An Exact Value For The Fine Structure Constant.

Don Blazys · August 30, 2011

To date, by far the most accurate measurement of the fine-structure constant

(measured at the scale of the electron mass) was made by Gerald Gabrielse

and colleagues from Harvard, Cornell and RIKEN.

Measuring the "magnetic moment" of a single electron in a "quantum cyclotron" and

inserting that value into state of the art QED equations, the value they determined is:

[math]\alpha^{-1}=137.035999084(51)[/math],

which means that the fine-structure constant lies somewhere in between:

[math]\alpha^{-1}=137.035999135[/math], and

[math]\alpha^{-1}=137.035999033[/math].

Now, these values were determined back in 2008, and since then,

no significant improvement in accuracy was ever accomplished,

despite enormous improvements in both the design of the equipment

and the QED equations themselves.

Thus, many scientists now suspect that further refinements in

the value of the fine-structure constant may not even be possible,

and that the last two values represent, for all practical purposes,

the actual lower and upper bounds of the fine-structure constant as

measured at the scale of the electron mass. Even Gabrielse himself

believes that the above values will hold for a long, long time to come.

Given the above facts, it now seems that in order to be correct,

any mathematical expression which results in the fine-structure constant

must not only match the above experimental value exactly, but must also include,

within it's form, some simple way of expressing, with the same degree of absolute accuracy,

those seemingly inherent lower and upper bounds.

Now, some of you may remember this article: http://donblazys.com/on_polygonal_numbers_3.pdf

which describes a finding that occured right here at Hypography many moons ago.

Well, thanks to a great fellow named Lars Blomberg, (who found it via the OEIS)

we now have values of [math]\varpi(x)[/math] to [math]x=10^{15}[/math].

This information was crucial in not only greatly improving the "counting function",

but also allowed me to derive these values of the fine-structure constant as well:

[math]\alpha^{-1}=137.035999084=(A^{-1}*\pi*e+e)*(\pi^{e}+e^{(\frac{-\pi}{2})})-\frac{1}{(6*\pi^{5}*e^{2}-2*e^{2})}[/math]

[math]\alpha^{-1}=137.035999135=(A^{-1}*\pi*e+e)*(\pi^{e}+e^{(\frac{-\pi}{2})})-\frac{1}{(6*\pi^{5}*e^{2}-2*e^{1})}[/math]

[math]\alpha^{-1}=137.035999033=(A^{-1}*\pi*e+e)*(\pi^{e}+e^{(\frac{-\pi}{2})})-\frac{1}{(\mu*e^{2}-2*e^{\frac{5}{2}})}[/math]

where [math]\mu=1836.15267247(80)[/math] is the "proton to electron mass ratio", and

[math]A=2.566543832171388844467529...[/math], is that very special "Blazys Constant"

which generates all of the prime numbers, in sequential order, by the following simple method:

Note that the whole number part is the first prime [math]2[/math], and that:

[math]((2.566543832171388844467529...)/2-1)^{-1}[/math]

is approximately:[math](3.530176989721365539402422...)[/math],

where the whole number part is the second prime [math]3[/math], and that:

[math]((3.530176989721365539402422...)/3-1)^{-1}[/math]

is approximately [math](5.658487746849688216649061...)[/math],

where the whole number part is the third prime [math]5[/math], and so on.

(In short, we divide the approximate number by it's whole number part, subtract [math]1[/math],

and take the reciprocal of the result to get the next approximate number whose whole number part is the next prime!)

That the fine-structure constant is thus related to the prime numbers was also discovered (independently) by Ke Xiao

who publised his findings in a paper entitled "Dimensionless Constants and Blackbody Radiation Laws" in

The Electronic Journal of Theoretical Physics. (It can be "Googled".)

I will post Lars Blomberg's determinations of [math]\varpi(x)[/math] in my next post,

and a revised "polygonal number counting function" in the post after that.

There's more... a lot more... but that's all for now.

It's good to be back.

Don.

CraigD · August 31, 2011

It's good to be back.

It’s good to see you’re still voraciously playing with finding expressions for physical constants (if [math]\alpha[/math] actual is a constant, which isn’t certain – see its wikipedia article for more).

To date, by far the most accurate measurement of the fine-structure constant (measured at the scale of the electron mass) was made by Gerald Gabrielse and colleagues from Harvard, Cornell and RIKEN.

Measuring the "magnetic moment" of a single electron in a "quantum cyclotron" and inserting that value into state of the art QED equations, the value they determined is:

[math]\alpha^{-1}=137.035999084(51)[/math],

Point of accuract: I understand from the above linked wikipedia article that a slightly more precise recommended value has been calculated from measurements of the other physical constants defining

[math]\alpha^{-1} = \frac{2 \varepsilon_0 h c}{e^2} = 137.035999074(44)[/math]

Now, these values were determined back in 2008, and since then, no significant improvement in accuracy was ever accomplished, despite enormous improvements in both the design of the equipment and the QED equations themselves.

Thus, many scientists now suspect that further refinements in the value of the fine-structure constant may not even be possible, and that the last two values represent, for all practical purposes, the actual lower and upper bounds of the fine-structure constant as measured at the scale of the electron mass. Even Gabrielse himself believes that the above values will hold for a long, long time to come.

Do you have a source for this, Don :QuestionM

My read of "New Measurement of the Electron Magnetic Moment and the Fine Structure Constant" (2008 D. Hanneke, S. Fogwell, G. Gabrielse) finds that the authors note the new [imath]\alpha[/imath] estimate is 2.7 times more accurate than the 2006 one it supersedes, supporting new physics experiments, but that like those that came before it, it can be improved with improved instrumentation and numeric approximation methods.

Given the above facts, it now seems that in order to be correct, any mathematical expression which results in the fine-structure constant must not only match the above experimental value exactly, but must also include, within it's form, some simple way of expressing, with the same degree of absolute accuracy, those seemingly inherent lower and upper bounds.

For the reasons I gave above, I don’t think this is true. Hanneke, Fogwell, and Gabrielse’s 2008 approximation is an approximate, not an exact value. Its error bounds are also approximate, and would be different if different statistical levels of confidence were assumed, or if they had simply collected and included more data in their analysis before publishing it.

[math]\alpha^{-1}=137.035999084=(A^{-1}*\pi*e+e)*(\pi^{e}+e^{(\frac{-\pi}{2})})-\frac{1}{(6*\pi^{5}*e^{2}-2*e^{2})}[/math]

[math]\alpha^{-1}=137.035999135=(A^{-1}*\pi*e+e)*(\pi^{e}+e^{(\frac{-\pi}{2})})-\frac{1}{(6*\pi^{5}*e^{2}-2*e^{1})}[/math]

[math]\alpha^{-1}=137.035999033=(A^{-1}*\pi*e+e)*(\pi^{e}+e^{(\frac{-\pi}{2})})-\frac{1}{(\mu*e^{2}-2*e^{\frac{5}{2}})}[/math]

where [math]\mu=1836.15267247(80)[/math] is the "proton to electron mass ratio" ...

Cool and fun expressions, but don’t think you should use the “exactly equals” symbol (“=”) rather than an “approximately equals” symbol (such as [imath]\dot=[/imath]). Unless you ad-hock define A to be the reciprocal of the rest of the expression times the given rational [imath]\alpha^{-1}[/imath] value, the expressions to the rightmost equal sign in those equations give irrational numbers with no exact finite length decimal number representation. As you note, You’ve also included a physical constant, [imath]\mu[/imath]

... and [math]A=2.566543832171388844467529...[/math], is that very special "Blazys Constant"
which generates all of the prime numbers, in sequential order, by the following simple method ...

I don’t think you should claim that the integer parts of

[math]A_n = \frac1{\frac{A_{n-1}}{\lfloor A_{n-1} \rfloor}-1}[/math]

give “all of the prime numbers” for some [math]A_1[/math]. As we showed in 2008 in the thread The Holy Grail Of Mathematics, no [imath]A_1[/imath] can generate more than the first 17 primes using this formula – as you put it, it’s “just a really wierd and incredible coincidence”, a “mere ‘curio’”.

It’s fun to be rereading and kicking this stuff around again, though. :)

Don Blazys · September 4, 2011

Quoting CraigD:

It’s good to see you’re still voraciously playing with finding expressions for physical constants.

