# Understanding Spinors & Tensors in QM & Algebraic Topology

• nateHI
In summary, the conversation discusses the difference between spinors and tensors in the context of algebraic topology and quantum mechanics. It is mentioned that there can be doubly connected and simply connected groups in the 3D parameterized space that describes rotations, and that these groups can be related to the concept of angular momentum in QM. There is also a connection between Lie algebras and algebraic topology, specifically in the context of Lie group representation theory.
nateHI
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?

Last edited:
nateHI said:
[...]the discreteness of spin (S) arises from the jump outside of r.s. that doubly connected groups undergo.

Since this is in the context of QM, I would have said that discreteness of angular momentum
eigenvalues arises because we're representing the generators of the angular momentum
Lie algebra as operators on a Hilbert space.

strangerep said:
Since this is in the context of QM, I would have said that discreteness of angular momentum
eigenvalues arises because we're representing the generators of the angular momentum
Lie algebra as operators on a Hilbert space.

OK, I agree. In that case, is there any connection between Lie Algebras and Algebraic Topology? If so, I suppose that would be my next area of interest.

It's rather a connection between Algebraic Topology and Lie Group Representation Theory.

## 1. What are spinors and tensors in quantum mechanics and algebraic topology?

Spinors and tensors are mathematical objects used to describe the properties of particles and fields in quantum mechanics and topology. Spinors represent the intrinsic angular momentum of particles, while tensors describe the geometric properties of spaces and fields.

## 2. How are spinors and tensors related to each other?

Spinors and tensors are both types of mathematical objects known as tensors. However, they have different transformation properties under rotations and reflections, making them useful for describing different physical phenomena.

## 3. What is the role of spinors and tensors in quantum mechanics?

In quantum mechanics, spinors and tensors are used to describe the properties of particles and fields, such as their spin, momentum, and energy. They are essential for understanding the behavior of subatomic particles and the interactions between them.

## 4. What is the significance of spinors and tensors in algebraic topology?

In algebraic topology, spinors and tensors are used to study the properties of topological spaces, such as their connectivity and curvature. They are important tools for understanding the structure of complex spaces and the relationships between them.

## 5. How can understanding spinors and tensors improve our understanding of the universe?

By using spinors and tensors, scientists are able to accurately describe and predict the behavior of particles and fields in the quantum world and the structure of spaces in the universe. This allows us to better understand the fundamental laws and principles that govern the behavior of our universe.

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