In the paper "Rotation as a Consequence of Norm Preservation in an Idempotent Basis," it was shown that representing a vector in the basis \(\{\ep,i\ep,\em,i\em\}\) naturally leads to the constancy of its norm. This property implies the orthogonality of the vector and its derivative \(J\cdot\dot J=0\), so a change of state occurs only in the direction tangent to the surface of constant norm. Thus, rotation arises not as an additional assumption, but as a direct consequence of norm preservation.

This paper considers the physical consequences of this geometric construction. It has been shown that differentiation of the rotational component leads to the time-dependent Schrödinger equation [1], with the idempotent \(\em\) itself acting as an algebraic projection onto the dynamic sector. The resulting energy scale \(W=\hbar\omega\) is related to the total relativistic energy of the internal motion, which allows us to obtain an expression for the rest mass of the particle in terms of its internal frequency: \[ m = \frac{\hbar\omega} {\gamma_{\mathrm{int}}c^2}. \] Therefore, conservation of the norm, rotation, the Schrödinger equation, internal energy, and the origin of mass turn out to be successive elements of a single mathematical construct.

Previously we considered a four-dimensional real basis \[ \tag{1} \left\{ \ep,\;i\ep,\;\em,\;i\em \right\}, \] generated by two mutually complementary idempotents \(\ep\) and \(\em\).

They satisfy the properties \[ \tag{2} \ep^2=\ep, \qquad \em^2=\em, \qquad \ep\em=0. \] Each idempotent direction forms its own complex plane: \[ \left\{\ep,i\ep\right\}, \qquad \left\{\em,i\em\right\}. \]

This representation allows us to separate the constant and rotational components of the motion. The component belonging to the direction \(\ep\) may remain unchanged, while the component in the plane \((\em,i\em)\) describes the internal periodic process.

In this paper, this result is used as the starting point for deriving the Schrödinger time-dependent equation and the relationship between the internal frequency and the particle mass. It is necessary to distinguish two levels of construction. The energy scale (hbar omega) arises directly from differentiation. The rest mass appears only after an additional physical assumption linking the total energy of internal motion with the relativistic expression (gamma_{mathrm{int}}mc^2).

Consider the internal motion vector \[ \tag{3} J(t) = \ep + \em e^{-i\omega t}, \] where \(\omega\) is the angular frequency of the internal periodic process.

Its norm is independent of time. The constant component \(\ep\) does not change, and the complex factor \(e^{-i\omega t}\) has unity absolute value. Therefore, time evolution changes only the direction of the rotational component, but not its magnitude.

Differentiating the square of the norm, we obtain \[ \tag{4} \frac{d}{dt}|J|^2 = 2\,\operatorname{Re} \left( J^{*}\cdot\dot J \right) = 0. \] Therefore, the real part of the scalar product of a vector and its derivative is zero. In the geometric representation of the basis under consideration, this condition is written as \[ \tag{5} J\cdot\dot J=0. \]

Condition (5) means that the derivative \(\dot J\) has no component along the vector \(J\) itself. Consequently, the change in state occurs only in the tangent direction to the surface of constant norm. In other words, conservation of the norm directly generates rotational evolution.

If the vector norm is interpreted as a complete energy invariant of the system, then its constancy expresses the law of conservation of total energy. Moreover, the equality \(J\cdot\dot J=0\) means that the energy does not change its total magnitude, but is only redistributed between the directions of the basis under consideration. This is why any permissible change in state can only be rotational: the radial component of the derivative is absent.

Expanding the complex exponential in expression (3) yields \[ \tag{6} J(t) = \ep + \em\cos\omega t - i\em\sin\omega t. \] Thus, the end of the second component moves along the unit circle in the complex plane \[ \left\{ \em,\;i\em \right\}. \]

The vector \(J(t)\) contains two fundamentally different components: a constant component \(\ep\) and a rotational component \(\eme^{-i\omega t}\). It is the rotational component that determines the internal temporal dynamics of the state under consideration.

In general, the rotational component can contain an arbitrary initial phase: \[ \tag{7} J_{\varphi}(t) = \ep + \em e^{-i(\omega t+\varphi)}, \] where \(\varphi\) is the initial phase of the internal motion.

