Noncommutative $\mathrm{Sp}\mathrm{}(2,R)\mathrm{}$ gauge theories: A field theory approach to two-time physics

Phase space and its relativistic extension is a natural space for realizing $\mathrm{Sp}\mathrm{}(2,R)\mathrm{}$ symmetry through canonical transformations. On a $(D\times 2)$-dimensional covariant phase space, we formulate noncommutative field theories, where $\mathrm{Sp}\mathrm{}(2,R)\mathrm{}$ plays a role as either a global or a gauge symmetry group. In both cases these field theories have potential applications, including certain aspects of string theories, M theory, as well as quantum field theories. If interpreted as living in lower dimensions, these theories realize Poincaré symmetry linearly in a way consistent with causality and unitarity. In case $\mathrm{Sp}\mathrm{}(2,R)\mathrm{}$ is a gauge symmetry, we show that the spacetime signature is determined dynamically as $(D-2,2).$ The resulting noncommutative $\mathrm{Sp}\mathrm{}(2,R)\mathrm{}$ gauge theory is proposed as a field theoretical formulation of two-time physics: classical field dynamics contains all known results of “two-time physics,” including the reduction of physical spacetime from D to $(D-2)$ dimensions, with the associated “holography” and “duality” properties. In particular, we show that the solution space of classical noncommutative field equations put all massless scalar, gauge, gravitational, and higher-spin fields in $(D-2)$ dimensions on equal footing, reminiscent of string excitations at zero and infinite tension limits.