Field theory of relativistic strings. I. Trees

We present an entirely new kind of field theory, a field theory quantized not at space-time points, but quantized along an extended set of multilocal points on a string. This represents a significant departure from the usual quantum field theory, whose free theory represents a definite set of elementary particles, because the field theory on relativistic strings can accommodate an infinite set of linearly rising Regge trajectories. In this paper, we (1) present canonical quantization and the Green's function of the free string, (2) introduce three-string interactions, (3) resolve the question of multiple counting, (4) complete the counting arguments for all $N$-point trees, and (5) introduce four-string interactions which yield a Yang-Mills structure when the zero-slope limit is taken. In future work, we will discuss loops and explicitly calculate Reggeon-Pomeron and Pomeron-Pomeron interactions.