What Are the Decoupled Chiral Lorentz Algebras?

[turin]

Homework Statement

Please bare with me; I don't know all of the terminology for this problem.

In many textbooks, the six-dimensional Lie algebra of the Lorentz group is identified with the three components of the rotation generator and the three components of the boost generator. Then, the claim is made that two decoupled chiral algebras can be formed from linear combinations of the generators. Two claims are made regarding these decoupled algebras:

1) They are exchanged with each other by complex conjugation.

2) There exist finite dimensional irreducible bases for faithful representations of the decoupled chiral groups, where the dimension is any whole number ≥2, in direct analogy to the representations of angular momentum.

Homework Equations

For example, from Peskin&Schroeder, Problem 3.1:
L is angular momentum, K is boost, and J is chiral generator.
[L_i,L_j]=iε_ijkL_k
[L_i,K_j]=iε_ijkK_k
[K_i,K_j]=-iε_ijkL_k
J^±_i=(1/2)(L_i±iK_i)
No specific representation is suggested. However, in the chapter, the Pauli matrices are used (Weyl spinor) or the gamma matrices are used (Dirac spinor), as well as what seems to be a completely different representation (4-vector) using a particular combination of products of metric tensors. The group is exponentiated from the generators with an "extra" i in the exponential.

For example, from Jackson, Section 11.7:
S is angular momentum, and K is boost. No mention is made of the decoupled chiral algebras.
S^*=S, K^*=K, S^T=-S, K^T=K
[S_i,S_j]=ε_ijkS_k
[S_i,K_j]=ε_ijkK_k
[K_i,K_j]=-ε_ijkS_k
The group is exponentiated from the generators directly.

For example, from Srednicki, Chapter 33:
J is angular momentum, K is boost, and N is chiral generator.
[J_i,J_j]=iε_ijkJ_k
[J_i,K_j]=iε_ijkK_k
[K_i,K_j]=-iε_ijkJ_k
N_i=(1/2)(J_i+iK_i), N_i^T*=(1/2)(J_i-iK_i)

The Attempt at a Solution

I have many questions/concerns.

Firstly, I am confused what exactly are the physical implications of the fact that the generators from the two chiral algebras commute. I still don't have a good physical intuition regarding Weyl fermions (or spinors in general). I suppose that the Lorentz transformation properties of the Weyl fermions are the paradigm for the distinction of these two chiral Lorentz groups, but I don't know how far I can extend this idea (e.g. to higher-dimensional representations). For instance, what measurement can I make to determine the chirality directly (AS OPPOSED TO INFERRING IT FROM THE HELICITY)?

Secondly, I am highly suspicious of the exchange under complex conjugation. Is this property specific to the representation/basis? It does not seem to be a fundamental property of the algebra. I don't think that it can be, because I can, for instance, redefine the structure constants (e.g. the difference between Peskin&Schroeder vs. Jackson). Furthermore, I don't understand what it means to take the complex conjugate of a generator in a Lie algebra. So, if my suspicion is correct, then how do I qualify the representation, and how can I determine the more general exchange property?

Thirdly, how can I prove that there are finite-dimensional faithful representations analogous to the finite representations of angular momentum? For instance, if I follow the approach in Shankar, then I cannot use the positive definiteness of the eigenvalues, because the generators are not generally Hermitian. So, there is no lower bound on the expectation value of N_x²+N_y².

 

[Avodyne]

I am confused what exactly are the physical implications of the fact that the generators from the two chiral algebras commute.

I don't think there are any. This only happen in d=4, but there are e.g. Weyl fermions in other numbers of dimensions.

Secondly, I am highly suspicious of the exchange under complex conjugation. Is this property specific to the representation/basis?

No.

It does not seem to be a fundamental property of the algebra. I don't think that it can be, because I can, for instance, redefine the structure constants (e.g. the difference between Peskin&Schroeder vs. Jackson).

Jackson's matrices are antihermtian, while Peskin's are hermitian. This accounts for the differing factors of i in various places, and is just a notational convention.

Thirdly, how can I prove that there are finite-dimensional faithful representations analogous to the finite representations of angular momentum? For instance, if I follow the approach in Shankar, then I cannot use the positive definiteness of the eigenvalues, because the generators are not generally Hermitian. So, there is no lower bound on the expectation value of N_x²+N_y².

Very good question! I have not seen this explained anywhere. But a few years ago I worked out an explanation for myself at a heuristic level.

