toucana Posted June 30 Posted June 30 A novel method of calculating the value of Pi has been found by an Indian physicist Aninda Sinha, Professor at the Centre for High Energy Physics (CHEP) - Indian Institute of Science, and co-author Arnab Saha, a post-doc assistant - published in Physical Review Letters https://www.iisc.ac.in/events/iisc-physicists-find-a-new-way-to-represent-pi/ The new method was stumbled upon by chance while the researchers were investigating how string theory can be used to explain certain physical phenomena: Quote “Our efforts, initially, were never to find a way to look at pi. All we were doing was studying high-energy physics in quantum theory and trying to develop a model with fewer and more accurate parameters to understand how particles interact. We were excited when we got a new way to look at pi,” Sinha says There are several well known methods of calculating Pi by using infinite series, notably the Wallis product formula first published by John Wallis in 1656, and the Gregory series discovered by Scottish mathematician James Gregory (1638-75). They can however be cumbersome to use. In the Gregory series you need to use 10,000 terms of the series to get 4 decimal places of π correct. The authors refer to a similar formula known in India as the Madhava series in recognition of a 14th-century Indian mathematician and astronomer Madhava of Sangamagrama who first discovered it - but while the Madhava series takes 5 billion terms to converge to ten decimal places, the new representation with λ between 10 and 100 takes 30 terms according to an appendix in the paper. https://www.iflscience.com/physicists-accidentally-discover-a-whole-new-way-to-write-pi-74768
exchemist Posted July 1 Posted July 1 10 hours ago, toucana said: A novel method of calculating the value of Pi has been found by an Indian physicist Aninda Sinha, Professor at the Centre for High Energy Physics (CHEP) - Indian Institute of Science, and co-author Arnab Saha, a post-doc assistant - published in Physical Review Letters https://www.iisc.ac.in/events/iisc-physicists-find-a-new-way-to-represent-pi/ The new method was stumbled upon by chance while the researchers were investigating how string theory can be used to explain certain physical phenomena: There are several well known methods of calculating Pi by using infinite series, notably the Wallis product formula first published by John Wallis in 1656, and the Gregory series discovered by Scottish mathematician James Gregory (1638-75). They can however be cumbersome to use. In the Gregory series you need to use 10,000 terms of the series to get 4 decimal places of π correct. The authors refer to a similar formula known in India as the Madhava series in recognition of a 14th-century Indian mathematician and astronomer Madhava of Sangamagrama who first discovered it - but while the Madhava series takes 5 billion terms to converge to ten decimal places, the new representation with λ between 10 and 100 takes 30 terms according to an appendix in the paper. https://www.iflscience.com/physicists-accidentally-discover-a-whole-new-way-to-write-pi-74768 What is λ?
toucana Posted July 1 Author Posted July 1 2 hours ago, exchemist said: What is λ? From the expanded text of the main article (§ QFT expectations) --> https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.132.221601 Quote The scattering amplitude computed now will depend on the field redefinition parameter 𝜆; in particular, the contact terms would depend on 𝜆. Similar arguments can be made with higher-spin massive exchanges.
exchemist Posted July 1 Posted July 1 12 minutes ago, toucana said: From the expanded text of the main article (§ QFT expectations) --> https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.132.221601 Hmm, but doesn't that make it a physical variable? Surely it has to have a particular exact value for the sum to yield a correct value for π?
sethoflagos Posted July 1 Posted July 1 51 minutes ago, exchemist said: Hmm, but doesn't that make it a physical variable? Surely it has to have a particular exact value for the sum to yield a correct value for π? I understand it correlates with the Riemann zeta function of spin value (?). Some detail at https://mathoverflow.net/questions/414594/scattering-amplitudes-and-the-riemann-zeta-function
Eise Posted July 1 Posted July 1 Sabine Hossenfelder has a video about it: I liked this comment of a user there: Quote Congratulations everone! That's the most practical application of the String Theory so far.
