The simplest non-associative generalization of supersymmetry
aa r X i v : . [ h e p - t h ] F e b The simplest nonassociative generalization of supersymmetry
Vladimir Dzhunushaliev ∗ Dept. Theor. and Nucl. Phys., KazNU, Almaty, 050040, Kazakhstan andIETP, Al-Farabi KazNU, Almaty, 050040, Kazakhstan
Nonassociative generalization of supersymmetry is suggested. 3- and 4-point associators for su-persymmetric generators are considered. On the basis of zero Jacobiators for three supersymmetricgenerators, we have obtained the simplest form of 3-point associators. The connection between3- and 4-point associators is considered. On the basis of this connection, 4-point associators areobtained. The Jacobiators for the product of four supersymmetric generators are calculated. Wediscuss the possible physical meaning of numerical coefficients presented on the right-hand sides ofassociators. The possible connection between supersymmetry, hidden variables, and nonassociativityis discussed.
PACS numbers: 11.30.Pb; 02.40.GhKeywords: supersymmetry, nonassociativity, hidden variables
I. INTRODUCTION
Supersymmetry is a well-defined mathematical theory that probably has the application in physics: it is a branch ofparticle physics that, using a proposed type of spacetime symmetry, relates two basic classes of elementary particles –bosons and fermions. In the standard approach supersymmetric generators are associative and anticommutative.Here we want to consider a nonassociative generalization of supersymmetry. We offer some 3-point associators forsupersymmetric generators Q a and Q ˙ a . Using some relation between 3- and 4-point associators, we obtain somelimitations on the possible form of 4-point associators. On the basis of these limitations, we offer 4-point associatorsfor supersymmetric generators.Nonassociative structures appear in: (a) quantum chromodynamics [1]; (b) Maxwell and Dirac equations [2, 3];(c) string theory [4]; (d) nonassociative quantum mechanics [5, 6]. For other ways of introducing nonassociativestructures into physics, see the monographs [7, 8].Here we would like to introduce nonassociative structures into supersymmetry and to discuss the physical conse-quences of such a procedure. II. THE SIMPLEST 3-POINT ASSOCIATORS
Recall the definition of associator [
A, B, C ] = ( AB ) C − A ( BC ) , (1)where A, B, C are nonassociative quantities. Now we want to demonstrate that it is possible to introduce the gener-alization of supersymmetry other than that given in Refs. [9, 10]. Let us define the following 3-point associators:[ Q a , Q b , Q c ] = α Q a ǫ bc + α Q b ǫ ac + α Q c ǫ ab , (2)[ Q ˙ a , Q b , Q c ] = β Q ˙ a ǫ bc , (3) (cid:2) Q a , Q ˙ b , Q c (cid:3) = β Q ˙ b ǫ ac , (4)[ Q a , Q b , Q ˙ c ] = β Q ˙ c ǫ ab , (5) (cid:2) Q a , Q ˙ b , Q ˙ c (cid:3) = γ Q a ǫ ˙ b ˙ c , (6)[ Q ˙ a , Q b , Q ˙ c ] = γ Q b ǫ ˙ a ˙ c , (7) (cid:2) Q ˙ a , Q ˙ b , Q c (cid:3) = γ Q c ǫ ˙ a ˙ b , (8) (cid:2) Q ˙ a , Q ˙ b , Q ˙ c (cid:3) = δ Q ˙ a ǫ ˙ b ˙ c + δ Q ˙ b ǫ ˙ a ˙ c + δ Q ˙ c ǫ ˙ a ˙ b , (9) ∗ Electronic address: [email protected] where ǫ ˙ a ˙ b = ǫ ab = (cid:18) − (cid:19) , (10) ǫ ab = ǫ ˙ a ˙ b = (cid:18) −
11 0 (cid:19) . (11)The dotted indices are lowered and raised by using ǫ ˙ a ˙ b , ǫ ˙ a ˙ b , and the undotted – by ǫ ab , ǫ ab .In order to introduce some limitations on the nonassociative algebra, we want to calculate the Jacobiator J ( x, y, z ) = [[ x, y ] , z ] + [[ y, z ] , x ] + [[ z, x ] , y ] = [ x, y, z ] + [ y, z, x ] + [ z, x, y ] − [ x, z, y ] − [ y, x, z ] − [ z, y, x ] , (12)where x, y, z are either Q a, ˙ a or their product. Let us calculate Jacobiators J ( Q a , Q b , Q c ) = 2 ( α − α + α ) Q a ǫ bc , (13) J ( Q ˙ a , Q b , Q c ) = 2 ( β − β + β ) Q ˙ a ǫ bc , (14) J ( Q a , Q ˙ b , Q c ) = 2 ( β − β + β ) Q ˙ b ǫ ca , (15) J ( Q a , Q b , Q ˙ c ) = 2 ( β − β + β ) Q ˙ c ǫ ab , (16) J ( Q a , Q ˙ b , Q ˙ c ) = 2 ( γ − γ + γ ) Q a ǫ ˙ b ˙ c , (17) J ( Q ˙ a , Q b , Q ˙ c ) = 2 ( γ − γ + γ ) Q b ǫ ˙ c ˙ a , (18) J ( Q ˙ a , Q ˙ b , Q c ) = 2 ( γ − γ + γ ) Q c ǫ ˙ a ˙ b , (19) J ( Q ˙ a , Q ˙ b , Q ˙ c ) = 2 ( δ − δ + δ ) Q ˙ a ǫ ˙ b ˙ c . (20)To have zero Jacobiators, we have to have the following equations for the parameters α, β, γ, δ : α − α + α = β − β + β = γ − γ + γ = δ − δ + δ = 0 . (21)Thus, we see that perhaps the most natural and the simplest choice of the parameters α, β, γ, δ is α = β = γ = δ = 0 , (22) | α | = | α | = | β | = | β | = | γ | = | γ | = | δ | = | δ | = ~ ℓ , (23)where the factor ~ /ℓ is introduced for equalizing the dimensionality of the left-hand sides and right-hand sides ofequations (2)-(9); ℓ is some characteristic length. Thus α = − α = ~ ℓ ζ , (24) β = − β = ~ ℓ ζ , (25) γ = − γ = ~ ℓ ζ , (26) δ = − δ = ~ ℓ ζ , (27)(28)where ζ , , , = ( either ± ± i . (29)Thus we have the following 3-point associators[ Q a , Q b , Q c ] = ~ ℓ ζ ( Q a ǫ bc − Q c ǫ ab ) , (30)[ Q ˙ a , Q b , Q c ] = ~ ℓ ζ Q ˙ a ǫ bc , (31) (cid:2) Q a , Q ˙ b , Q c (cid:3) = 0 , (32)[ Q a , Q b , Q ˙ c ] = − ~ ℓ ζ Q ˙ c ǫ ab , (33) (cid:2) Q a , Q ˙ b , Q ˙ c (cid:3) = ~ ℓ ζ Q a ǫ ˙ b ˙ c , (34)[ Q ˙ a , Q b , Q ˙ c ] = 0 , (35) (cid:2) Q ˙ a , Q ˙ b , Q c (cid:3) = − ~ ℓ ζ Q c ǫ ˙ a ˙ b , (36) (cid:2) Q ˙ a , Q ˙ b , Q ˙ c (cid:3) = ~ ℓ ζ (cid:0) Q ˙ a ǫ ˙ b ˙ c − Q ˙ c ǫ ˙ a ˙ b (cid:1) . (37) III. 4-POINT ASSOCIATORS
First, we want to consider the connection between 3- and 4-point associators. For example, the 4-point associator[ Q x Q y , Q z , Q w ] is [ Q x Q y , Q z , Q w ] = (( Q x Q y ) Q z ) Q w − ( Q x Q y ) ( Q z Q w ) , (38)where x, y, z, w are any combinations of dotted and undotted indices. The last term on the right-hand side of equation(38) is ( Q x Q y ) ( Q z Q w ) and it cannot be obtained by multiplying of any 3-point associator from the left-hand sidesof (2)-(9) neither by Q a nor Q ˙ a . Nevertheless, there is the relation between the 3- and 4-point associators:[ Q x Q y , Q z , Q w ] − [ Q x , Q y Q z , Q w ] + [ Q x , Q y , Q z Q w ] = [ Q x , Q y , Q z ] Q w − Q x [ Q y , Q z , Q w ] . (39) A. 4-point associators without dots
Let us assume the following 4-point associators[ Q a Q b , Q c , Q d ] = ρ , Q a Q b ǫ cd + ρ , Q a Q c ǫ bd + ρ , Q a Q d ǫ bc + ρ , Q b Q c ǫ ad + ρ , Q b Q d ǫ ac + ρ , Q c Q d ǫ ab , (40)[ Q a , Q b Q c , Q d ] = µ , Q a Q b ǫ cd + µ , Q a Q c ǫ bd + µ , Q a Q d ǫ bc + µ , Q b Q c ǫ ad + µ , Q b Q d ǫ ac + µ , Q c Q d ǫ ab , (41)[ Q a , Q b , Q c Q d ] = ν , Q a Q b ǫ cd + ν , Q a Q c ǫ bd + ν , Q a Q d ǫ bc + ν , Q b Q c ǫ ad + ν , Q b Q d ǫ ac + ν , Q c Q d ǫ ab . (42)Then the relation (39) gives us the following relations between 3- and 4-point nonassociative structure constants ρ , − µ , + ν , = α , (43) ρ , − µ , + ν , = α , (44) ρ , − µ , + ν , = α + α , (45) ρ , − µ , + ν , = 0 , (46) ρ , − µ , + ν , = α , (47) ρ , − µ , + ν , = α . (48)Perhaps the simplest limitations on the nonassociative structure constants ρ i, , µ i, , ν i, are as follows: ρ , = ρ , , µ , = µ , ν , = ν , ; (49) ρ , = ρ , + ρ , , µ , = µ , + µ , , ν , = ν , + ν , ; (50) ρ , − µ , + ν , = 0 . (51)Then the following solution of equations (49)-(51) that is compatible with (22) and (24) can be found: ρ , = − ρ , = ν , = − ν , = 12 ~ ℓ ζ ; (52) µ i, = 0 , i = 1 , , . . . , (53) ρ , = ρ , = ρ , = ρ , = ν , = ν , = ν , = ν , = 0 . (54)Finally, 4-point associators are [ Q a Q b , Q c , Q d ] = 12 ~ ℓ ζ ( Q a Q b ǫ cd − Q c Q d ǫ ab ) , (55)[ Q a , Q b Q c , Q d ] = 0 , (56)[ Q a , Q b , Q c Q d ] = 12 ~ ℓ ζ ( Q a Q b ǫ cd − Q c Q d ǫ ab ) . (57)One can immediately check that the Jacobiator J ( Q a Q b , Q c , Q d ) = 0 . (58) B. 4-point associators with one dot
In this section we consider 4-point associators with one dot moving from the left on the right side of associator.
1. 1-st case
We seek 4-point associators with one dot as follows:[ Q ˙ a Q b , Q c , Q d ] = ρ , ǫ cd Q ˙ a Q b + ρ , ǫ bd Q ˙ a Q c + ρ , ǫ bc Q ˙ a Q d + ρ , ( Q b Q c , x d ˙ a ) + ρ , ( Q b Q d , x c ˙ a ) + ρ , ( Q c Q d , x b ˙ a ) , (59)[ Q ˙ a , Q b Q c , Q d ] = µ , ǫ cd Q ˙ a Q b + µ , ǫ bd Q ˙ a Q c + µ , ǫ bc Q ˙ a Q d + µ , ( Q b Q c , x d ˙ a ) + µ , ( Q b Q d , x c ˙ a ) + µ , ( Q c Q d , x b ˙ a ) , (60)[ Q ˙ a , Q b , Q c Q d ] = ν , ǫ cd Q ˙ a Q b + ν , ǫ bd Q ˙ a Q c + ν , ǫ bc Q ˙ a Q d + ν , ( Q b Q c , x d ˙ a ) + ν , ( Q b Q d , x c ˙ a ) + ν , ( Q c Q d , x b ˙ a ) . (61)
