The idea that all matter exhibits wave properties is not new in physics. Louis de Broglie is generally considered to be the father of this idea, as his dissertation thesis Recherches sur la théorie des quanta (Research on the Theory of Quanta, [10]) was the first work to present it comprehensively. In that work, de Broglie built on previous discoveries by Max Planck and Albert Einstein and extended the concept of wave-particle duality, which until then was only considered and accepted for light, to all particles of matter. De Broglie received a Nobel Prize for Physics in 1929 for his research in this area [11].

De Broglie used the theory of relativity as a starting point. Namely, it was Einstein’s equation E = m c 2 , which relates the particle energy E to its mass m, and also the quantum “Planck-Einstein relation” E = h f , which relates the photon energy E to its frequency f. De Broglie concluded that the latter is generally valid and extended its use to all types of particles. In the above-mentioned relations, h is Planck’s constant.

For a particle at rest, de Broglie obtained the relation h f 0 = m 0 c 2 , where f ₀ and m ₀ are the particle’s frequency and mass at rest. However, this relation cannot be easily modified for a particle in motion.

De Broglie designated the “theorem of phase harmony” as the basic hypothesis of his theory, which he formulated as [10]:

The application of this theorem to particles of matter in motion led Louise de Broglie to the conclusion that every particle of matter is accompanied by a specific “phase wave” (also known as a “de Broglie wave” or “matter wave”) which has the above-stated properties, i.e., direction, velocity and frequency. The energy and momentum of this wave is equal to the energy and momentum of the particle. The particle and the wave are mutually inseparable: they represent two forms of the same quantum of energy.

While the mathematical validation of the initial theorem is not difficult to provide, finding an acceptable physical interpretation of de Broglie waves turned out to be very difficult. A de Broglie wave has very unusual properties. It has a different frequency compared to the particle it represents, it moves at a different velocity, and this velocity is always superluminal. The only success that has been achieved in this field was finding that velocity v _f is the phase velocity, and this is not the velocity at which wave energy moves. Instead, it moves at group velocity, which equals the velocity of the particle v. The energy and the particle thus move at the same speed.

The question arises whether the strange properties of the de Broglie waves are not just due to an incorrect choice of the space in which they are described. Maybe it suffices to choose a different space dimension or metric and the seemingly incomprehensible properties disappear. Now, an interesting opportunity arises to try out the bounded Euclidean space E₄-B, in which the Euclidean model of space and time (EMST) is formulated.

The EMST is fully equivalent competitive theory to Einstein’s SR which, although based on different space-time metric, leads to the same predictions (time dilation, length contraction, Lorentz transformation…) in ordinary three-dimensional space. The theory uses a four-dimensional Euclidean space with a metric signature (+,+,+,+), where all four dimensions are space-like. In the EMST, time is not considered as a separate dimension, it is only a linear parameter of motion.

Since the theory of EMST is quite extensive, I will limit myself to a summary of the basic information necessary to understand the paper. A detailed description can be found in [9].

The EMST is based on three fundamental presumptions (postulates):

• Space is four-dimensional Euclidean, with all four dimensions being space-like.

Important features of the EMST:

• The methods of time interval measurement, distance measurement and clocks synchronization are the same as in Einstein’s SR. Time is measured by “ideal clocks,” distances are calculated from the transit times of two-way light signals, and the same light signals are used for clocks synchronization. As in Einstein’s SR, coordinate time is maintained by a set of mutually synchronized clocks.

For the purposes of this article, it is necessary to recall some other relevant EMST relationships:

The magnitude of 4D velocity v _4D is given by formula v 4 D = v x 2 + v y 2 + v z 2 + v w 2 = v 3 D 2 + v w 2 = c (2) where v _x , v _y , v _z and v _w are components of the 4D velocity vector (in E₄-B) while v _3D is standard three-dimensional velocity (in ordinary E₃ space with coordinates x, y, z).