joigus Posted July 1 Posted July 1 (edited) 24 minutes ago, Eise said: Theory It probably has to do with a forward-scattering amplitude. Thereby the pi. Edited July 1 by joigus minor correction
exchemist Posted July 1 Posted July 1 3 hours ago, sethoflagos said: I understand it correlates with the Riemann zeta function of spin value (?). Some detail at https://mathoverflow.net/questions/414594/scattering-amplitudes-and-the-riemann-zeta-function OK, but looking at the equation in the OP, surely the value of the sum will depend on the exact value assigned to λ? Is λ then an exact number, i.e. a mathematical object with an exact definition, rather than a quantity derived from physical measurement?
studiot Posted July 1 Posted July 1 I really don't see the issue here. Many physical phenomena (perhaps most) have a spatial distribution so it is not suprising that something like scattering displays this characteristic, due to its statistical nature. And Pi is linked to spatial distributions both through the error function and the perfectly symmetrical ball in n dimensions. So any measurement of distribution will also include a measurement of Pi. But not this is only a measurement, not a mathematical derivation as in the Euler identity Since a perfectly spherically symmetrical distribution involves the volume, measuring the volume of an inflatable sphere is probably a much simpler way of achieving this end.
exchemist Posted July 1 Posted July 1 (edited) 11 minutes ago, studiot said: I really don't see the issue here. Many physical phenomena (perhaps most) have a spatial distribution so it is not suprising that something like scattering displays this characteristic, due to its statistical nature. And Pi is linked to spatial distributions both through the error function and the perfectly symmetrical ball in n dimensions. So any measurement of distribution will also include a measurement of Pi. But not this is only a measurement, not a mathematical derivation as in the Euler identity Since a perfectly spherically symmetrical distribution involves the volume, measuring the volume of an inflatable sphere is probably a much simpler way of achieving this end. So do you mean, in effect, that it all hinges on how accurate a determination of λ can be made? In which case it is not really a method of calculating π, so much as a way of estimating it by physical measurements. Edited July 1 by exchemist
John Cuthber Posted July 2 Posted July 2 From https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.132.221601 (with my emphasis) "The distinct plateau indicates ∂𝜆𝑀T≈0, as we would expect since the full amplitude is independent of 𝜆." On 7/1/2024 at 12:15 AM, toucana said: ... the new representation with λ between 10 and 100 takes 30 terms...
studiot Posted July 2 Posted July 2 1 hour ago, John Cuthber said: From https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.132.221601 (with my emphasis) "The distinct plateau indicates ∂𝜆𝑀T≈0, as we would expect since the full amplitude is independent of 𝜆." Thank you John If the series sum is truly independent of lambda , I thought I put in a value and have a go at summing it. But I hit a stumbling block. What does the subscript (n-1) mean at the end ?
toucana Posted July 2 Author Posted July 2 1 hour ago, studiot said: Thank you John If the series sum is truly independent of lambda , I thought I put in a value and have a go at summing it. But I hit a stumbling block. What does the subscript (n-1) mean at the end ? There is a reference to this subscript in the YT video by Sabine Hossenfelder which Eise linked to earlier (c.3:15 elapsed) It's apparently a Pochhammer Symbol - which she says is another type of Gamma Function.
studiot Posted July 2 Posted July 2 So you are saying that this new calculation is simply an application of this? from this paper https://www.google.co.uk/url?sa=t&source=web&rct=j&opi=89978449&url=https://www.rgnpublications.com/journals/index.php/jims/article/viewFile/252/229&ved=2ahUKEwjQ2fm934iHAxXgZEEAHdMYBrc4ChAWegQICBAB&usg=AOvVaw2wk5RR2G_aIiTmqkQoxzvZ In which case can we see the arithmetical / algebraic working (with proper definition of all parts) ?
toucana Posted July 2 Author Posted July 2 5 hours ago, studiot said: So you are saying that this new calculation is simply an application of this? from this paper https://www.google.co.uk/url?sa=t&source=web&rct=j&opi=89978449&url=https://www.rgnpublications.com/journals/index.php/jims/article/viewFile/252/229&ved=2ahUKEwjQ2fm934iHAxXgZEEAHdMYBrc4ChAWegQICBAB&usg=AOvVaw2wk5RR2G_aIiTmqkQoxzvZ In which case can we see the arithmetical / algebraic working (with proper definition of all parts) ? There is a worked example in the video previously cited at around 4.39 elapsed - (the video is only 7.09 long)- showing the convergence for λ = 20.
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