2026-07-08
Commutative algebra for a new Cartesian basis
Appendix to the article "A New Cartesian Basis in the Unit Space Model"
In the main work, the new Cartesian basis \(\{\ep, i\ep, \em, i\em\}\) was considered as the basis of a commutative algebra constructed on the idempotents \(\ep\) and \(\em\). This form is particularly convenient for geometric interpretation, since it immediately divides space into two independent complex planes.
This appendix considers another, equivalent form of the same commutative structure: the algebra \(\mathcal{A}\) with the basis \(\{1,i,j,ij\}\), where \(i^2=-1\), \(j^2=1\), and \(ij=ji\). It allows the same elements to be written not in terms of idempotent components, but in terms of the unit, imaginary unit, hyperbolic unit, and their product. Therefore, the goal of the application is not to replace a non-commutative algebra with a commutative one, but to compare two commutative representations and derive explicit transitions between them.
Two Commutative Forms of the Same Algebra
The first representation is idempotent. It uses the basis:
\[ \{\ep,\; i\ep,\; \em,\; i\em\}. \tag{1} \]
Here \(\ep\) and \(\em\) are idempotents, and \(i\) is the usual imaginary unit. The basic multiplication rules are:
\[ \ep^2=\ep,\qquad \em^2=\em,\qquad \ep\em=0,\qquad i^2=-1. \tag{2} \]
From these rules it immediately follows that the two components \(\ep\) and \(\em\) are not mixed. Therefore, any element can be written as the sum of two independent complex parts:
\[ Z=(a+ib)\ep+(c+id)\em, \qquad a,b,c,d\in\mathbb{R}. \tag{3} \]
The second representation is algebraic. It uses a basis:
\[ \{1,\; i,\; j,\; ij\}. \tag{4} \]
Multiplication in this basis is defined by the rules:
\[ i^2=-1,\qquad j^2=1,\qquad (ij)^2=-1,\qquad ij=ji. \tag{5} \]
Therefore, this is also a commutative associative algebra over \(\mathbb{R}\). An arbitrary element in it has the form:
\[ Z=A+Bi+Cj+D(ij), \qquad A,B,C,D\in\mathbb{R}. \tag{6} \]
Both notations describe the same structure, since the algebra is isomorphic to the direct sum of two copies of complex numbers:
\[ \mathcal{A}\cong\mathbb{C}\oplus\mathbb{C}. \tag{7} \]
This property is key: the space decomposes into two independent complex planes, and different bases merely reflect this decomposition differently.
Idempotents via the Hyperbolic Unit
The relationship between the two forms is defined by the hyperbolic unit \(j\), for which \(j^2=1\). Idempotents are expressed through it as follows:
\[ \ep=\frac{1+j}{2},\qquad \em=\frac{1-j}{2}. \tag{8} \]
Conversely, the hyperbolic unit is expressed through idempotents:
\[ j=\ep-\em, \qquad 1=\ep+\em. \tag{9} \]
For the product \(ij\), we obtain:
\[ ij=i\ep-i\em. \tag{10} \]
And for complex idempotent directions:
\[ i\ep=\frac{i+ij}{2}, \qquad i\em=\frac{i-ij}{2}. \tag{11} \]
Thus, the basis \(\{1,i,j,ij\}\) and the basis \(\{\ep,i\ep,\em,i\em\}\) are linearly expressed in terms of each other.
Transition between bases
The same element \(Z\) can be written in a new Cartesian basis:
\[ Z=ct\,\ep+x\,i\ep+y\,\em+z\,i\em, \tag{12} \]
or in an algebraic basis:
\[ Z=A\cdot 1+B\cdot i+C\cdot j+D\cdot(ij). \tag{13} \]
Substituting (8) and (11) into representation (12), we obtain:
\[ Z= \frac{ct+y}{2}\cdot 1 + \frac{x+z}{2}\cdot i + \frac{ct-y}{2}\cdot j + \frac{x-z}{2}\cdot(ij). \tag{14} \]
Therefore, the direct transition from coordinates \((ct,x,y,z)\) to coefficients \((A,B,C,D)\) has the form:
\[ \begin{cases} A=\dfrac{ct+y}{2},\\ B=\dfrac{x+z}{2},\\ C=\dfrac{ct-y}{2},\\ D=\dfrac{x-z}{2}. \end{cases} \tag{15} \]
Back transition:
\[ \begin{cases} ct=A+C,\\ x=B+D,\\ y=A-C,\\ z=B-D. \end{cases} \tag{16} \]
In matrix form, the direct transformation is written as follows:
\[ \begin{pmatrix} A \\B \\C \\D \end{pmatrix} = \frac12 \begin{pmatrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 0 & -1 & 0 \\ 0 & 1 & 0 & -1 \end{pmatrix} \begin{pmatrix} ct \\x \\y \\z \end{pmatrix}. \tag{17} \]
The inverse transformation is:
\[ \begin{pmatrix} ct \\x \\y \\z \end{pmatrix} = \begin{pmatrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 0 & -1 & 0 \\ 0 & 1 & 0 & -1 \end{pmatrix} \begin{pmatrix} A \\B \\C \\D \end{pmatrix}. \tag{18} \]
Isomorphism with \(\mathbb{C}\oplus\mathbb{C}\)
From formulas (8)–(11), it follows that the element \(Z=A+Bi+Cj+D(ij)\) can be represented as a pair of complex numbers:
\[ A+Bi+Cj+D(ij) \;\longleftrightarrow\; \left( (A+C)+i(B+D),\; (A-C)+i(B-D) \right). \tag{19} \]
The first complex component corresponds to the idempotent \(\ep\), and the second to the idempotent \(\em\):
\[ Z= \left[(A+C)+i(B+D)\right]\ep + \left[(A-C)+i(B-D)\right]\em. \tag{20} \]
If we use coordinates \((ct,x,y,z)\), this decomposition takes a particularly simple form:
\[ Z=(ct+ix)\ep+(y+iz)\em. \tag{21} \]
Formula (21) demonstrates the main advantage of an idempotent basis: the space-time element immediately decomposes into two complex planes \((ct,x)\) and \((y,z)\). Moreover, the basis \(\{1,i,j,ij\}\) reveals the same structure through a more symmetric algebraic notation.
