New derivation of pi links quantum physics and pure math | Hacker News Reader

New derivation of pi links quantum physics and pure math

aip.org

113 pointsbpolania20 comments

Many series that calculate pi are very simple, as a result, it can randomly appear in physical problems. See for instance http://ics.org.ru/doc?pdf=440&dir=e

That’s a nice definition of mathematical constants: Numbers whose calculation specifications have a Kolmogorov complexity that is low enough so that they reappear in different contexts with high probability.

More precisely it would be the algorithmic probability. Not only are there short programs that compute mathematical constants, but there are many such programs, so the total mass is even higher.

I'm not sure what you mean. Do you mean that a mathematical constant has multiple different shortest programs?

Not necessarily shortest, but it has many short programs.

The algorithmic probability of a sequence x is the sum over the set of all prefix-free programs u that calculate x of 2^-len(u). Think of the shortest description length as a MAP, while the algorithmic probability integrates over the full prior.

A constant which has many short programs can thus have a greater algorithmic probability than another constant with a slightly lower Kolmogorov complexity. To put it back in context, it's possible that a very short program computes some constant, but it's unlikely to be an important mathematical constants. What's particular about mathematical constants is that they keep appearing in many different situations.

A really interesting read, thank you!

Extended quote from original article...

"When Hagen started solving the problem himself, he immediately noticed a trend. The error of the variational approach was about 15 percent for the ground state of hydrogen, 10 percent for the first excited state, and kept getting smaller as the excited states grew larger. This was unusual, since the variational approach normally only gives good approximations for the lowest energy levels."

Rings bells from celestial mechanics. You can end up with things like a series solution converging on an elliptical or circular orbit when you apply series methods to simple (or 'simple' in this case - it is quantum mechanics!) systems.

I'll see if I can hack my way though their paper (linked to OA, downloadable in full text).

Pi appears in many many many quantum calculations, so the title of the press article is very bad.

The title of the research article is much better "Quantum mechanical derivation of the Wallis formula for pi" because as far as I know the Wallis formula has never appear before in a QM calculation.

The game of mafia (aka Werewolf) is linked to PI (my paper: http://arxiv.org/abs/1009.1031, for a summary: http://boardgamegeek.com/thread/1157558/polish-paper-werewol...), also via the Wallis formula. But so what?

Pi appears in physics roughly as often as 2 in finance. Even to describe the simples quantum states (ground states of the harmonic potential) you use Pi (as a normalization factor).

Of course there is nothing special about this result. The hyped "science" articles describing this are longer than the paper!

Lubos Motl wrote about it here: http://motls.blogspot.co.at/2015/11/pi-found-in-hydrogen-ato...

What's so special about this ? I was always under the impression that quantum physics is allready 99% pure math ;)

Esp. when they appear to be excited to see a physical phenomenon derived from the length of circle. I mean, really?

I felt the same. For example, gravity/electromagnetism force depends on distance, hence naturally lead to circular/spherical contours, which would involve Pi.

I don't know enough about quantum physics, but from what I understand, this pi seems to have arose as an artifact of the variational method used, rather than quantum mechanics itself.

be floored ^1.

https://en.wikipedia.org/wiki/Buffon%27s_needle

^1 and don't fall for phy PR

Dear Downvoter: can you tell when The Cat makes it through the crack?

It's quantum mechanics that enable the "variational method used" to give this result -- it wont appear in all calculations done with the same method in other domains.

Somebody should tell these guys about hbar.

Just kidding, it is interesting that this series showed up in QM. Terrible article title though...

Maybe because, not being a mathematician, and certainly not a quantum mechanic, I understand pi and circles as being lock-step mutually defining; from what I've read, this derivation of pi doesn't seem to be described as anything more than an interesting coincidence, and to call it a "link" is reaching.

Url changed from http://phys.org/news/2015-11-derivation-pi-links-quantum-phy..., which points to this.

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