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When the positions of two fermions are exchanged adiabatically in three-dimensional space, we know that the wave function gains a factor of $-1$. Is this related to Berry's phase?

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    $\begingroup$ Good question. I found this paper by Berry which connects the phase change under particle exchange to the spin of the particles. $\endgroup$ – Vijay Murthy May 6 '12 at 7:19
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The answer is yes. The angular momentum of a (scalar) particle moving in the background of a Dirac magnetic monopole of an odd magnetic charge becomes half integral. (The magnetic charge must be an integer by the Dirac quantization condition.)

Under these conditions, the phase acquired by the wavefunction through a 360 degrees rotation is $-1$. By the spin statistics theorem, the particle must be quantized as a fermion, even though its dynamics is described by bosonic coordinates.

This phase can be obtained as a Berry phase. This principle was used by Witten in his seminal paper "E. Witten, Current algebra, baryons, and quark confinement, Nucl. Phys. B 223 (1983) 433-444", to demonstrate that the Skyrmions are fermions.

Witten obtained the change of sign of the Skyrmion wave function through a Berry phase computation. Here, the magnetic monopole field stems from the Wess-Zumino-Witten term in the Skyrmion Lagrangian.

A clear exposition of Witten's work with the emphasis on the analogy with the motion in the background of a Dirac monopole is given in I.J.R. Atchinson's paper : "Berry phases, magnetic monopoles and Wess Zumino terms, or how the Skyrmion got its spin"

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