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This question follows a previous one of mine: Adiabatic theorem and Berry phase.

In his original paper [ M. V. Berry, Proc. R. Soc. Lond. A. Math. Phys. Sci. 392, 45 (1984) ], Berry discussed the possibility for an old-forgotten phase factor to arise after one loop is done in a parameter space, using the adiabatic theorem. This became famously known as a Berry's phase, see also Berry's curvature article on Wikipedia.

Soon after, [ Y. Aharonov and J. Anandan, Phys. Rev. Lett. 58, 1593 (1987) ], AA proposed to generalised this adiabatic phase to non-adiabatic -- still cyclic -- evolutions.

What confused me is the sentence:

[..] we regard $\beta$ [the AA geometric phase] as a geometric phase associated with a closed curve in the projective Hilbert space and not the parameter space [...]

I would like to make this sentence a bit more clear, especially regarding the degeneracy point discussed in the previous question Adiabatic theorem and Berry phase.

What is the space in which the degeneracy appears?

Any comment to improve this question is welcome.

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In Berry's paper, one considers a Hamiltonian depending on some external parameter $R$ (usually multidimensional); in the canonical example of a spin in an external magnetic field (Zeeman effect), the parameters are the 3 components of the magnetic field $\bf B$. For some values of the external parameter $R$, some eigenvalues may be degenerate. So degeneracies live in the abstract space of external parameter. For the spin example, the degeneracy appears at ${\bf B}=0$. The closed circuit discussed by Michael Berry takes place in the parameter space, and is driven externally and adiabatically.

In the case discussed by Aharonov and Anandan, the closed circuit considered takes place directly in the projective Hilbert space, i.e., the space of equivalence classes of Hilbert space vectors (two Hilbert space vectors which differ from each other only by a non-zero complex prefactor describe the same physical state, as they would give the same answer for the measurement of any observable operator); here, the reason of why and how fast the quantum state evolves is of no importance (this may be due to unitary evolution under the effect of the Hamiltonian, or externally driven as discussed by Berry, or even a non-unitary evolution due to a sequence of measurement operations). For this reason, the geometric phase of Aharonov and Anandan is considered to be more general concept than Berry's geometrical phase.

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