Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Geometric Phase and Modulo Relations for Probability Amplitudes as
  Functions on Complex Parameter Spaces   We investigate general differential relations connecting the respective
behavior s of the phase and modulo of probability amplitudes of the form
$\amp{\psi_f}{\psi}$, where $\ket{\psi_f}$ is a fixed state in Hilbert space
and $\ket{\psi}$ is a section of a holomorphic line bundle over some complex
parameter space. Amplitude functions on such bundles, while not strictly
holomorphic, nevertheless satisfy generalized Cauchy-Riemann conditions
involving the U(1) Berry-Simon connection on the parameter space. These
conditions entail invertible relations between the gradients of the phase and
modulo, therefore allowing for the reconstruction of the phase from the modulo
(or vice-versa) and other conditions on the behavior of either polar component
of the amplitude. As a special case, we consider amplitude functions valued on
the space of pure states, the ray space ${\cal R} = {\mathbb C}P^n$, where
transition probabilities have a geometric interpretation in terms of geodesic
distances as measured with the Fubini-Study metric. In conjunction with the
generalized Cauchy-Riemann conditions, this geodesic interpretation leads to
additional relations, in particular a novel connection between the modulus of
the amplitude and the phase gradient, somewhat reminiscent of the WKB formula.
Finally, a connection with geometric phases is established.

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The categories of this paper are: alg-geom
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The categories of this paper are: dg-ga
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The categories of this paper are: physics
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The categories of this paper are: atom-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: astro-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: cs
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-ex
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The categories of this paper are: q-bio
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The categories of this paper are: hep-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-lat
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The categories of this paper are: gr-qc
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The categories of this paper are: chem-ph