On the Geometry of the Hyperbolic Scator Space in 1+2 Dimensions

We consider the scator space in 1+2 dimensions—a hypercomplex, non-distributive hyperbolic algebra introduced by Fernández-Guasti and Zaldívar. We find a method for treating scators algebraically by embedding them into a distributive and commutative algebra. A notion of dual scators is introduced and discussed. We also study isometries of the scator space. It turns out that zero divisors cannot be avoided while dealing with these isometries. The scator algebra may be endowed with a nice physical interpretation, although it suffers from lack of some physically demanded important features. Despite that, there arise some open questions, e.g., whether hypothetical tachyons can be considered as usual particles possessing time-like trajectories.


Introduction
Following Fernández-Guasti and Zaldívar [1] we consider a commutative, nondistributive 1 + 2 dimensional algebra S, which is also associative provided that divisors of zero are excluded. The elements of this algebra will be called 1 + 2 dimensional scators [1]. Scators are denoted by o a = (a 0 ; a 1 , a 2 ), where components a 1 , a 2 are referred to as director components, and a 0 is usually called scalar component, or, in physical context, temporal component. The space of scators possesses the additive structure of the usual vector space and a specific non-distributive product. Scalars form a subset of the scator space closed under addition and multiplication. In this paper we confine ourselves to this definition of a scator, leaving aside earlier concepts of scators as objects generalizing scalars and vectors, see [2,3]. Dealing with scators, one has to be aware that they are defined in a fixed reference frame (although some analogue of the Lorentz transformation can be introduced).

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It was shown in [4] that this algebra may be given physical interpretation corresponding to some ideas of special relativity, although the metric in the scator space is different from the standard metric of Minkowski space. This scator metric (defined below) is called a scator-deformed Lorentz metric. It was emphasized that scators and deformed metrics can describe a kinematics of some kind of particles, including hypothetical tachyons.
In our paper we study metric properties of scators, paying special attention to proper definitions for causal realms appearing in this framework, what finally will lead to some convergence with work of Kapuścik [5].
To begin with, we reconsider some main points in reasoning leading to physical interpretation of scators, as introduced in [1]. The idea of describing physical phenomena along lines of the scator framework takes its origin in the fact that multiplication of scators mimics the shape of velocity addition formula known from special relativity (SR). Adapting popular convention c = 1 (speed of light being unity), we denote scators by overset o symbol, so that o a = (a 0 ; a 1 , a 2 ) = a 0 1; since multiplication by a scalar acts on scators in usual component-wise way. Here also we introduce notation β a1 = a1 a0 , β b1 = b1 b0 , etc. We will refer to objects of the form o a a0 = (1; β a1 , β a2 ), as velocity scators. We should show now that the composition rule for the velocity scators yields the Einstein law of addition of velocities. Moreover, properly applied, it recovers Lorentz transformation of physical measurements from one inertial frame to another.
Hence we define what reminds the formula for velocity addition in SR. Using these definitions we have to require a 0 = 0, b 0 = 0 (this is a crucial assumption).
Vol. 27 (2017) On the Geometry of the Hyperbolic Scator 1371 It is worth stressing that this structure agrees with SR in 1 + 1 dimensions only. Results produced by the multiplication formula for higher dimensions can be interpreted as a deformation of the standard Lorentz metric. The latter is obtained in the limit a 0 → ∞, b 0 → ∞ [4]. Therefore, the scator structure is compatible with the standard physical interpretation only in the 1 + 1 case. Higher-dimensional generalizations should be treated with care.
We emphasize the fact that multiplication acts in a non-distributive way, what is the hallmark of the structure. Therefore, computing , we easily see that in the generic case the right-hand side does not vanish, i.e., ( Hence, in general, the scator product is not distributive. By the way, the scator appearing on the right-hand side of (1.5) will be referred to as dual to o c (see Definition 2.1 below). Many properties of the scator product (1.2) were widely investigated in many contexts [1,8,9], also physical [4]. To gain some insight in possible physical interpretation of the scator algebra, we recall here some basic terminology from [1] and [9]: (1.6) Definition 1.2. We say that a scator is time-like, if a 2 0 > a 2 1 , and a 2 0 > a 2 2 , or a 2 0 < a 2 2 , and a 2 0 < a 2 1 , it is said to be space-like, if a 2 0 < a 2 1 , and a 2 0 > a 2 2 or a 2 0 < a 2 2 , and a 2 0 > a 2 1 , (1.8) and it is light-like, if The classification proposed above is analogous to what is well known from special relativity. Note that in both cases componentwise addition of scators or vectors does not preserve this classification. However, the norm of the product of two scators is just the product of their norms [9], which is very useful in this context. In this paper we present new results concerning the metric structure of the scator space, extending results of [4,9].
Note the base difference between what is proposed above and what was done in [8]: in the cited paper classification of scators was performed using components of the objects and quite sensible assumption that nothing observable can happen outside of the light bipyramid. As opposite, here we perform a classification in terms of the deformed metric (1.6) only, what has its formal consequences in the appearance of some new causal realms, with potential physical interpretations.  For instance, we indicate that the bipyramid considered in [4] has some kind of time-like "wings" around, described by the regime a 2 0 < a 2 1 and a 2 0 < a 2 2 , while, for example, in [9], there were considered mainly time-like events inside the light bipyramid (a 2 0 > a 2 1 and a 2 0 > a 2 2 ). Time-like region is marked as dark at Fig. 1.
We point out that the existence of such wings is not mentioned in the paper [4]. Probably this possibility was excluded from considerations by assumption. However, the wings seem to reveal some new aspects of causality in scator-deformed Lorentz metric (see Fig. 1) and in next sections we propose a possible physical interpretation.
A closely related notion, "super-restricted space conditions", was introduced in [9] without a direct relation to the type of considered events. Superrestricted space conditions define either time-like events (in even-dimensional spaces) or space-like events (in odd-dimensional spaces).
Note also important formal consequences of the proposed framework: Vol. 27 (2017) On the Geometry of the Hyperbolic Scator 1373 • The scator space is not invariant with respect to rotations (see Fig. 1). It may remind an archaic approach of Hamilton, where vector's components were treated as simple scalars. • In the scator framework we do not expect continuous symmetries properties of any kind and the summation convention have to be dropped. In this paper we introduce and study a number of discrete symmetries, instead. • In order to not affect physics by dilations, in the scator framework Lorentz boosts must be represented by unit-magnitude elements [4]. Good candidate for this is normalized velocity scator , where the proper time τ is defined in perfect analogy with this of SR, but here we use the scator-deformed Lorentz metric.
• In [4] it was also shown that under a boost we have where γ u is deformed Lorentz factor (the norm of the velocity scator corresponding to the normalized scator o U ), and the conjugation is needed in order to keep the standard sign convention.
• Note that the above defined o U must be time-like, so that it excludes superluminal boosts from the game.
The paper is organized as follows. In Sect. 2, we introduce basic objects and transformations responsible for isometries in the scator space S. Then, in Sects. 3 and 4, we propose and develop a new framework in which calculation proceeds in a more natural way, using a distributive product. In Sect. 5, we continue along these lines focusing on isometries. Sect. 6 is entirely devoted to the question of metric properties of scators; in particular, we obtain the closest analogue of a scalar product we can get, although it is not bilinear. The last section contains physical comments and conclusions.

