen atom’s spectrum is an example. Another is the weak interaction between the electron and proton, which generates very shortrange potentials.
Having chosen a particular , I wrote a simple computer code to numerically solve for the radial wavefunctions of energy eigenstates. I used this to compute a variety of binding energies, phase shifts and matrix elements. Some of the state binding energies are listed in Table 1. Note that the energies would have been given by , with , had there been no . Thus the 1 energy is more than doubled by the shortrange potential; is not a small perturbation. Also the shortrange nature of is evident in this data since the energies approach the Coulombic energies for the very lowenergy, large states.
Sample wave phase shifts are given in Table 2. These phase shifts depend upon the radius at which they are measured because of the long Coulomb tail in the potential ; I arbitrarily chose for my phaseshift measurements, this being much larger than the Bohr radius, , of my atom, and also much larger than the range of . (Alternatively, one can compute the phase shift with and without , and take the difference; the Coulomb divergence cancels in the difference.) I also computed for several states, as well as the wavefunctions at the origin; these are tabulated in later sections.
level  binding energy  level  binding energy 

1  1.28711542  6  0.0155492598 
2  0.183325753  …  
3  0.0703755485  10  0.00534541931 
4  0.0371495726  20  0.00129205010 
5  0.0229268241 
energy  phase shift  energy  phase shift 

0.000421343353  .03  1.232867297  
0.133227246  .07  0.579619620  
.001  1.319383451  .1  1.156444634 
.003  0.900186195  .3  0.106466466 
.007  0.146570028  .7  1.457426179 
.01  0.654835316  1  1.160634967 
The numerical analysis required to generate such data is minimal. The computation can easily be done using standard numerical analysis packages or symbolic manipulation programs on a personal computer. I strongly suggest that you invent your own and subject your own synthetic data to the following analysis.
Our challenge is to design a simple theory that reproduces our lowenergy data with arbitrarily high precision. We must do this using only the data and knowledge of the longrange structure of the theory (that is, we are given and ).
2.2 A Naive Approximation
A standard textbook approach to modelling our data is to approximate the unknown shortrange potential by a delta function, whose effect is computed using firstorder perturbation theory. The approximate hamiltonian is
(4) 
where is a parameter. Using firstorder perturbation theory, the energy levels in our approximate theory are
(5)  
Our approximation has only a single parameter, . This we must determine from the data. We do this by fitting formula (5) to our lowestenergy data, the 20 binding energy; this implies . We use the lowestenergy state because that is the state for which the replacement is most accurate.
In Figure 1 I show the relative errors in several wave binding energies, plotted versus binding energy, obtained using just a Coulomb potential () and using our approximate formula, Eq. (5), with . Both approximations become more accurate as the binding energy decreases, but adding the delta function in first order gives substantially better results. Our approximation to is quite successful.
The limitations of this approximation become evident if we seek greater accuracy. We have made two approximations. First we used only firstorder perturbation theory. There will be large contributions from second and higher orders in perturbation theory if the shortrange potential is strong. So we might wish to compute the secondorder contribution to :
(6) 
Unfortunately this expression gives an infinite shift; the sum over scattering eigenstates diverges as scattering momentum . The delta function is too singular to be meaningful beyond firstorder perturbation theory.
The second approximation is replacing by a delta function. The nature of this approximation can be appreciated by Fourier transforming . Since has a very short range, its transform depends only weakly on the momentum transfer . Thus we might try approximating by the first few terms in its Taylor expansion when calculating lowenergy matrix elements:
(7) 
This is an economical parameterization of the shortdistance dynamics since it replaces a function, , by a small set of numbers: , . Keeping just the first term is equivalent to approximating by a delta function. The additional terms correspond to derivatives of a delta function, and correct for the fact that the range of is not infinitely small — the expansion is in powers of times the range. This suggests that we might improve our approximation by adding a second parameter, again to be tuned to reproduce data:
(8) 
Unfortunately the additional firstorder shift in , , is infinite. Again the delta function is too singular at short distances for such matrix elements to make sense.