Velocity Vector and Trajectory of Motion
In the main article, the motion of a point is given by the velocity vector:
\[ V(t)=c\left(\ep+\em e^{i\omega t}\right), \tag{22} \]
where \(e^{i\omega t}=\cos\omega t+i\sin\omega t\). In an idempotent basis, this yields:
\[ \frac{V(t)}{c} = \ep + \cos\omega t\,\em + \sin\omega t\,i\em. \tag{23} \]
Moving to the basis \(\{1,i,j,ij\}\), we obtain:
\[ \frac{V(t)}{c} = \frac{1+\cos\omega t}{2}\cdot 1 + \frac{\sin\omega t}{2}\cdot i + \frac{1-\cos\omega t}{2}\cdot j - \frac{\sin\omega t}{2}\cdot(ij). \tag{24} \]
The same formula can be written in terms of half the angle:
\[ \frac{V(t)}{c} = \cos^2\frac{\omega t}{2} + \frac{\sin\omega t}{2}\,i + \sin^2\frac{\omega t}{2}\,j - \frac{\sin\omega t}{2}\,ij. \tag{25} \]
Integrating the velocity components over time and choosing the origin at \((0,0,0,0)\), we obtain:
\[ \begin{cases} ct=ct,\\ x=0,\\ y=\dfrac{c}{\omega}\sin\omega t,\\ z=\dfrac{c}{\omega}\left(1-\cos\omega t\right). \end{cases} \tag{26} \]
This is a helical line: along the \(ct\) axis, the point moves uniformly, and in the \(y,z\) plane, it makes a circular motion of radius \(c/\omega\). In a system of units where \(c=1\), this radius takes the form \(1/\omega\), but the time coordinate must then be written as \(T=t\), not as \(ct=t\). Both commutative forms of the algebra yield the same kinematic result.
Advantages of an idempotent representation
1. Visual geometry. The basis \(\{\ep,i\ep,\em,i\em\}\) immediately shows the decomposition of space into two complex planes: \((ct,x)\) and \((y,z)\).
2. Natural notation of motion. The velocity vector \(V(t)=c(\ep+\em e^{i\omega t})\) has a compact form: the first component specifies uniform motion along \(ct\), and the second specifies rotation in the \((y,z)\) plane.
3. Diagonal separation of components. Since \(\ep\em=0\), the two idempotent parts do not mix when multiplied. This makes transformations in independent planes particularly transparent.
4. Direct connection with coordinates. In this basis, the coordinates \(ct,x,y,z\) appear directly at the corresponding basis elements, so the geometric interpretation is as simple as possible.
Advantages of the \(\{1,i,j,ij\}\) Representation
1. Unified algebraic notation. The \(\{1,i,j,ij\}\) basis uses familiar units: real \(1\), complex \(i\), hyperbolic \(j\), and their product \(ij\).
2. Explicit connection between the complex and hyperbolic structures. This form clearly shows that two types of units are simultaneously present in a single commutative algebra: \(i^2=-1\) and \(j^2=1\).
3. Convenience for general transformations. The representation via \(A+Bi+Cj+Dij\) is convenient for deriving transition matrices, comparing with other algebras, and analyzing symmetries.
4. Preserving Commutativity. Unlike quaternion models, the order of the factors does not matter here: \(ij=ji\). This simplifies working with phase factors and power expressions.
5. Equivalence of \(\mathbb{C}\oplus\mathbb{C}\). The representation \(\{1,i,j,ij\}\) clearly shows that the entire construction can be viewed as two related but independent complex components.
Conclusions
This appendix shows that the new Cartesian basis \(\{\ep,i\ep,\em,i\em\}\) and the algebraic basis \(\{1,i,j,ij\}\) are two equivalent commutative representations of the same structure. Both bases lead to a decomposition of \(\mathcal{A}\cong\mathbb{C}\oplus\mathbb{C}\), but emphasize different aspects of this algebra.
The idempotent representation is convenient for geometric interpretation and description of motion, since it directly divides space into the \((ct,x)\) and \((y,z)\) planes. The representation \(\{1,i,j,ij\}\), in turn, emphasizes the unified algebraic nature of the construction, the connection between complex and hyperbolic units, and provides compact formulas for transitions between coordinates.
Thus, the application does not replace one algebra with another, but rather demonstrates the advantages of both notations. This allows the use of an idempotent basis where visual geometry is important, and the basis \(\{1,i,j,ij\}\) where it is more convenient to work with the general algebraic structure and transitions between representations.