- #1
nomadreid
Gold Member
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In "A Prime Case of Chaos" (http://www.ams.org/samplings/math-history/prime-chaos.pdf), the author states that "Physicists ... believe the zeroes of the zeta function can be interpreted as energy levels..." I have two problems with this:
(1) the non-trivial zeroes of the zeta function are complex numbers, and energy levels, are real numbers. So how could you interpret something like (1/2) + (14.134725)i as an energy level? Did he perhaps mean their probability amplitudes? But this would not give the same values, for example, as the Rydberg series.
(2) the three dots in the above quote say that this is tied to the zeta function being a trace formula. Following this lead, the best my very limited background in the mathematics needed for this physics could come up with, was in "Selberg trace formula and zeta functions" at http://empslocal.ex.ac.uk/people/staff/mrwatkin//zeta/physics4.htm, where it explains:
"Selberg trace formula ... relates ... the spectrum of the quantal motion on compact surfaces of negative curvature ...; however, it does not address itself ...to ... the detailed properties of the discrete energy spectrum ..."
which appears to lead to a dead end.
If a response gives any sources, could they please be on-line sources that are freely accessible?
Thanks for any help.
(1) the non-trivial zeroes of the zeta function are complex numbers, and energy levels, are real numbers. So how could you interpret something like (1/2) + (14.134725)i as an energy level? Did he perhaps mean their probability amplitudes? But this would not give the same values, for example, as the Rydberg series.
(2) the three dots in the above quote say that this is tied to the zeta function being a trace formula. Following this lead, the best my very limited background in the mathematics needed for this physics could come up with, was in "Selberg trace formula and zeta functions" at http://empslocal.ex.ac.uk/people/staff/mrwatkin//zeta/physics4.htm, where it explains:
"Selberg trace formula ... relates ... the spectrum of the quantal motion on compact surfaces of negative curvature ...; however, it does not address itself ...to ... the detailed properties of the discrete energy spectrum ..."
which appears to lead to a dead end.
If a response gives any sources, could they please be on-line sources that are freely accessible?
Thanks for any help.