We show that a simple $\mathrm{OSp}\mathrm{}(1/2)\mathrm{}$ world line gauge theory in 0-brane phase space $X^M\left(\tau {),P}^M\right(\tau )$ with spin degrees of freedom ${\psi }^M\left(\tau \right),$ formulated for a $\mathrm{}(d+2\mathrm{})\mathrm{}$-dimensional spacetime with two times $X^0\left(\tau {),X}^{0^{\prime }}\right(\tau ),$ unifies many physical systems which ordinarily are described by a one-time formulation. Different systems of one-time physics emerge by choosing gauges that embed ordinary time in $d+2\mathrm{}$ dimensions in different ways. The embeddings have different topology and geometry for the choice of time among the $d+2\mathrm{}$ dimensions. Thus, two-time physics unifies an infinite number of one-time physical interacting systems, and establishes a kind of duality among them. One manifestation of the two times is that all of these physical systems have the same quantum Hilbert space in the form of a unique representation of $\mathrm{SO}\mathrm{}(d,2\mathrm{})\mathrm{}$ with the same Casimir eigenvalues. By changing the number of spinning degrees of freedom ${\psi }_a^M\left(\tau \right),$ $a=1,2,\dots ,n$ (including no spin $n=0\mathrm{}$), the gauge group changes to $\mathrm{OSp}\mathrm{}(n/2\mathrm{}).$ Then the eigenvalue of the Casimir operators of $\mathrm{SO}\mathrm{}(d,2\mathrm{})\mathrm{}$ depend on $n$ and the content of the one-time physical systems that are unified in the same representation depend on $n.\mathrm{}$ The models we study raise new questions about the nature of spacetime.

• Received 18 June 1998

DOI:https://doi.org/10.1103/PhysRevD.58.106004