A general approach to quantum fields and strings on adeles

Abstract

We develop a general framework for constructing quantum theories on the space of adeles. We consider complementary approaches utilizing both the functional integral and generalized Poincaré (translation) invariant Hilbert space methods. As a starting point, we seek to construct the most general quantum theory on the rational numbers. Very general arguments land us in the adeles. Quantum amplitudes generically will not factorize into separate- p-adic sectors. The vacuum amplitude, in particular, factorizes only in a linear regime. We conclude with implications for the cosmological constant and adelic string sigma models.