Quaternions and Hilďert spaĐes By J.A.J. van Leunen Last modified: 10 oktober 2015 Abstract This is a compilation of quaternionic number systems, quaternionic function theory, quaternionic Hilbert spaces and Gelfand triples. The difference between quaternionic differential calculus and Maxwell based differential calculus is explained. Contents 1 Introduction..................................................................................................................................... 3 2 Quaternion geometry and arithmetic ............................................................................................. 3 2.1 Notation ..................................................................................................................................... 3 2.2 Sum ............................................................................................................................................ 4 2.3 Product ....................................................................................................................................... 4 2.4 Norm .......................................................................................................................................... 4 2.5 Quaternionic rotation ................................................................................................................ 4 3 The separable Hilbert space ℌ ........................................................................................................ 5 3.1 Notations and naming conventions ........................................................................................... 5 3.2 Quaternionic Hilbert space ........................................................................................................ 6 3.2.1 Ket vectors ....................................................................................................................... 6 3.2.2 Bra vectors ....................................................................................................................... 6 3.2.3 Scalar product.................................................................................................................. 7 3.2.4 Separable ......................................................................................................................... 8 3.2.5 Base vectors .................................................................................................................. 8 3.2.6 Operators ....................................................................................................................... 8 3.2.7 Unit sphere of ℌ ............................................................................................................ 15 3.2.8 Bra-ket in four dimensional space ................................................................................. 15 3.2.9 Closure ........................................................................................................................... 16 3.2.10 Canonical conjugate operator P .................................................................................... 16 3.2.11 Displacement generators .............................................................................................. 17 3.3 Quaternionic L² space .............................................................................................................. 17 4 Gelfand triple................................................................................................................................. 18 4.1 Understanding the Gelfand triple ............................................................................................ 18