What is a Spinor? Is it the Coefficients of a State Ket?

[captain]

is a spinor essentially the coeffients of the base kets of an abitrary state ket?

 

[genneth]

No. A spinor is a geometrical object, like a vector or a tensor.

 

[jostpuur]

If you choose basis kets to be the spin-up and spin-down states, then the coefficients form a spinor, but that is special case.

 

[StatusX]

Consider a spin-1/2 particle. The spin state of such a particle is determined by giving the (unique) direction along which a measurement of spin is guaranteed to give a value of \hbar/2. The space of such states is given by a 2 dimensional complex vector space (a Hilbert space). For example, we could take the state corresponding to the positive z-direction to be represented by (1,0) and the negative z-direction to be (0,1), and then all other directions would be represented by linear combinations of these to states.

These spin states correspond to geometric objects called spinors. Even though they seem to be nothing more than ordinary vectors, there is a slight difference, which is that if you were to rotate a spinor 360 degrees and return it to its original direction, it will not be the same as it started, but will have been negated. For example, as we rotate (1,0) around we get (0,1) after 180 degrees, then (-1,0) after 360. If we continue, we then get (0,-1), and finally (1,0). It takes two rotations for the spinor to return to its original form. Since states are only determined up to a constant anyway, there is no measurable difference between (1,0) and (-1,0), but this property has important consequences, for example, the pauli exclusion principle can be directly related to it.

 

[captain]

thanks for you help much appreciated

but how can SO(3) and SU(2) be rotation groups. I know that the SO(3) groups consists of the rotation group and the SU(2) group rotates the spinor, but how are those related.

or is the SU(2) groups considered as a rotation group only for spin 1/2 systems?

 

[StatusX]

SU(2) is the rotational group of the set of spinors, while SO(3) is the rotational group of ordinary 3 dimensional vectors. SU(2) is what is called the double cover of SO(3), which means there is a surjective, 2:1 homomorphism from SU(2) to SO(3). As I said, for spinors it takes two rotations to return a spinor to its original form. So a single rotation corresponds to a trivial element in SO(3), but to a non-trivial element in SU(2), in fact, the other element in SU(2) which maps under the above mentioned homomorphism to the identity in SO(3).

 

[dextercioby]

A spinor is a vector, i.e. an element of a finite dimensional vector space which carries a representation of the \mbox{Spin(n)} or \mbox{Spin(1,n)} groups, which are the universal covering groups of the proper rotation groups in n-dimensional Euclidean space (nonrel QM) or (n+1)-dimensional Minkowski space (rel QM), which are isomorphic to \mbox{SO(n)} and \mbox{SO}_{+}\mbox{(1,n)}, respectively. The isomorphisms \mbox{Spin(3)}\cong\mbox{SU(2)} and \mbox{Spin(1,3)}\cong\mbox{SL}(2,\mathbb{C}) are well known.

 

[genneth]

There's a slight unfortunateness in names. Spinors, vectors (tangent ones), covectors, forms, tensors, etc. are all vectors, in the sense that they form vector spaces -- perhaps better called a linear space, at least for this post. However, they have structure above and beyond that. They are all geometrical objects. Tangent vectors are a formal, algebraic realisation of what we intuitively know about little arrows. Covectors are like contour lines. I've heard of spinors being visualised as a flagpole with a flag on the end, but I've never really understood that. Anyway, when we impose a basis on these different objects, and do the same transforms, the components change in different ways -- we say that they are different representations of the transformation group.

So one way to motivate the existence of spinors, is by noting that in 3D, vectors, covectors, and tensors in general are (or carry) representations of the Lie group SO(3). However, it happens that the Lie algebra, so(3) is isomorphic to su(2). Furthermore, it happens that SU(2) is in fact "simpler" than SO(3). And weirdly, it seems that nature knows this, by making the spin of electrons elements of SU(2) (a spinor) rather than SO(3) (a vector).

 

[dextercioby]

Yes, you might say that nature "knows" this local isomorphism b/w SO(3) and SU(2), but let's not forget the fact that the human mind, by choosing the Hilbert space as the mathematical environment for QM and very cleverly making the identification {rays}={state vectors} instead of the probably more natural {vectors}={state vectors}, got both the existence and the description of the spin of particles as a direct result.

Of course, you might claim that the human mind is a part of nature And mother Nature then is really, really smart.