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  1. M

    I The problem with preons

    It's the coherence of the mixed state that's the issue, not the individual mass eigenstates. The three mass eigenstates are treated as particles, but nothing holds them together. They're just assumed to remain together to maintain coherence of the mixed state. But the universe is full of fields...
  2. M

    I The problem with preons

    The SM seems to side-step the issue. A basic assumption in the neutrino sector is that the mass eigenstates maintain coherence over very large distances. But no rigorous justification is given for this assumption. How mass eigenstates do this still seems to be an open question...
  3. M

    I The problem with preons

    I'm not advocating the preon models, but I was just wondering if you considered neutrinos as "composite" particles. They appear to be a mixed states of three mass eigenstates. So, a "composite" of three particles even if the particles aren't observable...
  4. M

    I Conserved quantities

    Before I forget, PFers might be interested to know that SO(3,3) space-time has two classes of spin one-half particle. One class of particles has spin angular momentum, as usual. The other class of "particles" has spin angular energy and the following two properties: (1) They are either...
  5. M

    I Conserved quantities

    Nice macroscopic example. I guess I would argue that we may not understand causality at the quantum scale. Take spin angular momentum. We know from Noether's theorem that it is associated with space rotations, but we are told that there is no actual physical rotation in space, it's an internal...
  6. M

    I Conserved quantities

    So, it turns out that fresh_42 was right: mathematically, there is a conserved quantity due to invariance under time rotations (https://doi.org/10.3390/sym12050817). This quantity has the same units of measure as the Planck constant. For lack of a better name, perhaps we can refer to it as...
  7. M

    I Leptons and the Lorentz Group O(3,3)

    Thanks jedishrfu. Just to be clear: O(3,3) space is a mathematical space and the linked article makes no claims about the physics in O(3,3). As mentioned above the mathematical properties and relationships of some group theory algebras in O(3,3) overlap with the mathematical properties and...
  8. M

    I Leptons and the Lorentz Group O(3,3)

    Here's a breakdown of O(3,3): O(3,3) has three O(3,1) subspaces (Minkowski) and three O(1,3) subspaces (associated with one space dimension and three time dimensions). Each of these subspaces is associated with a unique SU(2) x SU(2) subalgebra. A spinor transforms under the spin 1/2...
  9. M

    I Leptons and the Lorentz Group O(3,3)

    This is note about O(3,3) space-time. The related article is: https://doi.org/10.3390/sym12050817 Here's some background: In O(3,1) space-time (Minkowski), the six generators of rotations and boosts can form an SU(2) x SU(2) Lie algebra. This algebra is then used generically by all the...
  10. M

    I Conserved quantities

    I apologize for taking up so much of your time fresh_42. Thanks for all your replies. In case your interested, there is a really good walk through of Noether's theorem for normal space-time in Schwichtenberg, J. : Physics from Symmetry. 2nd ed., Springer 2017 Gotta go. Cheers, Marty4691
  11. M

    I Conserved quantities

    Ok, so we make it a purely mathematical exercise. We create a mathematical environment with two time coordinates and we assume that the conserved quantity doesn't have physical meaning. It seems possible that Noether's theorem might give us a mathematical expression for the conserved quantity...
  12. M

    I Conserved quantities

    I'm going to try and paraphrase your reply. I think you're saying that even if we come up with a conserved quantity via Noether's theorem, it "lives" in O(3,2) space-time and may not be physically defined in an O(3,1) space-time because the conserved value may dependent on two time variables...
  13. M

    I Conserved quantities

    If we just stick to the math, do you know if anyone has figured out Noether's theorem for time rotations?
  14. M

    I Conserved quantities

    Could we add another time dimension to Minkowski space-time to allow the rotation?
  15. M

    I Conserved quantities

    Hi, I have a question and I was hoping for some help. The reasoning goes something like this: There appears to be two fundamental types of coordinates x - space t - time and there appears to be three types of fundamental transformations - translations - rotations -...
  16. M

    Conserved quantity due to invariance under temporal rotations

    In the context of so(3,3), if the above is correct then there might be a second type of "spinor" representation. The normal spinor representation has the form {(J+iK), (J+iK), (J+iK)} where the Js are spatial rotation generators and the Ks are boost generators. The second type of...
  17. M

    Conserved quantity due to invariance under temporal rotations

    Hi, I was looking at the so(3,3) Lie algebra which has 3 temporal rotation generators as well as the normal 3 spatial rotation generators. When I try to use Noether's Theorem to determine what the conserved quantity is, due to invariance under temporal rotations, I seem to get an integral where...
  18. M

    I So(4) and QCD

    Thanks for your help.
  19. M

    I So(4) and QCD

    I guess my question is really: is there redundancy in QCD. In particular, is it possible to use so(4)=su(2)xsu(2) as the gauge group for QCD instead of su(3). I only have a couple of qualitative observations to suggest this speculation. The first is the relationship between su(3) and so(4). If...
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