2. 2-nd case
We seek 4-point associators with one dot as follows: (cid:2) Q a Q ˙ b , Q c , Q d (cid:3) = ρ , ǫ cd Q a Q ˙ b + ρ , (cid:0) Q a Q c , x d ˙ b (cid:1) + ρ , (cid:0) Q a Q d , x c ˙ b (cid:1) + ρ , ǫ ad Q ˙ b Q c + ρ , ǫ ac Q ˙ b Q d + ρ , (cid:0) Q c Q d , x a ˙ b (cid:1) , (62) (cid:2) Q a , Q ˙ b Q c , Q d (cid:3) = µ , ǫ cd Q a Q ˙ b + µ , (cid:0) Q a Q c , x d ˙ b (cid:1) + µ , (cid:0) Q a Q d , x c ˙ b (cid:1) + µ , ǫ ad Q ˙ b Q c + µ , ǫ ac Q ˙ b Q d + µ , (cid:0) Q c Q d , x a ˙ b (cid:1) , (63) (cid:2) Q a , Q ˙ b , Q c Q d (cid:3) = ν , ǫ cd Q a Q ˙ b + ν , (cid:0) Q a Q c , x d ˙ b (cid:1) + ν , (cid:0) Q a Q d , x c ˙ b (cid:1) + ν , ǫ ad Q ˙ b Q c + ν , ǫ ac Q ˙ b Q d + ν , (cid:0) Q c Q d , x a ˙ b (cid:1) . (64)
3. 3-rd case
We seek 4-point associators with one dot as follows:[ Q a Q b , Q ˙ c , Q d ] = ρ , ( Q a Q b , x d ˙ c ) + ρ , ǫ bd Q a Q ˙ c + ρ , ( Q a Q d , x b ˙ c ) + ρ , ǫ ad Q b Q ˙ c + ρ , ( Q b Q d , x a ˙ c ) + ρ , ǫ ab Q ˙ c Q d , (65)[ Q a , Q b Q ˙ c , Q d ] = µ , ( Q a Q b , x d ˙ c ) + µ , ǫ bd Q a Q ˙ c + µ , ( Q a Q d , x b ˙ c ) + µ , ǫ ad Q b Q ˙ c + µ , ( Q b Q d , x a ˙ c ) + µ , ǫ ab Q ˙ c Q d , (66)[ Q a , Q b , Q ˙ c Q d ] = ν , ( Q a Q b , x d ˙ c ) + ν , ǫ bd Q a Q ˙ c + ν , ( Q a Q d , x b ˙ c ) + ν , ǫ ad Q b Q ˙ c + ν , ( Q b Q d , x a ˙ c ) + ν , ǫ ab Q ˙ c Q d . (67)
4. 4-th case
We seek 4-point associators with one dot as follows: (cid:2) Q a Q b , Q c Q ˙ d (cid:3) = ρ , (cid:0) Q a Q b , x c ˙ d (cid:1) + ρ , (cid:0) Q a Q c , x b ˙ d (cid:1) + ρ , ǫ bc Q a Q ˙ d + ρ , (cid:0) Q b Q c , x a ˙ d (cid:1) + ρ , ǫ ac Q b Q ˙ d + ρ , ǫ ab Q c Q ˙ d , (68) (cid:2) Q a , Q b Q c , Q ˙ d (cid:3) = µ , (cid:0) Q a Q b , x c ˙ d (cid:1) + µ , (cid:0) Q a Q c , x b ˙ d (cid:1) + µ , ǫ bc Q a Q ˙ d + µ , (cid:0) Q b Q c , x a ˙ d (cid:1) + µ , ǫ ac Q b Q ˙ d + µ , ǫ ab Q c Q ˙ d , (69) (cid:2) Q a , Q b , Q c Q ˙ d (cid:3) = ν , (cid:0) Q a Q b , x c ˙ d (cid:1) + ν , (cid:0) Q a Q c , x b ˙ d (cid:1) + ν , ǫ bc Q a Q ˙ d + ν , (cid:0) Q b Q c , x a ˙ d (cid:1) + ν , ǫ ac Q b Q ˙ d + ν , ǫ ab Q c Q ˙ d . (70)
5. Final form of the associators. Jacobiators
Taking into account the relation (39) (as in Section III A), we obtain the following 4-point associators with one dot[ Q ˙ a Q b , Q c , Q d ] = − ~ ℓ ζ ǫ cd Q ˙ a Q b + 12 ~ ℓ ˜ ρ , ( x b ˙ a , Q c Q d ) , (71)[ Q ˙ a , Q b Q c , Q d ] = 0 , (72)[ Q ˙ a , Q b , Q c Q d ] = − ~ ℓ ζ ǫ cd Q ˙ a Q b − ~ ℓ ˜ ρ , ( x b ˙ a , Q c Q d ) , (73) (cid:2) Q a Q ˙ b , Q c , Q d (cid:3) = 12 ~ ℓ ζ ǫ cd Q a Q ˙ b + 12 ~ ℓ ˜ ρ , (cid:0) x a ˙ b , Q c Q d (cid:1) , (74) (cid:2) Q a , Q ˙ b Q c , Q d (cid:3) = 0 , (75) (cid:2) Q a , Q ˙ b , Q c Q d (cid:3) = 12 ~ ℓ ζ ǫ cd Q a Q ˙ b − ~ ℓ ˜ ρ , (cid:0) x a ˙ b , Q c Q d (cid:1) , (76)[ Q a Q b , Q ˙ c , Q d ] = 12 ~ ℓ ζ ǫ ab Q ˙ c Q d + 12 ~ ℓ ˜ ρ , ( x d ˙ c , Q a Q b ) , (77)[ Q a , Q b Q ˙ c , Q d ] = 0 , (78)[ Q a , Q b , Q ˙ c Q d ] = 12 ~ ℓ ζ ǫ ab Q ˙ c Q d − ~ ℓ ˜ ρ , ( x d ˙ c , Q a Q b ) , (79) (cid:2) Q a Q b , Q c , Q ˙ d (cid:3) = − ~ ℓ ζ ǫ ab Q c Q ˙ d + 12 ~ ℓ ˜ ρ , (cid:0) x c ˙ d , Q a Q b (cid:1) , (80) (cid:2) Q a , Q b Q c , Q ˙ d (cid:3) = 0 , (81) (cid:2) Q a , Q b , Q c Q ˙ d (cid:3) = − ~ ℓ ζ ǫ ab Q c Q ˙ d − ~ ℓ ˜ ρ , (cid:0) x c ˙ d , Q a Q b (cid:1) . (82)Considering the relation (39), we have found that ζ = − ζ , (83)and we set | ˜ ρ , | = | ˜ ρ , | = | ˜ ρ , | = | ˜ ρ , | = 1 . (84)One can check that the Jacobiators are J ( Q ˙ a Q b , Q c , Q d ) = 0 (85) J (cid:0) Q a Q ˙ b , Q c , Q d (cid:1) = 0 (86) J ( Q a Q b , Q ˙ c , Q d ) = 0 (87) J (cid:0) Q a Q b , Q c , Q ˙ d (cid:1) = 0 (88)if ˜ ρ , = ˜ ρ , , (89)˜ ρ , = ˜ ρ , . (90) C. 4-point associators with two dots
In this section we consider the case with two dots and with the different locations.