Dualities: Phenomenological Treatment
Now we turn our attention to the issue of isometries in the scator space S. We begin by defining some operations and then check their properties. The above statements follow from considerations similar to direct calculations included in [7]. Soon we will get a better understanding of these facts by applying a new approach which is both faster and simpler. Proposition 2.6. Operation of duality preserves the scator product: Proof. We present explicit calculation for the scalar component: Similar direct computation can be done for director components.

Proposition 2.7. Operation of duality is an isometry in 3-scator space.
Proof. For ordinary scator o a we have its norm Thus, for the dual scator ō a, we get Vol. 27 (2017) On the Geometry of the Hyperbolic Scator 1375 where the right-hand side is proportional to 1 1 1 (omitted for simplicity here and in many other places).     Proof. Denoting ō a i = (ā 0i ;ā 1i ,ā 2i ), we compute: Finally, we arrive at a very strong theorem providing some kind of translator between different kinds of duals. (2.20) Proof. One can perform lengthy straightforward calculation. However, applying a new approach of next sections, we will be able to present a very short proof in Theorem 5.3.   a 0 ; a 1 , a 2 , a 3 ), a 3 = a 1 a 2 a 0 .
We denote by {1 1 1, i i 3) The first three elements span the scator space S. The basis {1 1 1, i i i 1 , i i i 2 } is related to {1,ê 1 ,ê 2 } used in papers [1,8]. As in these papers, we demand that while the authors of [1,8] make different assumptions:ê 1ê2 =ê 2ê1 = 0. The last equalities can be neatly interpreted in our framework because Now we make our fundamental assumption about the space A. We assume that this is a commutative, associative and distributive algebra, compare [7]. We denote where we have written fourth components of scators explicitly in order to keep track of possible agreement with expected results. We have Next, we use distributivity and (3.4) (3.8) Then, due to (3.5), we obtain A. Kobus and J. L. Cieśliński Adv. Appl. Clifford Algebras We see that the obtained scalar component coincides with the scalar component of (1.2). In order to get the same director components we have to assume 10) which follows from (3.4) and (3.5). Finally, we get

Theorem 3.4. The fundamental embedding (3.1) is a multiplicative homomorphism of S andS:
Proof. We may check it by tedious straightforward computation, multiplying scalar and fourth component of (3.11) and comparing it with the product of director components. However, calculations can be avoided when we take into account the following factorization: We see immediately that F (  where λ is a real constant.

