A new approach to a $1 million mathematical enigma
Physicist translates the Riemann zeta function into quantum field theory
Date:
January 19, 2022
Source:
University of California - Santa Barbara
Summary:
Numbers like pi, e and phi often turn up in unexpected places
in science and mathematics. Pascal's triangle and the Fibonacci
sequence also seem inexplicably widespread in nature. Then there's
the Riemann zeta function, a deceptively straightforward function
that has perplexed mathematicians since the 19th century. The most
famous quandary, the Riemann hypothesis, is perhaps the greatest
unsolved question in mathematics, with the Clay Mathematics
Institute offering a $1 million prize for a correct proof.
FULL STORY ========================================================================== Numbers like p, e and f often turn up in unexpected places in science
and mathematics. Pascal's triangle and the Fibonacci sequence also seem inexplicably widespread in nature. Then there's the Riemann zeta function,
a deceptively straightforward function that has perplexed mathematicians
since the 19th century. The most famous quandary, the Riemann hypothesis,
is perhaps the greatest unsolved question in mathematics, with the Clay Mathematics Institute offering a $1 million prize for a correct proof.
==========================================================================
UC Santa Barbara physicist Grant Remmen believes he has a new approach
for exploring the quirks of the zeta function. He has found an analogue
that translates many of the function's important properties into quantum
field theory. This means that researchers can now leverage the tools
from this field of physics to investigate the enigmatic and oddly
ubiquitous zeta function. His work could even lead to a proof of the
Riemann hypothesis. Remmen lays out his approach in the journalPhysical
Review Letters.
"The Riemann zeta function is this famous and mysterious mathematical
function that comes up in number theory all over the place," said
Remmen, a postdoctoral scholar at UCSB's Kavli Institute for Theoretical Physics. "It's been studied for over 150 years." An outside perspective
Remmen generally doesn't work on cracking the biggest questions in
mathematics.
He's usually preoccupied chipping away at the biggest questions in
physics. As the fundamental physics fellow at UC Santa Barbara, he
normally devotes his attention to topics like particle physics, quantum gravity, string theory and black holes. "In modern high-energy theory,
the physics of the largest scales and smallest scales both hold the
deepest mysteries," he remarked.
One of his specialties is quantum field theory, which he describes as
a "triumph of 20th century physics." Most people have heard of quantum mechanics (subatomic particles, uncertainty, etc.) and special relativity
(time dilation, E=mc2, and so forth). "But with quantum field theory, physicists figured out how to combine special relativity and quantum
mechanics into a description of how particles moving at or near the
speed of light behave," he explained.
========================================================================== Quantum field theory is not exactly a single theory. It's more like
a collection of tools that scientists can use to describe any set of
particle interactions.
Remmen realized one of the concepts therein shares many characteristics
with the Riemann zeta function. It's called a scattering amplitude,
and it encodes the quantum mechanical probability that particles will
interact with each other. He was intrigued.
Scattering amplitudes often work well with momenta that are complex
numbers.
These numbers consist of a real part and an imaginary part -- a multiple
of SQRT-1, which mathematicians call i. Scattering amplitudes have
nice properties in the complex plane. For one, they're analytic (can be expressed as a series) around every point except a select set of poles,
which all lie along a line.
"That seemed similar to what's going on with the Riemann zeta function's
zeros, which all seem to lie on a line," said Remmen. "And so I
thought about how to determine whether this apparent similarity was
something real." The scattering amplitude poles correspond to particle production, where a physical event happens that generates a particle
with a momentum. The value of each pole corresponds with the mass of
the particle that's created. So it was a matter of finding a function
that behaves like a scattering amplitude and whose poles correspond to
the non-trivial zeros of the zeta function.
==========================================================================
With pen, paper and a computer to check his results, Remmen set to work devising a function that had all the relevant properties. "I had had
the idea of connecting the Riemann zeta function to amplitudes in the
back of my mind for a couple years," he said. "Once I set out to find
such a function, it took me about a week to construct it, and fully
exploring its properties and writing the paper took a couple months." Deceptively simple At its core, the zeta function generalizes the harmonic series: This series blows up to infinity when x <= 1, but it converges
to an actual number for every x > 1.
In 1859 Bernhard Riemann decided to consider what would happen when x
is a complex number. The function, now bearing the name Riemann zeta,
takes in one complex number and spits out another.
