Quantum theory and relativity

Elsewhere I describe personal encounters with George Mackey [31]. Those discussions often returned to a common set of questions: Does physics suggest a way to combine quantum theory, relativity, and interaction? To what extent can one formulate such quantum physics as mathematics? What is the present status of quantum field theory? What new mathematics emerges? What new insight does one apparently require to make further progress? We touch on these topics here, not only outlining some of the technical issues, but also attempting to address why one believes in their importance.