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B String theory, Calabi–Yau manifolds, complex dimensions

  1. Feb 14, 2018 #21

    bhobba

    Staff: Mentor

    Sure - that's one thing string theory suggested - I think it was Susskind that was a vocal proponent it was what string theory was trying to tell us.

    Thanks
    Bill
     
  2. Feb 14, 2018 #22

    bhobba

    Staff: Mentor

    In standard QM its 3D Euclidian - in QFT its 4D Minkowskian.

    That's utter gibberish like the rest of your post. As a mentor I can assure you if you keep it up action will be taken. To avoid that I STRONGLY suggest you learn some actual physics.

    Their is nothing at all strange about light waves. Its simply an elementary solution to Maxwell's equations in free space which in the Lorentz gauge is ∂u∂uAv =0 - its a very basic exercise in partial differential equations to show the simplest solutions are waves. Try it.

    Rest of misconceptions mercifully snipped.

    At the moment just a friendly warning - we generally only accept statements that can be backed up by peer reviewed papers or reputable textbooks. What you have been posting is, to be blunt, incoherent wild speculation. If you keep it up I can assure you stronger action will be taken.

    Thanks
    Bill
     
  3. Feb 14, 2018 #23

    bhobba

    Staff: Mentor

    Since EM has been mentioned in this thread lets see just whats really going on at a slightly non-rigorous level, but somewhat above B level. Have a look at good old F=ma. We will generalize this a bit to get something I call J and have it equal to k(d^2/dt^2)A where k is some constant and like J will not say anything yet about A. Now we will assume something about J - it is a 4 vector so you get Ju = k (d^2/dt^2) Au. This does not conform to the requirement of relativity - it should have the same form in all frames - the means (d^2/dt^2) should be ∂v∂v (of course the v should be raised in the second ∂v but I dot want to go to the trouble of Latex). The equation then is Ju = k ∂v∂v Au. We will assume Ju whatever it is is conserved ie obeys the equation of continuity ∂uJu = 0. This means ∂v∂v∂μAu = 0. To make this always true we will take ∂μAu = 0. The is called the Lorentz Gauge condition. Au is called the vector potential and Ju is called the 4 current. These are Maxwell's equations in the so called Lorentz guage - given special relativity they are not hard to come up with.

    K simply determines units so we will take it as 1 for simplicity.

    The equations then becomes ∂u∂u Av - ∂u∂vAv = Jv (1) since ∂vAv = 0.

    Take this equation and we notice something. If we substitute A'v = Av + ∂vΛ, where Λ is any function of the xi, for Av then we get exactly the same equation in A'v. In fact if we take ∂v∂vΛ= -∂vAv we have a differential equation in Λ, solve it for Λ, and you then get the Lorentz Gauge condition. Take (1) as our equation and we see it has this very interesting gauge symmetry. Its an example of whats called U(1) symmetry. I am not 100% who exactly showed this symmetry is in fact the rock bottom essence of EM - it may be Weyl or Schwinger or someone else - but physicists now know its really the essence of Maxwell's Equations.

    What I wrote is in some sense a derivation, but its validity depends on what went into it. It really is just fooling around with equations and seeing what you get - Dirac did this sort of thing a lot. Mostly it leads nowhere - but sometimes its reveals something quite interesting like here.

    Now how do we get the usual form of Maxwell's equations.

    Simply define Fuv = ∂uAv - ∂vAu and you get the normal electric and magnetic fields:
    https://quantummechanics.ucsd.edu/ph130a/130_notes/node451.html

    Forget the C; we will, for simplicity work in units C=1.

    We have ∂v∂u Fuv = 0 as can be easily seen. And ∂uFuv = Jv so, as we assumed ∂vJv =0 ie the 4 current is conserved. This is how the field equations are arrived at but how do charged particles move? For this we use Lagrangian Field Theory. Lagrangian's take the form of Lp + Li + Lf. Here Lf is the Lagrangian of the field. Li is the Lagrangian of interaction between the field and charged particle and Lp is the Lagrangian of a particle. We have the equation of the EM field - that's the equation when you have no charge - just the field and it gives ∂uFuv = 0. To get Jv we have the interaction term which must be -JvAv. You add that to the particle Lagrangian and you get how a charged particle moves ie the Lorentz Force law. The gory detail can be found here:
    http://www.phys.ufl.edu/~pjh/teaching/phy4605/notes/chargelagrangiannotes.pdf

    This is a beginner thread, and what I wrote is at least I level. Don't be worried if its above your level - that's not why I posted it. It was simply to show there is no mystery in Electromagnetism - it is so well understood you can almost pull it out of nothing.

    What string theory deals with are not issues like why we have negative charge - that isn't really a worry - string theory deals with much deeper issues - such as quantum gravity or like was mentioned before - suggesting that we have a massive number of spaces string theory can be compacted into. We live in a world that simply has one - we have to have one - the exact one we got isn't important as far as explanation is concercerned.

    Thanks
    Bill
     
    Last edited: Feb 15, 2018
  4. Feb 15, 2018 #24
    I am not sure what you are searching for but if you look at Holonomy group generated by covariant constant spinor, it gives you SU(N) (N=3 for the case you are considering) rather than SO(N). So the notion of Complex space rather than Real space need to be used. An even dimensional real manifold does not necessarily translates to Half the real dimension Complex manifold, in fact there would be a proper subset of Real manifolds which will translate to Complex Manifolds.
     
  5. Feb 15, 2018 #25
    Bill and others knowledgeable in superstring...

    I googled a lot about "superstrings without supersymmetry".. some come out but it is either not indepth or very complex.. even search within PF didn't produce a lot on this important viewpoints. So I picked up the dusty old book Not Even Wrong by Peter Woit and he gave some interesting mathematical history but some details were not given (my questions below). First to set the tone, Woit wrote that:

    "Another reason for being interested in supersymmetry was the hope that it might help with the problem of constructing a quantum field theory for gravity. One of the main principles of general relativity is what is called 'general coordinate invariance', which means that the theory doesn't depend on how one changes the coordinates one uses to label points in space and time. In some sense, general coordinate invariance is a local gauge symmetry corresponding to the global symmetry of space and time translations. One hope for supersymmetry was that one could some make a local symmetry out of it. This would be a gauge theory and might give a new version of general relativity, hopefully one whose quantum field theory would be less problematics."

    I'd like to know:
    1. Superstring theory without supersymmetry is supposed to be called bosonic string theory.. can't you create pure string theory without supersymmetry and supergravity, why?
    2. If a theory can produce bosons and fermions without strings and supersymmetry.. would it be viable? what BSM programmes do this (advanced version of LQG)?
     
  6. Feb 15, 2018 #26
    With your help, to understand more!

    So can I say that complex manifolds are a more diverse and rich group of manifolds then real manifolds?

    Thanks!
     
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