In this volume the author further develops his philosophy of quantum interpolation between the
real numbers and the p-adic numbers. The p-adic numbers contain the p-adic integers Zp which
are the inverse limit of the finite rings Z pn. This gives rise to a tree and probability
measures w on Zp correspond to Markov chains on this tree. From the tree structure one obtains
special basis for the Hilbert space L2(Zp w). The real analogue of the p-adic integers is the
interval [-1 1] and a probability measure w on it gives rise to a special basis for L2([-1 1]
w) - the orthogonal polynomials and to a Markov chain on finite approximations of [-1 1]. For
special (gamma and beta) measures there is a quantum or q-analogue Markov chain and a special
basis that within certain limits yield the real and the p-adic theories. This idea can be
generalized variously. In representation theory it is the quantum general linear group
GLn(q)that interpolates between the p-adic group GLn(Zp) and between its real (and complex)
analogue -the orthogonal On (and unitary Un )groups. There is a similar quantum interpolation
between the real and p-adic Fourier transform and between the real and p-adic (local unramified
part of) Tate thesis and Weil explicit sums.
