aa r X i v : . [ m a t h - ph ] J un On geometry of the scator space
Artur Kobus
Politechnika Bia lostocka, Wydzia l Budownictwa i In˙zynierii ´Srodowiskaul. Wiejska 45E, 15-351 Bia lystok, Poland
Jan L. Cie´sli´nski ∗ , Uniwersytet w Bia lymstoku, Wydzia l Fizykiul. Cio lkowskiego 1L, 15-245 Bia lystok, Poland
Abstract
We consider the scator space - a hypercomplex, non-distributivehyperbolic algebra introduced by Fern´andez-Guasti and Zald´ıvar. Wediscuss isometries of the scator space and find consequent method fortreating them algebraically, along with scators themselves. It occursthat introduction of zero divisors cannot be avoided while dealing withthese isometries. The scator algebra may be endowed with a nicephysical interpretation, although it suffers from lack of some physicallydemanded important features. Despite that, there arises some openquestions, e.g., whether hypothetical tachyons can be considered asusual particles possessing time-like trajectories.
MSC 2010 : 30G35; 20M14
Keywords :scators, tachyons, non-distributive algebras, hypercomplex numbers.
Following Fern´andez-Guasti and Zald´ıvar [1] we consider a commutative,non-distributive 1 + 2 dimensional algebra S , which is also associative pro-vided that divisors of zero are excluded. The elements of this algebra will becalled 1 + 2 dimensional scators [1]. Scators (a kind of hypercomplex num-bers) are denoted by o a = ( a ; a , a ), where components a , a are referredto as director components, and a is usually called scalar component, or, in ∗ E-mail: [email protected] temporal component. The space of scators possesses theadditive structure of usual vector space, with scalars being its subset, closedunder addition and multiplication. In this paper we confine ourselves tothis definition of a scator, leaving aside earlier concepts of scators as objectsgeneralizing scalars and vectors, see [2, 3].It was shown in [4] that this algebra may be given physical interpretationcorresponding to some ideas of special relativity, although metric in thescator space is different from the standard metric of Minkowski space. Thisscator metric (defined below) is called scator-deformed Lorentz metric. Itwas emphasized that scators and deformed metrics can describe a kinematicsof some kind of particles, including hypothetical tachyons.In our paper we study metric properties of scators, paying special at-tention to proper definitions for causal realms appearing in this framework,what finally will lead to some convergence with work of Kapu´scik [5].We emphasize the fact that multiplication acts in a non-distributive way,what is the hallmark of the structure. Indeed, we have o a o b = ( a ; a , a )( b ; b , b ) == (cid:18) a b + a b + a b + a a b b a b ; a b + a b + a a b a + a b b b ,a b + a b + a a b a + a b b b (cid:19) . (1.1)Therefore, computing∆( o a, o b ; o c ) := ( o a + o b ) o c − o a o c − o b o c = ( b a − a b )( a b − b a ) a b ( a + b ) (cid:18) c c c ; c , c (cid:19) , (1.2)where o c = ( c ; c , c ), we easily see that in the generic case the righ-handside does not vanish, i.e., ( o a + o b ) o c = o a o c + o b o c provided that b a = a b and a b = b a . Hence, in general, the scator product is not distributive. Bythe way, the scator appearing on the right-hand side of (1.2) will be referredto as dual to o c (see Definition 2.1 below).Many properties of the scator product (1.1) were widely investigated inmany contexts [1, 8, 9], also physical [4]. To gain some insight in possi-ble physical interpretation of the scator algebra, we recall here some basicterminology from [1] and [9], although modified a little:2 efinition 1.1. The modulus squared of a scator is given by k o a k = o a o a ∗ = a (cid:18) − a a (cid:19) (cid:18) − a a (cid:19) = a − a − a + a a a . (1.3) Definition 1.2.