All vectors of the family \(J_{\varphi}(t)\) differ only in the choice of the time reference. Indeed, when replacing \[ \tag{8} t' = t+ \frac{\varphi}{\omega} \] we obtain \[ \tag{9} e^{-i(\omega t+\varphi)} = e^{-i\omega t'}. \]

Therefore, changing the initial phase is equivalent to shifting the time reference and does not change the frequency, norm, or energy characteristics of the rotational component. Therefore, without loss of generality, we can assume \[ \tag{10} \varphi=0 \] and use the form \[ \tag{11} J(t) = \ep + \em e^{-i\omega t}. \]

The negative sign in the exponent corresponds to the standard time factor of stationary states in quantum mechanics and allows one to obtain the usual sign in the time-dependent Schrödinger equation. Choosing the opposite direction of rotation corresponds to a change in the sign of the frequency or complex conjugation of the solution.

Differentiating expression (11) with respect to time, we obtain \[ \tag{12} \frac{\partial J}{\partial t} = -i\omega\em e^{-i\omega t}. \] The constant component \(\ep\) disappears upon differentiation, so the derivative isolates only the rotational part of the vector.

The resulting derivative is directed tangentially to the rotational trajectory. This is directly consistent with the condition \[ J\cdot\dot J=0, \] obtained from the constancy of the norm. Thus, differentiation does not create a new independent direction, but describes a tangential change in an existing rotational component.

Multiply expression (12) by \(i\hbar\): \[ \tag{13} i\hbar \frac{\partial J}{\partial t} = \hbar\omega\em e^{-i\omega t}. \] The right-hand side has the dimension of energy and contains the standard quantum energy factor \[ \tag{14} W = \hbar\omega. \]

Therefore, differentiating the vector \(J(t)\) directly leads to the energy scale of the internal periodic process. However, the expression \(\hbar\omega\) currently characterizes the total energy of the rotational component, and not directly the rest energy of the particle.

Thanks to the properties of idempotents \[ \ep\em=0, \qquad \em^2=\em, \] multiplying a vector \(J(t)\) by \(\em\) recovers its rotational component: \[ \tag{15} \em J(t) = \em \left( \ep+ \em e^{-i\omega t} \right). \]

Expanding the product, we obtain \[ \tag{16} \em J(t) = \em\ep + \em^2e^{-i\omega t} = \em e^{-i\omega t}. \] Therefore, the idempotent itself \(\em\) acts as an algebraic projection onto the dynamic sector of the vector. No additional projection is required for this.

Taking into account expression (16), the result of differentiation can be rewritten as \[ \tag{17} \frac{\partial J}{\partial t} = -i\omega\em J. \] After multiplying by \(i\hbar\) we obtain \[ \tag{18} i\hbar \frac{\partial J}{\partial t} = \hbar\omega\em J. \]

Equation (18) represents the differential form of rotational evolution. Its left-hand side is determined by the tangential derivative of the state, while the right-hand side defines the action of the energy operator on the rotational sector.

The standard Schrödinger time-based equation [1] has the form \[ \tag{19} i\hbar \frac{\partial\Psi}{\partial t} = \widehat H\Psi, \] where \(\widehat H\) is the Hamiltonian of the system.

By comparing expressions (18) and (19), we determine the Hamiltonian of the internal motion: \[ \tag{20} \widehat H_{\mathrm{int}} = \hbar\omega\em. \] Then the equation for the vector \(J(t)\) takes the form \[ \tag{21} i\hbar \frac{\partial J}{\partial t} = \widehat H_{\mathrm{int}}J. \]

Substituting Hamiltonian (20) yields \[ \tag{22} i\hbar \frac{\partial J}{\partial t} = \hbar\omega\em J. \] This equation is identical in structure to the time-dependent Schrödinger equation, but has an important feature: the Hamiltonian arises directly from the geometry of the idempotent basis and the harmonic evolution of the rotational component.

In this interpretation, the Schrödinger time-domain equation describes the tangential motion of a state along a surface of constant norm. Norm conservation is the primary condition, rotation is its geometry.metric consequence, and the Schrödinger equation is a differential description of this rotational evolution.

If the constancy of the norm is associated with the constancy of the total energy invariant, then equation (21) can also be viewed as a differential form of the law of conservation of energy: the total magnitude of the state remains unchanged, while its direction evolves under the action of the Hamiltonian.

