Off-shell string physics

Recent advances in non-critical string theory allow a unique continuation, preserving conformal invariance, of critical Polyakov string amplitudes to off-shell momenta. These continuations possess unusual, apparently stringy, characteristics, which are unlikely to be reproduced in a string field theory. Thus our results may be an indication that some fundamentally new formulation, other than string field theory, will be required to extend our understanding of critical strings beyond the Polyakov path integral. Three-point functions are explicitly calculated. The tree-level effective potential is computed for the tachyon. (This preprint includes some computations used to arrive at results mentioned in hep-th/9211016.)