Re: A question about the spin 2 particle.

[Jack Uretsky <jlu@HEP.ANL.GOV>]

On Mon, 25 Dec 2000, Robert B Zannelli wrote:

> In a message dated 12/25/00 5:49:45 PM Eastern Standard Time, jlu@HEP.ANL.GOV
> writes:
>
> <<
>  I am always troubled by questions involving "someone told me that".  Also,
>  I don't understand what the purported quote has to do with the rest of the
>  question.
>                                  Regards,
>                                          Jack >>
> Jack As I explained the Feynman reference is from the book The Dirac Memorial
> Lectures, a book I have ordered but have not received yet. This information
> was provided to me by another very accomplished Physicist who is on another
> list.
        I try to dissuade my students from invoking authority to support
their views.  In any event, we are all of us "very accompished
physicists", and we are all of us prone to (a) making erroneous statements
and (b) having our brilliant statement misunderstood from time to time.

> Since I don't have the book yet all I have is someone told me. In any
> case the identification of particle exchange with particle rotation is a
> distraction (Though an interesting one) from my question which is related to
> rotations.
        Fine.  The "distraction" is part of the confusion.  So let's
forget the Feynamn quote, because it is irrelevant.

> There was fairly general agreement on that list after some debate,
> that for a graviton, the probability amplitude is returned after a one pi
> rotation.
        Physics questions are not decided by vote.  The issue is whether
you understand the answer well enough to reproduce the reasoning.  So
"agreement" is also irrelevant.

> All of that took place before the first response was ever received
> from this list. Also the members of that list tend not to write about me to
> third parties but rather respond directly to me. I just guess list etiquette
> differs from list to list.

        O.K.  My perception is that we are trying to model good teaching
practices.

        Here is an answer to what I understand to be your question.

        The spin 2 representations of the rotation group contains
eigenvectors of one of the generators (call it L_3) with eignvalues
M=_2,-1,0,1,2 (as you correctly wrote in your posting).  A rotation
of \phi around the 3-axis induces a change exp(iM\phi) in each respective
eigenvector.  Accordingly, the eigenvectors corresponding to M=+-1 will
change sign when subjected to a rotation of \pi.
        This has nothing to do with quantum mechanics.  I have used
only elementary Lie Group theory.
        The subject of gravitons takes us beyond the simple rotation
group into the mathematics of the Poincare group' (relativity now becomes
important).  Massless particles, it turns out for reasons beyond the scope
of this lecture, can only have M values along the direction of their
motion of +_L, where L is the "spin" of the particle.  See Weinberg, Vol.
1, pp. 253-4.
        A graviton eigenvector, therefore, is invariant under rotations
of \pi around the axis along its direction of motion.  We do not consider
rotations about other axes because they would change the direction of
motion.

        I hope this  helps.
                                        Regards,
                                                Jack