Our first attempt to refine the approximate model has failed. We replaced the true, but unknown shortdistance behavior of the potential by behavior that is pathologically singular as . Conventional wisdom, often to be seen in books and even recent conference proceedings, is that our approximations must be abandoned once infinities appear, and that only knowledge of the true potential can get us beyond such difficulties. This wisdom is incorrect, as I now show.
2.3 Effective Theory
The infinities discussed in the previous section are exactly analogous to those found in relativistic quantum field theories. Among other things, they indicate that even lowenergy processes are sensitive to physics at short distances. Modern renormalization theory, however, tells us that the lowenergy (infrared) behavior of a theory is independent of the details of the shortdistance (ultraviolet) dynamics. Insensitivity to the shortdistance details means that there are infinitely many theories that have the same lowenergy behavior; all are identical at large distances but each is quite different from the others at short distances. Thus we can generally replace the shortdistance dynamics of a theory by something different, and perhaps simpler, without changing the lowenergy behavior.
The freedom to redesign at shortdistances allows us to create effective theories that model arbitrary lowenergy data sets with arbitrary precision. There are three steps:

Incorporate the correct longrange behavior: The longrange behavior of the underlying theory must be known, and it must be built into the effective theory.

Introduce an ultraviolet cutoff to exclude highmomentum states, or, equivalently, to soften the shortdistance behavior: The cutoff has two effects. First it excludes highmomentum states, which are sensitive to the unknown shortdistance dynamics; only states that we understand are retained. Second, it makes all interactions regular at , thereby avoiding the infinities that plague the naive approach of the previous section.

Add local correction terms to the effective hamiltonian: These mimic the effects of the highmomentum states excluded by the cutoff in step 2. Each correction term consists of a theoryspecific coupling constant, a number, multiplied by a theoryindependent local operator. The correction terms systematically remove dependence on the cutoff. Their locality implies that only a finite number of corrections is needed to achieve any given level of precision.
We now apply this algorithm to design an effective hamiltonian that describes our data.
To begin, our effective theory is specified by a hamiltonian,
(9) 
where the effective potential, , must become Coulombic at large : , with , for large . We also need an ultraviolet cutoff. I chose to introduce a cutoff into the Coulomb potential through its Fourier transform:
(10)  
where
(11) 
is the standard error function. The new, regulated potential is finite at , but goes to for . It inhibits momentum transfers of order or larger. The exact form of the cutoff is irrelevant; there are infinitely many choices all of which give similar results.
It is not really necessary to regulate the Coulomb potential since it is only mildly singular at the origin. Nevertheless, in many applications, it is still a good idea. For example the potential and the wavefunctions are analytic at if we use the regulator described above. Numerical techniques are often much more accurate or convergent for analytic functions.
Shortdistance dynamics is explicitly excluded from the effective theory by the cutoff. We mimic the effects of the true shortdistance structure by adding local correction terms. As discussed in the previous section, the lowmomentum behavior of any shortrange potential is efficiently described in terms of the Taylor expansion in momentum space. Transforming back to coordinate space gives a series that is a polynomial in the momentum operator multiplied by a delta function. We need an ultraviolet cutoff to avoid infinities, and therefore we smear the delta function over a volume whose radius is approximately the cutoff distance . I chose a smeared delta function defined by
(12) 
but, again, the detailed structure of this function is irrelevant; other choices work just as well.
Remarkably, the structure of the correction terms is now completely determined, even though we have yet to examine the data. The effective potential must have the form
(13)  
where coupling constants , , …are dimensionless. It consists of a longrange part together with a series of local “contact” potentials. The contact terms are indistinguishable, to a lowmomentum particle, from the the true shortdistance potential, provided the coupling constants are properly tuned. The potential is nonrenormalizable in the traditional sense, but that is not a problem here since the cutoff prevents infinities.
