2026-07-14
Rotation as a Consequence of Norm Preservation in an Idempotent Basis
In this paper, we consider a vector in a basis of two complementary idempotents \(\{\ep,\em\}\) with complex coefficients. We show that this representation naturally generates a four-dimensional real basis \(\{\ep,i\ep,\em,i\em\}\), in which each idempotent direction forms its own complex plane.
For a vector of unit norm, we prove the central equality \(J\cdot\dot J=0\), which means the derivative has no longitudinal component. From this condition, a rotational differential equation is derived, the solution of which directly yields the vector \(J(t)=\ep+\em e^{i\omega t}\), without first introducing trigonometric functions or a complex exponential.
The result obtained is one of the key elements in constructing the general structure of the new basis. Unlike the traditional approach, rotation is considered here not as an initial assumption, but as a direct consequence of the preservation of the vector norm in the idempotent representation.
Idempotent decomposition
Consider two complementary idempotents \(\ep\) and \(\em\), possessing properties [1, 2]
\[\tag{1} \ep^2=\ep, \qquad \em^2=\em, \qquad \ep\em=0, \qquad \ep+\em=1. \] We represent the general vector of the algebra under consideration as follows:
\[\tag{2} J(t) = \ep X(t) + \em Y(t), \] where \(X(t)\) and \(Y(t)\) are complex functions of the real parameter \(t\). The condition \(\ep\em=0\) means that the two components belong to independent idempotent directions.
Automatic emergence of a four-dimensional basis
Vector (2) is written in a two-dimensional basis \(\{\ep,\em\}\) over the field of complex numbers. Let's decompose the complex coefficients into real and imaginary parts:
\[\tag{3} X(t) = x_0(t)+i x_1(t), \qquad Y(t) = x_2(t)+i x_3(t), \] where \(x_0,x_1,x_2,x_3\) are real functions. Substituting formula (3) into representation (2) yields
\[\tag{4} J = \ep \left( x_0+i x_1 \right) + \em \left( x_2+i x_3 \right). \] After expanding the parentheses, we obtain
\[\tag{5} \boxed{ J = x_0\ep + x_1 i\ep + x_2\em + x_3 i\em }. \] Therefore, a two-dimensional complex idempotent basis automatically generates a four-dimensional real basis.
\[\tag{6} \boxed{ \mathcal B_{\mathbb R} = \{ \ep, i\ep, \em, i\em \} }. \] Each idempotent defines its own complex plane:
\[\tag{7} \ep\mathbb C = \operatorname{span}_{\mathbb R} \{ \ep, i\ep \}, \qquad \em\mathbb C = \operatorname{span}_{\mathbb R} \{ \em, i\em \}. \] The complete space is represented as the direct sum of these planes:
\[\tag{8} \boxed{ \mathcal A = \ep\mathbb C \oplus \em\mathbb C }. \] Therefore, the same algebra can be viewed either as a two-dimensional space over \(\mathbb C\), or as a four-dimensional space over \(\mathbb R\):
\[\tag{9} \boxed{ \dim_{\mathbb C}\mathcal A=2, \qquad \dim_{\mathbb R}\mathcal A=4 }. \] Multiplication by a complex unit \(i\) maps each idempotent direction to a transverse direction of the same complex plane:
\[\tag{10} \ep \xrightarrow{\;i\;} i\ep, \qquad i\ep \xrightarrow{\;i\;} -\ep, \] \[\tag{11} \em \xrightarrow{\;i\;} i\em, \qquad i\em \xrightarrow{\;i\;} -\em. \] Thus, idempotents divide space into two independent components, and the complex unit expands each of them to the two-dimensional real plane.
Unit Vector
Next, consider a special case in which the first idempotent component remains constant:
\[\tag{12} X(t)=1. \] Then vector (2) takes the form
\[\tag{13} \boxed{ J(t) = \ep + \em Y(t) }. \] We define the real scalar norm of the general vector \(J=\ep X+\em Y\) by the expression
\[\tag{14} |J|^2 = \frac{ |X|^2+|Y|^2 }{2}. \] For vector (13), this definition takes the form
\[\tag{15} |J|^2 = \frac{ 1+|Y(t)|^2 }{2}. \] We require that the unit norm be preserved:
\[\tag{16} |J(t)|=1. \] From formulas (15) and (16), it immediately follows
\[\tag{17} \boxed{ |Y(t)|^2=1 }. \] Consequently, the variable component \(\em Y(t)\) can changeChange direction in the complex plane \(\{\em,i\em\}\), but cannot change its magnitude. The constant component remains directed along \(\ep\).
