Lab Notes for a Scientific Revolution (Physics)

February 29, 2008

Lab Note 2: Why the Compactified, Cylindrical Fifth Dimension in Kaluza Klein Theory may be the “Intrinsic Spin” Dimension


I am posting here a further excerpt from my paper at Kaluza-Klein Theory, Lorentz Force Geodesics and the Maxwell Tensor with QED. Notwithstanding some good discussion at sci.physics.relativity, I am coming to believe that the intrinsic spin interpreation of the compacified, hypercylindrical fifth dimension presented in section 4 of this paper may be compelling. The math isn’t too hard, and you can follow it below: The starting point for discussion equation is (3.2) below,

frac{dx^{5} }{dtau } equiv -frac{1}{b} frac{sqrt{hbar calpha } }{sqrt{G} m} =-frac{1}{b} frac{1}{sqrt{4pi G} } frac{q}{m} . (3.2)

which is used to connect the q/m ratio from the Lorentz law to geodesic motion in five dimensions, and b is a numeric constant of proportionality. Section 4 below picks up from this.

Excerpt from Section 4:

Transforming into an “at rest” frame, dx^{1} =dx^{2} =dx^{3} =0, the spacetime metric equation d/tau ^{2} =g_{/mu /nu } dx^{/mu } dx^{/nu } reduces to dtau =pm sqrt{g_{00} } dx^{0} , and (3.2) becomes:

frac{dx^{5} }{dx^{0} } =pm frac{1}{b} sqrt{frac{g_{00} }{4pi G} } frac{q}{m} . (4.1)

For a timelike fifth dimension, x^{5} may be drawn as a second axis orthogonal to x^{0} , and the physics ratio q/m (which, by the way, results in the q/m material body in an electromagnetic field actually “feeling” a Newtonian force in the sense of F=ma due to the inequivalence of electrical and inertial mass) measures the “angle” at which the material body moves through the x^{5} ,x^{0} “time plane.”

For a spacelikefifth dimension, where one may wish to employ a compactified, hyper-cylindrical x^{5} equiv Rphi (see [Sundrum, R., TASI 2004 Lectures: To the Fifth Dimension and Back, (2005).], Figure 1) and R is a constant radius (distinguish from the Ricci scalar by context), dx^{5} equiv Rdphi . Substituting this into (3.2), leaving in the pm ratio obtained in (4.1), and inserting c into the first term to maintain a dimensionless equation, then yields:

frac{Rdphi }{cdtau } =pm frac{1}{b} frac{sqrt{hbar calpha } }{sqrt{G} m} =pm frac{1}{b} frac{1}{sqrt{4pi G} } frac{q}{m} . (4.2)

We see that here, the physics ratio q/m measures an “angular frequency” of fifth-dimensional rotation. Interestingly, this frequency runs inversely to the mass, and by classical principles, this means that the angular momentum with fixed radius is independent of the mass, i.e., constant. If one doubles the mass, one halves the tangential velocity, and if the radius stays constant, then so too does the angular momentum. Together with the pm factor, one might suspect that this constant angular momentum is, by virtue of its constancy independently of mass, related to intrinsic spin. In fact, following this line of thought, one can arrive at an exact expression for the compactification radius R, in the following manner:

Assume that x^{5} is spacelike, casting one’s lot with the preponderance of those who study Kaluza-Klein theory. In (4.2), move the c away from the first term and move the m over to the first term. Then, multiply all terms by another R. Everything is now dimensioned as an angular momentum mcdot vcdot R, which we have just ascertained is constant irrespective of mass. So, set this all to pm {textstylefrac{1}{2}} nhbar , which for n=1, represents intrinsic spin. The result is as follows:

mfrac{Rdphi }{dtau } R=pm frac{1}{b} frac{sqrt{hbar c^{3} alpha } }{sqrt{G} } R=pm frac{1}{b} frac{c}{sqrt{4pi G} } qR=pm frac{1}{2} nhbar . (4.3)

Now, take the second and fourth terms, and solve for R with n=1, to yield:

R=frac{b}{2sqrt{alpha } } sqrt{frac{Ghbar }{c^{3} } } =frac{b}{2sqrt{alpha } } L_{P} , (4.4)