1. 1-st case
We seek 4-point associators with two dots as follows: (cid:2) Q ˙ a Q ˙ b , Q c , Q d (cid:3) = ρ , Q ˙ a Q ˙ b ǫ cd + ρ , (cid:0) Q ˙ a Q c , y d ˙ b (cid:1) + ρ , (cid:0) Q ˙ a Q d , y c ˙ b (cid:1) + ρ , (cid:0) Q ˙ b Q c , y d ˙ a (cid:1) + ρ , (cid:0) Q ˙ b Q d , y c ˙ a (cid:1) + ρ , Q c Q d ǫ ˙ a ˙ b + ρ , M ( c ˙ a,d ˙ b ) + ρ , M ( c ˙ b,d ˙ a ) , (91) (cid:2) Q ˙ a , Q ˙ b Q c , Q d (cid:3) = µ , Q ˙ a Q ˙ b ǫ cd + µ , (cid:0) Q ˙ a Q c , y d ˙ b (cid:1) + µ , (cid:0) Q ˙ a Q d , y c ˙ b (cid:1) + µ , (cid:0) Q ˙ b Q c , y d ˙ a (cid:1) + µ , (cid:0) Q ˙ b Q d , y c ˙ a (cid:1) + µ , Q c Q d ǫ ˙ a ˙ b + µ , M ( c ˙ a,d ˙ b ) + µ , M ( c ˙ b,d ˙ a ) , (92) (cid:2) Q ˙ a , Q ˙ b , Q c Q d (cid:3) = ν , Q ˙ a Q ˙ b ǫ cd + ν , (cid:0) Q ˙ a Q c , y d ˙ b (cid:1) + ν , (cid:0) Q ˙ a Q d , y c ˙ b (cid:1) + ν , (cid:0) Q ˙ b Q c , y d ˙ a (cid:1) + ν , (cid:0) Q ˙ b Q d , y c ˙ a (cid:1) + ν , Q c Q d ǫ ˙ a ˙ b + ν , M ( c ˙ a,d ˙ b ) + ν , M ( c ˙ b,d ˙ a ) . (93)(94)
2. 2-nd case
We seek 4-point associators with two dots as follows:[ Q ˙ a Q b , Q ˙ c , Q d ] = ρ , ( Q ˙ a Q b , x d ˙ c ) + ρ , Q ˙ a Q ˙ c ǫ bd + ρ , ( Q ˙ a Q d , x b ˙ c ) + ρ , ( Q b Q ˙ c , x d ˙ a ) + ρ , Q b Q d ǫ ˙ a ˙ c + ρ , ( Q ˙ c Q d , x b ˙ a ) + ρ , M ( b ˙ a,d ˙ c ) + ρ , M ( b ˙ c,d ˙ a ) , (95)[ Q ˙ a , Q b Q ˙ c , Q d ] = µ , ( Q ˙ a Q b , x d ˙ c ) + µ , Q ˙ a Q ˙ c ǫ bd + µ , ( Q ˙ a Q d , x b ˙ c ) + µ , ( Q b Q ˙ c , x d ˙ a ) + µ , Q b Q d ǫ ˙ a ˙ c + µ , ( Q ˙ c Q d , x b ˙ a ) + µ , M ( b ˙ a,d ˙ c ) + µ , M ( b ˙ c,d ˙ a ) , (96)[ Q ˙ a , Q b , Q ˙ c Q d ] = ν , ( Q ˙ a Q b , x d ˙ c ) + ν , Q ˙ a Q ˙ c ǫ bd + ν , ( Q ˙ a Q d , x b ˙ c ) + ν , ( Q b Q ˙ c , x d ˙ a ) + ν , Q b Q d ǫ ˙ a ˙ c + ν , ( Q ˙ c Q d , x b ˙ a ) + ν , M ( b ˙ a,d ˙ c ) + ν , M ( b ˙ c,d ˙ a ) . (97)
3. 3-rd case
We seek 4-point associators with two dots as follows: (cid:2) Q ˙ a Q b , Q c , Q ˙ d (cid:3) = ρ , (cid:0) Q ˙ a Q b , x c ˙ d (cid:1) + ρ , (cid:0) Q ˙ a Q c , x b ˙ d (cid:1) + ρ , Q ˙ a Q ˙ d ǫ bc + ρ , (cid:0) Q b Q c ǫ ˙ a ˙ d (cid:1) + ρ , (cid:0) Q b Q ˙ d , x c ˙ a (cid:1) + ρ , (cid:0) Q c Q ˙ d , x b ˙ a (cid:1) + ρ , M ( b ˙ a,c ˙ d ) + ρ , M ( b ˙ d,c ˙ a ) , (98) (cid:2) Q ˙ a , Q b Q c , Q ˙ d (cid:3) = µ , (cid:0) Q ˙ a Q b , x c ˙ d (cid:1) + µ , (cid:0) Q ˙ a Q c , x b ˙ d (cid:1) + µ , Q ˙ a Q ˙ d ǫ bc + µ , (cid:0) Q b Q c ǫ ˙ a ˙ d (cid:1) + µ , (cid:0) Q b Q ˙ d , x c ˙ a (cid:1) + µ , (cid:0) Q c Q ˙ d , x b ˙ a (cid:1) + µ , M ( b ˙ a,c ˙ d ) + µ , M ( b ˙ d,c ˙ a ) , (99) (cid:2) Q ˙ a , Q b , Q c Q ˙ d (cid:3) = ν , (cid:0) Q ˙ a Q b , x c ˙ d (cid:1) + ν , (cid:0) Q ˙ a Q c , x b ˙ d (cid:1) + ν , Q ˙ a Q ˙ d ǫ bc + ν , (cid:0) Q b Q c ǫ ˙ a ˙ d (cid:1) + ν , (cid:0) Q b Q ˙ d , x c ˙ a (cid:1) + ν , (cid:0) Q c Q ˙ d , x b ˙ a (cid:1) + ν , M ( b ˙ a,c ˙ d ) + ν , M ( b ˙ d,c ˙ a ) . (100)
4. 4-th case
We seek 4-point associators with two dots as follows: (cid:2) Q a Q ˙ b , Q ˙ c , Q d (cid:3) = ρ , (cid:0) Q a Q ˙ b , x d ˙ c (cid:1) + ρ , (cid:0) Q a Q ˙ c , x d ˙ b (cid:1) + ρ , ǫ ˙ b ˙ c Q a Q d + ρ , ǫ ad Q ˙ b Q ˙ c + ρ , (cid:0) Q ˙ b Q d , x a ˙ c (cid:1) + ρ , (cid:0) Q ˙ c Q d , x a ˙ b (cid:1) + ρ , M ( a ˙ b,d ˙ c ) + ρ , M ( a ˙ c,d ˙ b ) , (101) (cid:2) Q a , Q ˙ b Q ˙ c , Q d (cid:3) = µ , (cid:0) Q a Q ˙ b , x d ˙ c (cid:1) + µ , (cid:0) Q a Q ˙ c , x d ˙ b (cid:1) + µ , ǫ ˙ b ˙ c Q a Q d + µ , ǫ ad Q ˙ b Q ˙ c + µ , (cid:0) Q ˙ b Q d , x a ˙ c (cid:1) + µ , (cid:0) Q ˙ c Q d , x a ˙ b (cid:1) + µ , M ( a ˙ b,d ˙ c ) + µ , M ( a ˙ c,d ˙ b ) , (102) (cid:2) Q a , Q ˙ b , Q ˙ c Q d (cid:3) = ν , (cid:0) Q a Q ˙ b , x d ˙ c (cid:1) + ν , (cid:0) Q a Q ˙ c , x d ˙ b (cid:1) + ν , ǫ ˙ b ˙ c Q a Q d + ν , ǫ ad Q ˙ b Q ˙ c + ν , (cid:0) Q ˙ b Q d , x a ˙ c (cid:1) + ν , (cid:0) Q ˙ c Q d , x a ˙ b (cid:1) + ν , M ( a ˙ b,d ˙ c ) + ν , M ( a ˙ c,d ˙ b ) . (103)