The generators need to act in a space of states. Since the two sets of chiral generators commute, let me label each ket state with two indices: $|a_1,a_2\rangle$. One set of generators will act on one index, and the other set on the other index. [Eventually we will find that the first index stands for a pair $(\ell,m)$ with $\ell(\ell{+}1)$ being the eignvalue of $N_x^2+N_y^2+N_z^2$ and $m$ being the eigenvalue of $N_z$, and similarly for the second index, with $N_i$ replaced by $N_i^\dagger$.] Also define bra states $\langle a_1,a_2|$ that are orthogonal to the ket states:

$$\langle a_1,a_2|b_1,b_2\rangle = \delta_{a_1 b_1}\delta_{a_2 b_2}.~~~~~~\rm eq(1)$$

Let $N_i$ be one of the chiral generators. Then we have

$$\langle a_1,a_2|N_i|b_1,b_2\rangle = ({\cal N}_i)_{a_1 b_1}\delta_{a_2 b_2}~~~~~~\rm eq(2)$$

where ${\cal N}_i$ is a matrix (with row index $a_1$ and column index $b_1$) that must satisfy the commutation relations

$$[{\cal N}_i ,{\cal N}_j]=i\varepsilon_{ijk}{\cal N}_k .~~~~~~\rm eq(3)$$

Similarly, for $N_i^\dagger$ we have

$$\langle a_1,a_2|N^\dagger_i|b_1,b_2\rangle = \delta_{a_1 b_1}({\cal N}_i)_{a_2 b_2}~~~~~~\rm eq(4)$$

where ${\cal N}_i$ also satisfies the same commutation relations; given the symmetry between the two chiral algebras, I will assume that these matrices are numerically the same as the first set of matrices. (I think I could get away with them being the same up to a similarity transformation, but that would complicate the argument below.)

Now, hermitian conjugation is supposed to exchange the two algebras. I will therefore assume that

$$|a_1,a_2\rangle^\dagger = \langle a_2,a_1|.~~~~~~\rm eq(5)$$

Note the ordering of the indices compared with eq(1)!

Now take the hermitian conjugate of eq(2). We get

$$\langle b_2,b_1|N^\dagger_i|a_2,a_1\rangle = ({\cal N}_i)^*_{a_1 b_1}\delta_{a_2 b_2}.~~~~~~\rm eq(6)$$

Now compare this to eq(4). To do so, make the following replacements of dummy indices in eq(4): $a_1\to b_2,~a_2\to b_1,~b_1 \to a_2,~b_2\to a_1$. Eq(4) then reads

$$\langle b_2,b_1|N^\dagger_i|a_2,a_1\rangle = \delta_{b_2 a_2}({\cal N}_i)_{b_1 a_1}.~~~~~~\rm eq(7)$$

We see that eq(7) is compatible with eq(6) if and only if

$$({\cal N}_i)^*_{a_1 b_1} = ({\cal N}_i)_{b_1 a_1}.~~~~~~\rm eq(8)$$

That is, the matrix ${\cal N}_i$ must be hermitian.

Now that we know the ${\cal N}_i$ matrices must be hermitian, we can apply the usual derivation of the allowed representations of the algebra.

 

[turin]

Thank you very much for your elaborate reply. One thing, however, alarms me:

That is, the matrix ${\cal N}_i$ must be hermitian.

This indeed rules out some representations: for example, Jackson's or Sakurai's, in which N would be anti-Hermitian (if calculated from the rotation and boost generators as N=(1/2)(J+iK)).

Anyway, thank you again. You have guided my thoughts in a progressive direction.

 

[Avodyne]

This indeed rules out some representations: for example, Jackson's or Sakurai's, in which N would be anti-Hermitian

No! Jackson has different commutation relations because he has absorbed a factor of i into his angular-momentum generators (making them antihermetian rather than hermitian). If you use Jackson's commutation relations, then the correct condition is that the generators be antihermitian. If you use Peskin's, then the correct condition is that the generators be hermitian. These conditions are actually exactly the same once the differing convention regarding the factors of i is taken into account.

 

[turin]

I'm confused by what you're telling me. Are you telling me that I must define the chiral generators with an extra factor of i when I use Jackson's convention? Or maybe N=(1/2)(J+iK) in P&S but N=(1/2)(K+iJ) in Jackson? And in general, I must find the linear combination of J and K that is Hermitian, regardless of the Hermiticity of J and K themselves? Please elaborate.

Actually, I am uncertain about the consistency/meaning of your Equation (5). I was under the impression that, rather

|a₁,a₂>^*=|a₂,a₁>, whatever that ^* would mean in the abstract group-theoretic sense.

So, if your Equation (5) is true, then your Equations (2) and (4) imply that the generators themselves are what exchanges the chiral representations. In other words, I think that, given your Equation (5), the subscripts on the a's in Equations (1), (2) and (4) should be swapped. (And actually, I think Equation (5) should come first.) Please elaborate.