Substituting v 3 D = δ x 2 + δ y 2 + δ z 2 / δ t into Equation 1 gives c 2 δ τ 2 = c 2 δ t 2 − v 3 D 2 δ t 2 . Thus, δ τ = 1 − v 3 D 2 c 2 δ t (3) which, is the formula for time dilation known from SR.

4D momentum p _4D is introduced by relation p 4 D = p x 2 + p y 2 + p z 2 + p w 2 = p 3 D 2 + p w 2 where p _x , p _y , p _z and p _w are components of the 4D momentum vector and p _3D is standard three-dimensional momentum.

Four-dimensional Lorentz transformation is given by the following set of formulas: x = x ′ + u t ′ 1 − u 2 c 2 , y = y ′ , z = z ′ , w = w ′ , t = t ′ + u c 2 x ′ 1 − u 2 c 2 here u is three-dimensional velocity of coordinate system S′ (x′, y′, z′, w′) with regard to system S (x, y, z, w). The velocity u is parallel to x.

The following relations are also valid p 4 D = m c = E c (4) p w = m 0 c = E 0 c where m/m ₀ is the relativistic/rest mass of the body, E/E ₀ its total/rest energy and x stands for the absolute value of x.

The reasoning behind all the above statements and derivation of the formulas is given in [9].

In the first step, it suffices to study the motion of particle-waves in unbounded space E₄. The effect of the barriers in the fourth spatial dimension will be discussed later in Section 4.

Any wave can be understood as a phase ψ distributed in space. A certain value of phase ψ ∈ 〈 0 , 1 ) is allocated to each point in space E₄ at a given moment. Points with the same ψ value create planes, or to be more exact, three-dimensional hyperplanes in E₄. For a free particle represented by a harmonic plane wave, these hyperplanes (wave fronts) are mutually parallel and perpendicular to the direction of the wave motion. The distance between neighboring hyperplanes of the same phase equals the wavelength λ _4D , λ 4 D = c f

The 4D momentum of a particle can be derived from the relation Equation 4 by substitution for E, p 4 D = E c = h   f c = h λ 4 D (6)

A plane wave representing a particle with the wavelength λ _4D moves through space E₄ in any spatial direction s _4D , which makes an angle α to the axis w (Figure 2). We shall label s _3D the orthogonal projection of the direction s _4D into space E₃ with axes x, y, z. By ‘the orthogonal projection’ is meant the substitution of s _4D direction vector (x, y, z, w) by s _3D direction vector (x, y, z, 0). Direction s _3D will be used as an auxiliary axis in some considerations below.

The 3D velocity of a particle is determined by the formula v 3 D = c ⁡ sin ⁡ α the velocity in the fourth dimension analogically being v w = c ⁡ cos ⁡ α

It should be noted that sin ⁡ α = v 3 D c ,   cos ⁡ α = 1 − v 3 D 2 c 2

The wavelength of a particle λ _4D projects itself into the directions of axes s _3D and w as λ _3D and λ _w respectively, λ 3 D = λ 4 D sin ⁡ α = c v 3 D λ 4 D λ w = λ 4 D cos ⁡ α = λ 4 D 1 − v 3 D 2 c 2

The geometrical relation for the wavelengths 1 λ 4 D 2 = 1 λ 3 D 2 + 1 λ w 2 can easily be derived from the similarities of triangles OBA and COA (Figure 2).

The wavelength λ _3D can be further projected into the directions of the coordinate axes x, y and z. The values of λ _x , λ _y , λ _z are given by the relation 1 λ 4 D 2 = 1 λ x 2 + 1 λ y 2 + 1 λ z 2 + 1 λ w 2 (7)

If we multiply relation Equation 7 by the square of Planck’s constant, we obtain h 2 λ 4 D 2 = h 2 λ x 2 + h 2 λ y 2 + h 2 λ z 2 + h 2 λ w 2

This relationship is very important because it expresses the decomposition of the 4D momentum of the particle p 4 D = h / λ 4 D in the directions of the coordinate axes. It is thus equivalent to the relation p 4 D 2 = p x 2 + p y 2 + p z 2 + p w 2

It is clear that p x = h λ x , p y = h λ y , p z = h λ z , p w = h λ w (8) and also p 3 D = h λ 3 D (9)

The last relation is very interesting as it is the famous de Broglie relation for the momentum of an arbitrary particle [11]. This fundamental relation of all quantum mechanical theories is derived here within the framework of EMST, i.e., relativistic theory!