We say that a scator is time-like, if a > a , and a > a , or a < a , and a < a , (1.4) it is said to be space-like, if a < a , and a > a or a < a , and a > a , (1.5) and it is light-like, when a = a , or a = a . (1.6)The division proposed above is analogous to what is well known fromspecial relativity. Note that in both cases componentwise addition of scatorsor vectors does not preserve this division. On the other hand, the norm ofa product of two scators is just a product of their norms [9], which is veryuseful in this context. In this paper we present new results concerning themetric structure of the scator space, extending results of [4, 9].For instance, we indicate that bipyramid considered in [4] has some kindof time-like “wings” around, described by the regime a < a and a < a ,while, for example, in [9], there were considered mainly time-like eventsinside the light bipyramid ( a > a and a > a ). Time-like region ismarked as dark at Fig. 1.We underline that the existence of such wings has been overlooked inthe paper [4], what has its consequences in possible physical interpretationproposed in next sections. This seems to reveal some new aspects of causalityin scator-deformed Lorentz metric (see the picture at end of the paper).A closely related notion, “super super-restricted space conditions”, wasintroduced in [9] without a direct relation to the type of considered events.Super-restricted space conditions define either time-like events (in even-dimensional spaces) or space-like events (in odd-dimensional spaces).The paper is organized as follows. In section 2 we introduce basic objectsand transformations responsible for isometries in the scator space S . Then,in sections 3 and 4 we propose and develop a new framework in whichcalculation proceed in a more natural way, using a distributive product. Insection 5 we continue along these lines focusing on isometries. Section 6 isentirely devoted to the question of metric properties of scators; in particular,we obtain possible closest analogue of scalar product we can get, althoughit is even not bilinear. The last section contains physical comments andconclusions. 3igure 1: Time-like wings at a fixed time a ( a = 0). Duality operations(section 2) take us out from the bipyramid (represented in this cross-sectionby the inner square) to other causal realms, and conversely.4 Dualities - phenomenological treatment
Now we turn our attention to the issue of isometries in the scator space S .We begin with defining some operations and then check their properties. Definition 2.1.
If we have a 3-scator o a = ( a ; a , a ) , then we call thescator of the form o ¯ a = (cid:18) a a a , a , a (cid:19) (2.1) its dual (or ordinary dual) scator, and star denotes hypercomplex conjugate: o a ∗ = ( a ; − a , − a ) . (2.2) Lemma 2.2.
Operation of duality commutes with hypercomplex conjugation.Proof:
We have( o ¯ a ) ∗ = (cid:18) a a a , a , a (cid:19) ∗ = (cid:18) a a a , − a , − a (cid:19) (2.3)and o ( a ∗ ) = ( a ; − a , − a ) = (cid:18) a a a , − a , − a (cid:19) (2.4)which are evidently equal. Lemma 2.3.
Operation of duality is idempotent: o (¯ a ) = o a .Proof: This follows instantly by straightforward calculation.
Remark 2.4.
Hypercomplex conjugation is a homomorphism in S , i.e., ( o a o b ) ∗ = ( o a ) ∗ ( o b ) ∗ . (2.5) Operation of duality does not provide yet another homomorphic sctructurein S , i.e. o ab = o ¯ a o ¯ b . The above statements follow from considerations similar to direct cal-culations included in [7]. Soon we will get a better understanding of thesefacts by applying a new approach which is both faster and simpler.5 roposition 2.5.
Operation of duality preserves the scator product: o ¯ a o ¯ b = o a o b. (2.6) Proof:
We present explicit calculation for the scalar component:( o a o b ) = a b + a b + a b + a a b b a b , (2.7)and ( o ¯ a o ¯ b ) = ¯ a ¯ b + ¯ a ¯ b + ¯ a ¯ b + ¯ a ¯ a ¯ b ¯ b ¯ a ¯ b == a a b b a b + a b + a b + a a b b a a b b a b == a b + a b + a b + a a b b a b = ( o a o b ) . (2.8)Similar direct computation can be done for director components. Proposition 2.6.
