Time Evolution in String Field Theory and T-Duality

The time evolution operator (Schrödinger functional) of quantum field theory can be expressed in terms of first quantised particles moving on S1/Z2. We give a graphical derivation of this that generalises to second quantised string theory. Tduality then relates evolution through time t with evolution through 1/t and an interchange of string fields and backgrounds. Introduction In the mechanics of particles and fields it is natural to consider the evolution in time of arbitrary configurations. In second quantised string theory this is not so straightforward, for example in Witten’s theory [1] the natural time variable is that at the mid-point of the string rather than a global time for the whole string. In this letter we will construct the time evolution operator for second quantised strings by analogy with that for field theory. We begin by showing that when the field theory Schrödinger functional is written in terms of propagators expressed in first quantised form then these describe particles moving on a timelike orbifold S/Z2. The first quantised propagators have an immediate generalisation to string theory, suggesting that the Schrödinger functional for second quantised strings can be expressed in terms of first quantised strings moving on this orbifold. To strengthen the analogy we give a graphical construction of the field theory Schrödinger functional which extends to both open and closed string theory. This avoids using a Lagrangian formulation of string field theory. Finally we study the effect of T-duality on time evolution. [email protected] [email protected]