Product spaces and Nelson's inequality

Certain inequalities for unbounded linear operators in L2 extend to product spaces. This product space property is used to give a new proof that Gross's inequality for matrices implies Nelson's inequality for ordinary differential operators, and that this in turn implies Nelson's inequality for partial differential operators in infinitely many dimensions. A new proof of Gross's inequality is given. There is also a discussion of the physical meaning of Nelson's inequality in quantum field theory. The article may serve as an introduction for non-specialists to some of the recent mathematical work in this subject. (Auth.)