- #1
nateHI
- 145
- 4
I think I get the difference between spinors and tensors in the context of algebraic topology and QM but I want someone to scrutinize my understanding before I move on to another topic. I've never had a class in topology so I might be using some math terms incorrectly.
The 3D parameterized space (solid sphere) that describes rotations (let's call it r.s.) with arguments theta, sai (axis of rotation, n, in euclidean space (e.s.)), and kai ( the amount of that rotation in e.s. and the magnitude of the vector in r.s.) has doubly connected groups and simply connected groups. In r.s., the doubly connected groups arise from the fact that there is more than one way to describe rotations by an amount of pi. This non-uniqeness of some rotations equates to points on the surface of the r.s. sphere being equivalent to their polar opposite on the other side of the r.s. solid sphere.
-Simply connected groups are the more intuitive of the two groups and their paths can be shrunk to a point while not leaving the space of r.s.
-Doubly conencted groups are groups that are created by paths that exploit the non uniqness of rotations and leave the space r.s. Doubly connected groups have paths that can be shrunk to a point also but require you to travel double the path using a trick I can't describe in words.
What is the connection to Q.M. you ask? Total angular momentum which is the sum of spin and orbital angular momentum is related to the concept of r.s. because, well, simply put, anglular momentum consists of different types of rotations ( i.e. J=L+S).
My understanding starts to wain even more at this point but my intuition tells me that orbital angular momentum (tensors) is described by the group of simply connected paths in r.s. and spinor are rotations described by doubly connected paths in r.s. Also, the discreteness of spin (S) arises from the jump outside of r.s. that doubly connected groups undergo. Am I correct?
The 3D parameterized space (solid sphere) that describes rotations (let's call it r.s.) with arguments theta, sai (axis of rotation, n, in euclidean space (e.s.)), and kai ( the amount of that rotation in e.s. and the magnitude of the vector in r.s.) has doubly connected groups and simply connected groups. In r.s., the doubly connected groups arise from the fact that there is more than one way to describe rotations by an amount of pi. This non-uniqeness of some rotations equates to points on the surface of the r.s. sphere being equivalent to their polar opposite on the other side of the r.s. solid sphere.
-Simply connected groups are the more intuitive of the two groups and their paths can be shrunk to a point while not leaving the space of r.s.
-Doubly conencted groups are groups that are created by paths that exploit the non uniqness of rotations and leave the space r.s. Doubly connected groups have paths that can be shrunk to a point also but require you to travel double the path using a trick I can't describe in words.
What is the connection to Q.M. you ask? Total angular momentum which is the sum of spin and orbital angular momentum is related to the concept of r.s. because, well, simply put, anglular momentum consists of different types of rotations ( i.e. J=L+S).
My understanding starts to wain even more at this point but my intuition tells me that orbital angular momentum (tensors) is described by the group of simply connected paths in r.s. and spinor are rotations described by doubly connected paths in r.s. Also, the discreteness of spin (S) arises from the jump outside of r.s. that doubly connected groups undergo. Am I correct?
Last edited: