A Connection between Electrons/Positrons and Electromagnetic Waves?

[Dr Who]

I have been thinking recently about electrons and electromagnetic wave and have come up with the following idea. I was just wondering what you good people thought about it. (I know the information is presented in quite a technical way, but I tend to find that helps me to see any problems there maybe with the idea).

 

Introduction

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Both electrons (or positrons) and electromagnetic waves have always been thought of as separate entities, i.e. as particles and waves respectively. However it has been experimentally shown that both of these entities can display both particle and wave like behaviour. Moreover, electron-positron annihilations produce only electromagnetic waves and conversely electromagnetic waves can produce electron-positron pairs. These facts propose the possibility that there is some sort of connection between electromagnetic waves and electrons. (Note, the same connection would also hold for positrons (anti-matter electrons), but for simplicity, throughout the whole paper, we will only refer to one of them.) It is this connection that we will discuss within this paper.

 

Electrons and Electromagnetic Waves

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One of the central concepts of quantum mechanics is that all particles exhibit wave-particle duality, just as electromagnetic waves do. This implies that electrons will exhibit both wave and particle like properties. For example, Schrodinger modelled electrons as waves to produce a model of their structure around the nucleus of an atom. Conversely, we also know that an electron at rest is unable to move without an external force being applied to it, which is a particle like property. This is the opposite of an electromagnetic wave, which will always travel without any external forces acting upon it.

 

However, if we consider an electromagnetic wave then we know that its amplitude must become zero at its front and back. This implies that the total length of any electromagnetic wave must be a harmonic of its wavelength (i.e. its length is either 1/2, 1, 3/2, 2, 5/2, … times its wavelength). Now, if the length of the electromagnetic wave is one or more times its wavelength, then it must consist of at least one full wave. This means that the electromagnetic wave would contain all the ''information'' for it to propagate as a wave. Therefore, it would be able to travel and move as a wave through space. The question then, is what happens if the electromagnetic wave only consisted of half its wavelength? In this situation it would not have all the ''information'' required to propagate as a wave and would, therefore, most likely act in a particle like manner (i.e. if it was at rest, then it would remain in that state until acted on by a force, as defined by Newton.

 

Moreover, by definition electromagnetic waves consist of an oscillating electric and magnetic field. Therefore, if we looked at a single full length wave, then we could consider half the wave to be electrically positive, whilst the other half is electrically negative. This would still correlate with a complete wave being electrically neutral. Hence, in the case of an electromagnetic wave that is only half a wavelength long, then it would only consist of either the electrically positive half or the electrically negative half. These half waves would also each have a magnetic component attached to them, thus creating a magnetic monopole. However, this magnetic monopole would be extremely difficult to detect, since the magnetic field is weaker than the electric. Further adding to this difficultly, is the fact that an accelerating electric charge (e.g. this half wave) would create a secondary dipole magnetic field, which would help to mask the inherent magnetic monopole. Indeed these accelerations could be caused by the entire half wave physically moving up and down in space, due to it interacting with external energy (e.g. full propagating electromagnetic waves) in the local environment. This would imply that these monopoles would only be observed when the half wave was stationary or only moving at constant velocity in a vacuum environment that was close to absolute zero, and whose walls were sufficiently far away as to have no effect. Furthermore, we note that we are not the first to propose the existence of magnetic monopoles as Dirac proposed a general theory of magnetic monopoles back in 1948 and 't Hooft shows that these can exist within a quantum mechanical framework. Additionally, Morris (2009) wrote a paper where they stated that they had seen magnetic monopoles in spin ice Dy_2Ti_2O_7.

 

Also, if a positive and negative half wave were to collide with each other, then at that moment they would both have all the ''information'' required to propagate as a complete wave. Thus, this may be a mechanism whereby the two half waves would convert into two or more full waves, depending upon their momentum and angle of collision. This would imply though, that the two half waves were destroyed in the collision, leaving only full propagating waves behind. In fact this situation where two things collide, destroying each other and only leaving behind electromagnetic waves is every similar to electron, positron annihilations. Furthermore in the case of electron positron pairs, these can be produced from high energy electromagnetic waves.

 

Therefore, we have the following situations:

 

1. Half electromagnetic waves at rest cannot move without an external force, just like electrons at rest.

 

2. Half electromagnetic waves would have an electric charge associated with them, just like electrons.

 

3. When positive and negative half electromagnetic waves collide, they destroy each other, producing only full propagating electromagnetic waves, just as colliding positrons and electrons do.

 

This evidence would appear to imply that electrons consist of half electromagnetic waves!

