The Case for the Fine Structure Constant Approximation, α_∞

α ≈ α_∞ = cos(π/137)/137

by

James G. Gilson

Queen Mary College, University of London

October 14 2003

1. Introduction

I published the approximation formula α_∞, given above in the title, for the fine structure constant in the Journal Speculations in Science and Technology[1] in 1994. This new one parameter formula for the approximate value of α was derived using an alternative version of Schrödinger quantum theory. In 1996 in a paper called Calculating the Fine Structure Constant, I showed that this approximation could easily be obtained from the orthodox Schrödinger theory. In the same Physics Essays[2] paper, I derived a general two parameter formula of great accuracy which had α_∞ as a limit when the second parameter approached infinity. That the derivation could be made using the orthodox theory is perhaps not surprising as the alternative theory is not in conflict with orthodox theory at all but rather contains the orthodox theory as a subordinate lower dimensional theory. However, it is very surprising that it took over seventy years for this formula to emerge when it is clear that Sommerfeld[3] who first introduced α into physical theory in 1915-16 was himself in possession of almost all the necessary ideas that are required to find the formula, make a simple case for its validity and thus find at least an approximate theoretical value. This short paper will be used to give the simple derivation that he missed. Almost a decade has passed since I first published the approximation α_∞ for the fine structure constant and in those intervening years my understanding of the nature of the problem and of its solution has evolved greatly. In the following derivation this clearer image of what it is all about has, I hope, been incorporated.

2. Wave Capture

The Wave Capture Diagram below is a key tool in understanding the geometry of the fine structure constant. It is essentially a graphical description of a possible configuration of a length extended object in motion in a circular orbit about a center of attraction. It can be used to represent a circular path of an electron in a hydrogen atom and that is the context for the derivation to follow. Thus we are talking about the geometry of an equivalent wave rather than the geometry of a point particle representation. The idea of representing the electron as a length extended wave object is probably the missing idea that impeded Sommerfeld from getting at α's numerical value from the theory perspective. Here the moving object to be associated with the wave in the case of the approximation α_∞ is the straight line segment B₂A₁ instantaneously and approximately moving along its own length in motion in the direction of the arrow with leading end at the point A₁ and with the component velocity v = c cos(χ_N) the dominant component of actual vector velocity for small χ_N. That strictly speaking the moving segment has the very small but almost radial component of velocity v = c sin(χ_N) also is an aspect of the approximate nature of the interpretation of the diagram. It will be seen that this is compensated for by the way the argument is developed in the following discussion. From now on the straight line element B₂A₁ will be called the wave object. Strictly speaking, the inner circle Γ_o' is the path taken by the single point A₁. The outer circle is chosen so that the tail of the object B₂ moves with the speed of light c on and tangential to the circle Γ₀ as indicated on the diagram. Thus the construction is simple but in respect of the trailing end moving at the speed of light on the circle Γ_o it is an unusual configuration to be studied and in this sense the configuration has a geometrical limiting significance with the outer circle being special. On first encountering this diagram, it may impress as being weird because of the concentration on the outer circle being traversed at the speed of light. However, in any rotating system the object can be thought of as fixed at a point on a rotating radial line in a frame of reference rotating with it about the same center. The only additional construction here is that this radial line is thought of as extended to that point B₂ at which it will be found to have a transverse speed equal to c. Such an extension is in principle always possible. The generalised version of The Wave Capture Diagram for strong coupling or small values of the parameter N can be found here and with a theoretical discussion can be found in the file gil4.pdf.

The Wave Capture Diagram

The diagram can be regarded as representing the geometry of the relativistic length contraction with velocity effect[4] as can be seen from the Euclidean triangle formulae,

which can be recognised as the Lorentz contraction formula due to the velocity v and also where

is taken to be the velocity of theleading end A₁ of the wave object, r_o its rest length and L its contracted length due to the velocity v and ω its angular velocity about the origin 0. However, because the wave object L is rotating, v is strictly speaking not the mean or centroidal velocity which we will call v* and which in some sense needs to defined for the wave object. Rather it is a velocity of just the one end point A₁ of the wave object. However, v is a good approximation to that mean velocity when the length L is small which is the case for a high velocity v or small χ_N, conditions that do indeed hold in the fine structure constant evaluation context that is to be pursued in this article and this is partly the sense that the wave capture diagram in the above form refers specifically to the approximation α_∞. Another important characteristic of this diagram is that it is conformable to the quantum concept of projection quantization that usually appears in the angular momentum context. In the case of the wave capture diagram, it appears in the extended length context as follows. The projection of the radius r₀ through the angle χ_N gives the inner circle Γ₀' of radius r and this will be considered to be quantized as

where l_b can be regarded as a length quantum that is associated with the quantum number N. l_b can be read as being the quantum length associated with the inner boundary circle Γ₀'. This then implies a circumferential quantization of this circle determined by its circumference length,

