Lab Notes for a Scientific Revolution (Physics)

March 20, 2008

Derivation of Heisenberg Uncertainty from Kaluza Klein Geometry

For those who have followed my Kaluza-Klein (KK) work, I believe that it is now possible to derive not only intrinsic spin, but Heisenberg uncertainty directly from a fifth, compactified dimension in Kaluza Klein.  This would put canonical quantum mechanics on a strictly Riemannian geometric foundation which — as a side benefit — unites gravitation and electromagnetism.

I need to consolidate over the next few days and will of course make a more expanded post when I am ready, but here is the basic outline.  First, take a look at:

Intrinsic Spin and the Kaluza-Klein Fifth Dimension

where I show how intrinsic spin is a consequence of the compactified fifth dimension.  This paper, at present, goes so far as to show how the Pauli spin matrices emerge from KK.

Next, go to the two page file:

Spin to Uncertainty

This shows how one can pop Heisenberg out of the spin matrices.

Finally, go to the latest draft paper on KK generally, at:

Kaluza-Klein Theory and Lorentz Force Geodesics with Non-linear QED

This lays out the full context in which I am developing this work.  Please note that the discussion on intrinsic spin in the third link is superseded by the discussion thereof in the first link.

More to follow . . .

Jay.

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3 Comments »

  1. previous attempt failed

    I did a Google search on KK a year ago and found myself following this up (struggling more like) via Wikipedia and Physics Forum. Last week I started searching Google again and am delighted to see your work.

    I find your spin idea fascinating and your maths clear. I wish you well.

    I like your use of T and \tau. I think I may have made serious errors in not keeping these ideas clear.

    Relevant comment-1:
    intrinsic-spin-20 page 2 – I thought (see post 6 of ref below):
    g_{\mu,\nu}(5d) = g_{\mu,\nu}(4d) + \kappa^2 A_\mu A_\nu

    I found I needed the \kappa^2 A_\mu A_\nu term to get g(5)_{ab} g(5)^{bc} = \delta_a^c and (in post 15 – in progress) to get otherwise awkward terms to cancel in R1(5) of post 14. Perhaps you omitted this term as it is small. I don’t think this will be a big problem.

    Best wishes, Mike.

    P.S. Irrelevant comment-2:
    I do have a problem with the enormous d x^5 / d\tau = R d \phi / d\tau = u^5 which makes d\phi / d\tau even more enormous.
    But this is not your problem – it is built into KK (which started from GR).

    Where: u^5 = \lambda / \kappa = 10^{21} (approx) \lambda = q/m \kappa = your \bar \kappa and c = 1 = \hbar .

    I got round this by setting u^5 = 1 (as I started from elementary QM). Then I got errors of 10^{21} !!!! (see post 6 especially appendix 3). My struggle is recorded in the Physics Forum thread at http://www.physicsforums.com/showthread.php?t=171945

    (In post 13 I had a stab at reconciling this 10^21 error, and am working out how to tidy this up.)
    (In post 14 I asked for help in getting the sign right in R(5) = R(4) – EM. I think I have solved this. Half texts & papers have one sign, half the other. I don’t think it is to do with the sign of \eta_00, it may just be that some authors are more careful than others. I have prepared answer in post 15 in a word document, but am still mulling it over.)

    Comment by Mike Leggett — March 25, 2008 @ 1:58 pm | Reply

  2. Thanks for converting – I had 2 typos.

    g_{\mu\nu}(5d) = g_{\mu\nu}(4d) + \kappa^2 A_\mu A_\nu \;\;\&\;\; g(5)_{ab} g(5)^{bc} = \delta_a^c

    Comment by Mike Leggett — March 28, 2008 @ 2:04 pm | Reply

  3. Hi Mike, I fixed your typos in the original post. Jay.

    Comment by Jay R. Yablon — March 29, 2008 @ 11:29 am | Reply


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