ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

11. Mathematical Physics

1991 MS Classification: 05, 47, 81 Kisil, Vladimir V., Odessa State University, Ukraine Feynman Path Integral, Umbral Claculus, and Non-Commutative Combinatorics There are a lot of similar patterns in different areas of science. For example, the Feynman path integral in quantum mechanics and umbral calculus in combinatorics share many similar properties:

1. They provides an effective transition from relatively tractable classic language to more complicated quantum/combinatorial case.

2. The technique is not a formalism but merely a symbolism, namely the suggestive notations (not formal rules) form the contents of the method.

3. The approach is not an algorithm but a guide, which should be adjusted to need of every concrete problem.

4. A rigor mathematical basis was absent for a while. We demonstrate that the similarity is not just a coincidence. In fact the Feynman path integral could be viewed as an umbral calculus for noncommutative combinatorics. Reference Vladimir V. Kisil. The umbral calculus: a model from convoloids. page 25, 1997. E-print: funct-an/9704001.