Riemann also decided to extend the zeta function to numbers where the
real component was not greater than 1 by defining it in two parts:
the familiar definition holds in places where the function behaves, and another, implicit definition covers the places where it would normally
blow up to infinity.
Thanks to a theorem in complex analysis, mathematicians know there is
only one formulation for this new area that smoothly preserves the
properties of the original function. Unfortunately, no one has been
able to represent it in a form with finitely many terms, which is part
of the mystery surrounding this function.
Given the function's simplicity, it should have some nice features. "And
yet, those properties end up being fiendishly complicated to understand," Remmen said. For example, take the inputs where the function equals
zero. All the negative even numbers are mapped to zero, though this is
apparent -- or "trivial" as mathematicians say -- when the zeta function
is written in certain forms. What has perplexed mathematicians is that
all of the other, non-trivial zeros appear to lie along a line: Each of
them has a real component of 1/2.
Riemann hypothesized that this pattern holds for all of these non-trivial zeros, and the trend has been confirmed for the first few trillion
of them.
That said, there are conjectures that work for trillions of examples
and then fail at extremely large numbers. So mathematicians can't be
certain the hypothesis is true until it's proven.
But if it is true, the Riemann hypothesis has far-reaching
implications. "For various reasons it crops up all over the place in fundamental questions in mathematics," Remmen said. Postulates in fields
as distinct as computation theory, abstract algebra and number theory
hinge on the hypothesis holding true. For instance, proving it would
provide an accurate account of the distribution of prime numbers.
A physical analogue The scattering amplitude that Remmen found describes
two massless particles interacting by exchanging an infinite set of
massive particles, one at a time.
The function has a pole -- a point where it cannot be expressed
as a series - - corresponding to the mass of each intermediate
particle. Together, the infinite poles line up with the non-trivial
zeros of the Riemann zeta function.
What Remmen constructed is the leading component of the interaction. There
are infinitely more that each account for smaller and smaller aspects of
the interaction, describing processes involving the exchange of multiple massive particles at once. These "loop-level amplitudes" would be the
subject of future work.
The Riemann hypothesis posits that the zeta function's non-trivial zeros
all have a real component of 1/2. Translating this into Remmen's model:
All of the amplitude's poles are real numbers. This means that if someone
can prove that his function describes a consistent quantum field theory
-- namely, one where masses are real numbers, not imaginary -- then the
Riemann hypothesis will be proven.
This formulation brings the Riemann hypothesis into yet another
field of science and mathematics, one with powerful tools to offer mathematicians. "Not only is there this relation to the Riemann
hypothesis, but there's a whole list of other attributes of the Riemann
zeta function that correspond to something physical in the scattering amplitude," Remmen said. For instance, he has already discovered
unintuitive mathematical identities related to the zeta function using
methods from physics.
Remmen's work follows a tradition of researchers looking to physics to
shed light on mathematical quandaries. For instance, physicist Gabriele Veneziano asked a similar question in 1968: whether the Euler beta
function could be interpreted as a scattering amplitude. "Indeed it can," Remmen remarked, "and the amplitude that Veneziano constructed was one
of the first string theory amplitudes." Remmen hopes to leverage this amplitude to learn more about the zeta function.
"The fact that there are all these analogues means that there's something
going on here," he said.
And the approach sets up a path to possibly proving the centuries-old hypothesis. "The innovations necessary to prove that this amplitude
does come from a legitimate quantum field theory would, automatically,
give you the tools that you need to fully understand the zeta
function," Remmen said. "And it would probably give you more as well." ========================================================================== Story Source: Materials provided by
University_of_California_-_Santa_Barbara. Original written by Harrison
Tasoff. Note: Content may be edited for style and length.
========================================================================== Related Multimedia:
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Remmen's_scattering_amplitude_translates_the_Riemann_zeta_function_into
the_language_of_quantum_field_theory.
========================================================================== Journal Reference:
1. Grant N. Remmen. Amplitudes and the Riemann Zeta Function. Physical
Review Letters, 2021; 127 (24) DOI: 10.1103/PhysRevLett.127.241602 ==========================================================================
Link to news story:
https://www.sciencedaily.com/releases/2022/01/220119135044.htm
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