_{1}

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It was predicted by Einstein that energy and mass can be converted between each other. But why? Energy and mass are two very different physical concepts. How can they be exchanged with each other? We think the key to answer this question is to recall that a particle can behave like a wave. Particle properties like energy and momentum are known to be related to their corresponding wave properties (frequency and wave vector). Mass is clearly a particle property; is it also related to a wave property? This study suggests that it is. We found that mass and energy appear to share similar physical nature in the wave perspective. Both of them are related to the curvature of bending the vacuum medium during the propagation of the excitation wave. This similarity explains why they are convertible.

Recent studies of cosmology suggest that our universe is not only composed of visible matter; there are also dark matter and dark energy [

At this point, there are still many unanswered questions about the visible matter. For example, we know that energy and mass can be converted between each other. But what is its physical basis? This energy-mass conversion was attributed to the Special Theory of Relativity (STR) [^{2}. But why? Energy and mass are very different physical concepts. How can they be convertible? STR only proposed the energy-mass conversion rule; it did not explain the physical foundation of this rule. Thus, it will be interesting to explore ways to explain this rule on a physical basis.

We think the key to understand energy-mass conversion is to recognize that matter has wave properties. The concept of matter wave was proposed almost a century ago. Phenomenologically, it was found that a particle can behave like a wave or a corpuscular object depending on the situation [

Besides light wave, sound waves can also behave like particles. It is well known in condensed matter physics that excitation waves in a solid (or fluid) can be treated like particles (called “phonons”). Phonons can interact with other particles (like electrons) and play a major role in determining the electrical conductivity of many materials [

On the other hand, sub-atomic particles with mass can also behave like waves. It was shown in 1927 that electrons can be diffracted by a nickel crystal [

The physical nature of matter wave is still not clearly understood today. In most quantum mechanics textbooks, the particle is treated as a pointed object; the wave property is only associated with the probability of finding the particle at a particular space and time. (This is called the “Copenhagen interpretation”) [

Furthermore, we know particles can be created or annihilated in the vacuum. If the particle is a real object, how can it appear from nowhere or disappear suddenly with no trace? The only rational explanation is that the particle is an excitation wave, so that it can be excited by an energetic stimulation and it can be transformed from one type of wave into another type of wave. Therefore, we propose that the so-called “particle” is actually an excitation wave of the vacuum [

If the matter wave is a physical wave, then what is its carrying medium? In the 19^{th} century, the electro-magnetic field was thought to be carried by a medium called “aether” [^{th} century due to the following reasons:

1) The mechanical properties of this hypothetical aether were full of contradictions. Aether was supposed to fill all space between matters. Thus, it must be a highly fluidic substance (gas or liquid). Such fluidic medium, however, can only transmit low frequency dilational wave. In order to transmit high frequency transverse wave such as light, the aether must be a rigid solid. These two requirements are contradicting with each other.

2) The aether hypothesis could not explain why large astronomical objects like Earth or Mars can pass through it without experiencing any resistance. This is contradicting to the requirement that aether is a fluid or a solid.

3) The aether hypothesis was not supported by experiments. In late 19^{th} century, many experiments were conducted to detect the motion of aether using optical interferometers [

4) Later, the aether hypothesis was thought to be unnecessary. Finally in 1905, Einstein proposed the Special Theory of Relativity and showed that one can explain the null results easily without the assumption of aether [

Hence, people in the early 20^{th} century believed that the vacuum in our universe is totally empty. But such a view is no longer valid today. With the development of quantum electrodynamics, the vacuum is thought to contain the zero-point energy of all oscillations in the electromagnetic field [

In the more recent quantum field theory, a particle is regarded as the excitation of its field. Different types of particles are associated with different fields. However, these fields do not have the same meaning as classical fields. The physical meaning of quantum field is not clear at present. There is still a debate about what is more fundamental: Is it particle or field? Currently there is no agreement on it [

Thus, there is a dilemma now. On the one hand, the old concept of aether is unviable. On the other hand, with the development of quantum field theory, the vacuum can no longer be regarded as “emptiness”. What is the way out? We think this dilemma can be solved by assuming that the vacuum is a pre-existing medium in our universe and it fills all space; both matter waves and radiation waves are excitation waves of this medium [

This new model can overcome all problems of the previous aether hypothesis:

First, we can avoid contradicting mechanical requirements for the vacuum. Since the vacuum is a pre-existing medium occupying the entire universe, not just filling the space between matters, it does not need to be fluidic.

Second, the atoms making up the planet are all waves; there should be no friction between matters and the vacuum. Thus, planets can move within the vacuum medium without dragging.

