Sunday, April 01, 2007

The Quantum Vacuum, Loops and Lattice Artifacts

This post was written for a general audience, and hence is written in a rather more popular language than our usual fare at Life on the Lattice. If you are familiar with the basic ideas behind perturbative improvement, you may want to skip this post.

When we think about the vacuum in classical physics, we think of empty space unoccupied by any matter, through which particles can move unhindered and in which fields are free from any of the non-linear interaction effects which make e.g. electrodynamics in media so much more difficult.

In Quantum Field Theory, the vacuum turns out to be quite different from this inert stage on which things happen; in fact the vacuum itself is a non-linear medium, a foamy bubble bath of virtual particles popping into and out of existence at every moment, a very active participant in the strange dance of elementary particles that we call the universe.

A metaphor which may make this idea a little clearer could be to think of the vacuum as a sheet of paper on which you write with your pen. Looked at on a large scale, the paper is merely a perfectly flat surface on which the pen moves unhindered. On a smaller scale, the paper is actually a tangle of individual fibers going in all directions and against which the pen keeps hitting all the time, thus finding the necessary friction to allow efficient writing.

In the case where the paper is the vacuum, the analogue of the paper fibres are the bubbles of virtual particle pairs that are constantly being created and annihilated in the quantum vacuum, the analogue of the pen is a particle moving through the vacuum, and the analogue of friction is the modification of the particle's behavior as compared with the classical theory which happens as a result of the particle interacting with virtual particle pairs.

At first sight, this description of the vacuum may appear like wild speculation, but it has in fact very observable consequences. In Quantum Electrodynamics (QED), the famous Lamb shift is a consequence of the interactions of the electron in a hydrogen atom with virtual photons, as are the anomalous magnetic moment of the electron and the scattering of light by light in the vacuum. In fact, none of the amazingly accurate predictions of QED (the most accurate theory we have) would work without taking into account the effects of the quantum vacuum.

In lattice QCD, we care about the vacuum because it affects how the discrete lattice theory relates to its continuum limit. By discretising a continuum theory, we introduce a discretisation error: When comparing an observable Oa measured on a lattice with lattice spacing a with the same observable in the continuum O0, we find that they are related as

O_a=O_0+c_1(\mu a)+c_2(\mu a)^2+\dots

where μ is some energy scale that is typical of the reactions contributing to the observable O. In the classical theory (or at "tree level" as we say because the Feynman diagrams corresponding to classical physics have no loops in them), we can then tune the lattice theory so that as many of the ci as we want to get rid of become zero, and the discrepancy between lattice and continuum becomes small.

At the quantum level, however, we get Feynman diagrams with loops in them that describe how particles traveling through the quantum vacuum interact with virtual particles; the problem with these is that the virtual particles exist at very short distances and hence can have very large momenta by virtue of Heisenberg's uncertainty relation. At very large momenta, the deviation of the lattice theory from the continuum becomes very evident, and hence the loops on the lattice contribute terms that differ a lot from what the same loops would contribute in the continuum. And then we find that this difference reintroduces the a-dependence that we got rid of classically by tuning our theory!

This is clearly no good. What we need to do is to get rid of the a-dependence (up to some order in a) in the quantum theory, too. There are a number of ways how to go about this, but the one most commonly used is called perturbative improvement. In perturbative improvement, we calculate the effect of the virtual particle loops by evaluating Feynman diagrams (a Feynman diagram isn't just a pretty picture: there is a well-defined mathematical expression corresponding to each Feynman diagram) on the lattice and extracting their contribution to the lattice artifacts ci to some order in a. Once we have these contributions, we can then tune our theory again so that these contributions to the ci are cancelled, and the discrepancy between lattice and continuum becomes small again.

Unfortunately, evaluating Feynman diagrams on the lattice is much harder than in the continuum in many ways, so that we need some rather advanced methods to do this, and there aren't very many people doing it. So this is an area where progress has been slow for a while. The next post will tell you how a group of collaborators including myself recently made some pretty significant progress in this field.