2025-10-27
Hypercomplex structure of a plane electromagnetic wave and the quantum of action
A plane electromagnetic wave (PEW) is one of the fundamental solutions of Maxwell’s equations, describing the propagation of oscillations of the electric and magnetic fields in space and time. However, the traditional vector representation, while convenient for calculations, conceals the deep internal symmetry of the wave process. The use of hypercomplex numbers makes it possible to combine the electric, magnetic, spatial, and temporal components into a single analytical form that clearly reflects their interrelation. This approach reveals the geometric meaning of the wave as a unified four-dimensional entity rather than as a set of independent fields.
In this work, a plane electromagnetic wave is considered through a hypercomplex exponential expression containing the hyperbolic unit \(\jk\), which defines the direction of propagation and the internal structure of the wave. It is shown that from this representation, the longitudinal and transverse components of the wave naturally emerge, expressing not only the oscillatory but also the energetic aspects of the process. Furthermore, a connection between the hypercomplex phase and the concept of the quantum of action is revealed, allowing parallels to be drawn between classical electrodynamics and the principles of quantum mechanics.
Transverse Components of the Plane Electromagnetic Wave
As we established earlier, a plane electromagnetic wave can be represented through a hypercomplex exponential, in which the hyperbolic unit \(\jk\) defines the direction of propagation, and the coefficient \(c\) represents the speed of light in vacuum: \[\tag{1} V(t) = c\, \jm^{\v t} \\ c\,\jk^{\v t} = \s(t) + i\, x(t) + \jk\, z(t) + i\jk\, y(t) \] This expression describes the instantaneous values of four components of the wave process — temporal, electric, magnetic, and longitudinal — all combined into a single hypercomplex form.
Let us now consider separately the transverse components corresponding to the electric and magnetic fields. They can be written as (*): \[\tag{2} E(t) = {i \over c}\, E_0\, 2x(t) \] \[\tag{3} B(t) = {i \jk \over c^2}\, E_0\, 2y(t) \] \[\tag{4} H(t) = {i \jk \over \mu c^2}\, E_0\, 2y(t) \] where \(E_0\) is the amplitude of the electric field, and \(x(t)\) and \(y(t)\) are mutually perpendicular functions describing the oscillations of the electric and magnetic components. Since our wave, by definition, propagates in vacuum, the magnetic permeability equals its physical constant: \(\mu = \mu_0\).
The Umov–Poynting Vector
The energetic characteristics of the wave are described by the Umov–Poynting vector [1], which indicates the direction and density of the electromagnetic energy flow: \[\tag{5} S = E\, H = -{\jk \over \mu_0 c^3}\, E_0^2\, 4 x(t)\, y(t) \] In this expression, the hyperbolic unit \(\jk\) again plays the role of a directional indicator, pointing to the transfer of energy along the axis of wave propagation.
Substituting the time dependences \(x(t)\) and \(y(t)\), we obtain: \[\tag{6} S = {\jk \over \mu_0 c} E_0^2\, \sin^2 \wt \] Thus, the instantaneous energy flux density varies according to a harmonic law.
The mean value of the flux over one oscillation period is determined as: \[\tag{7} \left< S \right> = \jk {E_0^2\over 2\mu_0 c} \]
The hyperbolic unit \(\jk\) indicates the direction of energy propagation — along the coordinate \(z\). The resulting expression coincides in form with the standard expression for the mean energy flux density (intensity) of an electromagnetic wave. The only difference is the presence of the multiplier \(\jk\), which explicitly preserves the geometric directionality of the wave in the hypercomplex representation.
Longitudinal Components of the Plane Electromagnetic Wave
In the previous subsection, we examined the transverse components of the plane electromagnetic wave (PEW), which are associated with the oscillations of the electric and magnetic fields. Let us now turn to the longitudinal components that determine the directed propagation of the wave and describe its real (non-oscillatory) part.
To describe the longitudinal (real) part of the PEW, we introduce two interrelated components that define its internal structure along the direction of propagation.
1. Temporal (in-phase) component: \[\tag{8} v_t(t) = x(t) - z(t) = c\, \cos \wt \] which characterizes the harmonic variations of the wave phase in time, that is, oscillations of velocity along the temporal axis.
2. Spatial (longitudinal) component: \[\tag{9} v_z(t) = x(t) + z(t) = c \] defining the constant propagation velocity of the wave along the \(z\)-axis.
