Itzhak Bars

The physics that is traditionally formulated in one–time-physics (1T-physics) can also be formulated in two-time-physics (2T-physics). The physical phenomena in 1T or 2T physics are not different, but the spacetime formalism used to describe them is. The 2T description involves two extra dimensions (one time and one space), is more symmetric, and makes manifest many hidden features of 1T-physics. One such hidden feature is that families of apparently different 1T-dynamical systems in d dimensions holographically describe the same 2T system in d + 2 dimensions. In 2T-physics there are two timelike dimensions, but there is also a crucial gauge symmetry that thins out spacetime, thus making 2T-physics effectively equivalent to 1T-physics. The gauge symmetry is also responsible for ensuring causality and unitarity in a spacetime with two timelike dimensions. What is gained through 2T-physics is a unification of diverse 1T dynamics by making manifest hidden symmetries and relationships among them. Such symmetries and relationships is the evidence for the presence of the underlying higher dimensional spacetime structure. 2T-physics could be viewed as a device for gaining a better understanding of 1T-physics, but beyond this, 2T-physics offers new vistas in the search of the unified theory while raising deep questions about the meaning of spacetime. In these lectures, the recent developments in the gauge field theory formulation of 2T-physics will be described after a brief review of the results obtained so far in the worldline approach. A crucial element in the formulation of 2T-physics [1]-[10] is an Sp(2, R) gauge symmetry in phase space. All new phenomena in 2T-physics (including the two times) can be traced to the presence of this gauge symmetry and its generalizations. In the space of all worldline theories (i.e. all possible background fields) there is an additional symmetry that corresponds to all canonical transformations in the phase space of a particle. After describing the role of these two symmetries in the worldline formalism we will discuss field theory. In field theory these two symmetries combine and get promoted to noncommutative U ⋆ (1, 1) acting on fields as functions of noncommutative phase space. U ⋆ (1, 1) symmetry provides the foundation of 2T-physics in field theory, and leads to a unification of various gauge principles in ordinary field theory, including Maxwell, Einstein and high-spin gauge principles.