The Riemann ζ function (ζ, pronounced zeta) is a deceptively simple and ubiquitous function that has plagued mathematicians since the 19th century, and the associated Riemann conjecture has become one of the unsolved mysteries of the Millennium Grand Prix puzzle.
Recently, in a paper published in Physical Review Letters, physicist Grant Remmen elucidated his thinking in physics to develop a way to explore many of the oddities of ζ functions. In this new approach, he translated many important properties of Riemann's ζ functions into quantum field theory. This means that we can now use tools from the field of physics to study this mysterious and strange function, and may even lead us closer to the proof of the Riemann hypothesis.
Riemann conjecture
In the field of number theory, ζ function is a ubiquitous mathematical function that exists in the form of an infinite harmonic series.
ζ(s) function, s represents the exponential variable in the function, ζ (2) refers to the sum of squares of the series, ζ (4) is the sum of the quaternions of the series, and so on.
When s >1, the ζ function is convergent, it converges to a finite value.
In 1859, the mathematician Bernhard Riemann began to think: what would happen if the s in the ζ function were complex numbers. He extended ζ functions to the complex plane, finding that ζ functions converge only if the real part of s is greater than 1.
To extend the ζ function to the rest of the complex plane, Riemann used a technique in complex analysis called analytic extension. The key to analyzing extension is that there are actually two functions operating at the same time, one is the original ζ function, which operates in a limited range, that is, limited to the range of the real part of s greater than 1; the other is a completely new function with an extended domain, the Riemann ζ function: when the ζ function converges, the value of the Riemann ζ function is equivalent to the original ζ function; when the real part of s is less than 1, the value of the Riemann ζ function is equal to the analytic extension of the function defined by the series at s. (Note: ζ function cannot be extended to the entire complex plane, and it does not make sense when the real part of s is equal to 1.) ）
A ζ function that defines the domain is extended on the complex plane and can pass through the origin. | Credit: Grant Remmen, Harrison Tasoff
By extending the domain of the function, Riemann discovered that in the new domain, the function could pass through the origin. This means that when some specific value is entered into the function, the function value is 0, and these values are called ζ zero. For example, all negative even numbers are ζ zero.
However, these so-called "ordinary zeros" are not uninteresting to mathematicians, and Riemann is concerned with the inputs called "non-trivial zeros", because he notices that all non-trivial zeros seem to be in a straight line.
Functions ζ Riemann believes that all non-trivial zeros are in a straight line. | image reference: New Principles Institute
Thus, Riemann hypothesized that this pattern applied to all non-trivial zeros, specifically, the Riemann conjecture said that all non-trivial zeros were in a straight line, that is, in the complex plane where the real part of s was equal to 1/2.
This conjecture has now been confirmed in numerous examples, but it is still not enough to prove that the conjecture holds.
Quantum field theory provokes thought
Remmen's day-to-day research work is not focused on solving big problems in mathematics, but rather trying to solve big problems related to quantum gravity, string theory, and black holes. One of his specialties is quantum field theory, which combines special relativity and quantum mechanics to describe the behavior of particles moving at or near the speed of light.
Remmen realized that in quantum field theory, a concept called scattering amplitude has many of the same characteristics as Riemann ζ functions. The scattering amplitude encodes the probability of particle interactions in quantum mechanics. In general, it is suitable for cases where the momentum is complex, and it has good properties in the complex plane. In the complex plane, each point they surround is resolved (i.e., can be represented as a series), which seems to be similar to the non-trivial zeros of Riemann's ζ function on a straight line, except for a set of poles (points that cannot be represented by a series).
Shown in the figure is the scattering amplitude of Remmen, which translates the Riemann ζ function into the language of quantum field theory. | Image credit: Grant Remmen
So Remmen wondered if there was any real meaning behind this apparent similarity.
ζ quantum field theory version of the function
Remmen wanted to develop a way to map the scattered amplitudes in quantum systems to the mathematical features of Riemann ζ functions. He considers a quantum particle collision system that describes particles in terms of variables that cover their energy, momentum, and trajectory during scattering. Using these variables, he built a mathematical function with all the characteristics of the scattering amplitude on the basis of the ζ function.
Remmen's scattering amplitude describes two massless particles interacting by exchanging infinitely massive particles. In his function, the poles correspond to the mass of each intermediate particle, and an infinite number of such poles correspond to non-trivial zeros in the Riemann ζ function.
The Riemann hypothesis assumes that the non-trivial zeros of ζ function have real parts equal to 1/2, translating this into Remmen's model means that all amplitude poles are real numbers. That is, if one can prove that Remmen's function describes such a consistent quantum field theory that mass is a real number rather than an imaginary number, then he proves the Riemann hypothesis.
ζ quantum version of the function
Remmen's functions brought the Riemann hypothesis into another area of science and mathematics, providing mathematicians with powerful tools. It shows that not only is this correspondence present in the Riemann hypothesis, but there are many other properties in the Riemann ζ function that also correspond to the physical properties of the scattering amplitude. For example, Remmen has used physics methods to discover some mathematical identities related to ζ functions.
Now, Remmen hopes to use the scattering amplitude in physics to learn more about ζ function. This approach opens up a new path that may be possible to prove this great conjecture. Moreover, the new ideas that can be brought about when it comes to proving that this amplitude does indeed come from a reasonable quantum field theory will also provide us with the tools we need to fully understand ζ functions.
Now, Remmen constructs the most important components of the interaction, and there are an infinite number of parts that contain those small interactions, and those "ring amplitudes" will be the subject of future research.
#创作团队:
Author: Light rain
Design/Typography: Wenwen
#参考来源:
https://www.news.ucsb.edu/2022/020520/quantum-zeta-epiphany
https://physics.aps.org/articles/v14/s157
#图片来源:
Cover image: Institute of New Principles
First image: Grant Remmen