How do spinors differ from tensors?

[pellman]

In http://relativity.livingreviews.org/Articles/lrr-2004-2/" (section 2.1.5.2) the following is the first sentence in the section reviewing spinors:

"Spinors are representations of the Lorentz group only; as such they are related strictly to the tangent space of the space-time manifold."

The wording suggests that this is a way in which they differ from tensors. So in what sense are tensors not strictly related to the tangent space?

 

[bcrowell]

The big difference is that under a 360-degree rotation, a spinor reverses sign, whereas a tensor is left unchanged. That is, spinors behave like spin 1/2.

 

[arkajad]

The wording is indeed somewhat unfortunate, but already the next sentence of the paper makes it clear what is being meant:

"To see how spinor representations can be obtained, we must use the 2–1 homomorphism of the group SL(2,C) and the proper orthochronous Lorentz group, a subgroup of the full Lorentz group."

 

[pellman]

The wording is indeed somewhat unfortunate, but already the next sentence of the paper makes it clear what is being meant:

"To see how spinor representations can be obtained, we must use the 2–1 homomorphism of the group SL(2,C) and the proper orthochronous Lorentz group, a subgroup of the full Lorentz group."

I know that this statement is true... but how does it fit with the statement about being related strictly to the tangent space of the manifold.

Anyway, aren't tensors objects that "live" only on the tangent space as well?

 

[Fredrik]

A tensor field of type (k,l) is a section of the bundle of (k,l) tensors over spacetime. (Ask if you don't know what that means). A spinor field is a section of...some other kind of vector(?) bundle over spacetime. I'm embarrassed to admit that I don't know how it's defined. Perhaps someone else can tell both of us.

 

[arkajad]

A spinor field is a section of...some other kind of vector(?) bundle over spacetime. I'm embarrassed to admit that I don't know how it's defined. Perhaps someone else can tell both of us.

It's a little bit tricky. It can be done in several ways. For instance:

Let L be the connected component of the identity of the Lorentz group. Suppose we have spacetime M that is oriented and time oriented. Its orthonormal frames bundle is a principal bundle with L as its structure group. So far so good. We know we have 2:1 homomorphism SL(2,C) -> L.

We say that M admits spin structure if there exists an SL(2,C) bundle Of "spin frames" and 2:1 bundle morphism onto the orthonormal bundle that commutes with the group action. That is when you rotate spin frame by an SL(2,C) matrix A, then the corresponding to it Lorentz frame rotates by the corresponding Lorentz transformations.

There are theorems telling us which spacetimes admit spin structures and how many inequivalent. Now once you have spin frames - you define spinors as sections of associated vector bundles.

This is one method. Another method is to construct Clifford algebra bundle over spacetime and then look for for vector bundles that are "modules" for the Clifford algebra bundle. That is for vector bundles fibers of which are representations spaces for the Clifford algebra fibers. This is how physicists do it - using gamma matrices and their commutation relations without a priori specifying the space on which these matrices act.

Finally there is a third method - searching for spinor bundle as a subbundle of the Clifford algebra bundle - it should consist of "minimal ideals".

In each of these cases it is difficult to find a universal and constructive method of defining spinors, a method that would be valid for all topologically fancy spacetimes. At least I do not know such a method.

 

[George Jones]

The wording is indeed somewhat unfortunate, but already the next sentence of the paper makes it clear what is being meant:

"To see how spinor representations can be obtained, we must use the 2–1 homomorphism of the group SL(2,C) and the proper orthochronous Lorentz group, a subgroup of the full Lorentz group."

A tensor field of type (k,l) is a section of the bundle of (k,l) tensors over spacetime. (Ask if you don't know what that means). A spinor field is a section of...some other kind of vector(?) bundle over spacetime. I'm embarrassed to admit that I don't know how it's defined. Perhaps someone else can tell both of us.

This is the way I think of spinors. I have forgotten most of what I knew about Spin bundles, but see Fecko, Frankel, or Nakahara.

Maybe the paper means this: the Lorentz group is the symmetry group (if translation are neglected) of special relativity, and spinors are representation spaces of the double cover of the Lorentz group. What about representations of the double cover of the diffeomorphism group GR? Ne'eman did some famous work on this,

http://cdsweb.cern.ch/record/352618/files/9804037.pdf.

 

[dx]

A tensor field of type (k,l) is a section of the bundle of (k,l) tensors over spacetime. (Ask if you don't know what that means). A spinor field is a section of...some other kind of vector(?) bundle over spacetime. I'm embarrassed to admit that I don't know how it's defined. Perhaps someone else can tell both of us.

http://en.wikipedia.org/wiki/Spin_bundle

See also "Physics and Geometry" by Witten. Not too much detail on this specifc topic, but worth looking at.

 

[cesiumfrog]

It's a little bit tricky. It can be done in several ways. For instance:

Let L be the connected component of the identity of the Lorentz group. Suppose we have spacetime M that is oriented and time oriented. Its orthonormal frames bundle is a principal bundle with L as its structure group. So far so good. We know we have 2:1 homomorphism SL(2,C) -> L.

We say that M admits spin structure if there exists an SL(2,C) bundle Of "spin frames" and 2:1 bundle morphism onto the orthonormal bundle that commutes with the group action. That is when you rotate spin frame by an SL(2,C) matrix A, then the corresponding to it Lorentz frame rotates by the corresponding Lorentz transformations.

There are theorems telling us which spacetimes admit spin structures and how many inequivalent. Now once you have spin frames - you define spinors as sections of associated vector bundles.

This is one method. Another method is to construct Clifford algebra bundle over spacetime and then look for for vector bundles that are "modules" for the Clifford algebra bundle. That is for vector bundles fibers of which are representations spaces for the Clifford algebra fibers. This is how physicists do it - using gamma matrices and their commutation relations without a priori specifying the space on which these matrices act.

Finally there is a third method - searching for spinor bundle as a subbundle of the Clifford algebra bundle - it should consist of "minimal ideals".

In each of these cases it is difficult to find a universal and constructive method of defining spinors, a method that would be valid for all topologically fancy spacetimes. At least I do not know such a method.

Thankyou for that post.