It can be determined from the facts mentioned above that the de Broglie relation has a purely geometric origin. It is a result of the projection of a wave representing a particle in E₄ (or to be more exact in E₄-B) into E₃. In this projection the wavelength of the particle increases ( λ 3 D = λ 4 D / ⁡ sin ⁡ α ), while its momentum simultaneously decreases ( p 3 D = p 4 D ⁡ sin ⁡ α ).

In space E₄ the velocity of particle v _4D is always equal to the phase velocity of wave v _f,4D , which represents it, i.e., v 4 D = v f , 4 D = c . However, this is not true for its projection into E₃. While the 3D velocity of the particle is an orthogonal projection of the speed c into space E₃, i.e., v 3 D = c ⁡ sin ⁡ α (10) the phase velocity v f , 3 D is result of a different projection, see Figure 3. It holds that v f , 3 D = c sin ⁡ α (11)

We obtain a relationship between the group velocity and the phase velocity of a particle-wave in E₃ inserting Equation 10 into Equation 11, v f , 3 D = c 2 v 3 D (12)

The relation Equation 12 is part of de Broglie’s theorem of phase harmony, and as such an integral part of any relativistic quantum theory. It links the speed of a de Broglie wave with the speed of the relevant particle of mass.

In the EMST, just as in SR, transition between two mutually moving coordinate systems is achieved through Lorentz transformation. Transformation equations are linear with regard to spatial coordinates, and a plane wave thus transforms into a plane wave again. However, the transformed wave generally differs in its velocity, direction and wavelength from the original one.

The simplest situation occurs with the velocity. As mentioned above, the 4D speed of all objects equals c, which is applicable for any system of reference. It holds for a system S in rest as well as for system S′ in motion, v 4 D ′ = v 4 D = c .

The direction of propagation of waves is always perpendicular to the hyperplanes of the constant phase ψ. The orientation of these hyperplanes with regard to the coordinate system is given by the wavelengths λ _x , λ _y , λ _z and λ _w . For a hyperplane with phase ψ which passes through the origin of the coordinate system O(0, 0, 0, 0), a neighboring hyperplane with the same phase ψ passes through four points, which can be obtained by placing the wavelengths λ _x , λ _y , λ _z and λ _w on the relevant coordinate axes.

Although it would be possible to derive the changes in wavelengths λ _x , λ _y , λ _z and λ _w using the Lorentz transformation formulas, it will be easier to use the transformation formulas for the 4D momentum components [9]: p x ′ = p x − u c p 4 D 1 − u 2 c 2 , p y ′ = p y , p z ′ = p z , p w ′ = p w , p 4 D ′ = p 4 D − u c p x 1 − u 2 c 2 (13)

During the transition from S to S′ the momentum components in the directions perpendicular to u do not change. So it holds that λ y ′ = λ y , λ z ′ = λ z , λ w ′ = λ w . The value of λ _x is given by λ x = h p x .

It can be seen that the Lorentz transformation changes only one of the wavelengths determining the direction of the hyperplanes of the constant phase. It is the one in the direction of the velocity u. The transformation has a simple form from the geometric point of view. A few examples will be shown hereinafter. These are cases for wave motion in the x-w plane, i.e., those where at least two of the four components of plane wave velocity are zero. Such cases can be easily drawn on a two-dimensional sheet of paper.

Let us start with the case v _y = v _z = 0. It holds that λ y ′ = λ y = ∞ and λ z ′ = λ z = ∞ here.

