Quote:
|
Originally Posted by lumberjim
man, another brilliantly composed test wasted. oh, and you missed number ten. I'm sorry, I'll have to give you an 'incomplete'
|
Oops! Sorry about that. Here you go:
ten.
Note: Of course the above answer depends upon one's definition of "now." The value I have given in response to question (ten) was arrived at through the application of noncommutative geometry as an alternative hidden structure of Kaluza-Klein theory. This means that one leaves space-time as it is and one modifies only the extra dimensions; one replaces their algebra of functions by a noncommutative algebra, usually of finite dimension to avoid the infinite tower of massive states of traditional Kaluza-Klein theory. Because of this restriction and because the extra dimensions are purely algebraic in nature the length scale associated with them can be arbitrary, indeed as large as the Compton wave length of a typical massive particle.
The algebra of Kaluza-Klein theory is therefore, for example, a product algebra of the form:
A = C(V ) ⊗Mn.
Normally V would be chosen to be a manifold of dimension four, but since much
of the formalism is identical to that of the M(atrix)-theory of D-branes. For the simple models with a matrix extension one can use as gravitational action the Einstein-Hilbert action in ‘dimension’ 4+d, including possibly Gauss-Bonnet terms.
For a more detailed review, please refer to my lecture at the 5th Hellenic school in Corfu.