Unfortunately for me, "voracious playing" is more for younger folks, such as yourself,

who are still "full of beans" and have lots of "nervous energy" to spare. :)

Old guys like me just sit back and let the solutions come while playing with our

cute little grandchildren.

Besides, like I said before, I have very little interest in "physical constants",

and my main goal for now is to continue to investigate and develop

counting functions for polygonal numbers of order greater than 2.

That is something that is sorely needed in our mathematical literature,

and mine is the very first, and still the only one! :)

The extraordinarily accurate formula for [math]\alpha^{-1}[/math] that I have presented here came about solely as

a by-product of my research into counting functions for higher order polygonals.

(The similarity between it and the counting function on my website is quite apparent.)

...if [math]\alpha[/math] actually is a constant, which isn’t certain...

For me, the important thing is that [math]\alpha[/math] has a particular value when measured at a particular energy.

Measured at the "low energy" scale of the electron mass, it's value is [math]\alpha^{-1}=137.035999084(51)[/math].

Measured at the "high energy" scale of the W boson, its value is about [math]\alpha^{-1}=128[/math],

It's value at "zero energy" is really the biggest mystery of all, because that can't be measured,

but only deduced from theory. However, most theorists do agree that at zero energy,

the fine structure constant is somewhere around [math]\alpha^{-1}=137.03604[/math] or [math]\alpha^{-1}=137.03605[/math].

(Los Alamos National Laboratories uses the latter in all their calculations, while Wikipedia

doesn't even bother with the last two decimal places and gives [math]\alpha^{-1}=137.036[/math].)

Now, the most recently determined values of [math]\varpi(x)[/math], as determined by Lars Blomberg, are as follows:

[math]\varpi(10^{13})=6,403,626,146,905[/math]

[math]\varpi(10^{14})=64,036,270,046,655[/math]

[math]\varpi(10^{15})=640,362,727,589,917[/math]

and if we now use the above values to solve for [math]\alpha^{-1}[/math] in the counting function,

which for your convenience, can be found here: http://donblazys.com...l_numbers_3.pdf ,

we get, respectively:

[math]\alpha^{-1}=137.03603021541[/math].

[math]\alpha^{-1}=137.03604279074[/math].

[math]\alpha^{-1}=137.03604679116[/math].

Could it be that the counting function will give us that most mysterious number of all,

the fine-structure constant at zero energy, provided that we are able to determine sufficiently high

values of [math]\varpi(x)[/math]? Yes! It could be! It sure seems to be settling in that vicinity!

This is even better and more exiting than I ever thought possible, and here's the "icing on the cake"...

The value of [math]\alpha^{-1}[/math], as measured at the "high energy" scale of the W boson, is:

[math] \alpha^{-1}=128.08370052236=(A^{-1}*\pi*e+e)*(\pi^{e}+e^{(\frac{-\pi}{2})})-\frac{e^{\sqrt(137.035999084)}}{(6*\pi^{5}*e^{2}-2*e^{2})} [/math]

where [math]A=2.566543832171388844467529[/math] and [math]137.035999084[/math] is the "low energy"

value of [math]\alpha^{-1}[/math].

Quoting CraigD:

Point of accuract: I understand from the above linked wikipedia article that
a slightly more precise recommended value has been calculated from measurements of
the other physical constants defining [math] \alpha^{-1} = \frac{2 \varepsilon_0 h c}{e^2} = 137.035999074(44) [/math]

That "Codata" value is not from any particular experiment, but comes from multiple sources.

Most of its "weight" is still from the Gabrielse experiments, but it is also "tainted" by

other, much less reliable results. The last "Codata" value of [math]\alpha[/math], was off by more than

6 standard deviations, which is huge!

Quoting CraigD:

I don’t think you should claim that the integer parts of
[math] A_n = \frac1{\frac{A_{n-1}}{\lfloor A_{n-1} \rfloor}-1} [/math]

give “all of the prime numbers” for some .

Of course, I'm not claiming any such thing. At this point in time,

there is simply not enough information to know if the actual Blazys constant:

[math]2.566543832171388844467529...[/math] is involved or not.

