  Timelike T-duality of string theory appears as a symmetry of time evolution
in string field theory, exchanging evolution through times t and 1/t, and
exchanging boundary states with backgrounds. This is demonstrated by
constructing the string field Schrodinger functional, the generator of time
evolution, based on Feynman diagram arguments and in analogy with quantum field
theory. There, the functional can be described using only properties of first
quantised particles on a timelike orbifold. Using new sewing rules applicable
to both open and closed strings we generalise this approach to bosonic string
theory and express the string field Schrodinger functional in terms of strings
on S^1/Z_2.
