theories and methods originally developed by physicists in order to solve problems in economics, usually those including uncertainty or stochastic elements and nonlinear dynamics. Its application to the study of financial markets has also been termed statistical finance referring to its roots in statistical physics.
For more background, see this site or this blog. The payoff would be similar to the case of neuroeconomics--if you can link the economic problem to an analogous problem in the natural sciences that has been more thoroughly investigated, then the results of those investigations can be brought to bear in the economic problem as well. It may not be the most promising avenue of research, but academia thrives on experimentation and risk-taking in the realm of ideas.
The January 2008 issue of the Journal of Economic Dynamics and Control is a special issue on "Applications of Statistical Physics in Economics and Finance." In their introduction, J. Doyne Farmer and Thomas Lux discuss some of the reasons why the field has been slow to catch on among "mainstream" economists:
The contact between econophysics and economics has, however, been hampered by several factors. The very different culture of scientific publishing in physics and economics has generally prevented publications from econophysics in economics journals. This is partly a matter of style of presentation, but it also reflects fundamental differences in the epistemology of the two fields, in particular different views about the objectives of science. Physicists have a very different view about how work should be presented, and in particular about mathematical rigor (which they generally disdain). In addition, physics has a laissez-faire attitude about publication, believing that it is better to err on the side of letting as many new ideas in as possible, and to let the market eventually decide what is good and what is bad through a Darwinian process that selects what is useful and forgets what is not. As a result there are many econophysics papers of poor quality, which shocks economists. When combined with the fact that the best econophysics papers are published in journals that most economists never read, this body of work remains almost unknown outside the sphere of econophysics.
Communication between physicists and economists has been poor. Physicists are perhaps the only group of scientific professionals who are even more arrogant than economists, and in many cases the arrogance and emotions of both sides have been strongly on display. Many physicists have given the impression that they think that economists know little or nothing about their business, at the same time that they are asking for admission into their club. Many economists have reacted with apprehension to what they view as an attempted invasion by aliens, and have scornfully rejected any work by physicists out of hand, without bothering to have even a passing familiarity with it.
There seems to be a lot of truth in that assessment, and perhaps some of it is also applicable to the field of neuroeconomics as well. If you are interested in the links between economics and the sciences, the first article in that special issue, "Classical Thermodynamics and Economic General Equilibrium Theory," by Eric Smith and Duncan K. Foley, seems to make progress on establishing the parallels across economics and the relevant natural science. (See this working paper if you cannot access the journal directly.)