Originally posted by twhitehead
Or more generally:
https://en.wikipedia.org/wiki/Energy%E2%80%93momentum_relation
I believe that in quantum mechanics, the kinetic energy (mass x c^2 ) can be negative.
[youtube]B4VSqAB_JcU[/youtube]
No, kinetic energy cannot be negative. The energy is the timelike component of the four momentum. One finds this using a generalisation of Pythagoras' theorem. One the Euclidean plane the distance from the origin to some point with Cartesian coordinates x and y is given by:
d^2 = x^2 + y^2
So, given x and y, there are two values for d given by the positive and negative square roots. In normal geometry one would take the positive square root. But in the mass energy relationship and setting the speed of light to 1 we have:
m^2 = E^2 - p^2
Here m is the length of the vector, notice that we get a minus sign in front of the space like component of the four momentum. So there is an additional complication. If the four momentum is space like, corresponding to a faster than light particle, the mass is imaginary. Restricting ourselves to slower than light particles we can choose the comoving frame and set p = 0. This gives us:
E = +/- m
In classical physics one simply ignores the negative root. In quantum mechanics the situation is less simple. Using the naïve procedure one obtains something called the Klein-Gordon equation, but it predicts negative probability densities which meant it was rejected. This spurred Dirac to produce his equation, which predicts a four component wavefunction called a spinor, as the name suggests it naturally describes particles with spin and is the correct description for electrons. The probability densities are positive but the energy can still come out negative. His interpretation of this was to have the vacuum filled with negative energy particles, if one of the negative energy particles were excited into a positive energy state then it would leave a hole which would behave like a positive mass electron with positive charge. So in quantum theory these negative energy solutions correspond to positrons.
This isn't quite the end of the story because we have E^2 = m^2. So we could have both the mass and energy negative. This is being resolved at CERN in experiments on bunches of anti-atoms to see how they respond to gravitational fields. My feeling is that it's positive semi-definite and they won't see anything unusual, but physics is an empirical subject and it's experiment that will drive this.