Formula for Distributive Multiplication
A crucial point in our analysis is to express the difference (1.5) in terms of the fundamental embedding. We take into account distributivity of the algebra A assumed in the previous section. Note that F is a multiplicative homomorphism but is not additive, i.e., in general Vol. 27 (2017) On the Geometry of the Hyperbolic Scator 1379 Therefore, multiplication of scators has a geometric interpretation, while addition of such objects cannot be treated geometrically. In particular, But then we surely have because the product inS is distributive. We will try to express F ( On the other hand and this is exactly F (Δ(a, b; c)), as defined in (1.5)! Thus we have which looks suspiciously homomorphic. Equation (4.6) can be rewritten as where κ(   Proof. We multiply both sides of (4.8) by F (( o c) −1 ) and, taking into account that F is a homomorphism, we obtain the left-hand side of (4.10). Then we observe that which ends the proof. which manifestly shows that the scator algebra has numerous zero divisors.  a 0i ; a 1i , a 2i ), where i = 1, . . . , n, we have: We point out that F −1 is not a multiplicative homomorphism.
Counterexamples will be given in the next section.

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 that multiplication by the bivector i i i 12 acts on any element of the algebra A in a way similar to the Hodge operator producing its complementary element with respect to the "maximal form" i i i 12 . We have two pairs of complementary basis elements: the first one is 1 1 1 and i i i 12 and the second one is i i i 1 and i i i 2 . In order to unify proofs it is convenient to introduce the following notation, generalizing (5.1) and (5.2): and, taking into account that F is a bijection, we obtain (5.5). Now, we can give a simple proof for Theorem 2.17. We rewrite this theorem in a more convenient way, using the unification of dualities formulated in this section. Theorem 5.3. Duality operations have the following properties: Proof. Using (5.4) and (3.12) we obtain: which ends the proof.
and, finally, using (5.1), we get ). On the Geometry of the Hyperbolic Scator 1383

Metric Properties of Scators
We begin with a simple fact.

Lemma 6.1. Squared modulus of a scator satisfies:
where λ is a real constant.
Proof. Using Definition (1.6) we have by virtue of commutativity and associativity of the scator product. The second equality follows directly from (1.6).
Therefore, the squared modulus of the scator product factorizes like the product of complex numbers. Unfortunately, the squared modulus is not a quadratic form of the scator components, which means that there is not a corresponding bilinear form. However, we will introduce an analogue of the scalar product postulating the formula obeyed by quadratic forms.

Definition 6.2. Scalar product of scators is defined by
and, since we cannot safely proceed because of lack of distributivity in S, we move on toS, i.e., and from the formula (4.10) we get Hence, using (5.2) and Theorem 3.4, we obtain A. Kobus and J. L. Cieśliński Adv. Appl. Clifford Algebras To proceed further we need Remark 2.8 and the following properties of the hypercomplex conjugation and κ: Then from (6.8) and Definition 6.2 we have . We complete the proof by substituting (6.10).

Remark 6.5.
We easily see that We point out that this scalar product is non-defined if a scalar component of any factor is zero.

Underlying Physics and Conclusions
The scator metric (1.6) approaches the flat Minkowski metric of special relativity space-time in the limit a 0 → ∞. In the context of doubly-special relativity (DSR) [6], this limit means the restriction to times much larger than the Planck's time scale. Therefore the subject is not only of purely mathematical interest but may have interesting physical points, as well. The first question is, can we think of tachyons as para-particles possessing space-like trajectories? In the scator framework this would mean that at each instance of time tachyons have to be described by scators with negative squared modulus. 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 squared modulus would get reverted. It is hard to imagine an inertial, super-luminal observer that could be turned into another one with some kind of sub-luminal Lorentz-like transformation preserving scator metric. Hence it seems to us that a more natural way of understanding tachyons is that they are ordinary particles in a different causal domain and they cannot reach us because of the infinite-energy requirement to pass the bipyramidal light-barrier, inside which we are capable of taking the measurements.
Vol. 27 (2017) On the Geometry of the Hyperbolic Scator 1385 To sum up: we may suppose that tachyons are being super-luminal in a sense of belonging to a different sub-luminal causal realm, although then we look exactly the same way to them. Note that this hypothesis is surprisingly in accordance with recent remarks on the nature of tachyons [5].
Thus, some interesting formal effects arise in a quite natural fashion in the scator framework. Convergences with recent physical concepts [4,5], even if accidental, is intriguing, all the more so that this framework does not show any rotational symmetry properties we could expect on scientific and intuitive grounds.
The next question that appears in the physical context is whether the approach proposed in this paper works in the case of 1 + 3 dimensions? The answer is yes. It may be easily shown that in physical space-time of scators there is as opposed to (4.10), opening all doors needed [7]. In the above expression we have and We see that the generalization, although simple in principle, may lead to cumbersome calculations. This also implies existence of more dualities, associated with the basis of the A-space, now 8-dimesional. Here also dualities introduced for 1 + 2 dimensional case find their interpretation: the ordinary duality takes us into the realm of wings around the dipyramid, while internal and external dualities carry us out of the light bipyramid. Finally, we point out that the possibility for scators to be physically interpreted is strongly suppressed by the fact that the scator algebra does not possess rotational invariance. Fortunately, the considered dynamics does not provoke the appearence of absolute-rest frame [4], which leaves some hope for potential applications.
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