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 ((Formula presented.)), rather than starting with the position space representation of the momentum operator ((Formula presented.)). 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.