Wallis' Formula and Quantum Mechanics

[stevendaryl]

Does anybody know what the connection is between Wallis' formula for $\pi$ and quantum mechanics? There was an article about it:
https://www.eurekalert.org/pub_releases/2015-11/aiop-ndo110915.php
but like all articles for the lay public, all the details were left out.

Wallis' formula is:

$\frac{\pi}{2} = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} ...$

I've always thought there should be a geometric proof of this, say approximating a semicircle by rectangles of some sort, but the proofs I've seen are analytic.

 

[DrClaude]

The article can be found here: https://arxiv.org/abs/1510.07813

A geometric proof of Wallis' formula is given in http://liu.diva-portal.org/smash/get/diva2:375184/FULLTEXT01.pdf

 

[kith]

3Blue1Brown has a video about a geometric proof of the Wallis product:

 

[Matt Benesi]

3blue1brown rocks. Not sure about dropping the 0 though. Didn't get all the way through.

A while back (didn't finish it yet), I started working on a zeta/log chart, with a few switching formulas between them (and I was thinking about using it as a quantization structure for some art). The Wallis Product pops up (a few thingies down):

Since, $log(\frac{x}{y}) =\sum\limits_{n=1}^{\infty} \, (\frac{x-y}{x})^n \cdot \frac{1}{n}$, if you set y=x-1...

$$\begin{array}{|c|c|c|c|}<br /> \hline & \frac{\zeta(1)}{1} & \frac{\zeta(2)}{2}& \frac{\zeta(3)}{3}& \frac{\zeta(4)}{4} & \frac{\zeta(5)}{5}&\dots\\<br /> \hline \lim\limits_{x\to 1^+} log(\frac{x}{x-1}) & \frac{1}{1 \cdot 1^1} & \frac{1}{2 \cdot 1^2} & \frac{1}{3 \cdot 1^3}& \frac{1}{4 \cdot 1^4}& \frac{1}{5 \cdot 1^5}&\dots\\<br /> \hline log (\frac{2}{1}) & \frac{1}{1 \cdot 2^1} & \frac{1}{2 \cdot 2^2} & \frac{1}{3 \cdot 2^3}& \frac{1}{4 \cdot 2^4}& \frac{1}{5 \cdot 2^5}&\dots\\<br /> \hline log (\frac{3}{2}) & \frac{1}{1 \cdot 3^1} & \frac{1}{2 \cdot 3 ^2} & \frac{1}{3 \cdot 3^3}& \frac{1}{4 \cdot 3^4}& \frac{1}{5 \cdot 3^5} &\dots\\<br /> \hline log (\frac{4}{3}) & \frac{1}{1 \cdot 4^1} & \frac{1}{2 \cdot 4^2} & \frac{1}{3 \cdot 4^3}& \frac{1}{4 \cdot 4^4}& \frac{1}{5 \cdot 4^5} &\dots\\<br /> \hline log (\frac{5}{4}) & \frac{1}{1 \cdot 5^1} & \frac{1}{2 \cdot 5^2} & \frac{1}{3 \cdot 5^3}& \frac{1}{4 \cdot 5^4}& \frac{1}{5 \cdot 5^5} &\dots\\<br /> \hline \dots &\dots & \dots&\dots &\dots&\dots&\dots\\<br /> \hline<br /> \end{array}$$

A couple of functions for smooth movement between quantized log and zetas:

Spoiler

Log shift function, with x ? N a ? R : [0,1].. note that it's 2a/ 2n, so I drop the 2:

$log (\frac{x}{x-1}) \to log(\frac{x+1}{x}) \,= \, f(x,a) =\, \sum\limits_{n=1}^\infty \frac{1}{n\cdot x^n} - \frac{a}{n \cdot x^{2n}}$

