Momentum Gauge Fields and Non-Commutative Space-Time

In this work, we present a gauge principle that starts with the momentum space representation of the position operator ((x) over cap (i) = i (h) over bar partial derivative/partial derivative p(i)), rather than starting with the position space representation of the momentum operator ((p) over cap (i) = i (h) over bar partial derivative/partial derivative(xi)). This extension of the gauge principle can be seen as a dynamical version of Born's reciprocity theory, which exchanges position and momentum. 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, and (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.