What Is the Relationship Between the Lie-Algebra of SO(3,1) and SU(2)?

[Syrius]

Cheers everybody,

I have a question regarding the Lie-Algebra of the SO(3,1), i.e. the Lorentz group. The generators are denoted by $M_{\mu \nu}$ with $M_{\mu \nu} = - M_{\nu \mu}$ and fulfill the following commutation relations
$[M_{\mu \nu}, M_{\rho \sigma}] = g_{\mu \rho} M_{\nu \rho} + g_{\nu \sigma} M_{\mu \rho} - g_{\mu \sigma} M_{\nu \rho} - g_{\nu \rho} M_{\mu \sigma}$,
where $g_{\mu \nu}$ are metric coefficients.

I can now construct from these generators the operators
$\mathbf{J} = (M_{23},M_{31},M_{12}),\ \mathbf{K} = (M_{01},M_{02},M_{03})$,
which fulfill the relations
$[J_i,J_j] = i \epsilon_{ijk} J_k,\ [K_i,K_j]= -i \epsilon_{ijk} J_k,\ [J_i,K_j]=i \epsilon_{ijk} K_k$.

In a final step, I define
$\mathbf{J}^\prime = \frac{1}{2} ( \mathbf{J} + i \mathbf{K}),\ \mathbf{K}^\prime = \frac{1}{2} (\mathbf{J} - i \mathbf{K})$
with the commutation relations
$[J_i^\prime,J_j^\prime] = i \epsilon_{ijk} J_k^\prime,\ [K_i^\prime,K_j^\prime]= i \epsilon_{ijk} K_k^\prime,\ [J_i^\prime, K_j^\prime]=0$.

This means that I can construct generators that fulfill exactly the commutation relations of the Lie-Algebra of the SU(2). What does this mean for the group SO(3,1)? Can I deduce from this something like SU(2) x SU(2) is a representation of SO(3,1)?

Greetings, Syrius

 

[morphism]

Okay, as always, I find the physicist take on these matters to be confusing. What do you mean by SO(3,1), exactly? Since you're concerned with its Lie algebra, I assume you're referring to the identity component of SO(3,1) (i.e. what some people call the "restricted" Lorentz group), since it's all that matters here.

You're noticing the commutation relations of (the Lie algebra of) SU(2) because:
(1) SL(2,C) is the complexification of SU(2), and
(1) SL(2,C) covers (the identity component of) SO(3,1) in the topological sense,
(2) so the (complexified) Lie algebras of SU(2) (which is the complex Lie algebra of SL(2,C)) and (the identity component of) SO(3,1) are the same.

Generally, if a Lie group G covers a Lie group H (both assumed to be connected), then the Lie algebras will be isomorphic, and in fact the differential of the covering map at the identity will implement the isomorphism.

What you can say about representations is a bit delicate. First of all, you really need to clarify whether you're looking for representations of the group or of its Lie algebra.

 

[Syrius]

Hello morphism,

Okay, as always, I find the physicist take on these matters to be confusing. What do you mean by SO(3,1), exactly? Since you're concerned with its Lie algebra, I assume you're referring to the identity component of SO(3,1) (i.e. what some people call the "restricted" Lorentz group), since it's all that matters here.

Yes, I mean the identity component.

You're noticing the commutation relations of (the Lie algebra of) SU(2) because:
(1) SL(2,C) is the complexification of SU(2), and
(1) SL(2,C) covers (the identity component of) SO(3,1) in the topological sense,
(2) so the (complexified) Lie algebras of SU(2) (which is the complex Lie algebra of SL(2,C)) and (the identity component of) SO(3,1) are the same.

Thank you, that helps a lot. So in my own words, since the Lie algebra of SL(2,C) is nothing else than the direct sum of the SU(2) Lie algebra, and SL(2,C) is covering the identity component of SO(3,1), I am getting 6 generators which are equivalent to two times the generators of the SU(2).

Maybe it is a dumb question, but if we find that the Lie Algebra of SL(2,C) is just a direct sum of the Lie algebra of SU(2), what does that mean for the underlying groups. Is the SL(2,C) group then also just the direct sum of SU(2)'s?

And another question: How can I see that SL(2,C) is the complexification of SU(2)?

What you can say about representations is a bit delicate. First of all, you really need to clarify whether you're looking for representations of the group or of its Lie algebra.

I mean here representations of the group.

Many thanks in advance,
Syrius

 

[morphism]

Maybe it is a dumb question, but if we find that the Lie Algebra of SL(2,C) is just a direct sum of the Lie algebra of SU(2), what does that mean for the underlying groups. Is the SL(2,C) group then also just the direct sum of SU(2)'s?

No. An isomorphism of Lie algebras does not guarantee an isomorphism of the underlying Lie groups; all you get is a cover.

In this specific example it's easy to see that SL(2,C) isn't a direct sum of SU(2)'s: SL(2,C) isn't compact but SU(2) (and hence a direct sum of two SU(2)'s) is.

And another question: How can I see that SL(2,C) is the complexification of SU(2)?

There are a bunch of different definitions of "complexification", but here all we need to know is that SU(2) is a maximal compact subgroup of SL(2,C) and that at the Lie algebra level $\mathfrak{sl}(2,\mathbb C) = \mathbb C \otimes \mathfrak{su}(2)= \mathfrak{su}(2) \oplus i \mathfrak{su}(2)$.

I mean here representations of the group.

In that case, you generally can't "lift" representations from the Lie algebra to the group - not unless the group is simply connected. Generally you will get a representation of some covering group (you can always guarantee a representation of the universal covering group, but sometimes a "smaller" cover might suffice). Do you have a specific question you want answered? (I'm having trouble following what you said towards the end of your original post.)

 

[blue_raver22]

Dear Syrius,

Thank you for your question. The Lie-Algebra of the SO(3,1), also known as the Lorentz group, is a fundamental concept in physics and plays a crucial role in the study of space and time. The generators of this group, denoted by M_{\mu \nu}, represent the symmetries of the space-time manifold. These generators satisfy certain commutation relations, which are essential for understanding the behavior of physical systems under Lorentz transformations.

In your question, you have correctly pointed out that the generators of the Lorentz group can be expressed in terms of the generators of the SU(2) group, denoted by \mathbf{J} and \mathbf{K}. This means that the Lie-Algebra of the Lorentz group can be decomposed into two copies of the Lie-Algebra of the SU(2) group. This decomposition is known as the double cover of the Lorentz group and is often denoted as SO(3,1) = SU(2) x SU(2).

What this means is that the Lorentz group can be represented as a product of two SU(2) groups, and this has important implications in physics. For example, it allows us to use the mathematical tools developed for the SU(2) group to study the Lorentz group, which makes calculations and predictions easier. Moreover, this decomposition also helps us to understand certain physical phenomena, such as the spin of particles, in a more intuitive way.

In summary, the fact that the Lie-Algebra of the Lorentz group can be expressed in terms of the Lie-Algebra of the SU(2) group is a significant result in physics. It allows us to better understand the symmetries of space and time and provides us with powerful tools to study physical systems. I hope this helps to answer your question.

Best regards,