### Introduction

This paper is summary and enhancement of existing rather scattered literature regarding finite-dimensional quantum mechanics. In the later parts Feynman’s path summation is discussed.

Purpose of chapter 1 is to get familiar with finite-dimensional appoximation operator using discrete Fourier transformation as an example. In chapter 2 idea of inducing discrete kinematics using pair of mappings is discussed for special case of Schwinger approximation on flat configuration manifold R. In chapter 3 convergence question for defined Hilbert space imbedding of Swinger approximation on R is discussed. In chapter 4 most intuitive discrete-time evolution definitions are discussed. Special attention is paid to Feynman’s path integral. Feynman’s checkerboard problem closely connected to Feynman’s path integral is also included in this chapter. Read full Research Project here…

### Concluding remarks

Quantum mechanics can be effectively approximated on finite-dimensional Hilbert spaces, which is sometimes used for deriving propagators in Feynman’s path integral. Further research could be directed towards studying details of Feynman’s checkerboard problem, proving complementary theorems and applications on particular quantum systems. It might also be interesting investigating approximation mapping first N˜ discrete hermite functions to corresponding hermite functions and as reduction to map all hermite function to corresponding discrete hermite functions with scalar product defined such that first N˜ discrete hermite functions would be orthonormal basis.

### Author

Vaclav Kosar

### Acknowledgements

I would like to thank to prof. Jiri Tolar for support and guidance.