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Register a new userFailure of geometric electromagnetism in the adiabatic vector Kepler problem
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The magnetic moment of a particle orbiting a straight current-carrying wire may precess rapidly enough in the wires magnetic field to justify an adiabatic approximation, eliminating the rapid time dependence of the magnetic moment and leaving only the particle position as a slow degree of freedom. To zeroth order in the adiabatic expansion, the orbits of the particle in the plane perpendicular to the wire are Keplerian ellipses. Higher order post-adiabatic corrections make the orbits precess, but recent analysis of this `vector Kepler problem has shown that the effective Hamiltonian incorporating a post-adiabatic scalar potential (`geometric electromagnetism) fails to predict the precession correctly, while a heuristic alternative succeeds. In this paper we resolve the apparent failure of the post-adiabatic approximation, by pointing out that the correct second-order analysis produces a third Hamiltonian, in which geometric electromagnetism is supplemented by a tensor potential. The heuristic Hamiltonian of Schmiedmayer and Scrinzi is then shown to be a canonical transformation of the correct adiabatic Hamiltonian, to second order. The transformation has the important advantage of removing a $1/r^3$ singularity which is an artifact of the adiabatic approximation.
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Since quantum computers are known to break the vast majority of currently-used cryptographic protocols, a variety of new protocols are being developed that are conjectured, but not proven to be safe against quantum attacks. Among the most promising is lattice-based cryptography, where security relies upon problems like the shortest vector problem. We analyse the potential of adiabatic quantum computation for attacks on lattice-based cryptography, and give numerical evidence that even outside the adiabatic regime such methods can facilitate the solution of the shortest vector and similar problems.
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