Shift logs by a=1:

$$\begin{array}{|c|c|c|c|}<br /> \hline & \frac{\zeta(1)}{1} & -\frac{\zeta(2)}{2}& \frac{\zeta(3)}{3}& -\frac{\zeta(4)}{4} & \frac{\zeta(5)}{5}&\dots\\<br /> \hline log(\frac{2}{1}) & \frac{1}{1 \cdot 1^1} & -\frac{1}{2 \cdot 1^2} & \frac{1}{3 \cdot 1^3}& -\frac{1}{4 \cdot 1^4}& \frac{1}{5 \cdot 1^5}&\dots\\<br /> \hline log (\frac{3}{2}) & \frac{1}{1 \cdot 2^1} & -\frac{1}{2 \cdot 2^2} & \frac{1}{3 \cdot 2^3}& -\frac{1}{4 \cdot 2^4}& \frac{1}{5 \cdot 2^5}&\dots\\<br /> \hline log (\frac{4}{3}) & \frac{1}{1 \cdot 3^1} &- \frac{1}{2 \cdot 3 ^2} & \frac{1}{3 \cdot 3^3}& -\frac{1}{4 \cdot 3^4}& \frac{1}{5 \cdot 3^5} &\dots\\<br /> \hline log (\frac{5}{4}) & \frac{1}{1 \cdot 4^1} & -\frac{1}{2 \cdot 4^2} & \frac{1}{3 \cdot 4^3}& -\frac{1}{4 \cdot 4^4}& \frac{1}{5 \cdot 4^5} &\dots\\<br /> \hline log (\frac{6}{5}) & \frac{1}{1 \cdot 5^1} & -\frac{1}{2 \cdot 5^2} & \frac{1}{3 \cdot 5^3}& -\frac{1}{4 \cdot 5^4}& \frac{1}{5 \cdot 5^5} &\dots\\<br /> \hline \dots &\dots & \dots&\dots &\dots&\dots&\dots\\<br /> \hline<br /> \end{array}$$

zeta to eta shift function, s ?N, a ? R: [0,1], dropped the 2 from 2a/2n again!:

$\frac{\zeta(s)}{s} \to \frac{\eta(s)}{s} \, = g(s,a) = \, \sum\limits_{n=1}^\infty \frac{1}{s \cdot n ^ s} - \frac{a}{s \cdot (2n)^ s}$

Gets you here:
$$\begin{array}{|c|c|c|c|}<br /> \hline & \frac{\eta(1)}{1} & -\frac{\eta(2)}{2}& \frac{\eta(3)}{3}& -\frac{\eta(4)}{4} & \frac{\eta(5)}{5}&\dots\\<br /> \hline log(\frac{2}{1}) & \frac{1}{1 \cdot 1^1} & -\frac{1}{2 \cdot 1^2} & \frac{1}{3 \cdot 1^3}& -\frac{1}{4 \cdot 1^4}& \frac{1}{5 \cdot 1^5}&\dots\\<br /> \hline - log (\frac{3}{2}) & -\frac{1}{1 \cdot 2^1} & \frac{1}{2 \cdot 2^2} &- \frac{1}{3 \cdot 2^3}& \frac{1}{4 \cdot 2^4}& -\frac{1}{5 \cdot 2^5}&\dots\\<br /> \hline log (\frac{4}{3}) & \frac{1}{1 \cdot 3^1} &- \frac{1}{2 \cdot 3 ^2} & \frac{1}{3 \cdot 3^3}& -\frac{1}{4 \cdot 3^4}& \frac{1}{5 \cdot 3^5} &\dots\\<br /> \hline -log (\frac{5}{4}) &- \frac{1}{1 \cdot 4^1} & \frac{1}{2 \cdot 4^2} &- \frac{1}{3 \cdot 4^3}& \frac{1}{4 \cdot 4^4}& -\frac{1}{5 \cdot 4^5} &\dots\\<br /> \hline log (\frac{6}{5}) & \frac{1}{1 \cdot 5^1} & -\frac{1}{2 \cdot 5^2} & \frac{1}{3 \cdot 5^3}& -\frac{1}{4 \cdot 5^4}& \frac{1}{5 \cdot 5^5} &\dots\\<br /> \hline \dots &\dots & \dots&\dots &\dots&\dots&\dots\\<br /> \hline<br /> \end{array}$$

$$log(\frac{2}{1}) -log (\frac{3}{2}) + log (\frac{4}{3}) -log (\frac{5}{4}) +... \, = \,log(\frac{2}{1}) + log (\frac{2}{3}) + log (\frac{4}{3}) +log (\frac{4}{5}) +log (\frac{6}{5}) +...$$

