# Fundamental Concepts of Mathematics and Quantum Formalism

## Abstract

To understand the concept of quantum probability and its application to the cognitive domain, it is necessary to explain the basic concepts of quantum theory. Again, to understand the basic concepts of quantum theory one needs to understand the formalism of Hilbert space. There are several postulates in understanding quantum theory. These postulates are stated in this chapter in a simplistic manner without much mathematical rigor. Von Neumann introduced the projection postulate to understand the measurement process, and this postulate is discussed here in detail. Some experiments like Stern–Gerlach experiment play a crucial role in the development of this theory, a short description of which is given here for convenience. Since mathematical structure like Hilbert space is needed for the mathematical formulation based on these postulates, the various basic notions such as linear vector space, norm, inner product, etc. are defined here. The concept of observable is replaced by the self-adjoint operator in quantum theory. To make grasp those concepts, it is necessary to have some preliminary knowledge about the properties, especially those of self-adjoint operators which are elaborated here. Heisenberg’s uncertainty relationships in the context of unsharp observables are discussed. This may help in understanding the current status of research, as well as the developments of the problems related to cognitive science in a more realistic manner.

## Keywords

Quantum postulates Projection postulates Stern–Gerlach experiment Unsharp observable Heisenberg uncertainty relation Vector space Hilbert space Cognitive science## References

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