Natural Science Forum / Physics / Research / April 2008

Hi Gerard,

First, I'll note that I first learned general relativity from Dr.
Ohanian at RPI in the early 1980s.  Thanks for pointing out this work,
which Dr. Thomas Love at USC had earlier brought to my attention and
which I cited in a draft paper on this topic at
http://jayryablon.files.wordpress.com/2008/04/intrinsic-spin-30.pdf
(comments always welcome).  Hans DeVries also commented on your post
here, over at sci.physics.foundatons, see
http://groups.google.com/group/sci.physics.foundations/browse_frm/thread/c6fce6b
9b281dd3c/21da7a5375d8c5cc#21da7a5375d8c5cc.

Now, the above L = h R / (2 pi R) = h_bar is based  on tangential
integer wave numbers over a 2pi rotation.  But what if we go to
orientation / entanglement -- i.e., "version," as laid out in Misner /
Wheeler / Thorne in Gravitation, section 41.5.  Then we obtain a double
covering, and your equation above turns into:

L = h R / (4 pi R) = (1/2) h_bar

Might the above be the "more intuitive" way to which you refer?  Cross
reference to eqs. (5.3) and (5.4) in the file linked above.

Thanks,

Jay.

On Apr 13, 12:49 pm, "Jay R. Yablon" <jyab...@nycap.rr.com> wrote:

Isn't it just as simple as finding eigenvalues of L = r x p operator?
eigenvalues of L^2 are l(l+1) h-bar^2, with l an integer. There are no
other solutions to the e-value problem. Thus half-integer spin can't
come from an r x p operator.

Is there a problem with this line of argument?

Dan

[Yablon]
Not at all, Dan:

Finally, somebody has answered my question directly.  Let me rephrase
what I think you mean:

We generate a rotation through angle psi about the z-axis by acting on a
wavefunction phi via with the z-axis rotation generator m, according to:

exp[i psi m] phi.

Because psi has a period of 2 pi, the eigenvalues of m must be an
integer, not a half integer.  Am I correct so far?

If so, then my next question is this:  Go to section 41.5 of Misner,
Wheeler, Thorne, where they discuss orientation / entanglement.  There,
a rotation through 4pi is needed to restore not only orientation, but
the entanglement version.  So, if psi needs a period of 4pi rather than
2pi to restore the entanglement relationship with the surrounding
environment, then the Eigenvalues of m can be half integers, and can be
generated -- not by the "rotation group" -- but by what one might call
the "entanglement / rotation" group.  This would make it possible, I
believe, to create half-integer spin as well, from an rxp operator, with
the understanding that entanglement counts.  This would appear to pave
the way to place the fundamental representation SO(1,3) of the Lorentz
group, which uses the SU(2) generators, onto an rxp operator footing as
well.

Your further thoughts.

Thanks,

Jay.

On Apr 13, 12:49 pm, "Jay R. Yablon" <jyab...@nycap.rr.com> wrote:

Isn't it just as simple as finding eigenvalues of L = r x p operator?
eigenvalues of L^2 are l(l+1) h-bar^2, with l an integer. There are no
other solutions to the e-value problem. Thus half-integer spin can't
come from an r x p operator.

Is there a problem with this line of argument?

Dan

[Yablon]

What I replied with before is actually half of the answer.  The other
half is as follows:

As to the issue of whether rxp can be applied to intrinsic spin, I call
your attention to Ohanian's article "What is spin?," linked as
http://jayryablon.files.wordpress.com/2008/04/ohanian-what-is-spin.pdf.
Go to equation (16) and the discussion following.  With $ = integral,
and G=momentum density (see (10) through (13)), equation (16) can be
schematically rewritten as:

J = $ [(r x G)_orbital + (r x G)_spin] d^3 x

Then a few steps later after converting the triple cross product, the
second term turns directly into the expectation value of the sigma
operator, and by (19), we have spin 1/2.

Via the (r x G)_spin term in (16), it is clear that both spin and
orbital angular momentum *can* be treated with an rxp type operator,
contrary to what is often asserted.  That is *exactly* what Ohanian does
here.

If one then applies entanglement so that one has to cycle through 4pi
rather than 2pi, the distinction between orbital and spin angular
momentum in terms of group theory (fundamental versus non-fundamental
Lorentz group representations of SO(1,3)) can, it appears, be
eliminated, and each can be treated via and rxp analog to classical
theory.

BTW, I mentioned earlier that Ohanian taught me general relativity while
he was an adjunct professor over at nearby RPI (and also taught at Union
College here in Schenectady where I first met him) in the 1980s.  With
Wheeler's passing, I should also note if memory serves, that Ohanian is
a student of Wheeler's, and should be added to Wheeler's genealogies.

