String without strings

Scale invariance provides a principled reason for the physical importance o f Hilbert space, the Virasors algebra, the string mode expansion, canonical commutators and Schrodinger evolution of states, independent of the assump tions of string theory and quantum theory. The usual properties of dimensio nful fields imply an infinite, projective tower of conformal weights associ ated with the tangent space to scale-invariant spacetimes. convergence and measurability on this tangent tower are guaranteed using a scale-invariant norm, restricted to conformally self-dual vectors. Maps on the resulting Hi lbert space are correspondingly restricted to semi-definite conformal weigh t. We find the maximally- and minimally-commuting, complete Lie algebras of definite-weight operators. The projective symmetry of the tower gives thes e algebras central charges, giving the canonical commutator and quantum Vir asoro algebras, respectively. Using a continuous, m-parameter representatio n for rank-in tower tensors, we show that the parallel transport equation f or the momentum-vector of a particle is the Schrodinger equation, while the associated definite-weight operators obey canonical commutation relations. Generalizing to the set of integral curves of general timelike, self-dual vector-valued weight maps gives a lifting such that the action of the curve s parallel transports arbitrary tower vectors. We prove that the full set o f Schrodinger-lifted integral curves of a general self-dual map gives an im mersion of its 2-dim parameter space into spacetime, inducing a Lorentzian metric on the parameter space. This immersion is shown to satisfy the full variational equations of open string.