 
Summary: Does quantum field theory exist?
Final Lecture, Mathematics 189, Spring 1998
W. Arveson
We have seen how to quantize a finite dimensional classical mechanical system
(this will be reveiwed briefly below). The purpose of this lecture is to formulate a
mathematical problem associated with the quantization of interacting Boson fields,
to describe the serious mathematical problems assiciated with finding a (mathe
matically sound) solution to the problem, and to make some remarks about the
nature of the difficulties. The problem remains unsolved to this day.
Consider a nonlinear wave equation in four spacetime dimensions of the form
(1) + + 3
= 0.
Here, = (t, x, y, z) is a function defined on R×R3
, and are positive constants,
and is the wave operator
=
2
t2
 =