Operation of duality is an isometry in 3-scator space.Proof:
For ordinary scator o a we have its norm k o a k = o a o a ∗ = a (cid:18) − a a (cid:19) (cid:18) − a a (cid:19) . (2.9)Thus, for the dual scator o ¯ a , we get k o ¯ a k = o ¯ a o ¯ a ∗ = (cid:18) a a a , a , a (cid:19) (cid:18) a a a , − a , − a (cid:19) == a a a (cid:18) − a a a a (cid:19)(cid:18) − a a a a (cid:19) == a a a + a − a − a = a (cid:18) − a a (cid:19) (cid:18) − a a (cid:19) , (2.10)which exactly coincides with the norm of the original scator. Remark 2.7.
Taking into account (1.1), (2.2) and (2.1), we can easilyverify that o a ∗ o b + o a o b ∗ = 2 (cid:18) a b − a b − a b + a a b b a b (cid:19) , (2.11) where the right-hand side is proportional to (omitted for simplicity hereand in many other places). efinition 2.8. If we have a 3-scator o a = ( a ; a , a ) , then we call a scatorof the form o ¯ a i = (cid:18) a ; a , a a a (cid:19) (2.12) its internal dual scator, and a scator of the form o ¯ a e = (cid:18) a ; a a a , a (cid:19) (2.13) its external dual. Lemma 2.9.
Internal and external duality operations anti-commute withhypercomplex conjugation.Proof:
We have o (¯ a i ) ∗ = (cid:18) a ; a , a a a (cid:19) ∗ = (cid:18) a ; − a , − a a a (cid:19) (2.14)and o ( a ∗ ) i = ( a ; − a , − a ) i = (cid:18) − a ; a , a a a (cid:19) , (2.15)so that( o ¯ a ) ∗ i + o ( a ∗ ) i = 0 . (2.16)Similarly we have( o ¯ a ) ∗ e = ( a ; a a a , a ) ∗ = (cid:18) a ; − a a a , − a (cid:19) (2.17)and o ( a ∗ ) e = ( a ; − a , − a ) e = (cid:18) − a ; a a a , a (cid:19) , (2.18)so that( o ¯ a e ) ∗ + ( a ∗ ) e = 0 , (2.19)which ends the proof. 7 emma 2.10. Internal and external duality operations are idempotent.Proof:
It is enough to apply twice definitions of both operations.
Remark 2.11.
Operation of external and internal duality do not provideyet another homomorphic sctructures in S , so i.e. ( o ab ) i = o ¯ a io ¯ b i , ( o ab ) e = o ¯ a eo ¯ b e , (2.20) which follows from straightforwad calculation. Lemma 2.12.
Operations of internal and external duality preserve scatorproduct.Proof:
By direct computation, similar to the case of ordinary duality.
Definition 2.13.
A transformation that exchanges time-like events withspace-like events (and conversely) and leaves the type of light-like eventsunchanged is called a causality swap (or a pseudo-isometry).
Proposition 2.14.
Both internal and external duality operations are causal-ity swaps of 3-scator space.Proof:
Denoting o ¯ a i = (¯ a i ; ¯ a i , ¯ a i ), we compute: k o ¯ a i k = ¯ a i (cid:18) − ¯ a i ¯ a i (cid:19) (cid:18) − ¯ a i ¯ a i (cid:19) == ¯ a i (cid:18) − ¯ a i ¯ a i − ¯ a i ¯ a i + ¯ a i ¯ a i ¯ a i ¯ a i (cid:19) = a (cid:18) − a a (cid:19) (cid:18) − a a (cid:19) == − a (cid:18) − a a (cid:19) (cid:18) − a a (cid:19) = −k o a k (2.21)Therefore, if original scator represents a time-like event, then its internaldual has to represent a space-like event, and vice versa . Light-like scators donot change their type. Analogous computation, with the same consequences,can be done for the external dual. Corollary 2.15.
Duality operations commuting with hypercomplex conju-gation are isometries, while duality operations anti-commuting with hyper-complex conjugation are causality swaps.
Finally, we arrive at a very strong theorem providing some kind of trans-lator between different kinds of duals.8 heorem 2.16.
