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The note gives a watertight confirmation of the E-infinity Cantorian theory results for ordinary and dark cosmic energy density of the universe and respectively. The computation is fundamentally based on a golden mean fusion function that goes back to the highly original anyon proposal of F. Wilczek.

The exceedingly important role of anyons in physics started in earnest when F. Wilczek was able to show that the theoretical possibility of two dimensional exotic particles, [

Let us start by recalling the results of the ordinary and dark cosmic energy densities obtained previously using numerous methods [

γ ( O ) = ( ϕ ) 5 (1)

while for the uncorrelated and thus not directly measurable cosmic dark energy density one finds

γ ( D ) = 5 ϕ 2 (2)

The total density is therefore given by [

γ = γ ( O ) + γ ( D ) = ϕ 5 + 5 ϕ 2 = 2 (3)

Since it was established that Einstein’s E = m c 2 represents the maximal energy density possible, i.e. γ = 100 % corresponding to γ = 1 , then to bring the above result in line with E = m c 2 where m is the mass and c is the speed of light, then we simply interpret E = m c 2 as being E = ( 2 / 2 ) m c 2 [

E = ( ϕ 5 + 5 ϕ 2 2 ) m c 2 = ( ϕ 5 / 2 ) ( m c 2 ) + ( 5 ϕ 2 / 2 ) m c 2 = E ( O ) + E ( D ) = m c 2 (4)

where E(O) is the quantum particle ordinary energy and E(D) is that of the quantum wave dark energy [

E ≅ ( m c 2 ) / 22 + ( 21 / 22 ) m c 2 = E ( Einstein ) (5)

This result is in astounding agreement with accurate cosmic measurements and observations which assert that E(O) is about 4.5% and E(D) is the rest 95.5% of the total expected energy [

γ ( O ) = ϕ 5 / 2 = 0.04508497197 ≅ 1 / 22 ≅ 4.5 % (6)

for ordinary cosmic energy density and

γ ( D ) = 5 ϕ 2 / 2 = 0.9549150289 ≅ 21 / 22 ≅ 95.5 % (7)

for the dark energy section which as we know cannot be measured in a direct way without quantum wave non-demolition measuring devices that are technologically not yet available at the time of writing [

It is well known from topological quantum field theory and its relation to subfactors that there is a dimensional function for an explicit situation called 4-D fusion algebra given by [

d ( 1 ) = d ( ∈ ) = 1 (8)

and

d ( x ) = d ( β ) = 1 / ϕ (9)

Now, not so incidentally this 4-D function may be taken over to the two dimensional anyon where, as reasoned in the anionic theory [

V ( a ) = ( 2 ) ( 1 / ϕ ) = 2 ( 1 + ϕ ) = 2 + 1 + ϕ 3 = 3 + ϕ 3 (10)

On the other hand, the contribution of the anyon vacuum is given by the simple self explanatory equation

V ( v ) = ( 2 ) ( 1 ) = 2 (11)

The total is thus

V = 3 + ϕ 3 + 2 = 5 + ϕ 3 (12)

The corresponding Einstein maximal energy is therefore [

E = ( 5 + ϕ 3 5 + ϕ 3 ) m c 2 = m c 2 (13)

Dissecting 5 + ϕ 3 into the smooth (integer) part, i.e. 5 and the irrational transfinite fractal portion ϕ 3 which together form a self affine or a self similar fractal Kaluza-Klein spacetime dimension we may write [

E = ( ϕ 3 5 + ϕ 3 + 5 5 + ϕ 3 ) m c 2 (14)

The reader may attest for himself that ( ϕ 3 ) / ( 5 + ϕ 3 ) and 5 / ( 5 + ϕ 3 ) are nothing but exactly the same ordinary and dark cosmic energy densities which we found earlier on in numerous previous publications [

γ ( O ) = ϕ 3 5 + ϕ 3 = ϕ 5 / 2 (15)

and

γ ( D ) = 5 5 + ϕ 3 = 5 ϕ 2 / 2 (16)

exactly as should be [

d ( 1 ) + d ( ∈ ) + d ( x ) + d ( β ) = 1 + 1 + 1 / ϕ + 1 / ϕ = 2 + 3 = ϕ 3 = 5 + ϕ 3 (17)

It is more than gratifying and less than exhilarating to find that the profound anyons theory [

The author declares no conflicts of interest regarding the publication of this paper.

El Naschie, M.S. (2018) Golden Anyons for Cosmic Dark Energy Density. World Journal of Condensed Matter Physics, 8, 157-161. https://doi.org/10.4236/wjcmp.2018.84010