Third, the null results of the optical interferometer experiments reported by Michelson and Morley (1887) can be easily explained [

If particles are excitation waves, what is the physical nature of the vacuum medium? At this point, our knowledge about the vacuum is very limited. However, there is a very good hint from the work of Maxwell, who suggested that the vacuum behaves like a dielectric medium [

He later found a problem in it, because this equation would violate the condition of conservation of charge,

In order to fix this problem, Maxwell proposed to add a new term _{d}), which can also affect the magnetic field. So that in Equation (1), one should consider not only the externally applied current J, but also the internally induced displacement current J_{d}. That is

Thus, Maxwell thought that the correct form of Ampere’s law should be

This becomes a part of the final form of the Maxwell’s equations used today.

The addition of a displacement current term was a stroke of genius. This allowed Maxwell to construct his theory of light propagation. When Maxwell investigated the propagation of electro-magnetic waves in the vacuum, he argued that the external current J is zero, but the displacement current term is not zero. This is equivalent to say that he believed the vacuum is a dielectric medium. Thus, Equation (1A) in the vacuum becomes

Using this equation, one can easily derive the wave equation for the electro-magnetic wave,

where A is the vector potential and _{0} is permittivity and μ_{0} is permeability in free space). Thus, the key conceptual step in Maxwell’s theory of light propagation was to regard the vacuum as a dielectric medium. If one treats the vacuum as pure emptiness and remove the

If both the radiation waves and matter waves are excitation waves of the same vacuum medium, then waves representing different particles (with or without mass) should obey the same wave equation, which is determined by the physical properties of the vacuum medium. One may be surprised that this point is actually supported by the Special Theory of Relativity. STR predicts that no object can travel faster than the speed of light [

What is the basic form of this wave equation? At this point, we know the wave equation of at least one particle; that is the photon. This may provide a very useful hint. We know a photon is an oscillating wave of electro-magnetic fields, which are linear functions of the vector potential A^{μ} = (ϕ, A). If matter waves and radiation waves are both excitation waves of the same vacuum medium, we expect that the wave function of the matter wave ψ may also be a linear function of A^{μ}. This means that ψ should also obey the same wave function as shown in Equation (3), i.e.,

We propose that this is the basic wave equation for all types of excitation waves of the vacuum medium. ψ can have different solutions, which may represent different types of free particle. (Note: Even though Equation (4) does not look like any wave equations commonly used in quantum mechanics, it is not inconsistent with them. In fact, we can show later in this paper that Equation (4) can be reduced into the Klein-Gordon equation for particles of nonzero mass. By linearizing the Klein-Gordon equation, one can then derive the Dirac equation for electrons [

Now let us examine the different solutions of the basic wave equation. The simplest solution of Equation (4) is a plane wave

where

where ℓ is a fitting parameter which is subjected to the condition

By solving Equations (6A) and (6B) separately [

where J_{n} is Bessel function of the first kind; n is an integer or half integer; r and θ represent the radial distance and the azimuthal angle of the space vector in the transverse plane. (a is a normalizing constant.) As expected, the wave function of a free particle behaves like a travelling wave moving along the direction of its trajectory. But due to the presence of the Bessel function,

The wave function shown in Equation (8) contains four parameters, ω, k, ℓ and n. What are their physical meanings? Using the correspondence principle

At

This result makes good sense, since when one substitutes Equation (10) into Equation (7), and recalls that

which is identical to the energy-momentum relationship implied from STR [

At

Our finding that the rest mass m is related to the parameter ℓ in the wave function has important implication. From Equation (8), we see that the transverse component of the free particle wave function is described by a Bessel function, the asymptotic form of which is

Thus, ℓ can be regarded as the “transverse wave number” of the free particle, i.e., it is the inverse of the wavelength in the transverse oscillation,

This finding is very interesting; it is closely parallel to the Planck’s relation and the de Broglie relation, which show that the energy and momentum are related to the periodicity of oscillation of the vacuum medium. More specifically, E is related to the periodicity of oscillation in the time dimension, and p is related to the periodicity of oscillation in the spatial dimension along the direction of the trajectory. Our finding that the rest mass is associated with the oscillation periodicity in the transverse direction thus appears to make very good sense. In essence, our result suggests that energy, momentum and mass are all related to the curvature of bending the vacuum medium during the propagation of the excitation wave.

This result is not totally surprising, since both energy and mass must be created by “work”. Before the vacuum is excited by a stimulus, it is in the resting state (ground state). In order to trigger an excitation wave, one must create an energetic disturbance at a local region of the vacuum medium. Since it takes work to bend the vacuum medium, the more sharply the bending curvature, more work is required. This is true for bending the medium in all dimensions (spatial and temporal). Thus, a shorter wavelength of a propagating wave should always associate with a higher “energy state”, which may be reflected in an increase in energy, momentum or “mass” of the excitation wave.