Ordinary, internal, and external duality operations havethe following properties: o ¯ a o ¯ b i = o ¯ a io ¯ b = ( o ab ) e = o (¯ a ) eo b = o a ( o ¯ b ) e , o ¯ a o ¯ b e = o ¯ a eo ¯ b = ( o ab ) i = o (¯ a ) io b = o a ( o ¯ b ) io ¯ a eo ¯ b i = o ¯ a io ¯ b e = o ab = o ¯ a o b = o a o ¯ b. (2.22) Proof:
One can perform lenghty straightforward calculation. However, ap-plying a new approach of next sections we will be able to present a veryshort proof.
Let S ′ denotes the space of hyperbolic 3-scators with non-vanishing scalarcomponent ( o a ∈ S ′ if and only if a = 0). We are going to identify S ′ withobjects in a linear space A of dimension 4, where the fourth component isreserved for some geometric invariant. Definition 3.1.
The fundamental embedding of S ′ into A is given by themap F : S ′ → ˜ S ≡ F ( S ) ⊂ A , defined by S ′ ∋ o a = ( a ; a , a ) → F ( o a ) := ( a ; a , a , a ) , a = a a a . (3.1) Remark 3.2. F is a bijection between S ′ and ˜ S , so it has an inversion F − : ˜ S → S . We have also a natural projection π : A ∋ ( a ; a , a , a ) → ( a ; a , a ) ∈ S . (3.2)
Note that π = F − but π | ˜ S = F − . We denote by { , iii , iii , iii } a basis in the linear space A such that( a ; a , a , a ) = a
111 + a iii + a iii + a iii . (3.3)The first three elements span the scator space S . The basis { , iii , iii } isrelated to { , ˆ e , ˆ e } used in papers [1, 8]. As in these papers, we demandthat 111 = iii = iii = iii = 1 , (3.4)9ut we define iii = iii iii = iii iii . (3.5)while the Authors of [1, 8] make different assumptions: ˆ e ˆ e = ˆ e ˆ e = 0.The last equalities can be neatly interpreted in our framework because F − ( iii iii ) = F − ( iii ) = 0 . (3.6)Now we make our fundamental assumption about the space A . We assumethat this is a commutative, associative and distributive algebra, compare [7].We denote F ( o a ) = (cid:18) a ; a , a , a a a (cid:19) , F ( o b ) = (cid:18) b ; b , b , b b b (cid:19) , (3.7)where we have written fourth components of scators explicitly. in order tokeep track of possible agreement with expected results. We have F ( o a ) F ( o b ) = (cid:18) a + a iii + a iii + a a a iii (cid:19) (cid:18) b + b iii + b iii + b b b iii (cid:19) . Next, we use distributivity and (3.4) F ( o a ) F ( o b ) = a b + a b + a b + a a b b a b + iii ( a b + a b ) ++ iii iii a a a b + iii iii b b b a + iii ( a b + a b ) + iii iii a a a b ++ iii iii b b b a + iii (cid:18) a a a b + b b b a (cid:19) + iii iii ( a b + a b ) . (3.8)Then, due to (3.5), we obtain F ( o a ) F ( o b ) = a b + a b + a b + a a b b a b + iii ( a b + a b ) ++ iii iii a a a b + iii iii b b b a + iii ( a b + a b ) + iii iii a a a b ++ iii iii b b b a + iii (cid:18) a a a b + b b b a + a b + a b (cid:19) . (3.9)We see that the obtained scalar component coincides with the scalar com-ponent of (1.1). In order to get the same director components we have toassume iii iii = iii iii = iii , iii iii = iii iii = iii , (3.10)10hich follows from (3.4) and (3.5). Finally, we get F ( o a ) F ( o b ) = a b + a b + a b + a a b b a b + iii ( a b + a b ++ a a a b + b b b a ) + iii ( a b + a b + a a a b ++ b b b a ) + iii (cid:18) a a a b + b b b a + a b + a b (cid:19) . (3.11) Corollary 3.3.
From (3.11) we immediately see that o a o b = π ( F ( o a ) F ( o b )) . Theorem 3.4.
