it has been said that the electron is the fundamental representation of the Poincare group, with only two conmuting observables, $( \sigma , p_{\mu})$. This question regards what is usually called the superselection rules
what i don't understand why states that are superpositions of different $C$ like $e + \overline{e}$ (and electron and a positron superposition) are supressed and how. What about superpositions like $e + \mu$ (an electron and a muon)? and finally, why neutrino flavors are different and in this regard, can be mixed in superposition?
what about superpositions of particles with different mass? what are the currently accepted mechanisms for this supression? are there other alternatives worth pursuing? are there high-decoherence impedances in some state combinations that make them decohere quicker than others or are they fundamentally supressed?
Answer
The fact that combinations such as $(e^-+e^+)/\sqrt{2}$, (or other combinations such as a proton with an electron) are suppressed is a little mysterious. It falls outside of quantum mechanics. Any "theory of everything" needs to answer this sort of question.
We can't form combinations of electrons with positrons because they have different electric charges (and lepton number). Since electric charge is conserved, the formation of such a combination is impossible. If we could produce such a state, a measurement of it would result in the total charge of the universe changing from an undetermined state to one of two incompatible possibilities. Thus the superselection rule arises from an exact symmetry.
Note that it is possible to create a state where either an electron or a positron is present at a particular point. For example, the electron could have gone left and the positron right, or alternatively, the electron went right and the positron left. Those kind of combinations are possible and we write them as: $(e^-_Le^+_R + e^-_Re^+_L)/\sqrt{2}$. If we make a measurement on this state according to the natural basis, we either end up with the electron or the positron on the right, but the total charge is still zero.
The case of $e^-+\mu^-$, an electron and a muon (both negatively charged but with vastly different masses) allows linear superposition. There is no violation of charge symmetry, and unlike say a proton and a positron, there is not a violation of lepton or baryon number. In this case there is no superselection rule and these combinations can be obtained. They are observed in weak interactions. One sees these combinations in the MNS matrix.
An example of a weak interaction is a $W^-$ particle which decays into a combination of a charged lepton (an electron, muon or tau: $e^-, \mu^-,\tau^-$) and an anti-neutrino (i.e. $\bar{\nu}_1,\bar{\nu}_2,\bar{\nu}_3$). Which of these is produced is determined only by measurement so until then, one has a linear superposition of an electron, muon, and tau state.
For an introduction to the subject see "Neutrino Oscillations for Dummies" by Chris Waltham, Am.J.Phys. 72 (2004) 742-752 http://arxiv.org/abs/physics/0303116
A good question is "how long can superpositions last?" For the case of the superpositions of the anti-neutrinos, observations of anti-neutrinos emitted from the sun indicate that the superpositions last until they are observed on the earth. This is observed as the interference between anti-neutrinos (or neutrinos) with different masses and is usually described as "neutrino oscillation", but can also be described as "neutrino interference" between the three mass eignstates of neutrinos. So in this case at least, the superposition lasts about 8 minutes, but there is no theoretical maximum time for a superposition to last.
Now the mass differences between neutrinos are extremely small, around 0.05 eV or less than a millionth of the mass of an electron. So this isn't a counterexample to Penrose's proposal that quantum measurement results from differences in mass between different superpositions. See "Testing Gravity-Driven Collapse of the Wavefunction via Cosmogenic Neutrinos", Joy Christian, Phys.Rev.Lett. 95 (2005) 160403 / quant-ph/0503001 for a discussion. The abstract:
It is pointed out that the Diosi-Penrose ansatz for gravity-induced quantum state reduction can be tested by observing oscillations in the flavor ratios of neutrinos originated at cosmological distances. Since such a test would be almost free of environmental decoherence, testing the ansatz by means of a next generation neutrino detector such as IceCube would be much cleaner than by experiments proposed so far involving superpositions of macroscopic systems. The proposed microscopic test would also examine the universality of superposition principle at unprecedented cosmological scales.
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