the Gutzwiller trace formula
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"Gutzwiller gave a trace formula in the setting of quantum chaos which
relates the classical and quantum mechanical pictures. Given a chaotic
(classical) dynamical system, there will exist a dense set of periodic orbits,
and one side of the trace formula will be a sum over the lengths of these
orbits. On the other side will be a sum over the eigenvalues of the
Hamiltonian in the quantummechanical analog of the given classical dynamical
system.
This setup resembles the explicit formulas of prime number theory. In this
analogy, the lengths of the prime periodic orbits play the role of the
rational primes, while the eigenvalues of the Hamiltonian play the role of
the zeros of the zeta function. Based on this analogy and pearls mined from
Odlyzko's numerical evidence,
Sir Michael Berry
proposes that there exists a
classical dynamical system, asymmetric with respect to time reversal, the
lengths of whose periodic orbits correspond to the rational primes, and whose
quantummechanical analog has a Hamiltonian with zeros equal to the imaginary
parts of the nontrivial zeros of the zeta function. The search for such a
dynamical system is one approach to proving the Riemann hypothesis."
(Daniel Bump)
There are some excellent introductory notes regarding the GTF in
N. Snaith's thesis.
general notes on the number theoretic content
of Gutzwiller's papers
quantum chaology resources
back to trace formulae, explicit formulae and number theory page
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