The fundamental embedding (3.1) is a multiplicative homo-morphism of S and ˜ S : F ( o a ) F ( o b ) = F ( o a o b ) , hence o a o b = F − ( F ( o a ) F ( o b )) . (3.12) Proof:
We may check it by tedious straightforward computation, multiplyingscalar and fourth component of (3.11) and comparing it with the product ofdirector components. However, calculations can be avoided when we takeinto account the following factorization: F ( o a ) = a (cid:18) a a iii (cid:19) (cid:18) a a iii (cid:19) , F ( o b ) = b (cid:18) b b iii (cid:19) (cid:18) b b iii (cid:19) ,F ( o a ) F ( o b ) = a b (cid:18) a b a b + (cid:18) a a + b b (cid:19) iii (cid:19) (cid:18) a b a b + (cid:18) a a + b b (cid:19) iii (cid:19) . We see immediately that F ( o a ) F ( o b ) ∈ ˜ S which completes the proof. Remark 3.5.
We can easily check that F ( o a ∗ ) = ( F ( o a )) ∗ , F ( λ o a ) = λF ( o a ) , F − ( λF ( o a )) = λ o a, (3.13) where λ is a real constant. Formula for distributive multiplication
A crucial point in our analysis is to express the difference (1.2) in terms of thefundamental embedding. We take into account distributivity of the algebra A assumed in the previous section. Note that although F is a multiplicativehomomorphism but is not additive, i.e., in general F ( o a ) + F ( o b ) = F ( o a + o b ) . (4.1)Therefore, multiplication of scators has a geometric interpretation, whileaddition of such objects cannot be treated geometricly. In particular,( F ( o a + o b )) F ( o c ) = F ( o a ) F ( o c ) + F ( o b ) F ( o c ) . (4.2)But then we surely have F ( o a ) F ( o c ) + F ( o b ) F ( o c ) = ( F ( o a ) + F ( o b )) F ( o c ) , (4.3)because the product in ˜ S is distributive. We will try express F ( o a + o b ) interms of F ( o a ) and F ( o b ). First, we compute F ( o a + o b ) F ( o c ) == (cid:18) a ; a , a , a a a (cid:19) (cid:18) c ; c , c , c c c (cid:19) + (cid:18) b ; b , b , b b b (cid:19) (cid:18) c ; c , c , c c c (cid:19) = (cid:18) c ( a + b ) + c ( a + b ) + c ( a + b ) + ( a + b )( a + b ) a + b c c c ; c ( a + b ) + c ( a + b ) + c c c ( a + b ) + ( a + b )( a + b ) a + b c ,c ( a + b ) + c ( a + b ) + c c c ( a + b ) + ( a + b )( a + b ) a + b c ,c ( a + b ) + c ( a + b ) + c c c ( a + b ) + ( a + b )( a + b ) a + b c (cid:19) . (4.4)12n the other hand F ( o a ) F ( o c ) + F ( o b ) F ( o c ) == (cid:18) a ; a , a , a a a (cid:19) (cid:18) c ; c , c , c c c (cid:19) + (cid:18) b ; b , b , b b b (cid:19) (cid:18) c ; c , c , c c c (cid:19) = (cid:18) c ( a + b ) + c ( a + b ) + c ( a + b ) + (cid:18) a a a b b b (cid:19) c c c ; c ( a + b ) + c ( a + b ) + c c c ( a + b ) + (cid:18) a a a + b b b (cid:19) c ,c ( a + b ) + c ( a + b ) + c c c ( a + b ) + (cid:18) a a a + b b b (cid:19) c ,c ( a + b ) + c ( a + b ) + c c c ( a + b ) + (cid:18) a a a + b b b (cid:19) c (cid:19) . (4.5)Therefore F ( o a + o b ) F ( o c ) − F ( o a ) F ( o c ) − F ( o b ) F ( o c ) == (cid:18) ( a + b )( a + b ) a + b − a a a − b b b (cid:19) (cid:18) c c c ; c , c , c (cid:19) (4.6)and this is exactly F (∆( a, b ; c )), as defined in (1.2)! Thus we have F (( o a + o b ) o c − o a o c − o b o c ) = F ( o a + o b ) F ( o c ) − F ( o a ) F ( o c ) − F ( o b ) F ( o c ) , (4.7)which looks suspiciously homomorphic. Equation (4.6) can be rewritten as F ( o a + o b ) F ( o c ) − F ( o a ) F ( o c ) − F ( o b ) F ( o c ) = κ ( o a, o b ) F ( o ¯ c ) (4.8)where κ ( o a, o b ) is the scalar function standing before dual of o c in formulas(1.2) and (4.6). Remark 4.1.
