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# Momentum Gauge Fields and Non-Commutative Space-Time

Jun 2022

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

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