are holonomies arguably observables?



I ask this, since the way holonomies are defined, as closed paths,
makes one wonder how one would do in principle to get an operational
definition on a space-time. When the loop is the union of time-like
paths its possible to measure it, as long as one assumes that the
connection along a path orientation is minus the connection along the
opposite orientation, which may be fine if one doesnt want to add
additional 'dimensions' artifacts to solve for the hysteretic-like
behaviour, but even this leaves the question open how would one
physically measure holonomies like Riemann along space-space bivectors.


Well, its not like there is a big philosophical problem about
holonomies not being directly observable, since we use the A vector
field for EM, and obtain observable effects from its consequences and
behaviour. I'm just curious if there is a geometrically (and
physically) consistent way to directly measure holonomies, particularly
in space-space directions.

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