From the above, from the Wallis product for pi:
$$\sum\limits_{n=1}^\infty \,-1^{n+1} \, \cdot \frac{\eta{(n)}}{n} \, = \, log(\frac{\pi}{2})$$ I kept on going, because I thought that perhaps I could make a quantized framework for some art- connections between adjacent points in space would be mapped x to zeta, y to log, z to zeta/eta depth (determined by $1-2^{1-s}$)... not finished, because I have to do real work too.

Spoiler

When we do a zeta to eta conversion, we lose log(2), and are adding different logs together to get log (pi/4):

$$\begin{array}{|c|c|c|c|}<br /> \hline & 0\cdot \frac{ \zeta(1)}{1} & - \frac{1}{2} \cdot \frac{\zeta(2)}{2} & \frac{3}{4} \cdot \frac{\zeta(3)}{3} & -\frac{7}{8} \cdot \frac {\zeta(4)}{4} & \frac{15}{16} \cdot \frac{\zeta(5)}{5} & \dots\\<br /> \hline -log(\frac{9}{8}) & 0\cdot\frac{1}{1 \cdot 1^1} & - \frac{1}{2} \cdot\frac{1}{2 \cdot 1^2} &\frac{3}{4} \cdot \frac{1}{3 \cdot 1^3}& -\frac{7}{8} \cdot \frac{1}{4 \cdot 1^4}&\frac{15}{16} \cdot \frac{1}{5 \cdot 1^5}&\dots\\<br /> \hline -log (\frac{5^2}{24}) & 0\cdot\frac{1}{1 \cdot 2^1} & - \frac{1}{2} \cdot\frac{1}{2 \cdot 2^2} &\frac{3}{4} \cdot \frac{1}{3 \cdot 2^3}&-\frac{7}{8} \cdot \frac{1}{4 \cdot 2^4}& \frac{15}{16} \cdot \frac{1}{5 \cdot 2^5}&\dots\\<br /> \hline -log (\frac{7^2}{48}) &0\cdot \frac{1}{1 \cdot 3^1} &- \frac{1}{2} \cdot\frac{1}{2 \cdot 3 ^2} &\frac{3}{4} \cdot \frac{1}{3 \cdot 3^3}& -\frac{7}{8} \cdot \frac{1}{4 \cdot 3^4}& \frac{15}{16} \cdot \frac{1}{5 \cdot 3^5} &\dots\\<br /> \hline -log (\frac{9^2}{80}) &0\cdot \frac{1}{1 \cdot 4^1} &- \frac{1}{2} \cdot \frac{1}{2 \cdot 4^2} &\frac{3}{4} \cdot \frac{1}{3 \cdot 4^3}&-\frac{7}{8} \cdot \frac{1}{4 \cdot 4^4}& \frac{15}{16} \cdot \frac{1}{5 \cdot 4^5} &\dots\\<br /> \hline- log (\frac{11^2}{120}) &0\cdot \frac{1}{1 \cdot 5^1} & - \frac{1}{2} \cdot\frac{1}{2 \cdot 5^2} &\frac{3}{4} \cdot \frac{1}{3 \cdot 5^3}& -\frac{7}{8} \cdot \frac{1}{4 \cdot 5^4}& \frac{15}{16} \cdot \frac{1}{5 \cdot 5^5} &\dots\\<br /> \hline \dots &\dots & \dots&\dots &\dots&\dots&\dots\\<br /> \hline<br /> \end{array}$$

I got to the point where subsequent zeta/eta conversions add depth, have log charts, etc. so they can be stacked. It's written on paper though... and needs to be completed. But I ended up working on something else because this computer was a gift from Keith Anderson in the fractal entertainment industry (Fractaled Visions), so I feel like I should work on fractals. A bit.