Potential barriers in quantum field theory23 July 2014

For particles confined in potential wells, we would like to compare the rate for a particle to tunnel from one well to another with the strength of the interactions between particles confined in separate wells. Specifically, we would like to know whether there is a regime in which the time scale for the interaction to induce an O(1) phase shift can be parametrically small compared to the tunneling time.

One way to achieve such a separation in time scales would be to consider a model with two particle species, one heavy and one light, where the heavy particles are confined to the wells, and exchange of the light unconfined particles induces the phase shift. But before we resort to that option, we would like to see whether phase gates much faster than the tunneling time can be achieved in a model with just a single particle species.

In the case of phi^4 theory, the interaction arises from an exchange of two particles across the barrier between the wells, while tunneling is a single-particle process. We therefore expect the gate time to be much longer than the tunneling time when tunneling is a rare process. It seems more promising to include a phi^3 term, since in that case an interaction can be induced by exchange of a single particle. We expect in that case for the tunneling and the phase shift to be suppressed by similar exponential factors, so it will be hard to achieve a parametric separation, but let's do the calculation anyway to see what happens.

We'll compute to leading order in the phi^3 coupling but to all orders in the position-dependent source term which produces the confining potential.

To be consistent with Stephen's notation, we will consider a model with Lagrange density

Before we do the scattering calculation we consider tunneling. The first step is to relate J(x) to an external potential V(x) seen by the phi particle in the nonrelativistic limit. To find the relation between J(x) and V(x) we match the Born series for potential scattering to the Feynman diagram series for scattering in the field theory.

For a review of potential scattering and the Born series, see my class notes for Physics 164, Scattering Theory (1982) athttp://www.theory.caltech.edu/~preskill/notes.htmlespecially Chapters 4 and 5.

In Chapter 4 and 5 of the notes I derive the expression for the scattering amplitude in nonrelativistic potential scattering:

Another check on the relationship between J and the nonrelativistic potential:

According to my Scattering Theory notes, page 4.6

Okay ... now we want to compute the effective potential describing the interaction of two particles separated by a barrier. We do this by summing up all the Feynman diagrams to all orders in J, but sticking with lowest order in g. We only have to compute tree diagrams, but there are intermediate momenta to integrate over because J can transfer some momentum. The diagrams are:

In general this dressed propagator is not diagonal in momentum space, because J is not. But suppose we are interested in the case where the scattering conserves momentum, hence k = k', and the total momentum absorbed by the source is zero. Unfortunately, even in that case, although we only need to know the dressed propagator at k = k', it is not sufficient to consider only k = k' for the inverse of the dressed propagator. That is, we need to consider off diagonal matrix elements in momentum space for J in the diagrams, as it is only the total momentum transferred by the source that vanishes, not the momentum transferred at each vertex.

If J really were translation invariant, then it really would just be a shift in the mass squared, and our dressed propagator would become

This is how I would expect the answer to scale with J. By m I mean the mass of the particle to the left or right of the potential barrier, and m* = Sqrt{m^2 + 2J} is the mass inside the barrier. So in effect the particles interact by exchanging particles with mass m*.

In the analysis of tunneling, it was implicit that the wells are wide enough that we can approximate the binding energy by the well depth (I was using the approximation of infinite well width). So the tunneling rate agrees with WKB for a particle of mass m penetrating a barrier with width ell and height J / m.

So it seems like we can't win --- the exponential suppression is stronger for the gate than for the tunneling, becoming comparable in the limit of large J. Stephen reached qualitatively similar conclusions, though the details seem different.

In phi^4 theory we will be even worse off ... the interaction involves the exchange of two particles while tunneling is a one-particle process, so the gate will be even more heavily suppressed compared to the tunneling.

26 July 2014. Summing Diagrams

We are not really interested in computing 2 particle ---> 2 particle scattering here. Instead we want to compute the energy of a state in which particles are trapped in each of two wells, where the wells are separated by distance ell. Because they are confined in the wells, these particles do not have a definite momentum, and because our Lagrangian is not translation invariant, the interaction is not momentum conserving.

For phi^3 theory, we can capture the relevant situation well by considering an external classical source rho(x) coupled linearly to the field, and then ask how the energy depends on the source. (For the phi^4 theory, where the interaction arises from two-particle exchange, we would consider a source coupled to phi^2 instead. That is, we would compute the energy to all orders in J(x) but only to quadratic order in rho(x).)

I used this method on p 2.23 ff of my quantum field theory notes (Physics 205) to compute the Yukawa potential induced by exchange of a massive scalar. In that case the potential energy for sources localized at x and y was a function only of the difference x-y. Here, because we have a fixed background potential (no translation invariance), the potential energy is V(x,y), a function of both x and y. This function is obtained by Fourier transforming in both k and k' the expression I derived for the dressed two-point function in momentum space.

Let's briefly reprise the method:

For static classical sources, the potential energy is just the dressed propagator evaluated in position space, but at zero energy transfer. Of course, in e.g. phi^3 theory, this is just the tree approximation. Loops give the corrections higher order in the scalar self coupling, which we are ignoring for now.

If the theory also has spatial translation invariance, the propagator (and hence the potential) is a function of only the difference of position space coordinates.

(In two dimensions it's a Bessel function.)

