B936 said:
I do apologize if this is the wrong place to post this.
Why is it necessary to have 11 dimensions in M Theory? I've been at it for days emailing universities, Twitter, etc. No one has attempted to give me an answer and it's frustrating me.
I just want to know what the purpose of each of the dimensions of M Theory are and why it couldn't work with only 10 dimensions.
Thanks
Given a Lie group G and its maximal subgroup H, then the coset space G/H is the lowest-dimensional manifold which can have G as a symmetry group: \mbox{dim}(G/H) = \mbox{dim}(G) - \mbox{dim}(H). Now consider the following examples:
If G = ISO(1,3), the 10-dimensional Poincare group, H = SO(1,3) is the 6-dimensional Lorentz group. We can identify the 4-dimensional coset malifold ISO(1,3) /SO(1,3) = M^{4} with the Poincare’ symmetric
4D Minkowski space-time.
If G = SU(2): \mbox{dim}(SU(2) ) = 3, H = U(1): \mbox{dim}(U(1)) =1, then SU(2)/U(1) = S^{2}, where S^{2} is the SU(2)-
symmetric 2-dimensional sphere.
Now, take G = SU(3): \mbox{dim}(SU(3)) = 8, its maximal subgroup is the 4-dimensional group H = SU(2) \times U(1), and SU(3) / SU(2) \times U(1) = \mbox{CP}^{2} , is the SU(3)-
symmetric 4-dimensional (complex) projective space. We also know that the circle S^{1} is
one dimensional, U(1)-
symmetric space. Thus, the space \mathcal{V}^{7} = \mbox{CP}^{2} \times S^{2} \times S^{1} has
7 dimensions and admits the
symmetry group of the
SM, i.e. SU(3) \times SU(2) \times U(1). Clearly, M^{4} \times \mathcal{V}^{7} is
11-dimensional space
symmetric under
Poincare' and the
gauge group of the
SM: ISO(1,3) \times SU(3) \times SU(2) \times U(1).
So, if you want to formulate a
Kaluza-Klein type theory in which su(3) \times su(2) \times u(1)-valued gauge fields arise as components of the metric tensor, g_{AB}(x), in more than 4 (non-compact) space-time dimensions, you must have at least
7 extra dimensions, i.e., A, B = 1, 2, \cdots , 11. Thus
D = 11 is the
minimum number with which you can obtain the gauge fields of the gauge group SU(3) \times SU(2) \times U(1) by Kaluza-Klein method. Since, we have never been able to formulate a consistent field theory with gravity coupled to massless
spin > 2 particles. And, since D >11 supergravity contains such massless,
spin>2 particle, we conclude that
11 dimensions is the
maximum for consistent supergravity. It seems like a remarkable
coincidence that D=11, which is the
minimum number required by K-K procedure, is the
maximum number required by consistent supergravity.
Sam