Why are there 10 dimensions in string theory?

David Strayhorn (Neurology, Washington University in Saint Louis)

Published in physic.philica.com

Observation
String theory is one of several modern attempts at a theory of quantum gravity (QG), which is hoped to result in the unification of general relativity (GR) and quantum mechanics (QM). One aspect of most versions of string theory is that the physical world has T = 10 spacetime dimensions. 6 of these are presumably compacted microscopically, leaving the D = 4 macroscopic dimensions of our everyday experience. Why 10?

If D were, say, 5 rather than 4, then would string theory still predict T = 10 dimensions, with 5 instead of 6 being compacted? This may be the case. But the purpose of this observation is to suggest a different possibility: that T is a function of D. Specifically: I propose that T equals the number C of components of the metric tensor $g_{ij}$. Since the metric is symmetric, its specification requires only C = 10 independent components (rather than the full 4 x 4 = 16). Extension to arbitrary values of D yields the hypothesis: $T(D) = C(D) = \sum_{n=1}^{D} n$.

It may be a mere coincidence that T = C = 10. I have heard the explanation that T = 10 because string theory simply requires “that many degrees of freedom” to provide a full description of the laws and phenomena of the real world. But consider that the structure of string theory — as an attempt at QG — can reasonably be expected to be a function of the structure of GR and QM. And D is an important feature of the real world, incorporated (even if only in an implicit fashion) into the laws of GR. So if D were higher (or lower), then perhaps string theory would require more (or fewer) degrees of freedom, as proposed in the equation above.

Observation circumstances
This idea occurred to me during the course of my own (amateur!) research into QG, introduced in [1] and [2]. In my model, the quantum-mechanical observer is represented by a point in a space S. Each dimension of S is a coefficient in the power series expansion of the metric. So S is actually an infinite dimensional space; but it is “constructed” out of the ten components C, with C(D) calculated as above. So I simply speculated that there could be some sort of correspondence between the dimensionality of observer state space S in my model and the number of dimensions of string theory.

Note that I make no claim of expert knowledge of string theory; there could very well be a simple counterargument to my proposal, of which I am unaware!

References
[1] Strayhorn, D. (2006). PHILICA.COM Article number 27.
[2] Strayhorn, D. (2006). PHILICA.COM Article number 28.

Peer-review ratings as of 07:52:23 on 20th Feb 2018 (from 2 reviews, where a score of 100 is average):
Originality = 124.25, importance = 112.99, overall quality = 113.29

Published on Wednesday 11th October, 2006 at 22:28:14.

Peer review added 19th October, 2006 at 13:30:59

Very interesting article.

However, I have a question to ask?
You mentioned that T is a function of D. Is that meant in the sense as in “f is a function of x”? ie f(x)=…

If so, how can T = C = 10? Did you mean T(D)=C(D)=10 ?

Author comment added 19th October, 2006 at 22:31:37

In response to the reviewer’s questions:

“You mentioned that T is a function of D. Is that meant in the sense as in “f is a function of x”? ie f(x)=…”

Yes.

“If so, how can T = C = 10? Did you mean T(D)=C(D)=10 ?”

I was simply pointing out the following possibility: that my hypothesis is wrong, that T is not a function of D, but for whatever reason it is equal to 10, and it is just a coincidence that C(4) is also equal to 10.

But just to be clear: if my hypothesis is right, then in our world we have D = 4, and that gives us T(4) = C(4) = 10.

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