However, just because the "Holy Grail" thread turned out to involve

an incredible approximation to the Blazys Constant

rather than the Blazys Constant itself, it doesn't mean that

we should assume anything one way or the other in this particular case,

which is completely new, and totally different.

It's a pretty "cut and dry" proposition.

Either [math]\alpha^{-1}[/math] will appear as [math]137.035999083778...[/math]

or it won't.

Don.

LBg · September 15, 2011

Yes, I am the fellow that helped Don calculate w(10^15).

I have extended the calculations in Don's paper, and made some investigations on my own.

The results indicate that the presence of the "fine structure constant" in these data is rather speculative.

It is even doubtful that the form of the equation is the best one for approximating the data.

I have attached my investigations in PDF form.

/Lars

Investigation.pdf

Turtle · September 16, 2011

Yes, I am the fellow that helped Don calculate w(10^15).

a fine 10¹⁵ kudos on you lb for for your willingness & workmanship!! :thumbs_up :xparty:

I have attached my investigations in PDF form.

/Lars

may i quote from your tables in our thread on non-figurate (non-polygonal) numbers? thnx again and welcome to the hypography menagerie. :turtle:

Rade · September 16, 2011

From this link, the fine structure constant "error bars" may not be errors at all. The so-called constant may not be a constant but have different values at different times and different places in the universe:

http://www.physorg.com/news202921592.html

Don Blazys · September 24, 2011

As we have seen, for relatively low values of [math]x[/math], the function on my website gives exellent approximations of [math]\varpi(x)[/math]

when the inverse fine-structure constant [math]\alpha^{-1}[/math] assumes the "low energy" value: [math]\alpha^{-1}=137.035999084(51)[/math].

However, in order to maintain the uncanny accuracy of this counting function for very high values of [math]x[/math], (say, above [math]x=10^{12}[/math]),

we must remember that the fine-structure constant is actually a "running constant", and that it's "zero energy" value is [math]\alpha^{-1}\approx 137.03605(5)[/math],

which means that it can be viewed as a function whose range is [math]\alpha^{-1}\approx137.036[/math] to [math]\alpha^{-1}\approx137.0361[/math].

("Google searching" the number [math]137.03604(11)[/math] shows that this is by far the most widely used and

commonly accepted value of the fine structure constant. Incredibly, the range required by my counting function

falls exactly within that range, which is [math]\alpha^{-1}\approx137.03593[/math] to [math]\alpha^{-1}\approx137.03615[/math].)

The following polygonal number counting function is basically a simplified version of the one on my website:

[math]\varpi(x)\approx\left(\left(\sqrt{\left(\left(1-\frac{1}{\left(\alpha*\pi*e+e\right)}\right)*x\right)}-\frac{1}{4}\right)^{2}-\frac{1}{16}\right)*\left(1-\frac{\alpha}{\left(6*\pi^{5}-2*e\right)}\right)[/math]

where

[math]\alpha=\left(\left(A^{-1}*\pi*e+e\right)*\left(\pi^{e}+e^{\left(\frac{-\pi}{2}\right)}\right)-\frac{\left(\left(\pi^{e}+4+\frac{1}{16}\right)*\left(\ln\left(x\right)\right)^{-1}-\left(\sqrt{2}-1\right)\right)}{\left(6*\pi^{5}*e^{2}-2*e^{2}\right)}\right)^{-1}[/math]

and

[math]A=2.566543832171388844467529...[/math]

As you can see, the only real difference is that [math]\alpha[/math] is now being applied as the "running constant" that it actually is!

The results are stunning, and I will post them when I have more time.

Don.