Thus the circumferential quantum length πl_b is associated with the quantum number 2N on this circle and as the circumferential quantum length πl_b is also given by

it follows that the angular quantum χ_N is given by

That is to say the full circular angle 2 π is also quantized as

2a. The True Mean or Centroidal Velocity

It is not obvious a priori where the point moving with this mean velocity that will be denoted by v* will be found on the diagram. In constructing this wave capture scheme, the aim to to produce a relativistic quantization template against which rotating physical systems can be tested for conformity. Thus the specific values for v* and r* will be supplied by the system being tested. However, we can get some good ideas about it's location as follows. Because the wave object B₂A₁ length remains in a fixed orientation relative to and within the annulus between the circles Γ_o and Γ_o' in it's rotational motion and the speed v = c cos(χ_N) applies only to the leading end point A₁ of the wave object all the other points on the wave object move at higher speeds than v. Thus the mean velocity v* needed to represent the bulk motion of the wave object has a higher numerical value than v. That is to say, v* lies in value between v and the speed of light c or

In the above equations, ω is the angular velocity of the object about the center O and r* is the consequent radius, larger than r, of the circular path of the mean point. v*' = c sin(χ*) and v' = c sin(χ_N) represent the almost radial components of the velocities orthogonal to v* and v respectively. These components will be useful later. Thus if we retain the construction that the radius of the outer circle is equal to the rest length of the object, we see that the two end points of the object can be perceived as having shifted, A₁ upwards a distance r* - r and B₂ has moved anti-clockwise through a very small angle χ_N - χ* while keeping on the outer circle, all as a result of changing to the mean velocity v* perspective. Thus the perception of the position of the wave object is now just above the element L and very slightly shorter so that it terminates on the outer circle, Γ_o. This change is just the change of view that so often occurs in special relativity when one views moving extended objects from different velocity perspectives. This change takes into account the other component of velocity v*' = c sin(χ*) which is involved because the object is rotating. Thus the diagram with L representing the wave is approximate in the sense that this wave position and velocity are approximate. The position that the wave would be observed physically is the very slightly displaced position above the drawn element L. The use of the observed r* and observed v* compensates for the approximate aspect of the diagram as represented by the drawn line wave element and its geometrical position. Rearranging the last relation, we get

We define a quantity β_N measuring the ratio change from r to r* as,

β_N can be called the relativity velocity perspective change factor. In analogy with the quantization of the radial distance r, we can identify a quantization for the radial distance r* in terms of a quantum length l* associate with the integer N by using the equation,

Thus the relation between the observed l* and l_b is

and l* is the true quantum length associated with the projection process through the angle χ* as would be actually observed under the true mean velocity v*.

We can summarize the previous sections as follows. The wave capture process, graphically represented by the wave capture diagram, is a conceptual process that can be used to analyse the motion of an extended object in cyclical motion beteen two circular boundaries r and r₀. The process it represents is a new conceptual bridge between quantum aspects of motion and relativistic aspects of motion via quantization and relativistic length contraction. It can be used as a template to test length extended systems in rotational motion for conformity to a quantum-relativist bridging kinematical characteristic. In the next sections, this scheme will be used to test the motion of an electron about a hydrogen like atomic nucleus for such quantum relativistic bridging conformity.

3. The Hydrogen Like Family, {H_z}

To carry through this study of the fine structure constant, we need to consider material from Schrödinger quantum theory for electron states under the influence of the Coulomb potential,

The first function of r, V(r), equation (18) is the case for the potential between two different electric charges e₁ and e₂. The second function of r, equation (19) for V_Z(r), gives the case of the an electron with charge -e in the field of a nucleus with charge Ne. h' in (19) is here used to denote what is called hbar that is the Planck constant h divided by 2π.This last case equation (19) is that which applies to the set of hydrogen like atoms, atoms with Z protons in their nuclei and one orbiting electron. These potentials are just the classical Maxwell electrical potential that is effective between electric charges in the vacuum medium. Quantum theory only comes into this through the magnitude of the electron charge which itself is not given in any direct way by quantum theory at this moment in history. From the last equality, we get the physical definition of the fine structure constant in SI units. However, the combination of dimensioned physical quantities that make up α in (20) render it dimensionless.

