Some remarks on arithmetic physics *

There have been some recent speculations on connections between quantum theory and modern number theory. At the boldest level these suggest that there are two ways of viewing the quantum world, the usual and the arithmetic, which are in some sense complementary. At a more conservative level they suggest that there is much mathematical interest in examining structures which are important in quantum theory and analyze to what extent they make sense when the real and complex fields are replaced by the more unconventional fields and rings, like finite or nonarchimedean fields and adele rings, that arise in number theory. This paper explores some aspects of these questions.