Cosmological constant, supersymmetry, nonassociativity, and Big Numbers

The nonassociative generalization of supersymmetry is considered. It is shown that the associator of four supersymmetry generators has the coefficient $\sim \hbar/ \ell_0^2$ where $\ell_0$ is some characteristic length. Two cases are considered: (a) $\ell_0^{-2}$ coincides with the cosmological constant; (b) $\ell_0$ is the classical radius of electron. It is also shown that the scaled constant is of the order of $10^{-120}$ for the first case and $10^{-30}$ for the second case. The possible manifestation and smallness of nonassociativity is discussed.


I. INTRODUCTION
The observed value of the cosmological constant is smaller than that predicted by an effective local quantum field theory by a factor of 10 120 . This discrepancy has been called "the worst theoretical prediction in the history of physics" [1,2]. The great interest would be represented by the appearance of cosmological constant in any formula, even if obtained by some qualitative reasonings.
It is shown in Ref.
[3] that the idea of supersymmetry can be extended with the inclusion of nonassociativity into supersymmetry. The main idea presented there is that an associator of four supersymmetrical quantities Q a , Qȧ, Q b , Q˙b can be connected with the angular momentum operator.
In this Letter we want to consider the proportionality coefficient in this relation and, using some qualitative reasonings, to show that it may contain the factor which can be identified either with the cosmological constant or with the classical radius of electron.

II. NONASSOCIATIVE DECOMPOSITION OF THE ANGULAR MOMENTUM OPERATOR
In Ref.
[3], a nonassociative generalization of a supersymmetry algebra with the supersymmetry generators Q a , Qȧ (here a = 1, 2,ȧ =1,2) is considered. The simplest supersymmetry algebra considered there is (in this section we follow Ref. [3]): The anticommutator (1) connects the momentum operator P µ = −i∂ µ (here µ = 0, 1, 2, 3) and the generators Q a,ȧ . The Pauli matrices σ µ aȧ , σ aȧ µ are Let us define an associator as follows: It is assumed that the associator Q a , Qȧ, Q b Q˙b is where the operator is the angular momentum operator, and ζ is the still undefined numerical factor that equalizes the dimensions of the right-and left-hand sides of equation (8). Our main goal here is to derive this factor using some plausible physical arguments.

III. DEFINITION OF ζ
We see from equation (1) that the dimension of Q is From equation (8), one can find that We think that ζ should be constructed from fundamental constants. In this case one can consider following possibilities. The first one is that where c is the speed of light and G is the gravitational constant. But we think that the relation (8) is a quantum relation, and in some sense it should be similar to the commutation relation This means that the Planck constant should be included in ζ. For example, it can be done as where ℓ 0 is some characteristic length and should be constructed from physical constants. One can find following possibilities: is the classical radius of electron (15) Choosing ζ in the form (15) we can rewrite (8) in the dimensionless form where the quantities with tildes are dimensionless, and ζ = l 2 P l Λ ≈ 10 −120 ; where l P l = G c 3 is the Planck length; e, m e are the charge and mass of electron. Finally, write (8) in the form where the coefficient ζ from equation (8) is defined as ζ = ζ 0ζ and ζ 0 = ±1, ±i is now a dimensionless number. The inverse relation is We see that there are different possibilities for choosing ζ. Probably it may be due to the fact that the coefficient ζ in the relation (8) can be different for different physical situations. For example, on the large scales (∼ Universe) ℓ 0 ≈ 1/ √ Λ but on the micro scales (∼ r 0 ) ℓ 0 ≈ r 0 .

IV. DISCUSSION AND CONCLUSIONS
We have shown that the nonassociative generalization of supersymmetry has some coefficient that can be associated with some characteristic length. After scaling this dimensionless factor shows how small can be the manifestation of possible nonassociativity in physics. For the first case ζ ∼ c 3 /G and the dimensionlessζ ∼ 1 that is too much. For the second case the scaled value of this constant is the product of the squared Planck length and the cosmological constant ζ ∼ Λ, and consequently is ≈ 10 −120 . The possible manifestation of nonassociativity in this case become apparent in quantum gravity on large scales sinceζ contains , G and Λ. For the third case the scaled value of this constant is ζ ∼ Gm e e 2 /(c 3 2 ), and consequently is ≈ 10 −30 . The possible manifestation of nonassociativity in this case become apparent in quantum gravity + electrodynamics on micro scales sinceζ contains , G, e and m e .