A Quantum Approach to Relativity

L D HOWE

AEA Technology
Current Address: Serco Assurance,
B150, Harwell, Didcot, Oxon., OX11 0QJ, UK

PACS Numbers: 02.50.Wp, 03.65.Bz, 05.60.Gg

Abstract

Special Relativity depends on the speed of light being constant in space. This was originally founded in a wave model for electromagnetic radiation. If fundamental physics is viewed from the point of view of quanta in differing frames of reference, a new concept emerges. It is difficult to reconcile energy change when viewed from different frames of reference. The difficulty has been considered by using a tape recorder analogy. The nature of mass is considered with reference to the concept of a De Broglie vector. The concept of mass equivalence has been introduced.

1. Introduction

The concept of special relativity was grounded in the then current view of "waves through the Ether" (a medium that was hypothesised to carry light and other electromagnetic waves). It was assumed that the speed of light must be absolute and constant, irrespective of the frame of reference. This implies that, in a non-relativistic universe, different values for the speed of light should be measured when the Earth travels at differing speeds in differing directions (because the measured speed should be the addition of the two speeds). Michelson and Morley carried out an experiment [1] to detect the Earth's motion through the "Ether". However, they were unable to detect any difference in the speed of light when measured in different directions. Kennedy and Thorndyke [2] later verified that there was no observed change in the measured speed of light when the surface of the Earth travels in different directions. Thus Einstein conjectured that the speed of light was constant for all observers, leading to special relativity.

However, the conclusions drawn from the experiment have omitted a fundamental element. If the wave theory can be dispensed with completely, and a purely quantum approach adopted, the only conclusion that can be drawn from the results of the Michelson and Morley experiment is that the speed of photons is constant within a frame of reference. This does, of course, not mean that relativity is "wrong", any more than Newtonian mechanics or Galilean relativity are "wrong". Relativity may need revisiting and the author intends to do so in a forthcoming paper. However, it is useful here to explore frames of reference from the point of view of non-relativistic quanta.

2. Energy and the observer

Let us begin with an example that is totally non-relativistic in terms of Special Relativity. Consider four frames of reference, denoted by ⁰F₀, ¹F₁, ²F₂, and ³F₃, where the superscript denotes the frame of reference of the observer and the subscript the frame of reference of the observed particle or group of particles. (We will consider the term particle to include any co-acting group of particles). Let the relative velocities of the frames be ⁰V₁ = v ms^-1, ⁰V₂ = 2v ms^-1, ⁰V₃ = 3v ms^-1, with ¹V₂ = v ms^-1, ¹V₃ = 2v ms^-1 and ²V₃ = v ms^-1. A particle, P, mass m kg, moving in ²F₂ is observed in frame ⁰F₂ to have kinetic energy ⁰E₂ =2.0mv² J. If that particle accelerates to ⁰F₃, its kinetic energy is observed to increase to ⁰E₃, 4.5mv² J, representing an increase of 2.5 mv² J. However, as observed from ¹F₂, its kinetic energy increases from ¹E₂, 0.5mv² J, to ¹E₃, 2.0mv² J, representing an increase of 1.5mv² J. Thus the change in kinetic energy is different for the two observers. So the frame of reference of the observer determines not only the observed energy, but also the change in energy for the same observed phenomenon.

3. Frequency and the frame of reference

Consider the De Broglie frequency, f, of particle P, given by the formula e = hf. (Note the term “De Broglie wavelength” has been deliberately avoided to avoid association with “waves through the Ether”). If particle P leaves an object in ⁰F₀ travelling in ³F₃, it will be observed to have energy ⁰E₃, of 4.5mv² J, with a frequency of 4.5mv²/h Hz. It will be observed to arrive at an object in ²F₂ with energy ²E₃ of 0.5mv² J, with a frequency of 0.5mv²/h Hz, a red shift of 89%. However, it will be observed to arrive at an object in ¹F₁ with energy ¹E₃ of 2mv² J, with a frequency of 2.0mv²/h Hz, a red shift of 56%. Notice that the phenomenon of red shift is described here without the need for waves or relativistic effects.