From (1.3) it follows that the inverse of a scator (with respectto the scator product) is given by ( o c ) − = ( o c ) ∗ k o c k , (4.9) and light-like scators are not invertible. heorem 4.2. F ( o a + o b ) − F ( o a ) − F ( o b ) = κ ( o a, o b ) iii . (4.10) Proof:
We multiply both sides of (4.8) by F (( o c ) − ) and, taking into accountthat F is a homomorphism, we obtain the left-hand side of (4.10). Then weobserve that F ( o ¯ c ) = c c c + c iii + c iii + c iii = iii F ( o c ) . (4.11)Therefore F ( o ¯ c ) F (( o c ) − ) = iii F (( o c ) ∗ ) F ( o c ) k o c k = iii (4.12)which ends the proof. Remark 4.3.
As a direct consequence of (4.12) we get the following strangeresult o ¯ a ( o a ) − = 0 , (4.13) which manifestly shows that the scator algebra has numerous zero divisors. Corollary 4.4.
Theorem 4.2 implies that for a given set of scators, o a i =( a i ; a i , a i ) , where i = 1 , . . . , n , we have: F ( o a + . . . + o a n ) = F ( o a )+ F ( o a )+ . . . + F ( o a n )+ κ n ( o a , o a , . . . , o a n ) iii . (4.14) where κ n (a symmetric scalar function) can be expressed by κ : κ n ( o a , o a , . . . , o a n ) = n X j =2 κ ( o a + o a + . . . + o a j − , o a j ) . (4.15) Corollary 4.5.
From Theorem 4.2 it follows that the inverse of the funda-mental embedding is an additive homomorphism, i.e., F − ( F ( o a ) + F ( o b )) = F ( F ( o a ) + F − ( F ( o b )) ≡ o a + o b, (4.16) because F − ( iii ) = 0 . We point out that F − is not a multiplicative homomorphism. Counter-examples will be given in the next section.14 Dualities - systematic treatment
Here we also provide some back-up for what we have done earlier: at first,we see why we call duality operation a duality operation. First, we note thatmultiplication by bivector iii acts on any element of the algebra A in a waysimilar to the Hodge operator producing its complementary element withrespect to the “maximal form” iii . We have two pairs of complementarybasis elements: the first one is 111 and iii and the second one is iii and iii . Definition 5.1.
Duality operations are naturally given by: o ¯ a = F − ( iii F ( o a )) , o ¯ a i = F − ( iii F ( o a )) , o ¯ a e = F − ( iii F ( o a )) , (5.1) and, of course, o a = F − (111 F ( o a )) . Therefore, we have F ( o ¯ a ) = iii F ( o a ) , F ( o ¯ a i ) = iii F ( o a ) , F ( o ¯ a e ) = iii F ( o a ) . (5.2)Thus, taking into account F ( o a ) − ) = ( F ( o a )) − and (3.12), we get: o ¯ a o a − = 0 , o ¯ a i o a − = iii , o ¯ a e o a − = iii . (5.3)In order to unify proofs it is convenient to introduce the following nota-tion, generalizing (5.1) and (5.2): δ ddd ( o a ) = F − ( dddF ( o a )) , F ( δ ddd ( o a )) = dddF ( o a ) , F ( o a ) = dddF ( δ ddd ( o a )) , (5.4)where δ ddd denotes one of duality operations and ddd denotes the correspondingbasis element ( iii , iii or iii ). Note that ddd = 1. Lemma 5.2.