This geometrical interpretation of mass and energy suggests that the physical natures of energy and mass are very similar in the wave perspective. This explains why energy and mass can be converted between each other.

Based on the discussions above, we believe that both radiation wave and matter wave are excitation waves of the vacuum medium. The basic wave equation is determined by the physical property of the vacuum medium, which behaves like a dielectric system as suggested by Maxwell. The solution of this wave equation can represent particles of different types. We think the radiation wave is corresponding to the plane wave solution; while the matter waves are represented by the cylindrical wave solutions,

The various wave parameters in this equation represent different particle properties. The wave vector k is related to the momentum p; the frequency ω is related to the energy E; and the transverse wave number ℓ is related to the rest mass m. This wave function represents an excitation wave advancing in helical motion. This can be seen by examining the physical meaning of the parameter n. It appears that n is associated with the helicity of the free particle, since it is a quantum number conjugate to the angular coordinate θ. Dimensional analysis thus suggests that n is associated with some sort of angular momentum. Because of the added phase factor nθ in Equation (8), the wave function representing a free particle actually propagates in a helical fashion (see

We can demonstrate this point more explicitly. Let ω = ω_{1} + ω_{2}, substituting it into Equation (8), we have

The first exponential term _{2}/k can be adjusted at will. The wave function as a whole behaves like a moving vortex. The phase factor

In this work, we propose that all particles (with or without mass) are excitation waves of the vacuum medium. The basic wave equation is Equation (4). For particles without mass (i.e., radiation waves), there should be no problem to accept it, since this equation is identical to the wave equation of light. But for particles with rest mass (i.e., matter waves), can Equation (4) properly describe their excitation waves? In relativistic quantum mechanics, the motion of massive particles is usually described by the Klein-Gordon equation [

In the following, we will show that this is indeed the case. One should keep in mind that in describing the motion of a particle, there are two categories of wave equations. The first category is the basic wave equation which describes all types of excitation waves in the vacuum medium. Different solutions of this wave equation represent different types of particles. The second category is wave equations that describe the motion of a specific type of particle. We propose that Equation (4) belongs to the first category. Since it is applicable to all types of particles, the particle mass should not appear in the equation. For wave equations of the second category, the particle mass can appear in the wave equation, since it describes the motion of a single type of particle. We believe the Klein-Gordon equation or the Dirac equation belong to this category.

The question now is whether one can derive the Klein-Gordon equation or the Dirac equation from the basic wave equation of the vacuum medium (i.e., Equation (4)). As shown in Equation (5), the wave function representing an excitation wave in the vacuum medium has both a longitudinal component (ψ_{L}) and a transverse component (ψ_{T}). When one substitutes Equation (5) into Equation (4) and uses the technique of separation of variables, one can obtain

Since we know now that ℓ is connected with the rest mass m through Equation (10), the above equation becomes

This is identical to the Klein-Gordon equation if we regard ψ_{L} as the particle wave function ϕ. This analysis suggests that the wave function described by the Klein-Gordon equation only represents the longitudinal component of the travelling wave of a free particle with a specific mass m. This means that the Klein-Gordon equation can only describe the movement of a particle along its trajectory. Its wave function does not give any information on the oscillation of the vacuum medium along the transverse plane.

Once the Klein-Gordon equation is derived, one can then obtain the Dirac equation to describe the motion of an electron by linearizing Equation (14) [

Thus, our proposed basic wave equation is consistent with the commonly used quantum mechanical wave equations. The only difference is that the common quantum mechanical equations describe only the motion of a single type of particle. Furthermore, the wave functions in these quantum mechanical equations appear to describe only the motion of the excitation wave along the trajectory of the particle. The basic wave equation (Equation (4)), on the other hand, describes the complete wave function of the excitation wave of a free particle which may have different mass.

We may add that our discussion above only applies to the excitation wave of a free particle, such as a photon or an electron. For composite particles, such as protons or neutrons, the situation is much more complicated. One may need a further examination of the quark model. That will be far beyond the scope of this paper.

In conclusion, we believe all particles (with or without mass) are excitation waves of the vacuum medium. They behave like corpuscular objects only in the macroscopic view. Each of the particle properties and wave properties appears to have a one-to-one correspondence. Energy, momentum and mass are all related to the curvature of bending the vacuum medium during the propagation of the excitation wave. The similarity between the physical nature of energy and mass explains why the two can be converted between each other.

I thank the late Profs. John A. Wheeler and H. E. Rorschach for their encouragement when I started this work. I also thank Lan Fu for her assistance. The support of the Hong Kong University of Science and Technology to this work is acknowledged.

Donald C.Chang, (2016) Why Energy and Mass Can Be Converted between Each Other? A New Perspective Based on a Matter Wave Model. Journal of Modern Physics,07,395-403. doi: 10.4236/jmp.2016.74040