Momentum Gauge Fields and Non-Commutative Space–Time

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In this work, we present a gauge principle that starts with the momentum space representation of the position operator (x^i=iℏ∂∂pi), rather than starting with the position space representation of the momentum operator (p^i=−iℏ∂∂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.