Jay.

Other relevant papers are:

C. Van Winter, Annals of Physics 47, 232 (1968).
S.S. Sanikov, Nucl. Phys. 87, 834 (1967).

For an old discussion see
W. Pauli, Helv. Phys. Acta 12, 147 (1939).

Van Winter provides a very detailed discussion. He chooses
a set of spherical harmonics with half integer values of l,
and shows that orbital angular momentum operator is not
self-adjoint with respect to such a set of functions.
In the case of l = 1/2, his set of functions YW_{lm}
(up to a normalization constant which I omit here) is:

  YW_{1/2,1/2} =  sin^{1/2} \theta e^{i phi/2}
  YW_{1/2,-1/2} = sin^{1/2} \theta e^{-i phi/2}   (1)

But there is a problem with the above set of functions,
because the second function does not come from the first
by the application of the "ladder" operator L_. Namely,
L_ YW_{1/2,1/2} is not equal to YW_{1/2,-1/2}.
Instead, we have that L_ YW_{1/2,1/2} is proportional to
cos \theta sin^{-1/2} \theta e^{-i phi/2}.

Other authors, including E. Merzbacher, also use the above
set of functions (and its generalization for higher l and m
values), and consequently they do not obtain self adjoint
orbital angular momentum operator.

In this respect, it is more natural to take functions

  Y_{1/2,1/2} =  (i/\pi) sin^{1/2} \theta e^{i phi/2}
  Y_{1/2,-1/2} = -(i/\pi) cos \theta sin^{-1/2} \theta e^{-i phi/2}

for which it holds L_ Y_{1/2,1/2} = Y_{1/2,-1/2}.
We also have:

  L+ Y_{1/2,1/2} = 0
  L+ Y_{1/2,-1/2} = Y_{1/2,1/2}

But there is a catch: The action on L_ on the function
with lowest m value, i.e., m = -1/2 in our example,
we find

   L_ Y_{1/2,-1/2} = Y_{1/2,-3/2}

The result is not zero, as it should be. And yet, if we
act on the latter state with the raising operator L+,
we obtain zero:

  L+ Y_{1/2,-3/2} = 0.

The last result is crucial. It has for a consequence,
that the operators L_x, L_y, L_z are self adjoint,
provided that one employs a suitable renormalization
of the inner products between certain states. Then
everything appears to be OK, and the spherical harmonics
with half-integer l-values do form an irreducible
representation of the 3-dimensional rotation group.

All this has been discussed in papers
D. Pandres, J. Math. Phys. 6, 1098 (1965),
D. Pandres and D.A. Jacobson, J. Math. Phys. 9, 1401 (1968),

I followed  Pandres' approach in a paper of mine

"Rigid particle and its spin revisited"
Published in Found.Phys.37:40-79,2007.
http://arxiv.org/abs/hep-th/0412324
in which I tried to clarify some other aspects as well,
and also used a complementary functions Z_{lm} in
a superposition with Y_{lm}, in order to obtain
"good behavior" under rotations, parity and time reversal.
Pandres and I do not claim that the ordinary
orbital angular momentum can have half integer l-values.

A reason of why to reject Y_{lm} and Z_{lm}
with half-integer l-values in the description of orbital angular
momentum was given correctly by Dirac in his textbook on QM.
In the free case,  a complete  set of solutions to the
Schr\" odinger equation consists of plane waves,
which are single valued. The latter property has to be preserved
when we use  another representation, i.e., one with spherical
harmonics.
Hence, such Schr\" odinger basis for spinor representation of the
3-dimensional rotation group cannot refer to the ordinary configuration
space of positions, but to an internal space associated with
every point of the ordinary space.

Double valued spherical harmonics are admissible,
if they do not refer to the ordinary configuration space in
which the usual  quantum mechanical orbital angular momentum is defined,
but if they refer to an internal space in which a spin angular momentum
is defined. An example of such an internal space is the space of
velocities, associated  with the so called {\it rigid particle}
whose action contains the square of the
extrinsic curvature of a particle's  world line.

`Fermion' spherical harmonics are discussed also in papers:

G. Hunter, P. Ecimovic, I.M Walker, D. Beamish, S. Donev, M. Kowalski, A. Arslan
and S. Heck, J. Phys. A: Math. Gen. 32 795 (1999); G. Hunter and M. Emami-
Razavi, \Properties of Fermion Spherical Harmonics", [arXiv:quant-ph/0507006].