We investigate aspects of the relation between the quantum geometry of the
normal state (NS) and the superconducting phase, through the lens of
non-locality. By relating band theory to quantum estimation theory, we derive a
direct momentum-dependent relation between quantum geometry and the quantum
fluctuations of the position operator. We then investigate two effects of the
NS quantum geometry on superconductivity. On the one hand, we present a
physical interpretation of the conventional and geometric contributions to the
superfluid weight in terms of two different movements of the normal state
charge carriers forming the Cooper pairs. The first contribution stems from
their center-of-mass motion while the second stems from their zero-point
motion, thereby explaining its persistence in flat-band systems. On the other
hand, we phenomenologically derive an emergent Darwin term driven by the NS
quantum metric. We show its form in one and two-body problems, derive the
effective pairing potential in $s$-wave superconductors, and explicit its form
in the case of two-dimensional massive Dirac fermions. We thus show that the NS
quantum metric screens the pairing interaction and weakens superconductivity,
which could be tested experimentally by doping a superconductor. Our work
reveals the ambivalent relationship between non-interacting quantum geometry
and superconductivity, and possibly in other correlated phases.

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