Quantum phases and connections

Richard Montgomery

Aditya Gilra

Recently, it was proposed that the spin-statistics connection arises due to a quantum dynamical phase involving cosmic gravity (Unnikrishnan C S, gr-qc/0406043). There it was assumed that the gravitational quantum phase accumulated on the wavefunctions of each of two identical particles undergoing scattering in the presence of cosmic matter with the direction of their momentum changing by some angle is equivalent to the particles forward scattering with the entire universe rotating oppositely by the same angle. However, if one considers the gravitational phase accumulated by an electron in orbit to be analogous to that acquired by a scattering particle, the accumulated phase inferred from the fine-structure splitting in an atom is too small to explain the spin statistics connection. Though, the desirability of a dynamical proof of the spin-statistics connection remains.

J. M. Harrison, Jonathan P. Keating, J. M. Robbins +1 more

We develop a full characterization of abelian quantum statistics on graphs. We explain how the number of anyon phases is related to connectivity. For 2-connected graphs the independence of quantum statistics with respect to the number of particles is proven. For non-planar 3-connected graphs we identify bosons and fermions as the only possible statistics, whereas for planar 3-connected graphs we show that one anyon phase exists. Our approach also yields an alternative proof of the structure theorem for the first homology group of n -particle graph configuration spaces. Finally, we determine the topological gauge potentials for 2-connected graphs.

Paul Bracken

Abstract An introduction to several geometrical ideas which are of use to quantum mechanics is presented. The Aharonov‐Anandan phase is introduced and without reference to any dynamical equation, this phase is formulated by defining an appropriate connection on a specific fiber bundle. The holonomy element gives the phase. By introducing another connection, the Pancharatnam phase formula is derived following a different procedure. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008