Don Blazys · September 25, 2011

[math]x[/math]_______________________[math]\varpi(x)[/math]_________________[math]B(x_{F\alpha})[/math]__________Difference

10_______________________3______________________5___________________2

100______________________57_____________________60__________________3

1,000____________________622____________________628_________________6

10,000___________________6,357__________________6,364________________7

100,000__________________63,889_________________63,910_______________21

1,000,000________________639,946________________639,963______________17

10,000,000_______________6,402,325______________6,402,362_____________37

100,000,000______________64,032,121_____________64,032,273____________152

1,000,000,000____________640,349,979____________640,350,090____________111

10,000,000,000___________6,403,587,409__________6,403,587,408__________-1

100,000,000,000__________64,036,148,166_________64,036,147,619_________-547

1,000,000,000,000________640,362,343,980________640,362,340,964________-3016

10,000,000,000,000_______6,403,626,146,905______6,403,626,142,294_______-4611

100,000,000,000,000______64,036,270,046,655_____64,036,270,047,024_______369

200,000,000,000,000______128,072,542,422,652____128,072,542,422,825______173

300,000,000,000,000______192,108,815,175,881____192,108,815,179,004______3123

400,000,000,000,000______256,145,088,132,145____256,145,088,131,479_____-668

500,000,000,000,000______320,181,361,209,667____320,181,361,209,090_____-577

600,000,000,000,000______384,217,634,373,721____384,217,634,375,409______1688

700,000,000,000,000______448,253,907,613,837____448,253,907,608,821_____-5016

800,000,000,000,000______512,290,180,895,369____512,290,180,895,262_____-107

900,000,000,000,000______576,326,454,221,727____576,326,454,224,974______3247

1,000,000,000,000,000____640,362,727,589,917____640,362,727,590,859______942

LBg · September 25, 2011

As we have seen, for relatively low values of [math]x[/math], the function on my website gives exellent approximations of [math]\varpi(x)[/math]
when the inverse fine-structure constant [math]\alpha^{-1}[/math] assumes the "low energy" value: [math]\alpha^{-1}=137.035999084(51)[/math].

However, in order to maintain the uncanny accuracy of this counting function for very high values of [math]x[/math], (say, above [math]x=10^{12}[/math]),
we must remember that the fine-structure constant is actually a "running constant", and that it's "zero energy" value is [math]\alpha^{-1}\approx 137.03605(5)[/math],
which means that it can be viewed as a function whose range is [math]\alpha^{-1}\approx137.036[/math] to [math]\alpha^{-1}\approx137.0361[/math].

The initial hypothesis was that the counting function could be used to give a value of [math]\alpha^{-1}=137.035999084(51)[/math] which would become accurate as more data from the counting was obtained.

When it turns out that more counting data does not support this hypothesis it is abandoned and another one is constructed, using the concept of a "running constant". I don't see why [math]\alpha^{-1}=137.035999084(51)[/math] which is a number with a stated uncertainty should be regarded as a true constant whereas [math]\alpha^{-1}\approx 137.03605(5)[/math] which is another number with a stated uncertainty should be called "running". In any case, a given formula connecting the counting data with [math]\alpha^{-1}[/math] can only give a single value for [math]\alpha^{-1}[/math], not a range of values.

The following polygonal number counting function is basically a simplified version of the one on my website:

It is not a simplified version, it is more complicated in that [math]x[/math] is now included in the expression for [math]\alpha[/math].

The results are stunning, and I will post them when I have more time.

Yes, you are right, the new function matches the current counting data better.

/LBg

Don Blazys · September 25, 2011

Here:

http://www.integralworld.net/piacenza4.html

is a wonderful article about the fine structure constant that mentions the outstanding work of Professor Jose Alvarez Lopez.

His theory that all non-dimentional numbers can be determined by the number [math]137.03604[/math] is truly profound,

but what I found really interesting was this:

Quoting the linked article:

Professor Alvarez Lopez liked to point out that the existence of powers of 2 are common in atomic physics,
but told me that finding powers of 10, in what seem to be fundamental numbers for this discipline,
is "outstanding, previously unknown in nature and quite mysterious for orthodoxy".

Well, the table in my previous post shows that roughly the first half of the digits in every consecutive power of 10 are repeated and therefore

easily predictable. That's comparable to the prime counting function [math]Li(x)[/math], which predicts [math]\pi(x)[/math] to about the same degree of accuracy.

Quoting LBg:

The initial hypothesis was that the counting function could be used to give a value of [math]137.035999084(51)[/math]
which would become accurate as more data from the counting was obtained.

That hypothesis was made when the known values of [math]\varpi(x)[/math] were much, much, much less than they are now.

However, mathematics is full of surprises, and in the process of trying to verify that hypothesis, it was discovered that the expression:

[math](A^{-1}*\pi*e+e)*(\pi^{e}+e^{(\frac{-\pi}{2})})-\frac{q}{(6*\pi^{5}*e^{2}-2*e^{2})} [/math]

gives us all the significant values of the "fine-structure constant", provided that we choose proper values of [math]q[/math].

For instance, if we choose [math]q=1[/math], then the result is indeed [math]\alpha^{-1}=137.035999084[/math],

which matches Professor Gabrielse's results with uncanny precision! Then again...

Choosing [math]q=(\sqrt{2}+1)^{-1}[/math] results in [math]\alpha^{-1}=137.03604230759[/math], which is also significant in that it works very, very well

in my polygonal number counting function and falls right in the range of the best theoretical "zero energy" values of [math]\alpha[/math]. Then, yet again...

choosing [math]q=e^{\sqrt{137.03604...}}[/math] results in [math]\alpha^{-1}=128.08[/math] which is also very significant in that it is

exactly the value of the fine structure constant measured at the high energy value of the W boson!

Thus, what we found is a true "fine structure formula", that is also absolutely necessary if we are to

"make the math work" and closely approximate the purely mathematical function [math]\varpi(x)[/math].

Moreover, the counting function is now so good that it shows that the "random fluctuations" in [math]\varpi(x)[/math]

are either increasing very, very slowly, or not at all.

Best of all, the "fine structure formula" seems to involve the prime numbers via the "Blazys constant",

which may (or may not) lead to some construct that generates them.

Like I said, math is full of surprises, and you just never know...

Regardless of whether or not you are satisfied with the progress that has been made,

my sincere thanks goes out to You, Turtle, Phillip and everyone else who joined in and helped.

In my most humble opinion, the work that we all put in was well worth the effort.

Don.

Don Blazys · October 30, 2011

Quoting Rade:

From this link, the fine structure constant "error bars"
may not be errors at all. The so-called constant may not be
a constant but have different values at different times and
different places in the universe:

http://www.physorg.c...s202921592.html

I agree... and isn't it strange how my truly accurate counting function for

polygonal numbers of order greater than 2 actually requires some

function with a range of about [math]\alpha^{-1}=137.03604(11)[/math]?

Think about it. A true "theory of everything" would have to include values of

the fine structure constant measured at all energy levels and at all times

and places in the universe.

Thus, the purely mathematical model of the fine structure constant

that is slowly emerging as the result of our ongoing efforts to develop

the most accurate counting function possible for polygonal numbers

of order greater than 2 is interesting indeed, because it encompasses

all possible values of the fine structure constant and not just those

measured at or around the low energy value of the electron mass.

Don

Don Blazys · November 3, 2011

Quoting LBg:

It is even doubtful that the form of the equation
is the best one for approximating the data.

The form of the equation that I developed is correct.

However, the existing determinations of [math]\varpi(x)[/math] are still insufficient

for us to determine which constants are actually involved.

To illustrate, here is a counting function of the exact same form

as that in post #7, but which contains a few different constants...

[math] \varpi(x)\approx\left(\left(\sqrt{\left(\left(1-\frac{1}{\left(\alpha*\pi*e+e\right)}\right)*x\right)}-\frac{1}{4}\right)^{2}-\frac{1}{16}\right)*\left(1-\frac{\alpha}{\left(6*\pi^{5}-2*e\right)}\right) [/math]

where:

[math] \alpha=\left(\left(A^{-1}*\pi*e+e\right)*\left(\pi^{e}+e^{\left(\frac{-\pi}{2}\right)}\right)-\frac{\left(\left(\pi^e+\frac{\pi^{2}}{6}+\sqrt{2}+1\right)*\left(\ln\left(x\right)\right)^{-1}-\left(\sqrt{2}-1\right)\right)}{\left(6*\pi^{5}*e^{2}-2*e^{2}\right)}\right)^{-1} [/math]

which results in the following table:

[math]x[/math]_______________________[math]\varpi(x)[/math]_________________ [math] B(x_{F\alpha}) [/math]_________Difference