 

An implication of this connection would be, that since electrons have mass, then so would electromagnetic waves (at least under certain circumstances). In fact from relativity we know that E=m*c^2, where E is the energy, m is the relativistic mass and c is the speed of light in a vacuum. This equation not only links mass and energy together, but implies that mass is just a dense form of energy, since the speed of light squared is huge. Moreover, we know that for any particular electromagnetic wave, its velocity, v, is related to its wavelength, lambda, by v=f*lambda, where f is its frequency. From this equation, we have that the slower the waves travels (i.e. it is located in a non vacuum medium), the shorter its wavelength is. Thus, the slower the wave travels, the smaller the total volume of the wave, i.e. the energy becomes more and more dense. However, from the statement above, this would imply that as the wave slows down, the more mass it would portray! Additionally, the total energy of an electromagnetic wave would be split between mass energy and electromagnetic energy. Plus, these energies would transform from one to the other, dependent upon the wave's speed. Furthermore, relativity states that a particle with mass cannot travel at the speed of light and thus, an electromagnetic wave must be massless. Hence, it could be argued that all the energy within an electromagnetic wave in a vacuum, (i.e. travelling at the speed of light) is in the form of electromagnetic energy and thus has no mass. Indeed, if an electromagnetic wave travelling through a vacuum has no mass, then we can work out the total energy of an electromagnetic wave. This is done by calculating the wave's electromagnetic energy within a vacuum, which is given by E=h*f (i.e. Planck's relation), where h is Planck's constant. Thus, by equating this equation and E=m*c^2, we can find the amount of mass an electromagnetic wave of frequency f can portray is:

 

m = h*f / c^2. (1)

 

This is the relativistic mass though, and therefore if we convert this into rest mass, m_0, we obtain:

 

m_0 = (h*f / c^2) * square root(1 - v^2 / c^2), (2)

 

where v is the velocity of the wave. However, as velocity is only a relative quantity (i.e. it depends upon your reference frame), we will actually refer to the waves mass as m_v, meaning that:

 

m_v = (h*f / c^2) * square root(1 - v^2 / c^2). (3)

 

Therefore, equation (3) agrees that all electromagnetic waves travelling at the speed of light in a vacuum would be massless, since the speed of light is a constant for all observers (independent of their own speed). Also it states, that the maximum mass an electromagnetic wave can have, is proportional to its frequency and occurs when the wave's speed is zero (relative to the reference frame). We note though, that the wave's speed can only become zero when it has hit something and thus, at this point the wave is destroyed (i.e. the wave would cease to exist as all its energy has been transferred into the particle it impacted). This would correlate with the quantum mechanical view that when an electromagnetic wave fully impacts (as opposed to ``bouncing off'' or scattering) any of the sub-atomic particles it is actually destroyed and in doing so its energy has been transferred. Additionally, we can see this mass as the impact force on the object being impacted by the electromagnetic wave. Moreover, we stated previously that the velocity of a wave is directly proportional to its wavelength. Thus, relating this fact with equation (3), shows that as the wave slows down its mass increases and its length decreases (increasing the density of the wave's energy), which again correlates with our statement that mass is just a dense form of energy. Therefore, overall we would have the situation where the mass of the wave would be zero, when it is travelling at the speed of light. Then as the wave slows down its mass increases, at the expense of the electromagnetic energy, until a maximum mass is reacted at the point the wave impacts an object. This idea that electromagnetic waves can have mass when they are travelling slower than the speed of light, may also help to explain their wave-particle duality. Finally, we note that equation (3) shows us that even for an extremely high frequency wave (e.g. a gamma wave), its mass when stopped would only be comparable to an electron's mass. Thus, this may further explain why we have never been able to detect any mass associated with an electromagnetic wave. However, this implication that electromagnetic waves can have mass, would be one way of testing the original idea, of whether electrons can consist of half an electromagnetic wave. In particular, scientists would need to test whether equation (3) actually holds experimentally. Although, we should note that this experiment would be particularly difficult, since the mass of an electromagnetic wave would be extremely small, even for a high frequency, slow moving wave. Part of this experiment though has already been managed, as scientists have been able to slow electromagnetic waves down to several miles an hour. For example the Rowland Institute for Science managed to slow electromagnetic waves down to 38mph in 1999.

 

Furthermore, if electrons and electromagnetic waves are the same thing, then we should be able to calculate the properties of the electromagnetic wave that constitutes an electron. These electromagnetic wave properties would have to be calculated based upon the electron's experimentally known properties, (e.g. its mass). Now we know from relativity that mass and energy are equivalent, such that E=m_0*c^2 for a stationary particle. We also have Planck's relation, which states that the energy of an electromagnetic wave is proportional to its frequency, given by E=h*f. Thus, equating these two equations and rearranging for frequency, we obtain:

 

f= m_0*c^2 / h. (4)

 

This equation states what the frequency of an electromagnetic wave would be, if all the mass of a particle was converted into a single wave. Moreover, equation (4) is the same as de Broglie's or Comptons's equation, i.e.

 

lambda = h / (m_0*c) (5)

 

if we convert it from a description about wavelength to one of frequency, using c=f*lambda. We have the situation though, where it would appear that an electron is only half an electromagnetic wave. Thus, to obtain the correct frequency for the electromagnetic wave, we must double the electron's mass. Hence, the wave's frequency is:

 

f = 2*m_0*c^2 / h = 2.4*10^20 Hz (6)

 

based upon the rest mass of an electron being 9.1093826*10^-31 kg. Hence, it is possible to find a sensible frequency for half an electromagnetic wave that would correlate with the properties of an electron, although clearly more analysis is required. Interestingly however, equation (6) correlates with the (linear) Zitterbewegung frequency found in Dirac's equation, when it is applied to an electron. This further implies that there may be something to this idea.