So, now in the case where J(x) is static but spatially varying, we want to compute the dressed propagator in position space. Let's try to do this in one spatial dimension. The dressed propagator is

Therefore, in position space it may be feasible to compute all of the diagrams. Suppose x > ell / 2 and and y < -ell / 2. Then since z is in the interval -ell / 2 < z < ell / 2, we have|x - z | = x - z and |z - y| = z - y. That means that the z's in the exponential cancel, and the integrand does not depend on z at all; hence integrating over z just gives a factor of ell.

In higher orders it is more complicated, though. E.g. in the next order we integrate over z1 and z2, and the integrand has nontrivial dependence on the integration variables over a portion of the integration region.

We obtain a power series expansion in J ell / m, with an alternativing minus sign because both the propagator and the vertex contribute a factor of (-i).

As Stephen notes, since we are interested in the behavior for asymptotically large J, it is preferable to compute this Green function exactly, rather than as a power series expansion in J. But let's proceed a bit further with the expansion anyway, to see how it goes.

This term is suppressed by an additional factor of 1 / ( m * ell ) compared to the leading term. This will persist to higher orders in J. The dominant contribution in nth order comes from the interval satisfying z1 > z2 > z3 > .... > zn.This contribution has volume 1/n!. That means the leading terms exponentiate with an alternating sign when we sum up all orders in J. Therefore we will obtain:

With a little more work, I could check that the leading 1 / m*ell corrections as exponentiate in the right way. But it is really better to compute the Green function exactly, as Stephen has described.

When the tunneling rate is small, we can't make the interaction between the particles large in comparison. So we don't have a good way to get an O(1) phase shift without a large tunneling probability.

27 July 2014. The exact Green function

As Stephen points out, the Green function we need was computed by the Wronskian method in the solutions to Frank Porter's problem 5 (pages 8-14 in the problem solutions posted on our Google Doc page).

We need to translate his notation to ours.

Hooray! We did it!

27 July 2014. Why phi^4 theory doesn't help

I'd like to push back some more against Stephen's "ray of hope" remark in his July 22 notes. Regarding the one-loop contribution to the interaction potential in phi^4 theory he says, "many of the virtual particles have lots of energy, even enough to pass right over the barrier without tunneling at all. So maybe the suppression of interaction will be less in that case."

No, that is not how things work. The comment reflects a misunderstanding about the difference between real and virtual particles.

To understand how it really goes, it helps to examine Keith's calculation of the interaction potential in massive phi^4 theory, Appendix E in our long paper:http://arxiv.org/abs/1112.4833Stephen's reasoning might seem to apply in this case, too. There is a "barrier" for sending two particles --- it costs at least energy 2m, and so we find a potential suppressed by the exponential factor exp( -2mr ). Why not say that the virtual particles have enough energy to get over this barrier?

Keith cleverly changes the order of the integral over the Feynman parameter x, which arose in the evaluation of the one-loop diagram, and the integration over q, which arose from Fourier transforming the one-loop scattering amplitude to find the potential. To evaluate the q integral we distort the integration contour into the upper half plane, picking up a contribution from integrating over the branch cut of the logarithm on the imaginary q axis, starting at q = 2im. (In contrast, the leading order diagram in phi^3 theory is a tree diagram, and we only pick up the residue of the pole at q = im.)

Now look at the resulting expression in Eq. (274). Here we are doing a q integral (where the integration variable is the imaginary part of the original q), where there is an exponential factor exp( - qr ), weighted by a two-body density of states factor (because in the one-loop integral only two-particle intermediate states appear). Of course the integral is dominated by the lower endpoint of integration (q = 2m), so we get exp( - 2mr ) times a prefactor.

The only way to avoid that exponential suppression for large mr would be for the density of states to increase exponentially rather than as a power law with q, which never happens in a relativistic quantum field theory.

There are several ways to think about this:

-- There is a larger "barrier" for sending a particle with energy E as E increases, resulting in the exp( - Er ) suppression for crossing the barrier. It's harder, not easier, to exchange higher-energy "virtual" states.

-- More formally, think of the Euclidean (imaginary time) field theory, where the Schroedinger evolution operator for (imaginary) time T is exp( - HT ) instead of exp( -i HT ) as in the real-time theory. If we evaluate the "amplitude" for a state to propagate forward in time by T, summing over all intermediate energy eigenstates, the contribution from an intermediate state with energy E is suppressed by exp( - ET ). But in a Lorentz-invariant QFT, which becomes rotationally invariant in Euclidean space, there is no difference between propagating for time T or for distance r. So states of energy E make a contribution suppressed by exp( -Er ) to the interaction strength.

-- Suppose we have a square potential barrier. Deep inside the barrier (which means a distance from the edges of the barrier large compared to 1 / m ), the potential is constant, and behaves just like a contribution 2J to the mass squared of the particle. So we should be able to use Keith's calculation outside of a thin region near the edge, obtaining the same suppression factor e( -2mr ), but with m replaced bySqrt( m^2 + 2J ).

By that reasoning I knew that the answer had behave likeexp( - r Sqrt( m^2 + 2J ) )for a barrier of width r in phi^3 theory (which is right), and likewise I know that it should beexp( - 2r Sqrt( m^2 + 2J )in phi^4 theory. The tunneling rate still goes like exp( - r Sqrt( 2J ) ), so the interaction strength is hopelessly small in comparison.