The question arises as to the nature of De Broglie frequency. It has already been seen that the De Broglie frequency observed for a particle P will vary according to the frame of reference of the observer. However, two particles P₁ and P₂, both in the same frame of reference, with respective masses m kg and 2m kg, will always be observed to have a De Broglie frequency ratio of 1:2, irrespective of the frame of reference of the observer. This leads to the hypothesis that the De Broglie frequency consists of two components: the mass related part which is independent of the frame of reference of the observer and the velocity related part, which is independent of mass..

4. The tape recorder analogy

In order to understand this phenomenon, the simple, although rather inadequate, 1-dimensional analogy of a tape recorder may be used. Consider each frame of reference as a magnetic tape. All frames of reference move at constant speeds that differ from every other frame of reference. It is useful here to consider the concept of the De Broglie vector. Every particle will have a De Broglie vector, which on rotation in De Broglie space produces the De Broglie frequency associated with the particle. Every particle will be observed from within its own frame of reference to have a De Broglie frequency of zero. However, in this analogy, the phase of the De Broglie vector will be different at every position in this frame of reference. Imagine the phase shift, f, of the vector, l m along the tape is governed by the equation:

Now this equation is not dimensionally consistent, but considering that we are working in De Broglie space, the difficulty can be overcome by introducing a dimensional constant of 1 ms^-1. This may be thought of as the signature of the particle in its own frame of reference. Indeed, the value 2h/m may be regarded as the amplitude of the De Broglie vector. Hence, moving along the tape will result in the observation of a rotation of the vector and the generation of a De Broglie frequency. Now because the tape represents a singe frame of reference, it is impossible to move along it, because that involves changing to another frame of reference. This is equivalent to hopping onto another piece of magnetic tape travelling at a speed v ms^-1 relative to the particle's tape. But in this case

which is not an adequate description of the De Broglie frequency. However, consider the analogy of a video recorder, where the playback head rotates against the tape. If each tape (i.e. frame of reference) sees every other tape with its "playback head" rotating at a surface velocity of v(v-1) then we arrive at the standard equation

So in this analogy, 1-dimensional space consists of an infinite number of tapes, each travelling at a slightly different speed from its neighbour, each with its own playback head, used for monitoring the changing phase of De Broglie vectors of particles on other tapes. To agree with Special Relativity, each would have to see itself as having an equal number of tapes on each side, with the tapes on one side moving in the opposite direction to the tapes on the other side. The two peripheral tapes would be travelling at the speed of light, as seen by every tape. That is to say, the geometry of the cross section would be non-Euclidean, with the measurement of distance changing as one traverses the cross section.

Although this analogy is clearly inadequate for a model of what actually happens, it does give us an insight into the apparent differences in energy changes, as measured from different frames of reference. The idea that energy change depends on the observing mechanism in the frame of reference of the observer is a direct result of such an analogy. It clearly helps to explain the apparent discrepancy considered in Section 3 above.

5. The nature of mass

Mass has always been considered as the amount of "substance" in a particle, related to a universal, attractive force, gravity. However, if the tape recorder analogy is accepted, mass becomes identified with the De Broglie signature hypothesised in Section 4 above. It is normally assumed that photons have no mass. This in part stems from relativity, which proposes that no mass can achieve the speed of light. However, photons have a De Broglie frequency, and the energy is related by the standard equation (with f given in Hz)

Using the above analogy, an ultra violet photon, with f = 10^-15 would have a mass equivalence of 1.5  ´  10^-35 kg, which is more than 5 orders of magnitude less than that of an electron. This implies that high-energy gammas would have a mass equivalence similar to that of an electron. Notice the use of the term mass equivalence, rather than mass, because the hypothesis is that mass equivalence is the manifestation of the De Broglie signature, rather that the normally accepted reverse assumption. If this is the case, it may have an impact on our assumptions about General Relativity as well as Special Relativity.

6. Conclusions

A mechanism has been proposed to describe the motion and energy of a quantum particle, as observed from any frame of reference, by a single quantum property, the De Broglie vector. The vector and its rotation completely describe the mass and energy of the particle, as observed from that frame of reference. The rotation of the vector in a given frame of reference accounts for differing values of measured energy change, as observed from different reference frames.

References

[1] R S Shanckland: American Journal of Physics 32 p16
[2] R J Kennedy and E M Thorndyke Physical Review 42 p 400 (1932)

www.innovationgame.com/physics/relpaper.htm December 2000