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?

  • $\begingroup$ To clarify, this is a question about decoherence phenomena? So you're asking why are superselection rules important in decoherence? $\endgroup$ – BebopButUnsteady Jun 28 '11 at 20:03
  • $\begingroup$ no, i'm asking if decoherence is the underlying mechanism in superselection, or if there are other mechanisms at play $\endgroup$ – lurscher Jun 28 '11 at 20:27

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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    $\begingroup$ @Carl Brannen: I simply do not understand what is written here. Saying the superposition $(e+e^-)$ can't be formed because it violates conservation of charged is exactly like saying we can't create a state $|p=0> +|p=1>$ because it violates conservation of momentum. This is absurd. We see particles in states of reasonably well defined position all the time. Properly, we should write $e \otimes |U+> + e^-|U->$ where $|U+>$ is the state of the rest of the universe with an extra positive charge, but so what? Thats what we should always write and we don't. $\endgroup$ – BebopButUnsteady Jun 29 '11 at 16:28
  • $\begingroup$ --cont-- becuase its pointless. Further, by using the language of 'superpositions decaying' you make it sound as though there is something 'bad' about the state the state $e+e^-$ apart from the issues of measurement and decoherence, which was exactly the poster's original confusion. You say that we've measured $e^- + \mu^-$. Please correct me, but i know of no detector that will return a measurement of $e + \mu^-$. There are plenty that will return $e^-$ or $\mu^-$ with equal probability when applied to some decay products. But how is that any different than a decay that leads to -cont- $\endgroup$ – BebopButUnsteady Jun 29 '11 at 16:50
  • $\begingroup$ $e-$ or $e+$ (with other products to conserve charge)? $\endgroup$ – BebopButUnsteady Jun 29 '11 at 16:52
  • $\begingroup$ @Bebop; Re how you get states of uncertain momentum. The situation is analogous to how one obtains states of uncertain spin-1/2. That is, by your version of my argument, such a state should be impossible because it would violate conservation of momentum. If one measures spin along the x-axis, one obtains a state in which the spin measured along the z-axis becomes uncertain. Thus one can obtain such an uncertain state from states that are certain. This cannot be done with charge as it would require a rotation in charge space that cannot be physically realized. $\endgroup$ – Carl Brannen Jun 29 '11 at 18:25
  • $\begingroup$ @Carl Brannen: If a superposition of an electron and a positron is impossible because of the difference in charge, why is a superposition of an electron and a muon allowed as they differ in mass. Isn't that the exact same thing? $\endgroup$ – Kasper Jun 29 '11 at 18:36

The total electric flux at infinity will always decohere states with different charges exactly. This is the cause of the superselection rules.

  • $\begingroup$ so why this doesn't apply to states of different momenta and or spin eigenvalues? they still have different magnetic asymptotic fields, but they still can be superposed $\endgroup$ – lurscher Jun 29 '11 at 15:04
  • $\begingroup$ If the electron had always been moving with a certain 4-velocity in the past, this will lead to superselection rules. However, if the electron only started accelerating to a given 4-velocity some finite time ago, superselection rules no longer apply. $\endgroup$ – QGR Jun 29 '11 at 16:47

Superselection is about relative phases. In the particular case of $e+\bar{e}$, it concerns the U(1) symmetry. Both observables and states have to be invariant under symmetry transformations (for the symmetry to be a symmetry), but $e$ and $\bar{e}$ transform differently under U(1) transformations, so $e+\bar{e}$ cannot be invariant because it transforms to $\mathrm{e}^{i\alpha}e+\mathrm{e}^{-i\alpha}\bar{e}$.

The definition of different particle wave functions is that they transform differently under the symmetries of the system, so we can't take superpositions of different particles by definition. Of course we can construct tensor product states and mixtures.

A separate question, of course, is why one symmetry and not another, but once you introduce a symmetry into quantum mechanics, a lot follows.


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