Scientists at the Indian Institute of Science (IISc) have stumbled upon on a new series representation for the irrational number pi. It provides an easier way extract pi from calculations involved in deciphering processes like the quantum scattering of high-energy particles.
The new formula under a certain limit closely reaches the representation of pi suggested by Indian mathematician Sangamagrama Madhava in the 15th century, which was the first ever series for pi recorded in history. The study was carried out by Arnab Saha, a post-doc and Aninda Sinha, Professor at Centre for High Energy Physics (CHEP), and published in Physical Review Letters.
“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.
Sinha’s group is interested in string theory – the theoretical framework that presumes that all quantum processes in nature simply use different modes of vibrations plucked on a string. Their work focuses on how high energy particles interact with each other – such as protons smashing together in the Large Hadron Collider – and in what ways we can look at them using as few and as simple factors as possible. This way of representing complex interactions belongs to the category of “optimization problems.” Modeling such processes is not easy because there are several parameters that need to be taken into account for each moving particle – its mass, its vibrations, the degrees of freedom available for its movement, and so on.
Saha, who has been working on the optimization problem, was looking for ways to efficiently represent these particle interactions. To develop an efficient model, he and Sinha decided to club two mathematical tools: the Euler-Beta Function and the Feynman Diagram. Euler-Beta functions are mathematical functions used to solve problems in diverse areas of physics and engineering, including machine learning. The Feynman Diagram is a mathematical representation that explains the energy exchange that happens while two particles interact and scatter.
What the team found was not only an efficient model that could explain particle interaction, but also a series representation of pi.
In mathematics, a series is used to represent a parameter such as pi in its component form. If pi is the “dish” then the series is the “recipe”. Pi can be represented as a combination of many numbers of parameters (or ingredients). Finding the correct number and combination of these parameters to reach close to the exact value of pi rapidly has been a challenge. The series that Sinha and Saha have stumbled upon combines specific parameters in such a way that scientists can rapidly arrive at the value of pi, which can then be incorporated in calculations, like those involved in deciphering scattering of high-energy particles.
“Physicists (and mathematicians) have missed this so far since they did not have the right tools, which were only found through work we have been doing with collaborators over the last three years or so,” Sinha explains. “In the early 1970s, scientists briefly examined this line of research but quickly abandoned it since it was too complicated.”
Although the findings are theoretical at this stage, it is not impossible that they may lead to practical applications in the future. Sinha points to how Paul Dirac worked on the mathematics of the motion and existence of electrons in 1928, but never thought that his findings would later provide clues to the discovery of the positron, and then to the design of Positron Emission Tomography (PET) used to scan the body for diseases and abnormalities. “Doing this kind of work, although it may not see an immediate application in daily life, gives the pure pleasure of doing theory for the sake of doing it,” Sinha adds.
–Rashmi Kumari
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Here is the summary of the research paper published in Physical Review Letters:
New Representations of String Theory Amplitudes:
– Motivated by quantum field theory (QFT) considerations, new representations of the Euler-Beta function and tree-level string theory amplitudes have been presented.
– These new representations employ a two-channel, local, crossing symmetric dispersion relation, enabling analyticity everywhere except at the poles.
Truncation and Retention of String Amplitude Features:
– The ability to consider mass-level truncation while preserving all original amplitude features has been demonstrated.
– This allows for the retention of essential properties like exponentially soft high energy behavior and Regge behavior up to a specified energy scale.
Search for Field Theory Representation:
– Identification of a field theory representation for 2-2 string amplitudes has significant implications for truncation in mass levels while maintaining string amplitude properties.
– This representation aids in capturing all key stringy features up to a defined energy scale.
Link with S-Matrix Bootstrap and Crossing Symmetry:
– The modern 𝑆-matrix bootstrap technique utilizes a crossing symmetric basis for investigations truncating in spin and energy.
– Findings suggest the existence of a manifestly crossing symmetric representation of tree-level string amplitudes allowing for truncation in spin and mass levels.
Connection with Beta Function and String Amplitudes:
– The simplest string amplitude is associated with the beta function introduced by Euler, connecting to a string world-sheet picture.
– Expansion of the beta function in terms of poles or mass levels is a key aspect in understanding the birth of string theory.
QFT Expectations and Field Redefinition Parameters:
– Criteria for a representation of tree-level string amplitudes akin to QFT include explicit poles in all channels and capture of essential features within a specified range.
– The truncated representation may exhibit dependence on field redefinition parameters.
Deformations and Truncation Challenges:
– Demonstration of the difficulty in deforming away from the string amplitude while highlighting potentially interesting deformations with level truncation.
– Exploration of QFT-inspired parametric representations of the Zeta function showing fast convergence.
Background and String Amplitude Expansion:
– The beta function integral representation is foundational for understanding string amplitudes and their relation to pole expansions.
– Core concepts like the Pochhammer symbol and dual resonance models accentuate the distinction between string amplitudes and quantum field theories.
Introduction to Truncation in Field Theory:
– Truncating modes in field theory involves deleting terms involving specific variables to mimic level truncation.
– Computed scattering amplitudes depend on the field redefinition parameter, affecting contact terms.
Challenges in String Amplitude Representations:
– Analytic representations of string amplitudes are absent in the literature.
– Existing attempts fail to fulfill all established criteria and lack a convenient truncation method.
Parametric Representations and Truncation:
– Parametric representations of amplitudes are expected upon truncation.
– Suitable field redefinitions must allow for truncation respecting specific constraints.
Local Symmetric Dispersion Relation:
– Using the local dispersion relation yields series representations with poles in all channels.
– Contact interactions are generated, resembling Quantum Field Theory.
