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Re: Stephen Hawking's Theory Regarding Eleven Dimensions



Ann Marie Aoki wrote:

> How did Mr. Hawking theorize his eleven dimensions?

Hi,

Since no experts seems to answering your question let me try.

First of all, it wasn't Stephen Hawking who theorize 11 dimension.
Theory based on more than 4 dimensional spacetime was initiated
by Kaluza and Klein about 70 years ago. By assuming we lives in
five dimensional spacetime, where 'extra' spatial dimension looks
like tiny circle we can 'derive' 4 dimensional Einstein gravity and
electromagnetism together from 5 dimensional Einstein gravity.
Because of this aspect it was often called unified field theory, and
Einstein himself spent last years of his life on it without much of success.

Modern incarnation of theory began by many people, including
Freund, Cho (not me), DeWitt (as a homework problem in his
Les Houches summer school), Kerner, and Trautman and many
more. The idea is we can 'unify' not only gravity and
electromagnetism, but also with gauge theory (theory of forces
governing nuclear processes) if we use more than one extradimension.

Also, at the same time people developed theory called Super
Gravity(SUGRA). What is relevant here is;

1. There is a UNIQUE 11 dimensional SUGRA.
2. There is a sort of unique way to choose a vacuum (well,
     our universe - I hear screams from many of news group
     members.) called Freund-Rubin solution. What is cool about
     this solution is it nail down the fact we are living in the
     4-dimensional universe. In detail, it is SUSY that makes this
     solution works. So in some sense dimension of our universe
     - 4 - is nailed down by SUSY.
3. (Most relevant for us) 11 is MAXIMUM dimension for SUGRA.
     If we have more dimensions we are going to have a trouble in
     our 4-d universe. (In detail, when you reduce D=11, N=1
     SUGRA into 4-d SUGRA, you are end up with N=8, which
     is maximum consistent SUSY. This is because if N > 8 we
     are going to see particles with spin higher than 2, which
     believe to lead into inconsistency. - By the way, how tight this
     argument? Does anyone has comment?)

It was Ed Witten, who nail down 11 dimension in his classic paper
"Search for a Realistic Kaluza-Klein Theory". Nucl. Phys. B186
412-492 (1983). Here is a rough logic.
We demand that theory contains standard model. (Current theory
of all elementary particles), and be a SUGRA. He showed that to contain
standard model the dimension of extra space must be at
least 7. ( Space of lowest dimension with symmetry G act on it
is always a homogeneous space G/H with H maximal subgroup
of G. In the case of G= SU(3) x SU(2) x U(1), H = SU(2) x U(1)
x U(1). The dimension of G/H in this case is 7).

So, 11 dimension is very unique in a sense that this is a minimum
required dimension to contain standard model, and maximum
allowed dimension for SUGRA. Eventually, however, it was
Witten himself showed that there is a problem to get chiral fermion
from 11 dimension (using Atiyah - Hirzebruch index theorem for
dirac operator). So people eventually give up this 'naive' way of
doing it, and move on to string theory. - the point where my
confidence level begin falling down, so I am not going to talk about
it.

Cheers,

--
Demian H.J. Cho

Center for Gravitation and Cosmology
University of Wisconsin-Milwaukee