  We consider quantum field theory on a spacetime representing the Big
Crunch/Big Bang transition postulated in the ekpyrotic or cyclic cosmologies.
We show via several independent methods that an essentially unique matching
rule holds connecting the incoming state, in which a single extra dimension
shrinks to zero, to the outgoing state in which it re-expands at the same rate.
For free fields in our construction there is no particle production from the
incoming adiabatic vacuum. When interactions are included the total particle
production for fixed external momentum is finite at tree level. We discuss a
formal correspondence between our construction and quantum field theory on de
Sitter spacetime.
