- #1
QMrocks
- 85
- 0
i am learning Quaternion now for my EM course. Can someone enlighten me on the correspondence between Quaternion and Pauli Matrix algebra?
The Hamilton multiplication rules differ from the Pauli matrix rules only by a factor of i. It is possible to formulate special relativity with Hamilton quaternions having complex coefficients(called biquaternions) and indeed it was first done that way(Silberstein). It turns out that the formulae of general relativity are simpler with the Pauli quaternions. There is also a very interesting (and possibly significant) relation between the Pauli quaternions and three dimensional Clifford Algebra
A quaternion is a four-dimensional complex number system that extends the three-dimensional system of complex numbers. It is represented as q = a + bi + cj + dk, where a, b, c, and d are real numbers and i, j, and k are imaginary units. Quaternions are commonly used in physics, engineering, and computer graphics for their ability to represent rotations in three-dimensional space.
Quaternions have several important properties, including:
Pauli matrices are a set of three 2x2 matrices that are used to represent spin states in quantum mechanics. They are also used in physics and engineering for rotations and transformations. Quaternions and Pauli matrices are related through the concept of spinors, which are mathematical objects used to represent rotations in higher dimensions. Quaternions can be used to construct spinors in four dimensions, while Pauli matrices can be used to construct spinors in three dimensions.
Quaternions and Pauli matrices are commonly used in computer graphics to represent rotations and transformations in three-dimensional space. Quaternions are used to represent rotations and Pauli matrices are used to represent orientations. They are used in applications such as 3D animation, video games, and virtual reality to create realistic and smooth movements and transformations.
While Quaternions and Pauli matrices have many useful properties, they also have some limitations. One drawback is that they can be more difficult to visualize and understand compared to traditional three-dimensional representations. Additionally, their non-commutativity can make certain calculations and operations more complex. However, these drawbacks are outweighed by their many practical applications in science and engineering.