where L_{P} =sqrt{Ghbar /c^{3} } is the Planck length. This gives a definitive size for the compactification radius, and it is very close to the Planck length. (Keep in mind that we will eventually find in (10.14) infra that b^{2} =8, so (4.4) will become R=L_{P} sqrt{2/alpha } .) One point of interest, is that alpha is a runningcoupling. At low probe energies, where alpha to 1/137.036, R=5.853cdot bcdot L_{P} . However, this is just the apparentradius relative to the low probe energy. If one were to probe to a regime where alpha becomes large, say, of order unity, alpha =1 then R={textstylefrac{b}{2}} L_{P} is quite close to the Planck length of Wheeler’s geometrodynamic vacuum “foam.” [MTW Gravitation] at S 43.4, [Wheeler, J. A., On the Nature of Quantum Geometrodynamics, Annals of Physics: 2, 604-614 (1957).] **** By way of review, the Planck mass, defined from the term atop Newton’s law as a mass for which GM_{P} ^{2} =hbar c , is thus M_{P} =sqrt{hbar c/G} . In the geometrodynamic vacuum, the negative gravitational energy between Planck masses separated by the Planck length L_{P} =sqrt{Ghbar /c^{3} } precisely counterbalances and cancels the positive energy of the Planck masses themselves. The Schwarzschild radius of a Plank mass R_{S} =2GM_{P} /c^{2} =2sqrt{Ghbar /c^{3} } =2L_{P} .*** Since we have based the foregoing on a unit charge with spin ½, and since this is independent of the mass, the foregoing would appear to characterize the compactification radius R for all of the charged leptons, and to provide a geometric foundation for intrinsic spin. This suggests that for alpha =1 or on the order of unity, the compactification radius of the fifth dimension may become synonymous with the Planck length itself, or the Schwarzschild radius of the vacuum, or something close to one of both of these.

While (4.2) applies generally for a compactified spacelike fifth dimension, before proceeding too far with this intrinsic spin interpretation (4.3), however, it is worth noting that for a neutral body, q=0, such as the neutrino, we have dphi /dtau =0, and so there is no fifth-dimensional rotation. More generally, any electrically-neutral body must be considered to be non-moving through the x^{5} dimension, dx^{5} =0. This would suggest that the neutrino has no intrinsic spin, which is, of course, contradicted by empirical knowledge. So, (4.3), while intriguing, does need to be studied further. Also, the intrinsic spin interpretation (4.3) suggests conversely, that any elementary scalar particle which has no intrinsic spin, must be electrically neutral. This is, in fact, true of the hypothesized Higgs boson.

One other point should be made before returning to the main development, especially because we will later be compelled in (10.14) to regard the fifth dimension as spacelike, and because a primary discomfort which many physicists have with Kaluza-Klein theory emerges from the compactified, fifth spatial dimension, because this dimension does not appear to haveany physical manifestation. [ For example, see the discussion and conclusion in Häggblad, J., Kaluza-Klein Theory , (2003).]

Despite the above puzzle regarding the neutrino, *** This may be resolved if one considers Kaluza-Klein in a non-Abelian (Yang-Mills) SU(2)WxU(1)Y rather than the present abelian U(1)em context, because the neutrino will then have a non-zero weak isospin I^{3} =+{textstylefrac{1}{2}} to lay a geodesic foundation for its intrinsic spin, and by recognizing that in the context of U(1)em, one really cannot speak anyway, about any particles other than charged leptons and photons.*** the use of the term “intrinsic” to describe an inherent quantized angular momentum of elementary particles, covers up what is actually a deep ignorance of what “intrinsic spin” really means, geometrically. Why? For a material body to have an angular momentum, there must implicitly be a radius R with which that body circles about an origin. Even the smallest objects, if they have an angular momentum, must be rotating or spinning — at some finite spatial radius — about an origin. At the same time, nobody believes that intrinsic spin represents an angular momentum about a radius R in the three usual spatial dimensions. By associating intrinsic spin with motion through a fourth, compactified, hyper-cylindrical spatial dimension, one simultaneously makes sense of intrinsic spin and of a compact fourth spatial dimension. The material body now has a spatial radius R of rotation through a spatial dimension other than the usual three spatial dimensions to give meaning to its “intrinsic” spin, and the compactified fourth dimension now takes on real, physical meaning as something which is physically observed, via the phenomenon of intrinsic spin, and not merely a fictional idea that gives people pause about Kaluza-Klein theories specifically, and dimensional compactification in general. As a phenomonological matter, there are few things which have been better-established over the last century than that intrinsic spin permeates every aspect of particle physics. And yet, we there is no known geometric foundation for this pervasive phenomenon.

In sum, the understanding of intrinsic spin as cyclical motion through a fourth dimension of space which is curled up into a radius on the order of the Planck length, if this can be developed further and sustained, may be useful to overcome one of the most nagging objections about Kaluza-Klein theories, and would underscore a clearly-observed, physical manifestation of the fourth space dimension, rather than requiring one to reply, with some disingenuity, that the extra space dimension is too small so nobody will ever see it anyway. Thus, we conclude with the provisional hypothesis, that the fourth spatial dimension is best thought of as the “intrinsic spin dimension” of a real, physical, five-dimensional spacetime.


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