An Answer to a Question from
by
James G. Gilson j.g.gilson@qmul.ac.uk
The fine structure constant is dimensionless and if one could find a purely mathematical way of deriving it without reference to any physical measurement then it would be a calculable constant of nature similar to π and e with important consequences. It seems you have done this. Your formula is interesting in that the only integral value of n1 that enables the result to be close to the experimental value is 137 and the only integral value of n2 that fine tunes the result to exactly match the experimental value is 29. 28 or 30 does not do it. If non integral values for n1 or n2 had been allowed then the formula could always be adjusted to give an answer that matched the experimental value of the day and would have no deep significance.
However I am still at a loss to see how 137 and 29 can be deduced logically without reference to the experimental result or given the experimental result what meaning can be assigned to these numbers.
Ivor Brodie
Your question is valid and is a sticking point for some people who have looked at my theory and have become preoccupied with the same question. In fact that question has been asked in various forms for more than seventy years since α was first introduced by Arnold Sommerfeld in 1915-16. At the risk of seeming to be a politician, Let me firstly skirt your question and remind you of the way systems are analysed in orthodox quantum theory. To carry through this case let us introduce a typical but rather simplified orthodox quantum system involving angular momentum and its possible measured and theoretical values. Suppose this system has an angular momentum J of fixed and definite absolute magnitude value J=|J|.
Suppose further, that the unknown characteristic of this system is in what direction the angular momentum vector is pointing in ordinary three dimensional space relative to some fixed direction. According to quantum theory the usual situation is that to find out the direction we have to carry out a measurement and until that measurement is made we can have no idea what the answer is. Once the measurement has occurred which will have involved ascertaining the value of some parameter m, say, we can say yes the system was in that definite state nmeasured defined by the parameter m being measured. Often the measurement will involve determining a component of the vector J, in some special direction with the orthodox theory connotation that such components are given by projection quantization . Thus the theory construction could be
|
Jcos(θnmeasured) = nmeasuredj |
|
and a measurement of m = Jcos(θn), given that we know the unit quantity j, will yield the value n = nmeasured and thus the direction in space relative to the fixed direction can be calculated. However, the important and overriding point about all of this is that until the measurement is made we cannot by quantum theory or by any other means know what this directions is. We note that in this projection quantization context the situation can be represented by a circle of radius J with a fixed direction chosen, say the y-axis and the actual possible directions of the vector J marked in by a particular radius making the angle θnmeasured with the y-axis, a picture to be found in most quantum mechanics text books. Now you will recall that this situation is the source of most of the philosophical problems that quantum theory has thrown up. It has been extensively discussed for years and probably occupies more archived journal space than any other science philosophy subject. There are at the very least two points of view about this quantum dilemma. The first, we cannot get the value by theory because we do not know enough detail about the systems and the second, there is an intrinsic unknowability about the state of a quantum system other then by measurement. It is certainly the case that if we do have addition information about a system or the conditions under which a measurement has taken place we are better placed to make some sort of prediction about the outcome of the measurement though certainty is a rarity in quantum theory.
Now let us return to the actual question you asked. The physical system in which the coupling structure for quantum systems is held is quantum mechanical and involves projection quantization just as the example discussed above but in relation to geometrical lengths rather than angular momentum. Now why when we measure always by indirect methods the value of the fine structure constant do we always get the value a(n1,n2) associated with the eigen-number n1=137? The answer is that we don't always get that number. In recent years according to the orthodox theory, other value have been obtained agreeing with some running coupling value scheme, n1=128 occurs for example when measurements are made at higher energies. The reason for the popularity of n1=137 is that for years measurements were only made at low energies and even today most measurements are made at low energies when n1=137 does give the measured recommended low energy CODATA value. Cleary, the same type of argument can be applied to questions about n2. There are definite meanings associated with these numbers within my theory. The product n1n2 represents an orbital length quantization and n2 is an internal length quantization of the extended rather than point orbiting object.
To summarise, I suggest that the usual quantum interpretation for quantum numbers implies that they are not fixed in any theoretical way, other than that they might lie in certain finite or infinite sets of values, but rather depend on the circumstances of measurement. The question of finding a theoretic derivation for the numerical values for the pair of numbers (n1,n2) takes on an altogether different complexion in that context. The number n1=137
appears frequently because measurements are most common at low energies but this does not justify attaching a special significance to the value n1=137.
James Gilson,
For details of how the formula for alpha was obtained and the consequences arising from it visit the website:-
Fine Structure Constant, alpha