The action of the internal Hamiltonian on the constant component is \[ \tag{23} \widehat H_{\mathrm{int}}\ep = \hbar\omega\em\ep = 0, \] while the action on the rotational component is \[ \tag{24} \widehat H_{\mathrm{int}} \left( \em e^{-i\omega t} \right) = \hbar\omega \em e^{-i\omega t}. \]

Thus, the vector \(J(t)\) contains two energetically distinct sectors. The constant component \(\ep\) has zero energy with respect to the internal Hamiltonian, and the rotational component is an eigenstate with eigenvalue \[ \tag{25} W=\hbar\omega. \]

If we represent a vector in idempotent coordinates \[ \tag{26} J(t) \longleftrightarrow \begin{pmatrix} 1\\ e^{-i\omega t} \end{pmatrix}, \] then multiplication by \(\em\) corresponds to the matrix \[ \tag{27} \em \longleftrightarrow \begin{pmatrix} 0&0\\ 0&1 \end{pmatrix}. \]

Then the Hamiltonian of the internal motion has the form \[ \tag{28} \widehat H_{\mathrm{int}} = \begin{pmatrix} 0&0\\ 0&\hbar\omega \end{pmatrix}. \] The Schrödinger equation is written as \[ \tag{29} i\hbar \frac{\partial}{\partial t} \begin{pmatrix} 1\\ e^{-i\omega t} \end{pmatrix} = \begin{pmatrix} 0&0\\ 0&\hbar\omega \end{pmatrix} \begin{pmatrix} 1\\ e^{-i\omega t} \end{pmatrix}. \]

The left-hand side is equal to \[ \tag{30} i\hbar \begin{pmatrix} 0\\ -i\omega e^{-i\omega t} \end{pmatrix} = \begin{pmatrix} 0\\ \hbar\omega e^{-i\omega t} \end{pmatrix}, \] which is completely identical to the right-hand side. The matrix notation clearly shows the separation of the constant and dynamic sectors.

From expression (25), it follows that the rotational component is characterized by the total energy \[ \tag{31} W = \hbar\omega = h\nu, \] where \[ \tag{32} \omega=2\pi\nu. \] This is the standard quantum relationship between the frequency of a periodic process and its energy.

Within the model under consideration, it is assumed that this energy corresponds to the total relativistic energy of internal motion: \[ \tag{33} W = \gamma_{\mathrm{int}}mc^2, \] where \(m\) is the rest mass of the particle, and \(\gamma_{\mathrm{int}}\) is the hypothetical Lorentz factor of the internal dynamics.

Combining expressions (31) and (33), we obtain \[ \tag{34} \hbar\omega = \gamma_{\mathrm{int}}mc^2. \] Hence, the rest mass is equal to \[ \tag{35} m = \frac{\hbar\omega} {\gamma_{\mathrm{int}}c^2}. \]

Consequently, mass arises as a characteristic of an internal periodic process and is determined by its frequency, taking into account the ratio between the total internal energy and the rest energy. In this construction, frequency is the primary dynamic characteristic, and mass is the derived energy quantity.

In previous work, the relationship was proposed \[ \tag{36} \gamma_{\mathrm{int}} = \frac{1}{\alpha_{\mathrm{fs}}}, \] where \(\alpha_{\mathrm{fs}}\) is the fine structure constant, and \(\gamma_{\mathrm{int}}\) is the internal Lorentz factor.

Then the expression for mass takes the form \[ \tag{37} m = \frac{\alpha_{\mathrm{fs}}\hbar\omega} {c^2}. \] Accordingly, the rest energy is \[ \tag{38} mc^2 = \alpha_{\mathrm{fs}}\hbar\omega. \]

Since \[ \tag{39} W = \hbar\omega, \] we can write \[ \tag{40} mc^2 = \frac{W} {\gamma_{\mathrm{int}}} = \alpha_{\mathrm{fs}}W. \] Thus, the rest energy is the portion of the total energy of internal motion determined by the factor \(\alpha_{\mathrm{fs}}\).