5. 5-th case
We seek 4-point associators with two dots as follows: (cid:2) Q a Q ˙ b , Q c , Q ˙ d (cid:3) = ρ , (cid:0) Q a Q ˙ b , x c ˙ d (cid:1) + ρ , ǫ ˙ b ˙ d Q a Q c + ρ , (cid:0) Q a Q ˙ d , x c ˙ b (cid:1) + ρ , (cid:0) Q ˙ b Q c , x a ˙ d (cid:1) + ρ , ǫ ac Q ˙ b Q ˙ d + ρ , (cid:0) Q c Q ˙ d , x a ˙ b (cid:1) + ρ , M ( a ˙ b,c ˙ d ) + ρ M ( a ˙ d,c ˙ b ) , (104) (cid:2) Q a , Q ˙ b Q c , Q ˙ d (cid:3) = µ , (cid:0) Q a Q ˙ b , x c ˙ d (cid:1) + µ , ǫ ˙ b ˙ d Q a Q c + µ , (cid:0) Q a Q ˙ d , x c ˙ b (cid:1) + µ , (cid:0) Q ˙ b Q c , x a ˙ d (cid:1) + µ , ǫ ac Q ˙ b Q ˙ d + µ , (cid:0) Q c Q ˙ d , x a ˙ b (cid:1) + µ , M ( a ˙ b,c ˙ d ) + µ , M ( a ˙ d,c ˙ b ) , (105) (cid:2) Q a , Q ˙ b , Q c Q ˙ d (cid:3) = ν , (cid:0) Q a Q ˙ b , x c ˙ d (cid:1) + ν , ǫ ˙ b ˙ d Q a Q c + ν , (cid:0) Q a Q ˙ d , x c ˙ b (cid:1) + ν , (cid:0) Q ˙ b Q c , x a ˙ d (cid:1) + ν , ǫ ac Q ˙ b Q ˙ d + ν , (cid:0) Q c Q ˙ d , x a ˙ b (cid:1) + ν , M ( a ˙ b,c ˙ d ) + ν , M ( a ˙ d,c ˙ b ) . (106)
6. 6-th case
We seek 4-point associators with two dots as follows: (cid:2) Q a Q b , Q ˙ c , Q ˙ d (cid:3) = ρ , ǫ ˙ c ˙ d Q a Q b + ρ , (cid:0) Q a Q ˙ c , x b ˙ d (cid:1) + ρ , (cid:0) Q a Q ˙ d , x b ˙ c (cid:1) + ρ , (cid:0) Q b Q ˙ c , x a ˙ d (cid:1) + ρ , (cid:0) Q b Q ˙ d , x a ˙ c (cid:1) + ρ , ǫ ab Q ˙ c Q ˙ d + ρ , M ( a ˙ c,b ˙ d ) + ρ , M ( a ˙ d,b ˙ c ) , (107) (cid:2) Q a , Q b Q ˙ c , Q ˙ d (cid:3) = µ , ǫ ˙ c ˙ d Q a Q b + µ , (cid:0) Q a Q ˙ c , x b ˙ d (cid:1) + µ , (cid:0) Q a Q ˙ d , x b ˙ c (cid:1) + µ , (cid:0) Q b Q ˙ c , x a ˙ d (cid:1) + µ , (cid:0) Q b Q ˙ d , x a ˙ c (cid:1) + µ , ǫ ab Q ˙ c Q ˙ d + µ , M ( a ˙ c,b ˙ d ) + µ , M ( a ˙ d,b ˙ c ) , (108) (cid:2) Q a , Q b , Q ˙ c Q ˙ d (cid:3) = ν , ǫ ˙ c ˙ d Q a Q b + ν , (cid:0) Q a Q ˙ c , x b ˙ d (cid:1) + ν , (cid:0) Q a Q ˙ d , x b ˙ c (cid:1) + ν , (cid:0) Q b Q ˙ c , x a ˙ d (cid:1) + ν , (cid:0) Q b Q ˙ d , x a ˙ c (cid:1) + ν , ǫ ab Q ˙ c Q ˙ d + ν , M ( a ˙ c,b ˙ d ) + ν , M ( a ˙ d,b ˙ c ) . (109)
7. Final form of the associators. Jacobiators
Using the relation (39) for the connection between 3- and 4-point associators and the simplifications similar to thoseof used in section III A, we obtain (cid:2) Q ˙ a Q ˙ b , Q c , Q d (cid:3) = − ~ ℓ ζ Q ˙ a Q ˙ b ǫ cd + ~ ℓ ˜ ρ , M ( c ˙ a,d ˙ b ) + ~ ℓ ˜ ρ , M ( c ˙ b,d ˙ a ) , (110) (cid:2) Q ˙ a , Q ˙ b Q c , Q d (cid:3) = ~ ℓ ˜ µ , M ( c ˙ a,d ˙ b ) + ~ ℓ ˜ µ , M ( c ˙ b,d ˙ a ) , (111) (cid:2) Q ˙ a , Q ˙ b , Q c Q d (cid:3) = − ~ ℓ ζ Q c Q d ǫ ˙ a ˙ b + ~ ℓ ˜ ν , M ( c ˙ a,d ˙ b ) + ~ ℓ ˜ ν , M ( c ˙ b,d ˙ a ) , (112)[ Q ˙ a Q b , Q ˙ c , Q d ] = 12 ~ ℓ ˜ ρ , ( Q ˙ a Q b , x d ˙ c ) + 12 ~ ℓ ˜ ρ , ( x b ˙ a , Q ˙ c Q d ) + ~ ℓ ˜ ρ , M ( b ˙ a,d ˙ c ) + ~ ℓ ˜ ρ , M ( b ˙ c,d ˙ a ) , (113)[ Q ˙ a , Q b Q ˙ c , Q d ] = ~ ℓ ˜ µ , M ( b ˙ a,d ˙ c ) + ~ ℓ ˜ µ , M ( b ˙ c,d ˙ a ) , (114)[ Q ˙ a , Q b , Q ˙ c Q d ] = − ~ ℓ ˜ ρ , ( Q ˙ a Q b , x d ˙ c ) − ~ ℓ ˜ ρ , ( x b ˙ a , Q ˙ c Q d ) + ~ ℓ ˜ ν , M ( b ˙ a,d ˙ c ) + ~ ℓ ˜ ν , M ( b ˙ c,d ˙ a ) , (115) (cid:2) Q ˙ a Q b , Q c , Q ˙ d (cid:3) = ~ ℓ ˜ ρ , (cid:0) Q ˙ a Q b , x c ˙ d (cid:1) + ~ ℓ ζ Q ˙ a Q ˙ d ǫ bc + ~ ℓ ˜ ρ , ǫ ˙ a ˙ d Q b Q c + ~ ℓ ˜ ρ , (cid:0) Q c Q ˙ d , x b ˙ a (cid:1) + ~ ℓ ˜ ρ , M ( b ˙ a,c ˙ d ) + ~ ℓ ˜ ρ , M ( b ˙ d,c ˙ a ) , (116) (cid:2) Q ˙ a , Q b Q c , Q ˙ d (cid:3) = − ~ ℓ ˜ ρ , ǫ ˙ a ˙ d Q b Q c + ˜ µ , M ( b ˙ a,c ˙ d ) + ˜ µ , M ( b ˙ d,c ˙ a ) , (117) (cid:2) Q ˙ a , Q b , Q c Q ˙ d (cid:3) = − ~ ℓ ˜ ρ , (cid:0) Q ˙ a Q b , x c ˙ d (cid:1) + ~ ℓ ζ Q ˙ a Q ˙ d ǫ bc + ~ ℓ ˜ ρ , ǫ ˙ a ˙ d Q b Q c − ~ ℓ ˜ ρ , (cid:0) Q c Q ˙ d , x b ˙ a (cid:1) + ~ ℓ ˜ ν , M ( b ˙ a,c ˙ d ) + ~ ℓ ˜ ν , M ( b ˙ d,c ˙ a ) , (118) (cid:2) Q a Q ˙ b , Q ˙ c , Q d (cid:3) = ~ ℓ ˜ ρ , (cid:0) Q a Q ˙ b , x d ˙ c (cid:1) + ~ ℓ ζ ǫ ˙ b ˙ c Q a Q d + ~ ℓ ˜ ρ , ǫ ad Q ˙ b Q ˙ c + ~ ℓ ˜ ρ , (cid:0) Q ˙ c Q d , x a ˙ b (cid:1) + ~ ℓ ˜ ρ , M ( a ˙ b,d ˙ c ) + ~ ℓ ˜ ρ , M ( a ˙ c,d ˙ b ) , (119) (cid:2) Q a , Q ˙ b Q ˙ c , Q d (cid:3) = − ~ ℓ ˜ ρ , ǫ ad Q ˙ b Q ˙ c + ~ ℓ ˜ µ , M ( a ˙ b,d ˙ c ) + ~ ℓ ˜ µ , M ( a ˙ c,d ˙ b ) , (120) (cid:2) Q a , Q ˙ b , Q ˙ c Q d (cid:3) = − ~ ℓ ˜ ρ , (cid:0) Q a Q ˙ b , x d ˙ c (cid:1) + ~ ℓ ζ ǫ ˙ b ˙ c Q a Q d + ~ ℓ ˜ ρ , ǫ ad Q ˙ b Q ˙ c − ~ ℓ ˜ ρ , (cid:0) Q ˙ c Q d , x a ˙ b (cid:1) + ~ ℓ ˜ ν , M ( a ˙ b,d ˙ c ) + ~ ℓ ˜ ν , M ( a ˙ c,d ˙ b ) , (121) (cid:2) Q a Q ˙ b , Q c , Q ˙ d (cid:3) = ~ ℓ ˜ ρ , (cid:0) Q a Q ˙ b , x c ˙ d (cid:1) + ~ ℓ ˜ ρ , (cid:0) x a ˙ b , Q c Q ˙ d (cid:1) + ~ ℓ ˜ ρ , M ( a ˙ b,d ˙ c ) + ~ ℓ ˜ ρ , M ( a ˙ c,d ˙ b ) , (122) (cid:2) Q a , Q ˙ b Q c , Q ˙ d (cid:3) = ~ ℓ ˜ µ , M ( a ˙ b,c ˙ d ) + ~ ℓ ˜ µ , M ( a ˙ d,c ˙ b ) , (123) (cid:2) Q a , Q ˙ b , Q c Q ˙ d (cid:3) = − ~ ℓ ˜ ρ , (cid:0) Q a Q ˙ b , x c ˙ d (cid:1) − ~ ℓ ˜ ρ , (cid:0) x a ˙ b , Q c Q ˙ d (cid:1) + ~ ℓ ˜ ν , M ( a ˙ b,d ˙ c ) + ~ ℓ ˜ ν , M ( a ˙ c,d ˙ b ) , (124) (cid:2) Q a Q b , Q ˙ c , Q ˙ d (cid:3) = − ~ ℓ ζ ǫ ˙ c ˙ d Q a Q b + ~ ℓ ˜ ρ , M ( a ˙ c,b ˙ d ) + ~ ℓ ˜ ρ , M ( a ˙ d,b ˙ c ) , (125) (cid:2) Q a , Q b Q ˙ c , Q ˙ d (cid:3) = ~ ℓ ˜ µ , M ( a ˙ c,b ˙ d ) + ~ ℓ ˜ µ , M ( a ˙ d,b ˙ c ) , (126) (cid:2) Q a , Q b , Q ˙ c Q ˙ d (cid:3) = − ~ ℓ ζ ǫ ab Q ˙ c Q ˙ d + ~ ℓ ˜ ν , M ( a ˙ c,b ˙ d ) + ~ ℓ ˜ ν , M ( a ˙ d,b ˙ c ) , (127)where ( . . . , . . . ) is either commutator or anticommutator, and with the limitations˜ ρ , − ˜ µ , + ˜ ν , = 0 , (128)˜ ρ , − ˜ µ , + ˜ ν , = 0 , (129)˜ ρ , − ˜ µ , + ˜ ν , = 0 , (130)˜ ρ , − ˜ µ , + ˜ ν , = 0 , (131)˜ ρ , − ˜ µ , + ˜ ν , = 0 , (132)˜ ρ , − ˜ µ , + ˜ ν , = 0 , (133)˜ ρ , − ˜ µ , + ˜ ν , = 0 , (134)˜ ρ , − ˜ µ , + ˜ ν , = 0 , (135)˜ ρ , − ˜ µ , + ˜ ν , = 0 , (136)˜ ρ , − ˜ µ , + ˜ ν , = 0 , (137)˜ ρ , − ˜ µ , + ˜ ν , = 0 , (138)˜ ρ , − ˜ µ , + ˜ ν , = 0 . (139)One can check that the Jacobiators are (here we consider the simplest case x a, ˙ b = M ( a ˙ b,c ˙ d ) = 0) J (cid:0) Q ˙ a Q ˙ b , Q c , Q d (cid:1) = 0 , (140) J (cid:0) Q ˙ a , Q ˙ b Q c , Q d (cid:1) = − ~ ℓ ζ ǫ cd (cid:2) Q ˙ a , Q ˙ b (cid:3) − ~ ℓ ζ ǫ ˙ a ˙ b [ Q c , Q d ] , (141) J (cid:0) Q ˙ a , Q ˙ b , Q c Q d (cid:1) = 0 , (142) J ( Q ˙ a Q b , Q ˙ c , Q d ) = − ~ ℓ ζ ǫ bd [ Q ˙ a , Q ˙ c ] − ~ ℓ ζ ǫ ˙ a ˙ c [ Q b , Q d ] , (143) J ( Q ˙ a , Q b Q ˙ c , Q d ) = ~ ℓ ζ ǫ bd [ Q ˙ a , Q ˙ c ] + ~ ℓ ζ ǫ ˙ a ˙ c [ Q b , Q d ] , (144) J ( Q ˙ a , Q b , Q ˙ c Q d ) = − ~ ℓ ζ ǫ bd [ Q ˙ a , Q ˙ c ] − ~ ℓ ζ ǫ ˙ a ˙ c [ Q b , Q d ] , (145) J (cid:0) Q ˙ a Q b , Q c Q ˙ d (cid:1) = ~ ℓ ζ ǫ bc (cid:2) Q ˙ a , Q ˙ d (cid:3) + ~ ℓ ζ ǫ ˙ a ˙ d [ Q b , Q c ] , (146) J (cid:0) Q ˙ a , Q b Q c , Q ˙ d (cid:1) = 0 , (147) J (cid:0) Q ˙ a , Q b , Q c Q ˙ d (cid:1) = ~ ℓ ζ ǫ bc (cid:2) Q ˙ a , Q ˙ d (cid:3) + ~ ℓ ζ ǫ ˙ a ˙ d [ Q b , Q c ] , (148) J (cid:0) Q a Q ˙ b , Q ˙ c , Q d (cid:1) = ~ ℓ ζ ǫ ad (cid:2) Q ˙ b , Q ˙ c (cid:3) + ~ ℓ ζ ǫ ˙ b ˙ c [ Q a , Q d ] , (149) J (cid:0) Q a , Q ˙ b Q ˙ c , Q d (cid:1) = 0 , (150) J (cid:0) Q a , Q ˙ b , Q ˙ c Q d (cid:1) = ~ ℓ ζ ǫ ad (cid:2) Q ˙ b , Q ˙ c (cid:3) + ~ ℓ ζ ǫ ˙ b ˙ c [ Q a , Q d ] , (151) J (cid:0) Q a Q ˙ b , Q c , Q ˙ d (cid:1) = − ~ ℓ ζ ǫ ac (cid:2) Q ˙ b , Q ˙ d (cid:3) − ~ ℓ ζ ǫ ˙ b ˙ d [ Q a , Q c ] , (152) J (cid:0) Q a , Q ˙ b Q c , Q ˙ d (cid:1) = ~ ℓ ζ ǫ ac (cid:2) Q ˙ b , Q ˙ d (cid:3) + ~ ℓ ζ ǫ ˙ b ˙ d [ Q a , Q c ] , (153) J (cid:0) Q a , Q ˙ b , Q c Q ˙ d (cid:1) = − ~ ℓ ζ ǫ ac (cid:2) Q ˙ b , Q ˙ d (cid:3) − ~ ℓ ζ ǫ ˙ b ˙ d [ Q a , Q c ] , (154) J (cid:0) Q a Q b , Q ˙ c Q ˙ d (cid:1) = 0 , (155) J (cid:0) Q a , Q b Q ˙ c , Q ˙ d (cid:1) = − ~ ℓ ζ ǫ ab (cid:2) Q ˙ c , Q ˙ d (cid:3) − ~ ℓ ζ ǫ ˙ c ˙ d [ Q a , Q b ] , (156) J (cid:0) Q a , Q b , Q ˙ c Q ˙ d (cid:1) = 0 , (157)(158)0if ˜ ρ , = ζ , (159)˜ ρ , = ζ , (160) (cid:8) Q ˙ a , Q ˙ b (cid:9) = { Q a , Q b } = 0 . (161) D. 4-point associators with two dots
