I don't follow the comment about analyticity. I thought fields were real analytic, which meant one needed to use a power series to expand them, which has a pote

I don't follow the comment about analyticity. I thought fields were real analytic, which meant one needed to use a power series to expand them, which has a potentially infinite amount of information in the Taylor coefficients? So I don't see how the local data "effectively" captures non local information.