1. Introduction
The scator algebra was introduced by Fernández-Guasti and Zaldívar in a series of papers, starting from [
1]. The elliptic case can be considered as yet another approach to hypercomplex numbers [
2] with the corresponding theory of holomorphic functions [
3,
4], while the hyperbolic case has potential physical applications, usually related to deformations and generalizations of Lorentz symmetries of the special theory of relativity [
5,
6]; see also [
7,
8]. In this paper, we confine ourselves to the hyperbolic case, closely related to a specific deformation of the Lorentz symmetry. To be more precise, we consider a real linear space
with a fixed basis of unit vectors:
,
(their squares are assumed to be
). An element
is denoted as:
where
are real numbers (
is a scalar component, and
are referred to as director components). The unit scalar
is usually omitted; compare the last equality of (
1). The decomposition into scalar and director components is crucial for many properties of scators, including their characteristic nondistributive multiplication.
Scators form a large subset of
(denoted by
), which consists of all elements with a nonvanishing scalar component and lines along director components. In other words,
where:
Note that in our earlier papers,
was usually denoted by
; see [
9,
10].
Definition 1. The scator product of two scators, and , is denoted by . In the hyperbolic case, it is defined as follows [11,12]: For , Other cases are as follows:where is Kronecker’s delta.
The above definition implies the commutativity of the scator product. Another useful property is the compatibility of the scator product with the dilation, i.e.,
The scator product is nondistributive (i.e., usually, ) and nonassociative (although the associativity holds if all involved factors and their products have nonvanishing scalar components). We point out that scator multiplication admits zero divisors, for instance: the scator products of and are equal to zero.
The hypercomplex conjugation of scators maps
into
, namely:
Similarly, as in the case of the complex numbers, the scator norm is defined by a scator product of the scator and its hypercomplex conjugate:
Then, the application of Definition 1 yields:
In particular, for large
, the scator norm becomes close to the standard Minkowski metric, which is one of main reasons for considering scators as a peculiar extension of special relativity:
The definition of the scator product presented above is far from being obvious or natural. The first issue we would like to address in the next section is a clear and intuitive motivation for Definition 1.
In our recent paper, we proposed an extension of the scator product in
on the whole space
[
13]. Here, we follow an alternative path. We show that not only the definition of the scator product, but also its domain (
2) is, in a sense, natural.
2. Fundamental Embedding
Our main tool to understand the scator product and scator geometry is the so-called fundamental embedding, introduced in [
9]; see also [
10,
13]. The fundamental embedding
F maps the scator space
into
, where the space
is the algebra over
generated (using addition and multiplication, which is assumed to be commutative, associative, and distributive over addition) by elements
satisfying (in the hyperbolic case):
The basis elements of
are of the form
, where
J is a multi-index defined by a subset
of
, i.e.,
. In particular, we have the unit element
, vectors, bivectors, multivectors, and the element
:
A linear space
is isomorphic to Clifford and Grassmann algebras (although the multiplicative structures of all these spaces are totally different) [
14,
15].
In order to the motivate definition of the fundamental embedding, which appears at the end of this section (see Definition 2), we introduce some useful notions. First, we denote by
the natural projection of the vector space
on the space spanned by
(compare (
1)):
We consider the following subset of
:
where
and
are real parameters. Then, for
,
is of the form (
13), where:
We denote also:
where, obviously,
for
.
Theorem 1. There exists a one-to-one correspondence between and and a one-to-one correspondence between and .
Proof. Let us take
; see (
14). We have two cases. First,
, which implies
for
. In this case:
and we can rewrite
as:
Therefore, any element
is uniquely defined by its projection
; see (
15) and (
18). Hence, there is a bijection between
and
. Moreover, obviously, we can identify
with
. In other words, to any scator
with a nonvanishing scalar component, there corresponds exactly one element
such that
. Thus, a one-to-one correspondence between
and
is shown.
In the second case (
), the situation is more complicated. Note that
if and only if there exists
m such that
. Then, as a consequence,
and due to (
15),
reduces to:
Thus, for
,
has to be proportional to
. We point out that if
for any
, then
vanishes, and as a consequence,
. Therefore, the case
corresponds to the second part of the scator set (
2), which ends the proof. □
A one-to-one correspondence between and is realized by the projection . What is more, the projection maps the multiplicative structure of into the scator multiplication. In a sense, this fact can be treated as a derivation of the scator product.
