How could something have negative mass? With all the theories on how Neutrinos apparently broke the light barrier, there was one theory someone told me of how neutrinos might have less than zero mass, but she didn't explain how this was possible. So how could something have less than zero mass? What would it mean? What would something like that be like?
 A: I think your friend was confusing negative mass with imaginary mass. There are hypothetical particles called tachyons which would travel faster than light and also have mass $m$ such that $m^2 < 0$. As explained in the Wikipedia article, this is the case because the particles' energy has to be real, and
$$E = \frac{mc^2}{\sqrt{1 - \frac{v^2}{c^2}}}$$
When a particle travels faster than light, the denominator is the square root of a negative number, so it's imaginary. The numerator then also has to be imaginary to make the energy real. See also this question on tachyons and others on this site.
Some physicists have investigated how a particle with an actual negative mass might behave, and the results are quite strange. For example, a negative mass would be attracted to a positive mass but would in turn repel the positive mass, so that the two masses would accelerate continuously without any external input. Nothing like this has ever been observed, and there's no reason to expect that negative mass particles actually exist.
A: Imaginary mass (or negative mass squared) would lead to the effect that the potential energy in the Lagrangian has a maximum instead of a minimum which is unstable. So things with imaginary mass are rather called instabilities than particles. For negative mass (not squared) there is the example of the Dirac sea where particles below the sea level turn out to be antiparticles with postitive energy. This can be seen by considering the free space solutions of the Dirac equation where postitive mass solutions are interpreted as particles and negative mass solutions as antiparticles with positive energy.