Generally the effective potential reflects the symmetries of the true theory. Here our effective potential is rotationally invariant because our data are. When the data are not rotationally invariant, additional terms like must be included in , where now the coupling constants include vectors, tensors … that characterize the rotational asymmetries of the true theory.
One might worry that our Taylor expansion of the shortdistance dynamics would fail when we computed physical quantities like the scattering amplitude. This is because highmomentum states affect even lowenergy processes through quantum fluctuations. For example, in the true theory the scattering amplitude is
(14) 
The sums over intermediate states in secondorder and beyond include states with arbitrarily large momentum. A momentumspace Taylor expansion of the potential would not converge in matrix elements involving such states. Furthermore when we replace the exact potential by our effective potential , our ultraviolet cutoff in effect limits the sums to states with momenta less than the cutoff ; contributions from highmomentum intermediate states apparently are completely absent from our effective theory. Our effective theory is saved by the fact that highmomentum intermediate states are necessarily highly virtual if, as we assume, the initial and final states have a low energy. By the uncertainty principle, such states cannot propagate for long times or over large distances. Thus any contribution from these states is very local and can be incorporated into the effective theory using the same set of (smeared) deltafunction potentials already included in . In this way, the highmomentum states that are explicitly excluded, or badly distorted, by the cutoff are included implicitly through the coupling constants. Note that this means the coupling constants in our effective theory depend nonlinearly on the true potential .
While the true theory is obviously independent of the value of the cutoff , results computed in the effective theory are only approximately independent. The residual dependence in such a result is typically a power series in , where is the characteristic momentum associated with the process under study (for example, the initial momentum or the momentum transfer in a scattering amplitude). The contact terms remove these dependent errors orderbyorder in . Thus, for example, the term in removes errors of order and is the most important correction. Errors of order are removed by the terms, of which there may be only two since there are only two independent ways of combining two momentum operators with the smeared delta function. Obviously only a finite number of contact terms is needed to remove errors through any finite order in .
The coupling constants vary with since they must account for quantum fluctuations excluded by the cutoff. More or less is excluded as is increased or decreased, and therefore the coupling constants must be adjusted to compensate. They are said to be “running coupling constants.” When the shortdistance potential is weak, the coupling to highmomentum intermediate states is weak and the coupling constants change only slowly with .^{2}^{2}2This behavior is modified if the cutoff distance is very large. When is larger than the range of the longrange potential, or when there is no longrange potential, the coefficients of the contact operators tend to go to a constant. Thus for example, in will decrease like as increases so that the coefficient of the delta function operator becomes constant. On the other hand, the coupling constants are often very cutoff dependent when the shortdistance interactions are strong. The dependence of the effective theory on its coupling constants becomes highly nonlinear in this case. In fact this behavior can be so strong that the relative importance of different contact terms can change: an operator can act more like an operator, and vice versa. The physical systems I discuss here do not suffer from this complication, although it is easy to study the phenomenon by modifying the synthetic data appropriately.
Highly nonlinear behavior also results when the cutoff distance is made too small. From the previous section we know that taking is a bad idea. In general it makes little sense to reduce below the range of the true potential. By assumption, our data involves energies that are too low — wavelengths that are too long — to probe the true structure of the theory at distances as small as . When , highmomentum states are included that are sensitive to structure at distances smaller than . But the structure they see there is almost certainly wrong. Thus taking smaller than cannot improve results obtained from the effective theory. In fact, as the nonlinearities develop for small ’s, results often degrade, or, in more extreme cases, the theory may become unstable or untunable.