Figure 4 shows a change in the direction s _4D as well as in the wavelength λ _4D of a particle-wave as a result of the change in the reference system. During the transition from system S to system S′, the particle momentum components transform according to Formula 13. The wavelengths in the direction of the coordinate axes transform according to Equation 8, the wavelength λ _4D according to Equation 6. It is worth noticing that λ _w , i.e., the projection of the wavelength of the particle into the direction of axis w, never changes. This wavelength is an invariant of the Lorentz transformation.

For u equal to the 3D velocity of the particle ( u = v x ) it holds that v x ′ = 0 and λ x ′ = ∞ (Figure 4C). This situation describes the particle at rest in system S′. Its λ′ _4D is maximal and its momentum p _4D and total energy E minimal.

For u > v _x (Figure 4D), the velocity v′ _x has the opposite direction to v _x , λ′ _4D is shorter than in the previous case and the energy E increases again.

Another interesting case is a particle with zero rest mass m ₀ . Its momentum in the fourth dimension equals zero p w = m 0 c = 0 , and thus λ w = ∞ . The front of the particle-wave is parallel to the w-axis, i.e., the angle of its motion α = π/2. The particle velocity in the fourth dimension v w = 0 , and so v 3 D = v 4 D = c . According to the principle of the same 4D speed, such a particle must always move in E₃ at the speed of light. The Lorentz transformation changes its direction of propagation and wavelength (these phenomena are known as aberration of light and the Doppler effect), but never its speed.

The case for v _y = v _z = 0 is shown in Figure 5. In this special case, only the wavelength of the particle changes; the direction remains unchanged.

Let us examine another example in which both the wavelength and the direction of propagation of a particle with m ₀ = 0 changes.

Let us consider a system S with a photon with frequency f ₀ in it. The photon is moving in the direction of axis y, i.e., perpendicular to the direction in which system S′ moves. In system S it holds for the photon that v _y = c, v _x = v _z = v _w = 0 (Figure 6). In system S′, this photon moves with unchanged speed c, but in a different direction. It holds that v′ _z = v′ _w = 0 and v ′ x 2 + v ′ y 2 = c 2 where v x ′ = − u . The similarity of the triangles gives v y ′ / c = λ 4 D ′ / λ y ′ and thus λ 4 D ′ = v y ′ c λ y ′

Using λ y ′ = λ y = c / f 0 , λ 4 D ′ = c / f ′ and v ′ x 2 / c 2 + v ′ y 2 / c 2 = 1 we obtain c f ′ = 1 − v ′ x 2 c 2 c f 0 and after some editing f ′ = f 0 1 − u 2 c 2 (14)

Formula 14 gives the frequency f′ of the photon as measured in system S′. We have arrived at the formula for the transverse Doppler effect.

Let us modify the task. Now the wave with frequency f ₀ does not move in the direction of axis y, but in the direction of axis w. It does not represent a photon in motion but instead a massive particle at rest (v _w = c, v _x = v _y = v _z = 0). Its energy is given by the formula E ₀ = hf₀ .

From the viewpoint of system S′, the particle moves with velocity v x ′ = − u . Its energy is E′ = hf′.

The relationship between frequencies f ₀ and f′ remains the same as in the case of the transverse Doppler effect; the only difference is that axis w replaces axis y in the process of its derivation. Again, we obtain Formula 14. Between the mass of the same particle at rest (m ₀ ) and in motion (m′) the following formula applies m ′ = m 0 1 − v ′ x 2 c 2

It can be seen that the nature of relativistic mass increase, as well as of the transverse Doppler effect, is purely geometric. In principle, there is no difference between them.

Formula 13 describing the transformation of p _4D can be used to derive a general formula for the Doppler effect. Using the substitutions of p 4 D = h / λ 4 D = h f / c and p x = p 3 D ⁡ cos ⁡ φ , we obtain: p 4 D ′ = h f ′ c = h f c − u c h f c cos ⁡ φ 1 − u 2 c 2 where the quantity φ is the angle that the motion of the particle makes with the x-axis. For a photon p 3 D = p 4 D .

