Natural Science Forum / Physics / Relativity / January 2006

>  >>   You are trying to think classically here. There is nothing strange
>  >> about neutrino oscillations from the standpoint of quantum mechanics.
>  >> Operators which don't commute are diagnolized by the same states, so
>  >> regardless of what you choose for a set of basis states to diagonolize
>  >> one operator, the other will be a linear combination of those states.
>  >
>  >What I am trying to understand is what it would mean, if it wouldn't
>  >break physics, for a neutrino to oscillate between having mass and no
>  >mass.
>
>   Don't think of it that way. The mass eigenstates have a fixed mass.
> The oscillations between flavor eigenstates determine the relative
> probability of the type of neutrino it will be in a weak interaction.
>
>   By the way, the neutrinos in this model are but one of several
> possibilities. There is also the possibility that a neutrino can
> be its own anti-particle, in which case it's a majorana neutrino.
> Searches for those are conducted primarily by looking for neutrinoless
> double-beta decay (so far without success).
>
>  >This relates to what I wrote below because I feel it would provide some
>  >insight into the nature of mass if we had an example of something that
>  >had mass only sometimes. I'm not sure what insight would be provided,
>  >but I'm certain something would be learned.
>
>   Actually, the ``nature of mass'' is that mass is the m^2 in
> E^2 - p^2 = m^2. For example, consider -=- > (\hbar = c = 1), -=-

>       Which is just the result that e+ + e- -> \gamma\gamma
>       gives you two photons with the energy (m_pos + m_elec).
>              Well, this is not really the same thing,
>              but it illustrates a couple of points when dealing with
> relativist ic theories. First, if you are going to use a hamiltonian, then
> the mass is conserved, even if you end up with two massless particles from
> anihilation of two massive particles. The hamiltonian give you the total
> energy, which at a point in spacetime, happens to the mass of two massive
> particles. Second, the mass is conserved, but is not locally constant. -=-

>  -=-         Now this isn't really what we're talking
>              about with neutrino mixing, since obviously,
>    the mass is locally constant for the mass eigenstates (at least to
>    the extent that one can take the classical limit for the velocity).
>
>   However, this is a fairly involved question, so I located an
> article on neutrino phenomenology for you: hep-ph/9812360. The
> article might be rather dense, material wise,

>          but you can safely skip a number of sections,
>       in particular the sections on the see-saw mechanism
>                and effective lagrangians.

> Most of the stuff that pertains to your question will be in the first
> half of section 2, section 3 and the appendix on majorana neutrinos.
> It would probably be easier to ask questions where you get stuck
> (or since you previously mentioned that you had already looked
> at several articles, which didn't work for you, you could give
> me those references and post questions from those articles, which
> I will try to answer).
>
> [...]
>  >>   The flipping back and forth between velocity states is called
>  >> zitterbewegung. -=-

>  >> If we include the spin and note that fermions are left-handed,
>  >> then fermion must be entirely left-handed througout the
>  >> zitterbewegung. Constructing the correct eigenstates is beyond the scope
>  >> of this response, but the result is that the mass of the fermion is
>  >> obtained from a superposition of the longitudinal spin states.  In
>  >> general, longitudinal polarizations may be associated with particle
>  >> masses.
>  >
>  >It sounds like the neutrino would oscillate back and forth along its'
>  >direction of travel, but with a preferred direction.
>
>   Precisely, except the same thing applies to any massive particle.
>
>  >My first guess before doing a little research though, was that
>  >it travels in both directions at once.
>
>   Not really. It's an artifact of the uncertainty principle.
> A precise measurement of the velocity requires measuring an
> interval \Delta t in the limit that t -> 0, in which case the
> energy is infinite and so any precise measurement of the
> velocity must give the total energy, \gamma mc^2 = infinity,
> which implies that v = +/-c.
>
> [...]
>  >Do we know if there is a trigger for neutrino oscillation or if it
>  >happens "just because" ? (I like "just because" better than "random".)
>
>   None of the above. This is just a feature of quantum mechanics.
> Operators that are diagonal in some set of basis states will
> not be diagonal in a different set of basis states. If two operators
> do not commute, there is _no_ set of basis states which diagonalizes
> both operators (e.g., S_x and S_z). By choosing S_z, you leave S_x
> completely indeterminate and therefore if your particles are all
> in an eigenstate of S_z, a measurement of S_x is going to give you
> +x and -x randomly. Actually, there's a lot about random to like.
> If you imagine that every variable really has infinitely precise values
> and that the uncertainty principle is merely a difficulty in measurement,
> then the information required (by nature) to specify any interaction
> is infinite, etc. If instead you consider the possibility that there
> really is only a finite number of measurements one can perform to
> extract all of the information that exists, then the randomness is not
> very surprising. If the spin can have only two values, then it doesn't
> make any sense to try and measure 2 for each of S_x, S_y and S_z.
> Finding the spin along z necessarily leaves S_x and S_y not well-defined.