The resulting relationship between frequency and mass is \[ \tag{41} \boxed{ m = \frac{\hbar\omega} {c^2\gamma_{\mathrm{int}}} = \frac{\alpha_{\mathrm{fs}}\hbar\omega} {c^2} }. \]

The inverse expression allows us to determine the intrinsic angular frequency in terms of mass: \[ \tag{42} \omega = \frac{mc^2} {\hbar}\gamma_{\mathrm{int}} = \frac{mc^2} {\hbar\alpha_{\mathrm{fs}}}. \] For the usual frequency, we get \[ \tag{43} \nu = \frac{mc^2}{h}\gamma_{\mathrm{int}} = \frac{mc^2} {h\alpha_{\mathrm{fs}}}. \]

SinceThe Hamiltonian of internal motion is equal to \[ \widehat H_{\mathrm{int}} = \hbar\omega\em, \] the operator of the total energy mass can be defined: \[ \tag{44} \widehat M_{\mathrm{int}} = \frac{\widehat H_{\mathrm{int}}} {c^2}. \] Then \[ \tag{45} \widehat M_{\mathrm{int}} = \frac{\hbar\omega} {c^2}\em. \]

However, this operator corresponds to the total internal energy \(W\), and not directly to the rest mass. To obtain the rest mass operator, we must take into account the internal Lorentz factor: \[ \tag{46} \widehat M_0 = \frac{1}{\gamma_{\mathrm{int}}} \widehat M_{\mathrm{int}}. \] Hence \[ \tag{47} \widehat M_0 = \frac{\hbar\omega} {\gamma_{\mathrm{int}}c^2}\em. \]

Given the relation \[ \gamma_{\mathrm{int}} = \frac{1}{\alpha_{\mathrm{fs}}}, \] we obtain \[ \tag{48} \widehat M_0 = \frac{\alpha_{\mathrm{fs}}\hbar\omega} {c^2}\em. \] Its eigenvalue on the rotational component is equal to the rest mass: \[ \tag{49} \widehat M_0 \left( \em e^{-i\omega t} \right) = m\em e^{-i\omega t}. \]

The result obtained allows the following consistent interpretation. The constancy of the norm of the vector \(J(t)\) leads to the condition \[ \tag{50} J\cdot\dot J=0, \] which means that the derivative is directed tangent to the surface of constant norm. Therefore, the internal evolution has a rotational nature.

For the vector \[ J(t) = \ep + \em e^{-i\omega t} \] differentiation eliminates the constant component and isolates the internal periodic process: \[ \tag{51} \frac{\partial J}{\partial t} = -i\omega\em J. \]

Multiplicating the derivative by \(i\hbar\) converts the angular frequency to an energy scale: \[ \tag{52} i\hbar \frac{\partial J}{\partial t} = \hbar\omega\em J. \] Therefore The quantity \(\hbar\omega\) arises directly from the time evolution of the rotational component.

Idempotent \(\em\) is simultaneously an element of the basis and a projector onto the dynamical sector. This property allows us to write the Hamiltonian without introducing an additional operator: \[ \tag{53} \widehat H_{\mathrm{int}} = \hbar\omega\em. \]

Mass is not a direct result of differentiation alone. The total internal energy follows directly from the derivative \[ W=\hbar\omega. \] The transition to mass requires an additional physical relation \[ W=\gamma_{\mathrm{int}}mc^2. \] It is this assumption that links the intrinsic frequency with the observed rest mass.

Within this model, frequency is the primary characteristic of the internal motion, the total energy is defined as \(\hbar\omega\), and the rest mass is a fraction of this energy: \[ \tag{54} mc^2 = \frac{\hbar\omega} {\gamma_{\mathrm{int}}}. \]

The entire sequence can be represented as \[ \tag{55} \boxed{ |J|=\operatorname{const} \;\Longrightarrow\; J\cdot\dot J=0 \;\Longrightarrow\; \text{rotation} \;\Longrightarrow\; i\hbar\dot J=\widehat H_{\mathrm{int}}J \;\Longrightarrow\; W=\hbar\omega \;\Longrightarrow\; m= \frac{\hbar\omega} {\gamma_{\mathrm{int}}c^2} }. \]

The solution considered can be generalized by introducing constant complex amplitudes: \[ \tag{56} \Psi(t) = A\ep + B\em e^{-i\omega t}, \] where \(A\) and \(B\) are independent of time.

