${\mathrm{u}}_{\star }\mathrm{}(1,1)\mathrm{}$ noncommutative gauge theory as the foundation of two-time physics in field theory

A very simple field theory in noncommutative phase space ${(X}^M{,P}^M)$ in $d+2\mathrm{}$ dimensions, with a gauge symmetry based on noncommutative ${\mathrm{u}}_{\star }\mathrm{}(1,1),$ furnishes the foundation for the field theoretic formulation of two-time physics. This leads to a remarkable unification of several gauge principles in d dimensions, including Maxwell, Einstein and high spin gauge principles, packaged together into one of the simplest fundamental gauge symmetries in noncommutative quantum phase space in $d+2\mathrm{}$ dimensions. A gauge invariant action is constructed and its nonlinear equations of motion are analyzed. In addition to elegantly reproducing the first quantized worldline theory with all background fields, the field theory prescribes unique interactions among the gauge fields. A matrix version of the theory, with a large N limit, is also outlined.