The question of how to calculate the numerical value of the fine structure constant from theory was one of the most outstandingly difficult problems in mathematical physics for the greater part of the 20th century. There were many unsuccessful attempts by researchers including famous ones such as P. A. M. Dirac to find a formula for the fine structure constant. See John Baez's website page Open Questions in Physics and also the PhysLINK website. Substantial progress with this fundamental problem is outlined on this web page.
There is a general connection of the quantum coupling constants with π which was anticipated by R.P. Feynman in a remarkable intuitional leap some 40 years ago as can be seen from the following much quoted extract from one of Feynman's books.
There is a most profound and beautiful question associated with the observed coupling constant, e the amplitude for a real electron to emit or absorb a real photon. It is a simple number that has been experimentally determined to be close to -0.08542455. (My physicist friends won't recognize this number, because they like to remember it as the inverse of its square: about 137.03597 with about an uncertainty of about 2 in the last decimal place. It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it.) Immediately you would like to know where this number for a coupling comes from: is it related to pi or perhaps to the base of natural logarithms? Nobody knows. It's one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the "hand of God" wrote that number, and "we don't know how He pushed his pencil." We know what kind of a dance to do experimentally to measure this number very accurately, but we don't know what kind of dance to do on the computer to make this number come out, without putting it in secretly! ........Richard P. Feynman (1985). QED: The Strange Theory of Light and Matter. Princeton University Press. p. 129.
Feynman was right on both counts as α is related to π in a very specific way and indirectly also to e the base of natural logarithms. With hindsight there is an element of obviousness about these relations but it took a genius to recognise them at the time. The numerical values of a set of fundamental quantum coupling constants are given by the very simple formula depending on two integer parameters n1 and n2,
α(n1,n2) = n2cos(π/n1)tan(π/(n1× n2))/π.
The values of the quantum coupling constants are often denoted by the lower case Greek letter alpha, α with some specialising subscript. The fine structure constant itself is usually denoted by alpha with no subscript.
The numerical value of the fine structure constant α is given by the special
case n1 = 137 with n2 = 29,
α = α(137,29) = 29cos(π/137)tan(π/(137×29))/π = 0.00729735253186... .
The 1999 CODATA recommended experimental value for this quantity with the (± 27) uncertainty range centered on the last two digits (33) is,
α = 0.007297352533(27)
and the 2002 CODATA recommended experimental value for this quantity with the (± 24) uncertainty range centered on the last two digits (68) is,
α = 0.007297352568(24).
and the 2006 CODATA recommended experimental value for this quantity with the (± 24) uncertainty range centered on the last two digits (50) is,
α = 0.0072973525376(50).
A very accurate, 2006, New Value of the Fine Structure Constant from the Electron g-Value and QED, G. Gabrielse et al derive the value for this quantity as,
α =0.007297352569
Briefly, the hindsight obviousness of Feynman's conjecture is inherent in the simplest idea of coupling between two systems. Taking one of the objects as fixed in space then, if a second object moves in circles about the first, the two objects are coupled in some way. Circular or cyclical motion in two dimensions is the generic form taken by most simple coupled systems when two objects are involved and implies that π is involved and further two dimension circular motion maps onto unimodular complex numbers like ei θ with θ arbitrary and so e the base of natural logarithms is also involved. Such unimodular complex numbers are also the basic constituent of the mathematical description for quantum states in general. Yes certainly obvious, but not so fifty years ago when a smokescreen of deep complicated and little understood theoretical structures obscured the obvious except for the very few individuals such as Feynman with penetrating insight.
The connection of α with π is much deeper than the fact that π accurs in the formula for α(n1,n2) given above. We can define two simple but very significant generalizations of π , πi(n) and πo(n) which will be called π-in and π-out and having values which depend on an integer parameter n. π can be defined as the ratio of the circumference of a circle to twice its radius, C/(2r). Given any n-sided equilateral polygon P(n) it will have a small radius ri(n), the distance from its center to the center of a side, and a large radius ro(n), the distance from its center to a vertex. Thus analogously to the way π is defined, two generalizations of π can be defined by dividing the perimeter length of the polygon by 2ri(n) or 2ro(n). This gives the two integer dependent generalizations of π with properties following,
πi(n) = n tan(π/n),
πo(n) = n sin(π/n),
πo(n) < π < πi(n),
πi(∞) = πo(∞) = π.
Inspection of the formula for α(n1,n2) reveals that it can be expressed in terms of the first generalized π as,
α(n1,n2) = cos(π/n1)πi(n1 × n2)/(n1π).
Thus if n2 goes to infinity
α(n1,∞) = cos(π/n1)/n1.
This last formula gives a very accurate first approximation for the values of the coupling constants. It was in fact discovered before the two parameter exact formula was found.
Thus the relation between the coupling constants and the generalized π can be put into the more tidy form,
α(n1,n2)π = α(n1,∞)πi(n1 × n2). ...........................†
The important part played by polygons in this theory is a consequence of quantization. Motion round a polygon must occur with directional jumps, whereas motion round a circle can be taking place classically with a continuously changing direction of motion.
Much more detail about this area of research and the formula †, the very important part played by special relativity and the implications for high energy physics together with downloadable files on the subject can be found on my website in the Mathematics Department at Queen Mary College London. The downloadable files on my website contain accounts of how some of the very important results from what is called the standard model for particle physics can easily be obtained from this theory. A very simple theoretical formula can be obtained for Weinberg's weak-mixing angle θW. The mass ratio of the W and Z gauge bosons can be obtained. Another application is the use of the accurate formula for the fine structure constant to produce a method for finite renormalization.
An interesting recent contribution to finding an accurate value for α has been presented by Michael Wales. He claims that there are good reasons for the ratio of an electron's time in a Bohr orbit to an internal electronic time to have the definite integral value NW = 2573380, such that
α = NW -1/3 ≈ α(137,25).
The approximation above is a value I used in earlier work based on CODATA's 1986 value. This approximation is very close indeed to the Wales' value. The Wales' number is used by Manfred Geilhaupt in his work on the fundamental constants. Some of the many past attempts at finding a formula for α are listed on Ivan Gorelik's web page Fine Structure Constant CollectionYou may have to download the utf-8 character set from Alan Wood or from Unicode Consortium in order to read these pages.
click here to read Belorussian translation (provided by Galina Miklosic)
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