  Trip flow between areas is a fundamental metric for human mobility research.
Given its identification with travel demand and its relevance for
transportation and urban planning, many models have been developed for its
estimation. These models focus on flow intensity, disregarding the information
provided by the local mobility orientation. A field-theoretic approach can
overcome this issue and handling both intensity and direction at once. Here we
propose a general vector-field representation starting from individuals'
trajectories valid for any type of mobility. By introducing four models of
spatial exploration, we show how individuals' elections determine the
mesoscopic properties of the mobility field. Distance optimization in long
displacements and random-like local exploration are necessary to reproduce
empirical field features observed in Chinese logistic data and in New York City
Foursquare check-ins. Our framework is an essential tool to capture hidden
symmetries in mesoscopic urban mobility, it establishes a benchmark to test the
validity of mobility models and opens the doors to the use of field theory in a
wide spectrum of applications.
