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## 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

_{ijk}L

_{k}

[L

_{i},K

_{j}]=iε

_{ijk}K

_{k}

[K

_{i},K

_{j}]=-iε

_{ijk}L

_{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

_{ijk}S

_{k}

[S

_{i},K

_{j}]=ε

_{ijk}K

_{k}

[K

_{i},K

_{j}]=-ε

_{ijk}S

_{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

_{ijk}J

_{k}

[J

_{i},K

_{j}]=iε

_{ijk}K

_{k}

[K

_{i},K

_{j}]=-iε

_{ijk}J

_{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

_{x}

^{2}+N

_{y}

^{2}.