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S6g. What are interpolating fields?
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Traditional QFT has rules for computing reasonable approximations
to the S-matrix of a field theory. The S-matrix describes the behavior
of a state of the system under a transition from time t=-inf to time 
t=+inf. But in a complete dynamical theory, one would like to be able 
know what happens in-between at finite times. In nonrelativistic QM,
this information is given by the Schroedinger equation. In QFT it is
given by the interpolating field - called interpolation since it 
interpolates between the infinite limiting times.

More precisely, the dynamical information about the interpolating 
field is represented mathematically in the Wightman functions,
which give the (renormalized) vacuum expectations of field products
at arbitrary combination of space-time points.

Unfortunately, no one knows how to compute the latter in relativistic
$D quantum field theories. However, Wightman functions have been
constructed rigorously in lower dimension (more precisely 
in certain superrenormalizable theories in 2 and 3 dimensions).