 

As we have already mentioned, relativity states that the mass of a particle increases with its velocity. In particular this relationship between mass and velocity is given by:

 

m = m_0 / square root(1 - v^2 / c^2), (7)

 

where m_0 is the particle's rest mass. Therefore, since we know that a force is required to accelerate a particle and a force requires energy, then this implies that some of that energy actually goes into increasing the mass. This again appears to be creating another link between energy and mass, in this case the more kinetic energy a particle has the more mass it has. In particular, let us consider a particle accelerator that contains a vacuum and has all of its walls at absolute zero. In this case there would be nothing (no matter or energy) inside the particle accelerator. Now let us assume that we ''place'' an electron inside it, and accelerate this electron close to the speed of light, using an electric field. In this situation, we have from relativity that the mass of the electron would have been greatly increased, since it is travelling close to the speed of light. However, this leaves the question, where did the electron gain its mass or energy from, since there is no mass or energy inside the particle accelerator, apart from the surrounding electric field? The answer comes from the fact that, as the electron is accelerated it generates a back electromagnetic field (similar to the motor), which has the effect of blue shifting the electron's inherent half electromagnetic wave. (Putting that another way, the electron transfers energy out of the field accelerating it and into its inherent half electromagnetic wave, blue shifting it.) Now since, mass is directly proportional to frequency, any blueshift occurring will also increase the mass by the same amount. Conversely, if the electron is decelerated, then its electromagnetic or gravitational field must do work against the external field that is causing the electron's velocity to change. Thus, this work causes the electron's inherent half electromagnetic wave to redshift and hence, the electron would lose mass. Additionally, as the change in mass is purely related to velocity and energy cannot be destroyed, then the rest mass of the electron would remain constant, independent of how many times it was accelerated and decelerated. Lastly, if the electron moved at a constant velocity, then none of its fields would be working with or against any external fields and thus, it would neither gain nor lose mass. In fact this relationship between mass and frequency may help to explain why mass, length and time (which at first glance all appear independent properties) all change with velocity by the Lorentz factor (or the reciprocal of it)

 

We note however, that this relationship between mass and velocity given by relativity is different from the relationship between mass and velocity of an electromagnetic wave that was discussed above (given by equation (3)). The reason for this is due to the fact that the total energy of the electromagnetic wave remains constant, whereas for the particle, its energy is continuously increasing with its speed. Therefore, the particle will gain more and more mass as its speed increases, but the mass of the electromagnetic wave will increase as the wave slows down, due to the energy transfer from electromagnetic to mass. Furthermore, let us assume that an electron consists of half an electromagnetic wave. If this is indeed the case, then the inherent velocity of the half electromagnetic wave (i.e. the velocity it would travel at, if it was a full wave), must be slower than the speed of light, otherwise the electron would have no mass.

 

Discussion

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In this paper we have investigated whether there is a connection between electrons and electromagnetic waves. What we have found is that electromagnetic waves whose length is half their wavelength have very similar properties to electrons or positrons. In particular:

 

1: Half electromagnetic waves at rest cannot move without an external force, just like electrons at rest.

 

2: Half electromagnetic waves would have an electric charge associated with them, just like electrons.

 

3: When positive and negative half electromagnetic waves collide, they destroy each other, producing only full propagating electromagnetic waves, just as colliding positrons and electrons do.

 

Furthermore, when investigating what frequency half an electromagnetic wave would be, to give it the same properties as an electron, it correlated with the Zitterbewegung frequency found in Dirac's equation. However, these connections also implied that electromagnetic waves would have (or at least portray) mass under the correct circumstances. In fact E=m*c^2 directly implies that mass is just a dense form of energy and from this came the idea that electromagnetic waves do have mass when they are travelling slower than the speed of light. Indeed the slower they travelled the more mass they had, until the wave speed become zero, at which point they had their maximum amount of mass. This maximum mass was directly related to the frequency of the wave (and the number of waves). Additionally, we noted that at this point the wave would be destroyed, since for its speed to be zero, it must have hit something (i.e. the wave had cease to exist as all its energy has been transferred into the particle it impacted), which correlates with quantum mechanics. Moreover, this implication that electromagnetic waves have mass, would be one way of testing whether electrons can consist of half an electromagnetic wave. Although, an issue with this experiment is that each wave has an extremely small amount of mass, even for high frequency, slow moving waves. Lastly, we explained why the rate of change of mass with velocity for a particle and an electromagnetic wave would be different. This was due to the fact that the total energy of the electromagnetic wave remained constant, whereas the energy of the particle continuously increased.

 

Finally, if these half electromagnetic waves and electrons were found to be equivalent, then it would explain why both electromagnetic waves and electrons have a wave-particle duality to them. The question is though, are they the same?

 

(If you are interested I can give you some of the references where I got this information from. However, I left them out currently, to try and make it easier to read and understand. Also I hope that you can understand the equations that I have used, but if not then let me know, and I will re-write them for you.)

 

Thank you for your time and I look forward to hearing what you have got to say.