W. Pauli: The connection between spin and statistics

[jostpuur]

http://prola.aps.org/abstract/PR/v58/i8/p716_1

I'm trying to read this, and it's not going very well!

On the second page:

A tensor or spinor which transforms irreducibly under this group can be characterized by two integral positive numbers $(p,q)$. (The corresponding "angular momentum quantum number" $(j,k)$ are then given by $p=2j+1$, $q=2k+1$, with integral or half-integral $j$ and $k$.)

What two numbers is Pauli talking about? Isn't a spinor of a particle usually characterized by a one number?

 

[dextercioby]

Relativistic spinors are characterized by 2 semiinteger positive numbers, corresponding to the nonunitary, finite dimensional representations of the SL(2, C) group.

 

[jostpuur]

Is this right:

If a spinor of a non-relativistic particle is characterized by a number $j$ (which is integer or half-integer), then the wave function has $2j+1$ components, that means it is of form

$$\psi(x) = \left(\begin{array}{c}<br /> \psi_{j}(x) \\ \psi_{j-1}(x) \\ \vdots \\ \psi_{-j}(x) \\<br /> \end{array}\right)$$

If that was right, how is the situation changed when we give up the assumption of non-relativisticness?

 

[dextercioby]

We basically go, in group theory language, from the double cover of SO(3) = SU(2) (Galilei case) to the double cover of $SO_{o} (1,3)$ = SL(2,C).

In term of Lie algebras (= stands for isomorphisms),

so(3)=su(2), finite dimensional irreds described by one parameter.

so(1,3) = sl(2,c) = su(2) $\oplus$ su(2), thus in the special relativistic case the numbers of parameters describing the irreds is double (2).

 

[jostpuur]

If I want to have a transformation group that transforms some objects under rotations, the group will need to be parametrized with three variables. If I then want to extend the transformation to be applied with (relativistic) boosts too, the amount of parameters must be increased to six. Is this, what the two indices are all about?

That doesn't make fully sense. The amount of parameters in rotations and boosts are always going to be three in both. How is this related to the $j$, which is related to the amount of components in spinor?

That almost sounds as if a particle could transform like spin-1/2 particle in rotations, and like spin-1 particle in boosts. Wouldn't make any sense!

 

[element4]

See also this http://en.wikipedia.org/wiki/Representations_of_the_Lorentz_Group" page.

 

[element4]

If I want to have a transformation group that transforms some objects under rotations, the group will need to be parametrized with three variables. If I then want to extend the transformation to be applied with (relativistic) boosts too, the amount of parameters must be increased to six. Is this, what the two indices are all about?

That doesn't make fully sense. The amount of parameters in rotations and boosts are always going to be three in both. How is this related to the $j$, which is related to the amount of components in spinor?

That almost sounds as if a particle could transform like spin-1/2 particle in rotations, and like spin-1 particle in boosts. Wouldn't make any sense!

I think you are confusing the parameters needed for the group manifold, and the parameters labeling the irreducible representations.

As bigubau said, the irreducible representations of the Lorentz group can be labeled by a pair of "$$SU(2)$$ labels": $$(i,j)$$. For a given $$(i,j)$$, the dimension of the representation is (2j+1)(2i+1). A right and left handed Weyl spinor are labeled as $$(0, 1/2)$$ and $$(1/2, 0)$$, respectively. While a Dirac spinor, which is reducible, is labeled as $$(1/2,0)\oplus (0,1/2)$$ and has 4 components.

 

[jostpuur]

See also this http://en.wikipedia.org/wiki/Representations_of_the_Lorentz_Group" page.

This link turned out to be helpful. And this:

$$\mathfrak{so}(1,3)\otimes\mathbb{C} = \mathfrak{sl}(2,\mathbb{C})\oplus\mathfrak{sl}(2,\mathbb{C})$$

I'll return to this thread later.