3 ms·

> I have never seen someone try to use Pauli matrices to solve a trigonometry problem, but it can certainly be done. The Pauli matrices in this context are iso

> I have never seen someone try to use Pauli matrices to solve a trigonometry problem, but it can certainly be done. The Pauli matrices in this context are isomorphic to quaternions, which certainly have been used to solve geometric problems in 3 dimensions (although not necessarily by physicists), as has been discussed many many times here on HN. The property of describing spin-1/2 particles (i.e. generating SU(2)) is precisely the same property that makes the quaternions amenable for use in reasoning about 3D rotations!

thatcherc 5y ago
The relationship between 3D rotations, spin-1/2 particles, and unit quaternions appears on one of the best-named Wikipedia pages: Exceptional Isomorphisms! [0] [0] - https://en.wikipedia.org/wiki/Exceptional_isomorphism#Spin_groups https://en.wikipedia.org/wiki/Exceptional_isomorphism#Spin_g...
jacobolus 5y ago
> describing spin-1/2 particles (i.e. generating SU(2)) This has the relationship backwards. Particles have this spin group because they can rotate in 3-dimensional space, not because they have some mysterious association with the complex plane. That we can represent rotations using 2x2 unitary matrices of unit determinant is just a coincidence. There are a bunch of such coincidental isomorphism between groups. The explanation is comparable to the “strong law of small numbers” https://en.wikipedia.org/wiki/Strong_law_of_small_numbers https://en.wikipedia.org/wiki/Strong_law_of_small_numbers (The unitary matrices come about when we take points on the 2-sphere and stereographically project them onto the plane, representing points in the plane by complex numbers. Then concentric rotations of 3-space of correspond to particular Möbius transformations of the complex plane, which can be represented as the special unitary matrices. This works out cleanly when projecting onto a 2-dimensional plane, but representing rotations gets trickier in higher dimensions.) * * * But you are missing Jason’s point. There are clearly many possible ways of representing the same relationships. The underlying relationships don’t change if you e.g. use classical spherical trigonometry. The question he is trying to ask instead is: which representation is most conceptually clear and intuitive to work with (after some experience), has the nicest and easiest to manipulate notation, etc. There are valid reasons to use a stereographic representation of points on the sphere. (For instance, it involves 2 coordinates instead of 3.) And from there, representing spherical rotations as Möbius transformations is convenient and effective. But conceptually, if trying to solve arbitrary problems with pen and paper, representing points as vectors and rotations as scalar+bivector "quaternions" is a lot more natural. Especially if you have a problem where some parts are not confined to the sphere. For more on representing spherical geometry stereographically, cf. https://observablehq.com/@jrus/planisphere https://observablehq.com/@jrus/planisphere – there are some tools even here where we get leverage out of treating stereographically projected points as 2D vectors rather than as complex numbers, and separating the concepts of scalar, vector, and bivecor.

thatcherc 5y ago

The relationship between 3D rotations, spin-1/2 particles, and unit quaternions appears on one of the best-named Wikipedia pages: Exceptional Isomorphisms! [0] [0] - https://en.wikipedia.org/wiki/Exceptional_isomorphism#Spin_groups https://en.wikipedia.org/wiki/Exceptional_isomorphism#Spin_g...

jacobolus 5y ago

> describing spin-1/2 particles (i.e. generating SU(2)) This has the relationship backwards. Particles have this spin group because they can rotate in 3-dimensional space, not because they have some mysterious association with the complex plane. That we can represent rotations using 2x2 unitary matrices of unit determinant is just a coincidence. There are a bunch of such coincidental isomorphism between groups. The explanation is comparable to the “strong law of small numbers” https://en.wikipedia.org/wiki/Strong_law_of_small_numbers https://en.wikipedia.org/wiki/Strong_law_of_small_numbers (The unitary matrices come about when we take points on the 2-sphere and stereographically project them onto the plane, representing points in the plane by complex numbers. Then concentric rotations of 3-space of correspond to particular Möbius transformations of the complex plane, which can be represented as the special unitary matrices. This works out cleanly when projecting onto a 2-dimensional plane, but representing rotations gets trickier in higher dimensions.) * * * But you are missing Jason’s point. There are clearly many possible ways of representing the same relationships. The underlying relationships don’t change if you e.g. use classical spherical trigonometry. The question he is trying to ask instead is: which representation is most conceptually clear and intuitive to work with (after some experience), has the nicest and easiest to manipulate notation, etc. There are valid reasons to use a stereographic representation of points on the sphere. (For instance, it involves 2 coordinates instead of 3.) And from there, representing spherical rotations as Möbius transformations is convenient and effective. But conceptually, if trying to solve arbitrary problems with pen and paper, representing points as vectors and rotations as scalar+bivector "quaternions" is a lot more natural. Especially if you have a problem where some parts are not confined to the sphere. For more on representing spherical geometry stereographically, cf. https://observablehq.com/@jrus/planisphere https://observablehq.com/@jrus/planisphere – there are some tools even here where we get leverage out of treating stereographically projected points as 2D vectors rather than as complex numbers, and separating the concepts of scalar, vector, and bivecor.