After some modifications we obtain f ′ = f 1 − u c cos ⁡ φ 1 − u 2 c 2 (15)

The formula gives the frequency f′ of a photon that we would measure in system S′ if it has frequency f in system S and moves at an angle φ with respect to the x-axis. The quantity u is the velocity at which system S′ moves relative to system S. The formula is identical to the formula derived by Einstein in the SR framework, see [1]. The Formula 14 derived earlier is a special case of Equation 15 for φ = π/2.

It was shown above that the value of λ _w is an invariant of the Lorentz transformation. The wavelength λ _w of a particle-wave is related to its rest mass m ₀ by formula λ w = h m 0 c (16)

It is obvious that λ _w can only change with a change in the rest mass of the particle. Thus, it does not change during the Lorentz transformation, nor during any of the physical interactions in which the rest mass of the particle is conserved.

It is not without interest that Formula 16 is used in particle physics to determine the so-called Compton wavelength. For that, an identical formula holds: λ c = h / m 0 c .

The quantity λ _w is thus numerically equal to λ _c , but has a clear geometric meaning. The quantity λ _w is not only an alternative way to express the inertial properties of a particle (its rest mass), it is also projection of the real wavelength of that particle into one of the spatial dimensions.

A particle-wave with m ₀ ≠ 0 moving through space E₄-B periodically reflects from the barriers in the fourth dimension, and the incident and reflected waves compose a superposed wave. As the projections λ _w of the wavelength λ _4D of both waves are the same, the superposed wave has the character of a standing wave (normal-mode vibration) in the fourth dimension. In the remaining three dimensions nothing disrupts the free propagation of the composed wave, and so it forms a travelling wave (Figure 7). The 2D variant of this phenomenon is described in detail in [12].

In principle, space E₄-B is a specific kind of four-dimensional waveguide which only restricts the propagation of waves in one of the dimensions (flat waveguide). The side ratio x:y:z:w of this “world waveguide” is ∞ : ∞ : ∞ : w ¯ / 2 .

Just as we do not know the nature of the waves representing matter in E₄-B, so we do not know the nature of the barriers that confine them. In principle, there are two types of barriers enclosing a waveguide. They could be fixed boundaries with zero wave amplitude on them (nodal surfaces) or free boundaries with non-zero amplitude (with antinodes) on them. An example of the first type is an electromagnetic waveguide, an example of the second is mechanical waves in a long steel rod. It is reasonable to assume that both barriers enclosing space E₄-B are of the same type.

Regardless of the type of the barriers, the incident and reflected waves interfere inside the world waveguide. Just like other waveguides, this one also has its modes. They differ in terms of the values of the wavelength λ _w . Based on the fact that constructive interference should occur during the superposition of waves and both barriers are of the same type, the distance between the barriers w ¯ / 2 must be the whole multiple of half of the wavelength λ _w . Otherwise, any wave motion between the barriers would spontaneously disappear as a result of destructive wave interference.

The wave must fulfil the condition n λ w 2 = w ¯ 2 (17) where n is a nonnegative integer which defines the relevant mode of the wave motion.

This condition must be fulfilled by all particles with the lifetime longer than 1 × 10⁻²⁰ s. Particles with the shorter lifetime will not travel a distance significantly longer than w ¯ and thus will not produce standing waves between the E₄-B barriers.

Relationships between λ _w and other quantities such as p _w , E ₀ and m ₀ imply that condition Equation 17 limits the set of acceptable rest masses of long-lived particles. Their masses can thus only gain certain selected values. Values λ _x , λ _y and λ _z can change continuously, as can the momentum p _3D and the kinetic energy of particles.