Theorem 2. The multiplication in the space (induced from the natural commutative product in ), mapped by the projection π, yields the scator product in the scator space ; see (4). In other words, can be treated as a definition of the scator product for elements from the space .
Proof. Elements of
are given by Formula (
18). Straightforward computation yields:
Then, the coefficients of
by
yield the Formula (
4), which ends the proof. □
The scator product is defined for all scators with a nonvanishing scalar component. However, as shown below, the result of this multiplication can be outside of this set.
The projection
is not an isomorphism between
and
, because neither
nor
are closed with respect to the multiplication. In the one-dimensional case, we have:
Therefore, in the case
, the above result is proportional to
, and as a consequence, it does not belong to
. In the general case (
22), the situation is analogous. If
, then
is proportional to
:
Finally, we would like to address the problem of extending Theorem 2 on all elements of
, including elements of the form (
24). Note that:
for any values of
parameters
(
), which means that the preimage of
under
is very large.
Fortunately enough, choosing the simplest element in this preimage, namely , we obtain the required extension of Theorem 2. In other words, we embed the scator space into in the way leading uniquely to the scator product of Definition 1.
Definition 2. The fundamental embedding is defined as: Corollary 1. For any and , we have: Remark 1. F is a bijection between and . What is more, π restricted to coincides with , which means that in this case, .
The main advantage of the fundamental embedding is a natural motivation for the definition of the scator product proposed by Fernández-Guasti. (Definition 1). Indeed, we can present the following alternative definition of the scator product; see [
9].
Theorem 3. is equivalent to Definition 1 of the scator product.
Proof. The first part of Definition 1, given by Equation (
4), follows from (
28) directly by Theorem 2. The second part, given by Equation (
5), can be directly computed, as follows.
where we took into account that:
In these computations, we used the relations (
11). Note that the terms replaced by dots are bivectors or multivectors of higher order. Finally,
which ends the proof. □
Remark 2. The scator product is, in general, nonassociative. Indeed, Both expressions would be identical, equating , provided that . The last equality is true only when restricted to ; compare Remark 1. For instance, we have: In this case, we see clearly that .
Nonassociativity is usually related to the quantum aspects of physical systems (see [
16,
17,
18]). An attempt to involve quantum effects has been made also in the case of scators [
19].
3. Group Structure of the Embedded Scator Space
The set
is closed under multiplication. Indeed, for any
, we have:
In order to determine the group structure, we have to consider the invertibility of the elements of
. As usual, conjugate elements (compare (
7)) are useful in this context. Hypercomplex conjugation at the level of the space
is realized by the reflection
(
. Thus:
Hence, if
for all
, then:
The following elementary equality is very helpful.
Therefore, the product of an element proportional to (i.e., such that ) by any element of the set is proportional to , as well.
Corollary 2. An element of is noninvertible if and only if it is proportional to for at least one value of k ().
The next corollary is another direct consequence of (
37).
Corollary 3. The product of two invertible elements is invertible. Invertible elements of form a multiplicative group, which is denoted by .
Theorem 4. is a commutative, simply connected, multiplicative group.
Proof. We compute the product of two elements of the form of (
38) (parameters corresponding to the second element are marked with a prime).
It is convenient to use a bijection:
Taking into account that:
we conclude that
is of the form of (
38).
The inverse element always exists and is computed as a special case of (
36):
Simple connectedness follows immediately if we consider the following homotopy:
where
. □
Remark 3. One can easily see that:where: is closely related to the “restricted space subset” [11] (scalar components of scators from the restricted subset can be both positive and negative). The bijection (
40) suggests a convenient parameterization of the group
using the exponential representation. Indeed, taking into account (
11), we compute:
In other words,
and finally,
Thus, any scator from
can be represented as:
Theorem 5. Any element can be represented as:where and J is a multi-index. Proof. Given an element
of
, we use the following identity,
wherever the coefficient
. Thus, the element
can be expressed, up to the sign, as a product of some number of basis vectors (i.e., shortly,
) multiplied by an element of
. Hence, we obtain (
53). □
4. Embedded Scator Space as an Intersection of Quadrics in a Higher-Dimensional Space
We showed that the set of scators,
, is embedded in the space
of dimension
. In this section, we study the geometry of the embedding
, assuming tacitly that the basis (
12) is orthonormal (i.e.,
). The coordinates in the space
are denoted by
, where
J is a multi-index.
