Wednesday, February 23, 2005

Millenium Math Problem; Existence of Mass Gap in Yang Mills Theory

The problem of existence of mass gap in the Yang Mills Theory is one of the 7 Millenium Math Problems of the Cray Institute.

My research has solved the mass gap problem! The answer, surprisingly, is that the mass gap does NOT exist in Yang-Mills theory. Not only I have proved positively that in mathematics, Yang-Mills theory does not yield a definite mass gap, I have also proved that even if Yang Mills Theory does yield a definite mass gap mathematically, it still does NOT result in the physical mass gap we see in the real world.

My conclusion is that the mass gap DOES EXIST in the physical world. But also that the physical mass gap does NOT come from Yang Mills Theory, which is a theory based on infinite spacetime.

The real physical mass gap arises from the FINITENESS of spacetime, i.e., it is only because our universe is finite and enclosed, that we observe a mass gap, and strong and weak interactions therefore has a limited range.

The proof that mass gap does not exist in Yang Mills theory is actually extremely simple. All you need to do is a little bit scale transformation. i.e., try to like say scale space coordinates X into beta*X, you will find that the beta can lump into a separable coefficient and be removed, so you result in exactly the same Yang Mills Theory. But the original Yang Mills theory would lead to a mass gap of Delta, and the new form of Yang Mills Theory will lead to (Delta/beta), with everything else equal. So Delta = Delta/beta for arbitrary beta, therefore Delta === 0.

The insight of why Yang Mills, or for that purpose ANY existing theory, does NOT yield a mass gap, is that all of these theories lack of an essential constant without which it is impossible to lead to a definite calculation of the value of a mass gap, therefore they can not possibly lead to a definite and none-zero mass gap.

Yang Mills and all existing quantum theory takes these two constants as input: light speed C, and Planck constant hbar. It is very easy to show that using just these two constants, there is NO way you can put up a formula which combines these two constants and calculates a value of the unit of mass, or energy.

You need a further input into the theory, a characteristic mass, or a characteristic length scale, to be able to arrive at a definite mass gap value. GUITAR is the only theory capable of doing it.


GUITAR Predictions of Muon Mass

Previously I disclosed that GUITAR leads to precise prediction of neutron to electron mass ratio, accurate to 10 decimal places and agrees with the experimental value completely. See:

The same reasoning connecting decay lifetime to particle mass can be applied to Muon as well, and it also leads to amazingly precise results.

Before we start let me emphasis again that I am using the natural unit set where hbar = c = 1, and electron mass Me = alpha. In the natural unit, one time unit is
T0 = 9.399637148x10^-24 seconds

Also, recall that the observational "age" of the universe is:
Tu = PI*N
N = PI * exp(2/(3*alpha)) = 1.4898x10^40

First, the Muon decay lifetime is:
ln (Tau) = 40 - alpha, with alpha = fine structure constant.
Which gives:
Tau = exp(40-alpha) = exp (40 - 1/137.03599911) = 2.3367383x10^17
Certainly, keep in mind our results are in natural unit set, to convert back to MKS unit:
Tau = 2.3367383x10^17 * T0 = 2.3367383x10^17 * 9.399637148x10^-24 seconds
Tau = 2.19645 x10^-6 seconds
That agrees excellently wiht experimental value of Tau = 2.19703x10^-6 seconds.

Now let's calculate the Muon mass:

Mu = beta,
beta^2 = ln (Tu)/ln(PI*Tau)
beta^2 = ln (PI*N) /ln (PI*2.3367383x10^17)
beta^2 = ln (PI*1.4898253x10^40)/ln(PI*2.3367383x10^17)
beta^2 = 2.27645
beta = sqrt(2.27645) = 1.5088
Mu = 1.5088

That is it. That is the result. We have found the Mu mass to be 1.5088.

How come? remember we are using the natural unit set, in which Me = alpha.
Mu/Me = 1.5088/alpha = 1.5088 * 137.03599911 = 206.760

The most precise experimental value of muon to electron mass ratio is:
Mu/Me = 206.768

So my theoretical value of muon mass has at least 6 effective decimal places. Not bad at all.