The waves in space E₄-B have a somewhat different character than in E₄, but the main parameters that characterize them, i.e., frequency, velocity and angle α, remain unchanged. All the relationships presented in Section 3 remain valid.

The value n = 1 from Formula 17 corresponds to the fundamental mode of the waveguide, i.e., the mode with the largest wavelength. For this the relation λ w = w ¯ holds.

It is possible to accept the assumption that the fundamental mode of the waveguide corresponds to the wavelength of the electron, i.e., the lightest particle with non-zero rest mass (m_e = 9.11.10^–13 kg). According to Equation 16, its wavelength is equal to λ _w,e = 2.43 × 10⁻¹² m. The barrier distance w ¯ / 2 in this case comes out to be 1.21 × 10⁻¹² m. It would be too early to consider this value as a proven barrier distance, but it can be considered as an initial estimate.

It may be mentioned that the electron wavelength λ _w,e given above is equal to the circumference of the (fictitious) first Bohr orbit of the hydrogen atom (with radius a ₀ = 5.3 × 10⁻¹¹ m) multiplied by the fine structure constant α ¯ . 2 π a 0 α ¯ = 2.43   ×   10 − 12   m

It was mentioned in the Introduction that de Broglie assumed two different frequencies, f and f ₁ , for each particle of matter in motion. Their relationship to the frequency f ₀ of a particle at rest is f 0 = f 1 1 − v 3 D 2 c 2 = f 1 − v 3 D 2 c 2 (18)

In the submitted four-dimensional concept of particle-waves, every particle of matter in motion is represented by a wave with the frequency f = f 0 / 1 − v 3 D 2 / c 2 , which is consistent with de Broglie’s assumption. It holds that f = c / λ 4 D .

However, the particle-wave also moves in the space between the barriers. It reflects from them and the frequency of motion between the barriers (cyclic frequency) is different from the own frequency of the particle-wave. This frequency is a function of the path of the particle-wave between individual reflections. Its path Δs _4D increases with the growing 3D velocity of the particle in reference system S, and the cyclic frequency decreases reciprocally (Figure 8). For a particle at rest in S it holds that f 0 = c / 2   Δ s 4 D = c / w ¯ , while for a particle in motion f 1 = c / 2   Δ s 4 D = c / w ¯ 1 / ⁡ cos ⁡ α = c / w ¯ v w / c = c / w ¯ 1 − v 3 D 2 / c 2 . So, it holds that f 1 = f 0 1 − v 3 D 2 / c 2 . The presented example is for a particle-wave with λ w = w ¯ .

Comparison with Equation 18 shows that the second frequency f ₁ predicted by de Broglie also has a physical meaning. It expresses the periodicity of particle-wave cyclic motion between the barriers, and is thus related to the flow of proper time. The resulting effect is in full accordance with description of time dilation and Formula 3 above.

It remains to show that for an observer moving with a particle, both frequencies are the same and equal to f ₀ . For such an observer, the cyclic frequency f ₁ is apparently higher than for an observer at rest. This is caused by the use of slower reference clocks in the system in motion S′ f 1 ′ = f 1 1 − v 3 D 2 c 2 (19)

The frequency of waves f is also affected by the slowing down of moving clocks, but it is also affected by the Doppler effect. The observer is moving in the same direction as the waves and the frequency will thus decrease f ′ = f 1 − v 3 D 2 c 2 v f , 3 D − v 3 D v f , 3 D

The formula can be modified by the use of substitution for v _f,3D from Equation 12 f ′ = f 1 − v 3 D 2 c 2 c 2 v 3 D − v 3 D c 2 v 3 D = f 1 − v 3 D 2 c 2 c 2 v 3 D 1 − v 3 D 2 c 2 c 2 v 3 D = f 1 − v 3 D 2 c 2 (20)

Comparison of Formulas 19, 20 with Formula 18 shows that f 0 = f 1 ′ = f ′ . Therefore, it is not possible to distinguish a system in motion from a stationary one by observing the mentioned frequencies.