Euler-Beta Function Representation:
– Utilizing a generalized Euler-Beta function allows for amplitude representations with simple poles at mass levels.
– Choosing appropriate parameters satisfies necessary criteria for amplitude representations.
Open Superstring Amplitude:
– Setting specific values for parameters leads to representations describing massless poles at the lowest level.
– Requirement of polynomial residues dictates the uniqueness of certain parameter values.
Nontrivial Check of Amplitude Representation:
– The 𝑛=0 term gives 1/(𝑠1𝑠2), with no 𝜆 dependence, and the 𝜆 dependence in the other levels cancels out when summing over all poles, which indicates the correctness of the representation.
Residues over Poles:
– The residues over the poles are written in a particular form that manifests the spectrum of exchanges when expressed in terms of Gegenbauer polynomials in 𝑑 dimensions, ensuring tree-level unitarity of the amplitude.
Efficient Truncation Demonstration:
– An efficient truncation of the series representation is demonstrated, showing it is possible to truncate the sum to the number of terms of 𝒪(𝑅), capturing all the essential features of the actual amplitude in a suitable range of 𝜆.
Numerical Investigations:
– Numerical investigations suggest that for 𝜆≳1 with 𝑁=20 terms in the truncated sum, the agreement with the actual result is more than 99%, and the convergence of the sum improves dramatically when 𝜆≠0.
Truncated Expansion Agreement:
– Evidence is presented that the truncated expansion, for a suitable range of 𝜆, captures all the essential features of the actual amplitude, both in the physical and unphysical regions.
Deformations and Unitarity:
– A certain deformation of the amplitude is considered, retaining crossing symmetry and having the same spectrum, which changes the residues at the poles, and unitarity restricts the deforming parameter 𝑐≥1.
High Energy Limits and Deformations:
– The analysis of the high energy limits suggests that as soon as the deforming parameter 𝑐≠1, the Regge behavior is ruined beyond some energy scale.
Limits of String Theory:
– The representations presented can be used to study an intermediate limit where only some of the higher spin massive modes are effectively massless, and this analysis is made possible through the formulas.
String Field Theory:
– Results obtained for the open string also hold for the closed string.
– Explicit representation given for ℳcl(𝑠1,𝑠2)=[Γ(−𝑠1)Γ(−𝑠2)Γ(−𝑠3)/Γ(1+𝑠1)Γ(1+𝑠2)Γ(1+𝑠3)] is ℳcl(𝑠1,𝑠2)=−1𝑠1𝑠2𝑠3+∞∑𝑛=11(𝑛!)2(1𝑠1−𝑛+1𝑠2−𝑛+1𝑠3−𝑛+1𝑛)(1−𝑛2+𝑛2√1−4𝑦n3)2𝑛−1.
Prospects:
– The new representations hold potential for reexamining experimental data for hadron scattering.
– The representations can be used to extend existing on shell techniques to incorporate contact diagrams for higher point functions.
Appendix – Crossing symmetric dispersion relation:
– A review of the two-channel crossing symmetric dispersion relation for identical scalar 2-2 scattering provided.
– A local dispersion relation is established to avoid nonlocal terms in the representation.
– The paper discusses null constraints and the use of different representations for efficient truncations.
Behavior at Limit:
– Amplitude behavior as s1 approaches infinity with s2 fixed: -s^-1 + s21Γ(-s2){sin[π(s1+s2)]/sin(πs1)}
– Truncated representation accurately capturing the behavior up to the truncation
String Theory Characteristics:
– Exponentially soft high energy, fixed angle scattering behavior characteristic of string theory
– Truncated representation faithfully capturing this behavior
Closed-String Amplitude:
– Series representations for massless scattering in closed-string theory with s1+s2+s3=0
– One-parameter family of representations obtained for the Virasoro-Shapiro amplitude
Local Dispersion Relation:
– Parametric representation of the amplitude using the two-channel symmetric local dispersion relation
– Crossing symmetry requirements for the massless pole and locality constraints
Manifest Crossing Symmetry:
– Regrouping terms to manifest full crossing symmetry levelwise
– Fully crossing symmetric dispersion relation delineated for specific conditions
Representation Simplification:
– Pochhammer simplification for integer values and form of the simplified expression
– Transformation of the representations to polynomials in Mandelstam variables
Low Energy Expansion:
– Expansion of the Gamma function form at y=0 and comparison of powers
– Expression for ζ(j) in terms of the terms yielding the series expansion
Jacobi Polynomial:
– The blog discusses the Jacobi polynomial and its properties such as (𝛼,𝛽)(𝑥) and the values of 𝛼𝑘 and 𝛽𝑘.
Generating Function for Zeta Function:
– A generating function for the zeta function in terms of the harmonic number is presented as 𝑍(𝑥)=∞∑𝑛=1𝑥2𝑛!(2𝑛−𝑥𝑛2−𝑛𝑥−1𝜆+𝑛)(1+𝜆(𝑥−𝑛)𝜆+𝑛)𝑛−1.
Formula for 𝜋:
– A formula for 𝜋 is given as 𝜋=4+∞∑𝑛=11𝑛!(1𝑛+𝜆−42𝑛+1)((2𝑛+1)24(𝑛+𝜆)−𝑛)𝑛−1.
Madhava Series:
– In the 𝜆→∞ limit, the summand of the series goes over to (−1)4/(2𝑛+1), known as the Madhava series.
Convergence of Series:
– While the original series takes five billion terms to converge to ten decimal places, a new representation with 𝜆 between 10 and 100 takes only 30 terms for convergence.
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For original article, please click here:
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.132.221601