The intriguing phrase in the title

Diffusion, Quantum Theory, and Radically Elementary Mathematics, edited by William G. Faris. Mathematical notes 47. Princeton, NJ : Princeton University Press, 2006.

TC Mathematics Library QC21.3 .D54 2006 Link to MnCat Record

turns out not to be new:

Radically Elementary Probability Theory, by Edward Nelson. Annals of mathematics studies no. 117. Princeton, N.J.: Princeton University Press, 1987. TC Mathematics Library QA1 .A626 no.117 Link to MnCat Record

The connection between the books is explicit, since the more recent one resulted from a conference in honor of the author of the older one, Edward Nelson (Princeton).

From the former:

"This volume explains diffusion motion and its relation to both nonrelativistic quantum theory and quantum field theory. It also shows how diffusive motion concepts lead to a radical reexamination of the structure of mathematical analysis. . . an infinitesimal approach to diffusion and related probability topics that is radically elementary in the sense that it relies only on simple logical principles."

From the latter:

"This work is an attempt to lay new foundations for probability theory, using a tiny bit of nonstandard analysis. The mathematical background required is little more than that which is taught in high school, and it is my hope that it will make deep results from the modern theory of stochastic processes readily available to anyone who can add, multiply, and reason. What makes this possible is the decision to leave the results in nonstandard form. . . .Mathematicians are quite reightly conservative and suspicious of new ideas. They will ask whether the results developed here are as powerful as the conventional results, and whether it is worth their while to learn nonstandard methods. These questions are addressed in an appendix. . . . The purpose of this appendix is to demonstrate that theorems of the conventional theory of stochastic processes can be derived from their elementary analogues by arguments of the type usually described as generalized nonsense; there is no probabiliistic reasoning in this appendix. This shows that the elementary nonstandard theory of stochastic processes can be used to derive conventional results; on the other hand, it shows that neither the elaborate machinery of the conventional theory nor the devices from the full theory of nonstandard analysis, needed to prove the equivalence of the elementary results with their conventional forms, add anything of significance: the elementary theory has the same scientific content as the conventional theory. This is intended as a self-destructing appendix."