Differentiation yields \[ \tag{57} i\hbar \frac{\partial\Psi}{\partial t} = \hbar\omega B\em e^{-i\omega t}. \] On the other hand, \[ \tag{58} \em\Psi = B\em e^{-i\omega t}. \] Therefore, \[ \tag{59} i\hbar \frac{\partial\Psi}{\partial t} = \hbar\omega\em\Psi. \]

Therefore, the equation \[ \tag{60} i\hbar \frac{\partial\Psi}{\partial t} = \widehat H_{\mathrm{int}}\Psi, \qquad \widehat H_{\mathrm{int}} = \hbar\omega\em, \] is satisfied not only for the normalized vector \(J(t)\), but also for the entire family of states consisting of a constant \(\ep\)-component and a harmonic \(\em\)-component.

If the amplitudes \(A\) and \(B\) are constant, then the norm of such a state also does not change over time. Consequently, its evolution remains rotational, and the Hamiltonian acts only on the dynamical idempotent sector.

The resulting equation describes only the time evolution of the internal periodic component. It does not yet include spatial derivatives, potential energy, or interaction with external fields. Therefore, the expression \[ i\hbar \frac{\partial J}{\partial t} = \hbar\omega\em J \] should be considered as a time equationfor a free internal state with a fixed frequency.

To construct a more general equation, it will be necessary to define a spatial operator in the same idempotent basis and establish a relationship between frequency, momentum, and external interactions. Only then can the resulting construction be compared with the full space-time Schrödinger equation.

It should also be emphasized that the equality \(W=\gamma_{\mathrm{int}}mc^2\) and the relationship \(\gamma_{\mathrm{int}}=1/\alpha_{\mathrm{fs}}\) are additional physical assumptions of the model under consideration. They do not follow solely from the algebraic properties of the idempotent basis and require independent physical justification.

Nevertheless, even at this stage it is clear that the standard time structure of quantum mechanics arises directly from the harmonic dependence of the rotational component and the condition of norm constancy.

In the paper "Rotation as a Consequence of Norm Conservation in an Idempotent Basis" it was shown that the constancy of the vector norm leads to orthogonality \(J\cdot\dot J=0\). This condition implies the absence of a radial component of the derivative, so the change of state occurs only along the tangent to the surface of constant norm. Thus, rotation arises as a direct geometric consequence of the law of norm conservation.

For the vector \[ J(t) = \ep + \em e^{-i\omega t} \] differentiation isolates the rotational component and leads to the expression \[ i\hbar \frac{\partial J}{\partial t} = \hbar\omega\em J. \] Since the idempotent \(\em\) cancels the constant component \(\ep\) and preserves the rotational component, it itself acts as a projector onto the dynamic sector.

On this basis, the Hamiltonian of the internal motion is determined without introducing additional projectors: \[ \widehat H_{\mathrm{int}} = \hbar\omega\em. \] The resulting equation \[ i\hbar \frac{\partial J}{\partial t} = \widehat H_{\mathrm{int}}J \] coincides in form with the time-dependent Schrödinger equation. Thus, the Schrödinger equation is interpreted as a differential description of the rotational evolution of a state with a constant norm.

If the norm is considered a complete energy invariant, then its conservation expresses the law of conservation of total energy. In this case, the Schrödinger equation describes not a change in the total energy, but a change in the direction of the state and the redistribution of its components with the energy invariant remaining constant.

Differentiation directly determines the energy scale of the internal rotation: \[ W=\hbar\omega. \] After accepting the physical assumption \[ \hbar\omega = \gamma_{\mathrm{int}}mc^2 \] we obtain an expression for the rest mass: \[ m = \frac{\hbar\omega} {c^2\gamma_{\mathrm{int}}}. \] Therefore, the mass turns out not to be an independent initial parameter, but a derived characteristic of an internal periodic process.

As a result, the two works form a single logical sequence: conservation of the norm generates rotation, rotational evolution leads to the Schrödinger time equation and the energy scale \(\hbar\omega\), and the relationship between the total internal energy and the rest energy allows us to obtain the particle's mass. The new idempotent basis thus allows us to consider norm conservation as a fundamental geometric principle, from which rotational evolution, the Schrödinger time equation, the energy relation E=ℏω, and the expression for the particle mass follow sequentially.