The embedded scator space seems to consist of two parts. The first part, defined by the condition
, contains scators parameterized by
in the following way:
We denote by
the coefficient of
by
, and in general, the coefficient of
by
is denoted by
. Thus:
and we may shortly write down:
The norm of
is identified with the norm of the corresponding scator (i.e.,
), but the formula
holds, as well:
Therefore,
and taking into account (
56),
where
denotes the cardinality (number of elements) of the multi-index
J.
Corollary 4. The scator metric in coincides with the pseudo-Euclidean metric (60) in the -dimensional space . The metric (60) has signature zero. The second part of the embedded scator space, corresponding to , apparently consists of n coordinate axes (lines along ). However, we suggest another interpretation (planes instead of lines), motivated by the following low-dimensional cases.
4.1. Scator Transformations for N = 2 as Isometries in a Four-Dimensional Space of Zero Signature
Let us consider the case
. An element
of
, given by:
is an embedded scator if and only if one of the following possibilities hold:
It is tempting to replace conditions (
62) by one equation
, defining a quadric in
:
The equation
is not equivalent to (
62), because the constraint
is not its necessary consequence. However, we conjecture that the quadric (
63) can be more fundamental than (
62). Therefore, we find transformations preserving the following quadratic constraints:
where
C is a constant. This system of two equations can be rewritten in the following, equivalent, form:
which means that this is an intersection of two hyperbolic cylinders. The most general linear transformation (modulo reflections) preserving these two quadrics is a system of two hyperbolic “rotations” (boosts):
where
and
are constant parameters. It corresponds to the following linear transformation in the space
:
where:
Equation (
67) can be shortly written as
. Note that
A is symmetric (
). Moreover,
which means that
. We can verify in a straightforward way that
, which means that
.
Theorem 6. The orthogonal transformation (67) can be realized as a multiplication by the unit scator , where ():where (). Proof. Taking into account (
61) and performing the multiplication on the right-hand side of (
70), we obtain:
or in the matrix form:
Matrices (
67) and (
72) are identical if:
Computing
and
from (
73), we obtain the inverse transformation:
One can check by straightforward computation that the other two equations resulting from (
73), namely:
are then identically satisfied (taking into account
and
). Therefore, substituting (
74) into (
70), we obtain the matrix
A given by (
67). □
4.2. Embedded Scators as the Intersection of Quadrics in the Case N = 3
If
, then the general element of
, given by:
is an embedded scator if the last four coordinates are parameterized by the four first coordinates as follows:
Equation (
77) implies that the scator norm is a pseudo-Euclidean norm in eight-dimensional space
:
The system (
77) can be rewritten as the intersection of nine quadrics:
These equations are not independent, of course. Now, we can ask about the consequences of (
79) in the case of
. First, it follows from the three equations on the left that at least two of the three coordinates
vanish. Suppose that
. Then:
Hence, either
and
or
. Analogous results follows if we take
or
. Thus, we arrive at the following set of general solutions to the system (
79):
Note that the solution
(when
) is included as a special case of the second equation of (
81). Therefore, the subset
reduces to the union of three two-dimensional quadrics; compare (
63). We conjecture that a similar property holds in higher dimensions, as well.
6. Conclusions
In the generic case, the scator product, proposed by Fernández-Guasti and Zaldívar [
1], is induced by another product (in another space, namely
), which is commutative, associative, and distributive over addition. The space
, spanned by vectors
(
) and their products, may be understood as a commutative analogue of the geometric algebra or the Clifford algebra [
14,
15]. In this context, we can interpret
as bivectors and
(where
J is a multi-index) as multivectors. The set
(the fundamental embedding of scators with nonvanishing scalar component) has a natural group structure reminiscent of a commutative analogue of the Clifford (or Lipschitz) group [
23]. Theorems 1 and 2 show that this group is, indeed, a natural model of the scator space. Another interesting feature of the space
is the metric structure induced by the scator metric. It turns out that this is a pseudo-Cartesian metric (the squared norm of basis multivectors
is one for even multiple indices and
for odd multiple indices).
Section 4 presents a new interpretation of the scator product as an isometry (orthogonal transformation) in this space.