This case is identical to subsection III B after replacing ˙ a ↔ a . IV. THE CONNECTION WITH SUPERSYMMETRY
Now we want to pounce on supersymmetry. In this case the operators Q a, ˙ b obey the following anticommutators { Q a , Q b } = (cid:8) Q ˙ a , Q ˙ b (cid:9) = 0 , (162) { Q a , Q ˙ a } = Q a Q ˙ a + Q ˙ a Q a = 2 σ µa ˙ a P µ , (163)where the operator P can be a nonassociative generalization of standard accosiative operator − i ~ ∂ µ ; the Pauli matrices σ µa ˙ a , σ a ˙ aµ are defined in the standard way σ µa ˙ a = (cid:26)(cid:18) (cid:19) , (cid:18) (cid:19) , (cid:18) − ii (cid:19) , (cid:18) − (cid:19)(cid:27) (164) σ a ˙ aµ = (cid:26)(cid:18) (cid:19) , (cid:18) (cid:19) , (cid:18) i − i (cid:19) , (cid:18) − (cid:19)(cid:27) (165)with the orthogonality relations for the Pauli matrices σ a ˙ aµ σ νa ˙ a = 2 δ νµ , σ a ˙ aµ σ µb ˙ b = 2 δ ab δ ˙ a ˙ b . (166)Let us note the following interesting relation which follows from (30)( Q a Q b ) Q b = − ~ ℓ ζ Q b ǫ ab . (167)Roughly speaking, one can say that this expression “destroys” sometimes Q a = 0 property of Grassmann numbers.This results in distinctions between supersymmetries based on associative and nonassociative generators. But thisdifference will be (in the dimensionless form) of the order of l P l /ℓ , where ℓ is some characteristic length. Forexample, if ℓ = Λ − / (where Λ is the cosmological constant) then this difference will be ≈ − . V. SUPERSYMMETRY, HIDDEN VARIABLES, AND NONASSOCIATIVITY
In this section we want to consider a possible connection between supersymmetry, hidden variables, and nonasso-ciativity.First of all we want to remind what is the hidden variables theory. In Wiki [11] one can find the following definitionof hidden variables theories ”. . . hidden variable theories were espoused by some physicists who argued that the stateof a physical system, as formulated by quantum mechanics, does not give a complete description for the system; i.e.,that quantum mechanics is ultimately incomplete, and that a complete theory would provide descriptive categoriesto account for all observable behavior and thus avoid any indeterminism. . . .. . . In 1964, John Bell showed that if local hidden variables exist, certain experiments could be performed involvingquantum entanglement where the result would satisfy a Bell inequality. . . .Physicists such as Alain Aspect [12] and Paul Kwiat [13] have performed experiments that have found violations ofthese inequalities up to 242 standard deviations[14] (excellent scientific certainty). This rules out local hidden variabletheories.. . . Gerard ’t Hooft [14, 15] has disputed the validity of Bell’s theorem on the basis of the superdeterminism loopholeand proposed some ideas to construct local deterministic models.“1We want to pay attention to what we talked about the associative observables. That is natural for physical quantitiesin quantum mechanics. But in Ref. [16] the possibility of consideration of nonassociative hidden variables is discussed.In this case these quantities are unobservable ones.Let us consider what happens in our case. We have supersymmetric decomposition of (probably generalized)momentum operator (163). The constituents Q a, ˙ a are unobservable according to the nonassociative properties (30)-(37) and (55)-(57). Following this way, we can say that we have unobservable nonassociative operators Q a, ˙ a that aresimilar to hidden variables. The main difference compared with the standard hidden variables is unobservability ofthe nonassociative hidden-like variables. VI. DISCUSSION AND CONCLUSIONS