Any duality operation commutes with inversion, i.e., δ ddd (( o a ) − ) = ( δ ddd ( o a )) − . (5.5) Proof:
Using F ( o a ) = dddF ( δ ddd ( o a )) and 1 = F ( o a ) F (( o a ) − ) (because F is ahomomorphism) we obtain1 = dddF ( δ ddd ( o a )) F (( o a ) − ) = F ( δ ddd ( o a )) dddF (( o a ) − ) = F ( δ ddd ( o a )) F ( δ ddd (( o a ) − )) . On the othe hand, we have 1 = F ( δ ddd ( o a )) F (( δ ddd ( o a )) − ). Therefore, F ( δ ddd (( o a ) − )) = F (( δ ddd ( o a )) − ) 15nd, taking into account that F is a bijection, we obtain (5.5).Now, we can give a simple the proof for Theorem 2.16. We rewritethis theorem in a more convenient way, using the unification of dualitiesformulated in this section. Theorem 5.3.
Duality operations have the following properties: δ ppp ( o a o b ) = δ ppp ( o a ) o b = o aδ ppp ( o b ) ,δ ppp ( o a ) δ qqq ( o b ) = δ qqq ( o a ) δ ppp ( o b ) = δ pqpqpq ( o a o b ) , (5.6) where ppp, qqq ∈ { iii , iii , iii } .Proof: : Using (5.4) and (3.12) we obtain: δ ppp ( o a o b ) = F − ( pppF ( o a o b )) = F − ( pppF ( o a ) F ( o b )) = F − ( F ( o a ) pppF ( o b )) . (5.7) F − ( pppF ( o a ) F ( o b )) = F − ( F ( δ ppp ( o a )) F ( o b )) = δ ppp ( o a ) o b,F − ( F ( o a ) pppF ( o b )) = F − ( F ( o a ) F ( δ ppp ( o b ))) = o aδ ppp ( o b ) , (5.8)which ends the proof. Corollary 5.4.
The preservation of the scator product by any duality is aspecial case of Theorem 5.3 for qqq = ppp , namely: δ ppp ( o a ) δ ppp ( o b ) = o a o b , (5.9) because δ is the identity operation. We begin witha simple fact.
Lemma 6.1.
Modulus squared of a scator satisfies: k o a o b k = k o a k k o b k , k λ o a k = | λ |k o a k . (6.1) where λ is a real constant. roof: Using the definition (1.3) we have k o a o b k = ( o a o b )( o a o b ) ∗ = ( o a o a ∗ )( o b o b ∗ ) = k o a k k o b k , (6.2)by virtue of commutativity and associativity of the scator product. Thesecond equality follows directly from (1.3).Therefore, the modulus squared of the scator product factorizes like theproduct of complex numbers. Unfortunatelly, the modulus squared is nota quadratic form of the scator components, which means that there is notcorresponding bilinear form. However, we will introduce an analogue of thescalar product postulating the formula obeyed by quadratic forms. Definition 6.2.
Scalar product of scators is defined by o a · o b = 12 (cid:18) k o a + o b k − k o a k − k o b k (cid:19) . (6.3) Corollary 6.3. o a · o a = k o a k . (6.4) Theorem 6.4.
The scalar product in S is explicitly given by o a · o b = a b − a b − a b + ( a + b ) ( a + b ) a + b ) − a a a − b b b . (6.5) Proof:
We start from k o a + o b k = ( o a + o b )( o a + o b ) ∗ = ( o a + o b )( o a ∗ + o b ∗ ) , (6.6)and, since we cannot safely proceed because of lack of distributivity in S ,we move on to ˜ S , i.e., k o a + o b k = F − ( F ( o a + o b ) F ( o a ∗ + o b ∗ )) , (6.7)and from the formula (4.10) we get k o a + o b k = F − (( F ( o a ) + F ( o b ) + κ ( o a, o b ) iii )( F ( o a ∗ ) + F ( o b ∗ ) + κ ( o a ∗ , o b ∗ ) iii )) . Hence, using (5.2) and Theorem 3.4, we obtain k o a + o b k = o a o a ∗ + o b o b ∗ + o a o b ∗ + o a ∗ o b ++ κ ( o a, o b )( o ¯ a ∗ + o ¯ b ∗ ) + κ ( o a ∗ , o b ∗ )( o ¯ a + o ¯ b ) + κ ( o a, o b ) κ ( o a ∗ , o b ∗ ) . (6.8)17o proceed further we need Remark 2.7 and the following properties of thehypercomplex conjugation and κ : o ¯ a ∗ + o ¯ a = 2 a a a , (6.9) κ ( o a ∗ , o b ∗ ) = κ ( o a, o b ) = ( a + b )( a + b )( a + b ) − a a a − b b b . (6.10)Then from (6.8) and Definition 6.2 we have2 o a · o b = κ + 2 κ (cid:18) a a a + b b b (cid:19) + 2 (cid:18) a b − a b − a b + a a b b a b (cid:19) = (cid:18) κ + a a a + b b b (cid:19) − a a a − b b b + 2( a b − a b − a b ) , where κ = κ ( o a, o b ). We complete the proof by substituting (6.10). Remark 6.5.
