This week I have been explaining quantum tunneling in quantum mechanics and quantum field theory for my class. It is a fun subject and when I was educated in it a few details were left out. Happily nowadays its easier to find that information. I’m particularly happy with the description of this set of phenomena in the book by Tom Banks: “Modern Quantum Field Theory” , where he actually goes very carefully about how instantons and such are related to the WKB approximation in quantum mechanics, so that one has a rather strong intuitive sense of how these ideas interpolate between the abstract Euclidean formulation of the path integral and a phenomenon that we can understand in other ways.
The rest of this article is a rant on technical details without equations. Now that I have advertised where I’m getting my information from, I could stop here. However…
There are in the end some kinematic factors that end up missing from most discussions. For example, a tunneling amplitude (or more precisely the Euclidean action) does not automatically take into account the dimensional part of the answer: a tunneling amplitude should be converted to a tunneling rate, or to an effective term in the Hamiltonian that encodes the new phenomena that tunneling brings with it. This is what you would do if you are calculating nuclear decays by using tunneling estimates.
These extra factors are usually hard to compute but they are subleading when one takes the logarithm. However, knowing that they have units lets one fix them by ‘hand’ so to speak.
The most typical thing that ones needs for such physics is a flux of incident particles, or a sufficiently similar idea. If one starts in a vacuum that can be described as a harmonic oscillator with frequency f, the parameter f tells us how often a classical particle in such an oscillator would hit the wall. Using Plancks constant this can be turned also into an energy scale to put in a Hamiltonian for tunneling between two ground states. This is a semiclassical reasoning: the calculation is essentially governed by putting together classical solutions to euclidean equations of motion plus a classical analysis of a stable configuration. Basically one is using f to determine the flux of incident particles in a classical situation with one particle only.
In quantum field theories one always gets that instantons tunnel between configurations, but many times the configurations will have different topology. They lead to a rate of decay per unit volume in a vanilla scenario. If the theory is simple there is essentially only one mass scale in the problem, and this mass scale can be used to fix the dimensionality. In setups with fermions, the change of topology brings with it fermion zero modes.
These zero modes tell us that in the tunneling process one changes the number of fermions, so one does not get a vanilla answer. Instead one gets a rate of production of fermions that violate some classical conservation law with corresponding form factors from the extended nature of the configuration in Euclidean space.
In real calculations these factors arise from summing over the moduli of the instantons. This is what leads to effects that are proportional to the volume times time. This is why one ends up getting a well defined calculation as a rate per unit volume. The proper normalization depends on a complicated one loop calculation which is why this is usually left out. My problem right now is finding a place where some of these one loop calculations are done in such a way that it would take me less than an hour to explain such a computation.
If someone out there knows one such calculation that can be done in a reasonable amount of time, I would be very grateful for the tip.