Thus, the pair \((v_t, v_z)\) describes the longitudinal structure of a plane electromagnetic wave. The temporal component \(v_t\) reflects the phase oscillations of the wave, while the spatial component \(v_z\) corresponds to the steady transfer of energy along the direction of propagation.
From these relations follows an important conclusion: the transverse components of the PEW appear only in the oscillatory process and do not participate in the transfer of energy through space. In contrast, the longitudinal components are responsible for the propagation itself: the temporal part \(v_t\) describes the exchange between temporal and spatial forms of energy, while the spatial component \(v_z\) ensures the continuous transfer of energy along the propagation axis.
Quantum of Action
In this section, we will show several remarkable correspondences that link the formalism of hyperbolic numbers not only with classical physics but also with quantum mechanics. Under certain conditions, the hyperbolic unit can be regarded as a carrier of an elementary quantum of action — that is, as a mathematical analogue of the very concept of quantization of phase space.
The Bohr–Sommerfeld integral [2], one of the key conditions of the old quantum theory, expresses the quantization of action over a closed cycle in phase space. It can be written as follows: \[\tag{10} \oint p\, dq = \hbar \left(2\pi n - i m \ln \jk \right) = 2\pi \hbar \left(n + {m \over 4} \right) \] Here it is assumed that the quantity \(\ln \jk^{\hbar}\) corresponds to the action accumulated by the system over one full period of motion — in other words, it effectively defines the energy level. This form of the expression shows that the quantization of action can be interpreted as the periodic repetition of a phase multiplier, where the hyperbolic phase \(\ln \jk\) serves as the “phase logarithm” of the cycle.
In a more general form, the Bohr–Sommerfeld integral can be written as a condition of phase self-consistency: \[\tag{11} \exp \left({i \over \hbar} \oint p\, dq \right) = \jm^{m}, \] which reflects the requirement of single-valuedness of the wave function after one complete traversal of a closed path in phase space. Here \(m\) is the Maslov index.
In quantum mechanics, the phase of the wave function is directly related to the action \(S\) [3]: \[\tag{12} \psi = e^{iS / \hbar} \] Here \(\psi\) is the wave function, and the exponential \(S / \hbar\) indicates that the phase of the wave is proportional to the action, while the action itself measures how far the system has “advanced” or “rotated” in phase space over a given time interval. Thus, the action acts as a phase parameter defining the internal rhythm of the wave state.
Let us now return to the hyperbolic unit and note that for \(n = 0, m = 1\) the action integral takes the form: \[\tag{13} S = \oint p\, dq = -i\hbar \ln\jk, \] This means that the action is proportional to the logarithm of the hyperbolic phase. Substituting this expression into the general formula for the wave function gives: \[\tag{14} \psi = e^{iS / \hbar} = \jk \] Thus, \(\jk\) represents a “pure phase” of the quantum of action — a wave state generated by its own action. The logarithm of the hyperbolic phase \(\ln \jk\) can be regarded as a measure of rotation or displacement of the system in phase space: it determines the number of quanta of action accumulated over a period and specifies the energy level of the state.
Conclusion
The analysis carried out has shown that the hypercomplex representation of a plane electromagnetic wave makes it possible to combine its temporal, spatial, electric, and magnetic components into a single coherent structure. Such a description not only reveals the internal symmetry of the wave but also demonstrates the distinct physical roles of its components. The transverse components participate in the oscillatory process, forming the electric and magnetic fields, whereas the longitudinal ones determine the directed transfer of energy and maintain the constancy of the propagation velocity.
Thus, the hypercomplex model of the plane electromagnetic wave provides a deeper understanding of the interaction between time and space in the wave process. It shows that wave propagation can be viewed as a dynamic exchange between temporal and spatial forms of energy, where each phase of motion is accompanied by a steady transfer of action along the propagation axis. This approach opens the way to interpreting the electromagnetic wave as a quantum of organized action, bridging the classical and quantum descriptions of physical reality.
(*) Unlike quaternion algebra, bicomplex algebra is commutative \[ i \jk = \jk i, \] associative, and unital (with the unit \(1\)), which provides significantly more possibilities for analytical and geometric constructions. Due to commutativity, bicomplex algebra allows operations such as differentiation and integration with respect to a bicomplex variable, analogous to the complex case.
References
- Wikipedia. Poynting vector.
- Wikipedia. Bohr–Sommerfeld model.
- Wikipedia. Wave function.