From the Coulomb potential (19), we get the definition of the classical electron radius here denoted by l_c,

where m is the electron rest mass. Another useful quantity related to the classical electronic radius l_c is the Compton wave length of the electron divided by 2 π. Here it will be denoted by 2l₀. It is defined by

Even the value of the electron rest mass is not obtainable from quantum theory at this moment in history. The quantities we require that do come from quantum theory are the formulae for what is called the first Bohr radii and the electron's velocities on these orbits in the system of hydrogen like atoms. They will be denoted by r_B,Z and v_B,Z and are given by

These are the only quantity that are strictly quantum mechanical in origin that will be neededin the following work. We are here looking for a theoretical method for deriving the value of the physically defined quantity the fine structure constant α above. However, we do a priori physically have some limited but definite physical information about the range of possible values within which α resides. All significant measurements over the last seventy plus years have given that

Thus it essential that our theoretical derivation conforms to that inequality. In fact, we can make use of this physically implied constraint by taking α to have the numerical form,

with χ assumed to be small but not zero, because this form automatically satisfies the inequality above as a result of the cosine function always being less than or equal to unity.

4. The Numerical Value of α_∞

let us now take r* to have a value from one of the values r_B,Z from among the set of first atomic Bohr orbit values,

and the velocity v* in this orbit to be the velocity of the electron on the same first Bohr orbt,

Using these values we can find the radius r₀ of the outer circle Γ₀ by noting that

by construction. It follows from (22) that

The radius r of the inner circle, Γ₀', is thus

It follows using (28) and (32) that the ratio measure β_N is, in the hydrogen like atomic set context, given by

It follows from equations (14) and (28) that

and using (31) we get

Thus if N is taken to be equal to Z ≤ 137, we get

Thus we obtain a formula for the fine structure constant, α with only the first factor β numerically undetermined. The second factor is clearly the first approximation to α,

α∞ = cos(π/137)/137 = 0.0072973510109..... .

39

that we seek. The measured value of α is,

according to CODATA[5]. The numerical value of β₁₃₇ has been found from theory and is in fact given by

The same value is found from measurement. However, here we are studying the approximation, α_∞ and its significance. The detailed derivation of β_N for a general integer value N = n₁ can be found in the article called Calculating The Fine Structure Constant in the file gilg.pdf. Here we can get by simply by noting how very close to 1 its value is and consequently what a very good approximation α_∞ is to the physically measured value of α. The value to be associated with the quantum length l* in the hydrogen-137 context can be obtained from equation (4) with N = 137,

having used (17) so that by (36) we have

5. Conclusions

The Wave Capture Diagram is a graphical representation of a new conceptual link between quantum theory and relativity theory which, in this work, I call a quantum-relativity bridge. It applies to length extended objects in cyclical motion and is based on projection quantization and relativity length contraction. It is a kinematic-geometric template construction which accepts specific values from a given system under test and then yields the information as to whether the system conforms to the quantum-relativity bridge characteristic. In this article, the first Bohr orbits of the quantum hydrogen-like systems are tested. The conclusion is that these orbits do conform accurately. This conclusion is reinforced by the test yielding an accurate value for the fine structure constant. Inspection of the wave capture diagram and noting the quantized nature of the angle χ_N = π/N involved, it is clear that the diagram refers to an N-sided polygon. This is the clue to how quantum system coupling involves the geometry of polygons in a fundamental way. Such polygons are also basic to the mathematics of representations of the cyclic group. In the case studied here, the group order is N = 137. In the above, the cycling process shows up clearly by the wave in motion being confined between the circles of radii r and r₀. From equations (12), (15), (37) and (38) we get a clear explanation of the physical role of the β₁₃₇ velocity perspective change factor. It simply converts the approximate α_∞ which involves the boundary almost radial velocity v' = c sin(π/137) with the contraction factor (1 - (v'/c)²)^1/2 into the measured α with almost radial mean velocity component v*' = c sin(χ*) and contraction factor (1 - (v*'/c)²)^1/2 by multiplicative substitution. Thus from the approximate α_#8734; boundary perspective the observed or measured α perspective is generated. It is thus that the rotational motion over and above the translational v motion relativity contraction of the rest length L₀ is taken into account. Extension further into the mathematics of cyclical group theory, leads to an accurate theoretical evaluation of β₁₃₇ for the hydrogen-137 atom with the full cyclical group involved then being of order 29 × 137. However, this is another story and can be found elsewhere on this website. Detailed information about the numerical values of physical constants can be found in reference [5].

Appendix

6. References

Acknowledgments

Fine Structure Constant, alpha