  We derive a scheme for systematically enumerating the responses of gapped as
well as gapless systems of free fermions to electromagnetic and strain fields
starting from a common parent theory. Using the fact that position operators in
the lowest Landau level of a quantum Hall state are canonically conjugate, we
consider a massive Dirac fermion in $2n$ spatial dimensions under $n$ mutually
orthogonal magnetic fields and reinterpret physical space in the resulting
zeroth Landau level as phase space in $n$ spatial dimensions. The bulk
topological responses of the parent Dirac fermion, given by a Chern-Simons
theory, translate into quantized insulator responses, while its edge anomalies
characterize the response of gapless systems. Moreover, various physically
different responses are seen to be related by the interchange of position and
momentum variables. We derive many well-known responses, and demonstrate the
utility of our theory by predicting spectral flow along dislocations in Weyl
semimetals.