We easily see that ( λ o a ) · ( λ o b ) = λ ( o a · o b ) . (6.11) but, in general, λ ( o a · o b ) = ( λ o a ) · o b = o a · ( λ o b ) = λ ( o a · o b ) and ( o a + o b ) · o c = o a · o c + o b · o c. (6.12) We point out that this scalar product is ill-defined if a scalar component ofany factor is zero.
We point out that the scator metric (1.3) approaches the flat Minkowskimetric of special relativity space-time in the limit a → ∞ . It was shownin [4] that the framework of scators excludes possibility of existence of anabsolute rest frame, so it is consistent with the principle of relativity [6].Therefore the subject is not only of purely mathematical interest but mayhave interesting physical points, as well.The first question is, can we think of tachyons as para-particles possess-ing space-like trajectories? In the scator framework this would mean that ateach instance of time tachyons have to be described by scators with negative18odulus squared. Thus understood tachyons, in order to remain tachyons,need to experience sub-luminal Lorentz boosts at that each instance of time,since otherwise the sign of the modulus squared would get reverted. It ishard to imagine inertial, super-luminal observer that could be turned intoanother one with some kind of sub-luminal Lorentz-like transformation pre-serving scator metric. Hence it seems to us that a more natural way ofunderstanding tachyons is that they are ordinary particles in a differentcausal domain and they cannot reach us because of the infinite-energy re-quirement to pass the bipyramidal light-barrier, inside which we are capableof taking the measurements.To sum up: we may suppose that tachyons are being super-luminal in asense of belonging to a different sub-luminal causal realm, although then welook exactly the same way to them. Note that this hypothesis is surprisinglyin accordance with recent remarks on the nature of tachyons [5].Next question that appears in the physical context is whether the ap-proach proposed in this paper works in the case of 1 + 3 dimensions? Theanswer is yes . It may be easily shown that in physical space-time of scatorsthere is F ( o a + o b ) = F ( o a ) + F ( o b ) + c , iii iii + c , iii iii + c , iii ˆ e + c , , iii iii iii , (7.1)as opposed to (4.10), opening all doors needed [7]. In the above expressionwe have c i,j ( o a, o b ) = ( a i + b i )( a j + b j ) a + b − a i a j a − b i b j b (7.2)and c , , ( o a, o b ) = ( a + b )( a + b )( a + b )( a + b ) − a a a a − b b b b . (7.3)We see that the generalization, although simple in principle, may lead tocumbersome calculations. This also implies existence of more dualities, as-sociated with the basis of the A -space, now 8-dimesional. Here also dualitiesintroduced for 1 + 2 dimensional case find their interpretation: the ordinaryduality takes us into the realm of wings around the dipyramid, while internaland external dualities carry us out of the light bipyramid.We point out, however, that possibilities for scators to be physicallyinterpreted is strongly suppressed by the fact that the scator algebra doesnot possess rotational invariance. Fortunatelly, the considered dynamicsdoes not provoke the appearance of absolute-rest frame [4], which leavessome hope for potential potential applications.19 eferences [1] M.Fern´andez-Guasti, F.Zald´ıvar: A Hyperbolic Non-Distributive Algebra in 1+2dimensions, Adv. Appl. Clifford Algebras
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