Thus we have considered a nonassociative generalization of supersymmetry. We have shown that: (a) one can choosesuch a form of 3-point associators that the corresponding Jacobiators are zero; (b) there is the relation between 3-and 4-point associators; (c) using these expressions, one can find 4-point associators.We have seen that in all definitions of associators there is the Planck constant and some characteristic length ℓ .The presence of the Planck constant permits us to make natural assumptions that these associators can be regardedas a nonassociative generalization of the Heisenberg uncertainty principle. In this case the characteristic length ℓ will be a new fundamental constant, and the corrections arising in this case have the order of ~ /ℓ . For example, if ℓ ≈ Λ − / (where Λ is the cosmological constant) then the dimensionless corrections ≈ − , i.e., are negligible.Instead of introducing a fundamental length ℓ , we can introduce a fundamental momentum P = ~ /ℓ . Physicalconsequences of introducing new fundamental quantities ℓ or P (that are consequences of nonassociativity) are: • There appears a minimum momentum P . • There appears a maximum length ℓ . • The appearance of the maximum length ℓ leads to the fact that the curvature is bounded below: R min ≈ /ℓ . • The minimum momentum P and the maximum length ℓ are connected by the Heisenberg uncertainty principle: P ℓ ≈ ~ . • The experimental manifestation of possible nonaccosiativity can arise only for a physical phenomenon wheneither the momentum p ≈ P ≈ − kg · m · s − or on the scales l ≈ ℓ ≈ Λ − / ≈ m (if the fundamentallength ℓ ≈ Λ − / ).We have also discussed a possible interpretation of nonassociative supersymmetric generators Q a, ˙ a as hidden-likevariables in quantum theory. The main idea here is that the nonassociativity leads to unobservability of thesevariables. Acknowledgements
This work was supported by a grant Φ . [1] M. Gunaydin and F. Gursey. Quark statistics and octonions. Phys. Rev., D9 (1974) 3387-3391.[2] M. Gogberashvili. Octonionic electrodynamics, J. Phys. A (2006) 7099-7104; [arXiv:hep-th/0512258].[3] M. Gogberashvili. Octonionic version of Dirac equations, Int. J. Mod. Phys. A (2006) 3513-3524, [arXiv:hep-th/0505101].[4] M. Gnaydin and D. Minic, Fortsch. Phys. , 873 (2013) [arXiv:1304.0410 [hep-th]].[5] M. Bojowald, S. Brahma and U. Buyukcam, Phys. Rev. Lett. (2015) 22, 220402; doi:10.1103/PhysRevLett.115.220402,[arXiv:1510.07559 [quant-ph]].[6] V. Dzhunushaliev, Found. Phys. Lett. , 157 (2006); doi:10.1007/s10702-006-0373-2, [hep-th/0502216].[7] S. Okubo, “Introduction to octonion and other nonassociative algebras in physics,” Cambridge University Press, 1995.[8] F. G¨ursey and C-H Tze, “On the Role of Division, Jordan, and Related Algebras in Particle Physics,” World Scientific,1996.[9] V. Dzhunushaliev, “Three-point non – associative supersymmetry generalization and weak version of non – commutativecoordinates,” arXiv:1504.00573 [hep-th]. [10] V. Dzhunushaliev, “Nonassociative Generalization of Supersymmetry,” Adv. Appl. Clifford Algebras, 2015 Springer Basel,DOI 10.1007/s00006-015-0580-7; arXiv:1302.0346 [math-ph].[11] https://en.wikipedia.org/wiki/Hidden_variable_theory .[12] A. Aspect, P. Grangier and G. Roger, Phys. Rev. Lett. , 91 (1982).[13] P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum and P. H. Eberhard, Phys. Rev. A60