Derivative of a unit vector
We define the derivative of the vector using the standard limit procedure:
\[\tag{18} \dot J(t) = \lim_{\Delta t\to0} \frac{ J(t+\Delta t)-J(t) }{ \Delta t }. \] Since idempotents are time-independent, formula (13) implies
\[\tag{19} \dot J(t) = \em\dot Y(t). \] For two vectors
\[\tag{20} A = \ep A_{+} + \em A_{-}, \qquad B = \ep B_{+} + \em B_{-} \] Introduce the real scalar product
\[\tag{21} A\cdot B = \frac12 \operatorname{Re} \left( \overline{A_{+}}B_{+} + \overline{A_{-}}B_{-} \right). \] With this definition,
\[\tag{22} J\cdot J = |J|^2. \] Since the vector norm is preserved,
\[\tag{23} \frac{d}{dt}|J|^2=0. \] Taking into account formula (22), we obtain
\[\tag{24} \frac{d}{dt} \left( J\cdot J \right) = 2J\cdot\dot J = 0. \] Therefore,
\[\tag{25} \boxed{ J\cdot\dot J=0 }. \] Formula (25) is the central result of this paper. It shows that preserving the norm automatically eliminates any component of the derivative directed along the vector itself. Consequently, a change in the unit vector can only occur in the transverse direction. It is this property that further leads to the rotation of the complex component and the emergence of an exponential solution.
In other words, a derivative cannot increase or decrease the length of a vector. It only describes a change in its direction. For the variable component \(\em Y(t)\), belonging to the plane \(\{\em,i\em\}\), such a transverse change is a rotation within the given complex plane.
For vector (13), equality (25) takes the form
\[\tag{26} J\cdot\dot J = \frac12 \operatorname{Re} \left( \overline{Y}\dot Y \right) = 0. \] Therefore,
\[\tag{27} \boxed{ \operatorname{Re} \left( \overline{Y}\dot Y \right) = 0 }. \] The same condition can be obtained by direct differentiation of the equality \(Y\overline Y=1\):
\[\tag{28} \frac{d}{dt} \left( Y\overline Y \right) = \dot Y\overline Y + Y\dot{\overline Y} = 0. \] Since the two terms in formula (28) are complex conjugates, we have
\[\tag{29} 2\operatorname{Re} \left( \overline Y\dot Y \right) = 0. \] Thus, the constancy of the norm and the orthogonality of the vector of its derivative are two equivalent forms of the same condition.
Logarithmic Derivative
From the condition \(|Y|=1\) it follows that \(Y(t)\neq0\). Therefore, we can define the logarithmic derivative
\[\tag{30} q(t) = \frac{\dot Y(t)}{Y(t)}. \] Since \(Y\overline Y=1\), we have
\[\tag{31} \frac{\dot Y}{Y} = \overline Y\dot Y. \] From formula (27) it follows
\[\tag{32} \operatorname{Re}q(t)=0. \] Therefore, the logarithmic derivative is purely imaginary. Therefore, there exists a real function \(\omega(t)\), such that
\[\tag{33} \boxed{ q(t)=i\omega(t) }. \] Substituting definition (30), we obtain the differential equation
\[\tag{34} \boxed{ \dot Y(t) = i\omega(t)Y(t) }. \] The factor \(i\) does not arise here as an additional assumption. It is a consequence of the fact that the logarithmic derivative of a constant absolute value function has no real part. In a four-dimensional real basis, multiplication by \(i\) transforms the direction \(\em\) into the direction \(i\em\), defining a transverse change within the second idempotent plane.
Equation of Motion of the Total Vector
From formula (13) it follows
\[\tag{35} J-\ep = \em Y. \] Using formulas (19) and (34), we obtain
\[\tag{36} \dot J = \em\dot Y = i\omega(t)\em Y. \] Therefore,
\[\tag{37} \boxed{ \dot J = i\omega(t) \left( J-\ep \right) }. \] Equation (37) describes the change in the total vector. The component \(\ep\) remains constant, and the variable part \(J-\ep=\em Y\) rotates within the complex plane \(\{\em,i\em\}\).
Multiplication by \(i\) transforms the variable component into the transverse direction of the same plane, and the real quantityand \(\omega(t)\) determines the rate of change of its direction.
