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Oh, I'm probably wrong and just arguing from my prejudices. I obviously haven't tried reading their paper. What is their argument for why the strategy would wor

by jordigh 10y ago

Oh, I'm probably wrong and just arguing from my prejudices. I obviously haven't tried reading their paper. What is their argument for why the strategy would work?

tgb 10y ago
Uh, I'm out of my field so I'm not sure I can summarize it better than the article did, but I can try. They prove that the Riemann hypothesis is equivalent to a property of a specific operator, however they can't prove that property (that's it's self-adjoint under some inner product) since they don't know what the appropriate domain of the operator is. But the operator does do several things: first, it's classical version is what was expected. Secondly, they suggest an approach towards getting an inner product that would the operator self-adjoint. Lastly, the authors conclude "The fact that iH [the operator in question] is PT symmetric, with a broken PT symmetry, offers a fresh and optimistic outlook" which I don't understand well enough to add to.
- bermanoid 10y ago
  Taking this down to an even more basic level that doesn't require much knowledge of this stuff - the basic strategy of this program is to find a quantum system that they can prove (rigorously) has energy levels that map directly to the Riemann zeta zeros (well, the non-trivial ones) under a simple shift/rotation in the complex plane. Since energy levels of these systems are always real (and in a straight line on the complex plane), this would establish that the zeros are all in a straight line as well, namely the one where Re(z) = 1/2. The tough bit is finding that system. These researchers seem to think they've made headway; I'm withholding judgment for now, since by my reading the whole motivation for reaching for a physical system (real eigenvalues) is exactly what they still need to establish about their proposed system. They might have just succeeded in recasting the Riemann hypothesis in a different but no more tractable form, we'll have to see.

by jordigh 10y ago

Oh, I'm probably wrong and just arguing from my prejudices. I obviously haven't tried reading their paper. What is their argument for why the strategy would work?

tgb 10y ago

Uh, I'm out of my field so I'm not sure I can summarize it better than the article did, but I can try. They prove that the Riemann hypothesis is equivalent to a property of a specific operator, however they can't prove that property (that's it's self-adjoint under some inner product) since they don't know what the appropriate domain of the operator is. But the operator does do several things: first, it's classical version is what was expected. Secondly, they suggest an approach towards getting an inner product that would the operator self-adjoint. Lastly, the authors conclude "The fact that iH [the operator in question] is PT symmetric, with a broken PT symmetry, offers a fresh and optimistic outlook" which I don't understand well enough to add to.

bermanoid 10y ago
Taking this down to an even more basic level that doesn't require much knowledge of this stuff - the basic strategy of this program is to find a quantum system that they can prove (rigorously) has energy levels that map directly to the Riemann zeta zeros (well, the non-trivial ones) under a simple shift/rotation in the complex plane. Since energy levels of these systems are always real (and in a straight line on the complex plane), this would establish that the zeros are all in a straight line as well, namely the one where Re(z) = 1/2. The tough bit is finding that system. These researchers seem to think they've made headway; I'm withholding judgment for now, since by my reading the whole motivation for reaching for a physical system (real eigenvalues) is exactly what they still need to establish about their proposed system. They might have just succeeded in recasting the Riemann hypothesis in a different but no more tractable form, we'll have to see.