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?
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2 Answers
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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.
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$\begingroup$ $i^4=1$ contains two assignments:$i^2=1$ and $i^2=-1$. We may supress the first assignment by the power of definition.What happens if we don't do that and accept a mathematical formulation where $i^2=1$ could be used in certain situations?[This is a speculative notion and I am not staking a claim]. Regarding negative mass:The quantum mechanical mass of a particle may be negative . The quantum mechanical mass of an electron may be negative--this information is well known. We might have an analogous situation with the neurtrinos[ a speculative suggestion] $\endgroup$ Commented Dec 31, 2011 at 8:05
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2$\begingroup$ @Anamitra Palit: this is a valid (but unusual) algebraic idea--- people generally do this sort of thing in physics by using the Pauli matrix $\sigma_x$ which is (0,1;1,0), which is separate from "i", which can be taken to be (0,-1;1,0) or $i\sigma_y$ (this is a real matrix). You find both objects used all over. Mathematically, "i" is more elegant because it produces a division algebra which is algebraically complete. Neither is useful for the neutrino, because tachyonic neutrinos don't explain OPERA. $\endgroup$ Commented Dec 31, 2011 at 10:43
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$\begingroup$ @David Zaslavsky:The Quantum Mechanical mass of a particle is given by:$m*=\hbar^2(\frac{d^2 E}{dk^2})^{-1}$.This may have a negative value and has nothing to do with antiparticles.The negative value of QM mass finds use in the band theory of semiconductors. How would you relate this to the non-observance of negative mass in your answer? $\endgroup$ Commented Dec 31, 2011 at 11:40
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$\begingroup$ @Ron Maimon:If i=+1 [in a restricted manner]there is a possibility of spacelike intervals getting converted to timelike ones.How would you take that?We may write:$i^2 \times i^2 \times i^2 \times i^2\times i^2 \times i^2 $=(-1)*(-1)* (-1)*(-1)*(-1)*(-1)=1$=> $i^12=1$=> $[i^12]^{1/4}=(1)^(1/4)$=>$i^3=[1,-1,i,-i]$ =>$i^2\times i=[1,-1,i,-i]$ There is a possibility of i=1 in this formulation $\endgroup$ Commented Dec 31, 2011 at 12:07
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$\begingroup$ @Anamitra: If you set $i^2 = 1$ but $i \ne 1$, then you have simply given a new name to $-1$... Note that for such a "definition": $(i-1)(i+1) = i^2 - 1 = 0$. So $i=\pm 1$. Also $i$ is usually not defined by $i^4 = 1$ (or rather can not be defined in such a way), but to be a solution to $X^2 + 1 = 0$ (which itself has some degeneracy, but not in an essential way). $\endgroup$– SamCommented Jan 1, 2012 at 8:57
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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.