General Solution
Equation (34) can be written in separable form:
\[\tag{38} \frac{dY}{Y} = i\omega(t)\,dt. \] Integrating from the initial moment \(t_0\) to the moment \(t\), we obtain
\[\tag{39} \ln \frac{Y(t)}{Y(t_0)} = i \int_{t_0}^{t} \omega(\tau)\,d\tau. \] Therefore,
\[\tag{40} Y(t) = Y(t_0) \exp \left[ i \int_{t_0}^{t} \omega(\tau)\,d\tau \right]. \] The initial value \(Y(t_0)\) has unit absolute value and determines only the initial direction of the variable component. We choose
\[\tag{41} t_0=0, \qquad Y(0)=1. \] Then
\[\tag{42} Y(t) = \exp \left[ i \int_{0}^{t} \omega(\tau)\,d\tau \right]. \] The full vector takes the form
\[\tag{43} \boxed{ J(t) = \ep + \em \exp \left[ i \int_{0}^{t} \omega(\tau)\,d\tau \right] }. \] Formula (43) describes the most general case of rotation of the variable component with an arbitrary dependence \(\omega(t)\). Preservation of the norm is ensured by a purely imaginary exponent.
Constant Frequency
Consider the case of constant frequency:
\[\tag{44} \omega(t) = \omega = \operatorname{const}. \] Then
\[\tag{45} \int_{0}^{t} \omega\,d\tau = \omega t. \] From formula (42) it follows
\[\tag{46} Y(t)=e^{i\omega t}. \] Substituting the found solution into formula (13), we obtain the main result:
\[\tag{47} \boxed{ J(t) = \ep + \em e^{i\omega t} }. \] The complex exponential was not introduced here as the initial form of motion. It arose as the solution of a differential equation obtained from the conservation of the norm and the orthogonality of the vector to its derivative.
Relationship of the solution to a four-dimensional basis
In general, the vector \(J=\ep X+\em Y\) has four real coordinates in the basis \(\{\ep,i\ep,\em,i\em\}\). For solution (47), the constant component belongs to the direction \(\ep\), and the variable component moves in the plane \(\{\em,i\em\}\).
Therefore, the specific trajectory (47) belongs to the real subspace
\[\tag{48} \operatorname{span}_{\mathbb R} \{ \ep, \em, i\em \}. \] The coordinate along \(i\ep\) for the chosen solution is zero. However, the direction \(i\ep\) remains a full-fledged element of the general four-dimensional basis and arises from complex modulation of the first idempotent component.
If both complex components are allowed to vary,
\[\tag{49} J(t) = \ep X(t) + \em Y(t), \] then motion can occur simultaneously in two independent planes:
\[\tag{50} \operatorname{span}_{\mathbb R} \{ \ep, i\ep \}, \qquad \operatorname{span}_{\mathbb R} \{ \em, i\em \}. \] Thus, a complete four-dimensional basis arises before a specific law of motion is chosen, while the expression \(J=\ep+\em e^{i\omega t}\) describes a special trajectory within one of its three-dimensional subspaces.
Logical Sequence of Output
The main result of the work can be represented as a sequence
\[\tag{51} \{\ep,\em\}_{\mathbb C} \quad\Longrightarrow\quad \{ \ep, i\ep, \em, i\em \}_{\mathbb R}, \] \[\tag{52} |J|=1 \quad\Longrightarrow\quad J\cdot\dot J=0 \quad\Longrightarrow\quad \operatorname{Re} \left( \frac{\dot Y}{Y} \right) =0, \] \[\tag{53} \frac{\dot Y}{Y} = i\omega \quad\Longrightarrow\quad Y=e^{i\omega t} \quad\Longrightarrow\quad J=\ep+\em e^{i\omega t}. \] Thus, the four-dimensional real basis arises from the complex extension of two idempotent directions, and the rotation arises from the preservation of the norm inside one of the resulting complex planes.
Conclusions
This paper shows that a two-dimensional complex idempotent basis \(\{\ep,\em\}\) automatically generates a four-dimensional real basis \(\{\ep,i\ep,\em,i\em\}\). Each idempotent forms its own complex plane, and the complete space is represented as the direct sum of these planes. Therefore, the four-dimensional structure is not introduced as an additional assumption, but arises directly from the complex nature of the coefficients of the idempotent decomposition.
For a special vector \(J=\ep+\em Y(t)\) the condition\(|J|=1\) leads to a constant modulus \(Y(t)\) and to the central equality \(J\cdot\dot J=0\). This equality excludes the longitudinal component of the derivative and allows only transverse variation of the variable component in the plane \(\{\em,i\em\}\).
Orthogonality implies that the logarithmic derivative of the variable component is purely imaginary: \(\dot Y/Y=i\omega\). For a constant \(\omega\) solution is \(Y=e^{i\omega t}\), and the full vector takes the form \(J=\ep+\em e^{i\omega t}\). Therefore, both the new four-dimensional basis and the rotation within its idempotent plane follow naturally from the idempotent decomposition, complex structure, and norm preservation.
Materials used
- Wikipedia. Idempotence.
- Wikipedia. Idempotent (ring theory).