Exploring Direct Sums of Lorentz Group Representations

[tomdodd4598]

Hey there,

I've suddenly found myself trying to learn about the Lorentz group and its representations, or really the representations of its double-cover. I have now got to the stage where the 'complexified' Lie algebra is being explored, linear combinations of the generators of the rotations and boosts have been defined, and the pairs of eigenvalues (m,n) of these linear combinations label the representations, where m and n are half-integers.

Now, all of this I think is fine (maybe I have some of the details wrong), but I've read something that has confused me a little on the Wikipedia article:
"Since for any irreducible representation for which m ≠ n it is essential to operate over the field of complex numbers, the direct sum of representations (m, n) and (n, m) have particular relevance to physics, since it permits to use linear operators over real numbers."

Could someone explain this in a little more detail? In particular, why do the irreducible representations for which m ≠ n not have relevance to physics, such as the (0, ½) or (½, 0)?

 

[PeterDonis]

why do the irreducible representations for which m ≠ n not have relevance to physics, such as the (0, ½) or (½, 0)?

The article doesn't say irreducible representations with $m \neq n$ do not have any relevance for physics. In the paragraph just above the one you quoted, it gives a possible physical interpretation of $(0, \frac{1}{2})$ and $(\frac{1}{2}, 0)$ (though not one we're likely to find in experiments any time soon given the current LHC results).

What the article says is that the direct sum representations $(m, n) \oplus (n, m)$, for $m \neq n$, have particular relevance for physics (i.e., get used a lot), because you can avoid complex numbers--in other words, the things that transform under these representations can have components that are all real numbers, which is nice if you want those components to describe things you can physically observe.

 

[tomdodd4598]

What the article says is that the direct sum representations $(m, n) \oplus (n, m)$, for $m \neq n$, have particular relevance for physics (i.e., get used a lot), because you can avoid complex numbers--in other words, the things that transform under these representations can have components that are all real numbers, which is nice if you want those components to describe things you can physically observe.

Ah, ok, so I just misunderstood that part - thanks. One last thing, hopefully: you say these symmetric sums, such as $(\frac{1}{2}, 0) \oplus (0, \frac{1}{2})$, can act on vectors which are composed of real numbers. Very often I see bispinors with complex components, so does the above imply that there is always some change of basis to one in which the components are all real, and if so, using this basis, does the above necessarily mean the representation is also composed of real numbers?

 

[PeterDonis]

Very often I see bispinors with complex components, so does the above imply that there is always some change of basis to one in which the components are all real, and if so, using this basis, does the above necessarily mean the representation is also composed of real numbers?

AFAIK, yes, as long as you're only talking about classical objects and classical transformations. This gets a little dicey with spinors since they don't have all the properties one would expect "classical objects" to have, and they only really appear in the theory when you're trying to construct a framework that you're going to end up using for a quantum field theory of fermions.

 

[tomdodd4598]

AFAIK, yes, as long as you're only talking about classical objects and classical transformations.

Ok, gotcha - thanks!