Finally I should comment on the physical significance of the cutoff. A common practice in applications of potential models, for example to nuclear or atomic physics, is to use a cutoff like our’s to account for some physical effect, for example finite nuclear size. Then one must worry about whether the functional form of the cutoff is physically correct. We are not doing this here. Our cutoff is only a cutoff; physical effects like finite nuclear size are incorporated systematically through the contact terms. We need never worry, for example, about what the true charge distribution is inside a nucleus. Indeed this is the great strength of the effective theory. The effective theory gives us a simple, universal parameterization for the effects of shortdistance structure. The form of the contact terms is theory independent. Only the numerical values of the coupling constants , …are theory specific; for our (lowenergy) purposes, everything that we need to know about the shortdistance dynamics of the true theory is contained in these coupling constants. As discussed earlier, this situation is conceptually similar to a multipole analysis of a classical field. The couplings here are the analogues of the multipole moments, while the deltafunction potentials are analogous to the multipoles that generate the field.
The exact form of the cutoff, and the exact definition of the smeared delta function are irrelevant. I encourage you to make up your own versions of these and repeat the analysis given here. You will get similar results for binding energies and phase shifts, although all of your coupling constants will probably be different, and possibly quite different.
2.4 Tuning the Effective Theory; Results
We now tune the parameters of our effective theory so that it reproduces our lowenergy data through order . For simplicity, we examine only wave properties. Thus we need only the and terms in , Eq. (13); we can drop the term in since it is important only for wave states.^{3}^{3}3It is obvious that the term couples only to waves in the limit . When is nonzero, however, this term has a small residual coupling to waves. This results in an interaction of order , which may be ignored to the order we are working here. One can easily design contact terms that contribute only to a single channel in orbital angular momentum. For example, our smeared delta function could be replaced by a local potential that is separable.
To generate the results discussed here I first chose a reasonable value for the cutoff distance: to begin with, , the Bohr radius for the Coulombic part of the interaction. Then I varied the coupling constants and until the wave phase shifts from the effective hamiltonian agreed with the data at energies and . I found and . Having found the coupling constants, I then generated binding energies, phase shifts, and matrix elements using the effective theory. I also generated results with only the correction (); the effective theory requires a different value of in this case, since the two contact terms affect each other.
I tuned the couplings using phase shifts at very low energies. This was to minimize the effect of the errors, which arise because we truncated our effective potential at order . These “truncation” errors are smallest for the lowestenergy data, since they are generally proportional to raised to some positive power. In general one should use the most infrared data available when tuning. Alternatively one might attempt a global fit to all of the data; but then it is crucial to give greater weight in the fit to lowenergy data than to highenergy data: some estimate of the error due to truncation must be added to the experimental errors used to weight the data in the fit.[2]
So long as , is a simple, nonsingular function of . Consequently the effective theory is simple to solve numerically. To compute the results here, I reused the code that I wrote to generate the synthetic data, but with the true potential replaced by the effective potential . The effective theory is no harder to solve than the other; in particular, because of the cutoff, there are no infinities.
In Figure 2 I compare the errors in the binding energies obtained from the tuned effective theory with those obtained in Section 2.2 from firstorder perturbation theory. Even with only the correction there is a sizable reduction in errors, due to contributions from second and higher orders in the correction — our numerical solution of the effective theory is nonperturbative, and so includes all orders. The accuracy at low energies is further enhanced by the correction, which begins to account for the finite range of the shortrange interaction. The crossing of the curves for the and theories around has no significance; the error in the theory changes sign there, passing through zero.
The phase shifts tell a similar story; see Figure 3. At low energies, the errors decrease steadily as the correction terms are added, orderbyorder in . The slope of the error curve changes as each new correction is added, getting steeper by one power of each time the order of the error is increased by , as expected. Similar behavior is apparent in the previous figure, which shows the errors in the binding energies.
Both of these figures show that the correction terms have little effect at energies . At these energies, the particle’s wavelength is sufficiently short that the particle can probe the detailed structure of . Effective theories are useless in this limit. To go much beyond in this example, one must somehow uncover the true shortdistance structure of the theory.