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just a little bummed the single number wasn't " 42
One thing I love philosophically is the further you delve into studying things at a quantum level, the faster the physics crashes headlong into metaphysics.
Friedmann did not set out to look for pi nor for the Wallis formula. The discovery began in a quantum mechanics course taught by Carl Hagen, a professor of physics at the University of Rochester.
While the quantum calculations developed by Danish physicist Niels Bohr in the early twentieth century give accurate values for the energy states of hydrogen, Hagen wanted his students to use an alternate method—called the variational principle—to approximate the value for the ground state of the hydrogen atom.
They could then calculate the values for the different energy states and compare them with the values obtained by Bohr almost a century ago. This enabled them to determine how the ratio of the Bohr values to the values obtained with the “tweaked” variational principle changed as higher and higher energy levels were taken into account.
And they were surprised to see that the ratio yielded—effectively—the Wallis formula for pi.
One of the most helpful clues for proving the Riemann hypothesis has come from function theory, which reveals that the values of the imaginary part, t, at which the function vanishes are discrete numbers. This suggests that the nontrivial zeros form a set of real and discrete numbers, which is just like the eigenvalues of another function called a differential operator, which is widely used in physics.
In the early 1900s, this similarity led some mathematicians to wonder whether there really exists a differential operator whose eigenvalues correspond exactly to the nontrivial zeros of the Riemann zeta function. Today this idea is called the Hilbert-Pólya conjecture, named after David Hilbert and George Pólya—despite the fact that neither of them published anything about it.
One of the interesting things about the newly discovered operator is that it has close ties with quantum physics.
In 1999, when mathematical physicists Michael Berry and Jonathan Keating were investigating the Hilbert-Pólya conjecture, they made another important conjecture. If such an operator exists, they said, then it should correspond to a theoretical quantum system with particular properties. This is now called the Berry-Keating conjecture. But no one has ever found such a system before now, and this is a second important aspect of the new work.
"We have identified a quantization condition for the Berry-Keating Hamiltonian, thus essentially verifying the validity of the Berry-Keating conjecture," Brody said.
Normally, physicists describe quantum systems using highly symmetric mathematical matrices whose solutions, or “eigenvalues,” correspond to the system’s energy levels. The symmetries of these matrices usually guarantee that imaginary numbers cancel out and the eigenvalues are real, so that these matrices make sense as descriptions of physical systems. But for 20 years, Bender and Brody have studied matrix descriptions of quantum systems that relax the usual symmetry requirements and respect a weaker property called parity-time (or PT) symmetry. Following a 2015 conversation with Müller, they discovered that they could write down a PT-symmetric matrix whose eigenvalues correspond to the nontrivial zeros of the Riemann zeta function. “This came as a real surprise to us,” Brody said. However, because the matrix was only PT-symmetric, instead of following the usual stricter symmetries, it isn’t guaranteed to have real eigenvalues — the property that would ensure that corresponding zeros have real parts equal to ½.
The researchers spelled out several arguments for why the eigenvalues of their matrix are probably real, and why, in that case, the Riemann hypothesis is probably correct, but they came short of proving it. “Whether it will be difficult or easy to fill in the missing steps, at this point we cannot speculate,” said Brody. “Further work is needed to get a better feeling as to the scale of difficulty involved.”