  In this work we present a gauge principle that starts with the momentum space
representation of the position operator (${\hat x}_i = i \hbar
\frac{\partial}{\partial p_i}$) rather than starting with the position space
representation of the momentum operator (${\hat p}_i = -i \hbar
\frac{\partial}{\partial x_i}$). We discuss some simple examples with this new
type of gauge theory: (i) analog solutions from ordinary gauge theory in this
momentum gauge theory, (ii) Landau levels using momentum gauge fields, (iii)
the emergence of non-commutative space-times from the momentum gauge fields. We
find that the non-commutative space-time parameter can be momentum dependent,
and one can construct a model where space-time is commutative at low momentum
but becomes non-commutative at high momentum.
