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A contravariant vector $v^\mu$ transform as the fundamental representation of the Lorentz group as $$v^\mu=\Lambda^\mu_{~~~\nu}v^\nu$$ which is tensor of rank-one. A rank-two tensor transforms as $$t^{\mu\nu}=\Lambda^\mu_{~~~\rho}\Lambda^\nu_{~~~\sigma}t^{\rho\sigma}.$$ Similarly, an $SU(2)$ doublet is a set of two fields $\psi^i$ $(i=1,2)$ which transforms under $SU(2)$ group as $$\psi^i=U^i_{~j}\psi^i.$$ This is the fundamental representation of $SU(2)$ group.

Therefore, the doublets $\psi^i$'s for the $SU(2)$ group are analogs of $v^\mu$ for the Lorentz group in the sense that both are fundamental representations of the respective groups. Compare first equation with third.

Now my question is, which representations of the Lorentz group are the analogs of an $SU(2)$ triplet $\phi^i$ ($i=1,2,3$) which is a set of three fields or any $n$-plet $\phi^i$ ($i=1,2,3,..n$) which a set of $n$ fields? I hope the question is clear. Please help me with an answer as less technical as possible.

Qmechanic
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Solidification
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  • (https://en.wikipedia.org/wiki/Representation_theory_of_the_Lorentz_group#Common_representations). – Cosmas Zachos Nov 09 '19 at 12:34
  • The $n$-dimensional rep of SU(2) has isospin $n=2j+1$. Similarly, an $n$-dimensional rep of Lorentz $SU(2)^2$ has isospins $n=(2j_1+1)(2j_2+1)$. Given some $n$, just solve (non-uniquely) for $(j_1,j_2)$. – AccidentalFourierTransform Nov 09 '19 at 14:15
  • @AccidentalFourierTransform Actually, roughly speaking, various representations of $SO(3,1)$ transform by 0,1,2,.. etc factors of $\Lambda$ matrix. A scalar does not. A 4-vector by 1 factor of $\Lambda$, a rank-2 tensor by 2 factors of $\Lambda$ etc. I was asking whether singlets, doublets, triplets of $SU(2)$ can be classified similarly. If no then why such singlet, double, triplet etc representations don't exist for $SO(3,1)$? I think, I could ask the same thing for SO(3) and SU(2) for not to complicate matters. – Solidification Nov 09 '19 at 14:29

2 Answers2

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    • The double-cover of the 3D rotation group $SO(3;\mathbb{R})$ is isomorphic to $SU(2)$.

    • The double-cover of the restricted Lorentz group $SO^+(1,3;\mathbb{R})$ is isomorphic to the complex Lie group $SL(2,\mathbb{C})$.

    The former pair are subgroups of the latter pair, respectively, cf. the diagram below.

    Returning to OP's question, the $n$-dimensional irrep of $SL(2,\mathbb{C})$ restricts to the $n$-dimensional irrep of $SU(2)$, which is the $n$-dimensional projective irrep of $SO(3;\mathbb{R})$.

$$ \begin{array}{ccccc} SU(2) & \subseteq & SL(2,\mathbb{C})& \subseteq & SL(2,\mathbb{C})_L\times SL(2,\mathbb{C})_R \cr \pi\downarrow && \pi\downarrow && \pi\downarrow\cr SO(3;\mathbb{R}) & \subseteq & SO^+(1,3;\mathbb{R}) & \subseteq & SO(1,3;\mathbb{C})\end{array}$$

  1. Similarly, the double-cover of the complexified proper Lorentz group $SO(1,3;\mathbb{C})$ is isomorphic to $SL(2,\mathbb{C})_L\times SL(2,\mathbb{C})_R$, which has $SU(2)_L\times SU(2)_R$ as a subgroup. Again an irrep $(n_L,n_R)$ of the bigger group restricts to a corresponding irrep $(n_L,n_R)$ of the subgroup.

  2. See e.g. this and this related Phys.SE posts for more details.

Qmechanic
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  • Physically, what is an $n$-dimensional irrep of SL(2,C)? I mean, 4-vector, tensor or what? – Solidification Nov 09 '19 at 08:42
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    The $n$-dim irrep of the complex Lie group $SL(2,\mathbb{C})$ is a complex holomorphic representation, which is not the case for the Lorentz 4-vector representation (and tensor product representations thereof). The classification of Lorentz irreps instead uses tensor products of left and right Weyl spinor representations. – Qmechanic Nov 09 '19 at 09:30
  • @Qmechanic, by "tensor products of left and right...", do you mean the cartesian product instead? AFAIK, tensor product of groups doesn't make sense. – Cham Nov 10 '19 at 16:39
  • @Cham: I mean tensor products of representations, i.e. tensor products of the pertinent vector spaces, not tensor products of groups. – Qmechanic Nov 10 '19 at 16:45
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The way you are seeking analogs, you can only find analogs for certain specifically defined representations. For example, as you do, you can say that the fundamental of $SU(2)$ is analogous to the fundamental of $SO(1,3)$ as in that they are both fundamentals of the respective groups. Similarly, you can say that the adjoint of $SU(2)$ is analogous to the adjoint of $SO(1,3)$. But, this is almost a word-game. There is no actual correspondence between representations compared in such a manner. To wit, the "analogous" representations are not even of the same dimension.

A more useful way to go about comparing the representations of $SU(2)$ and $SO(1,3)$ is via realizing that $\mathfrak{su}(2)\times\mathfrak{su}(2)\cong\mathfrak{so}(1,3)$. So, a particular pair of representations of the two $SU(2)$s corresponds to some particular representation of $SO(1,3)$. For example, the $\text{Spin }\frac{1}{2}\times\text{Spin }0$ representation (i.e., the $\text{Doublet}\times\text{Singlet}$ representation) of $SU(2)\times SU(2)$, commonly denoted as the $(\frac{1}{2},0)$ representation, corresponds to the left-handed (Weyl) spinor representation of $SO(1,3)$. Similarly, the $(\frac{1}{2},0)\oplus(0,\frac{1}{2})$ representation of $SU(2)\times SU(2)$ corresponds to the bispinor Dirac representation of $SO(1,3)$. The fundamental representation of $SO(1,3)$, i.e., the vector representation, corresponds to the $(\frac{1}{2},\frac{1}{2})$ representation of $SU(2)\times SU(2)$. See: https://en.wikipedia.org/wiki/Representation_theory_of_the_Lorentz_group

  • In your scheme, how will you compare an SU(2) triplet with a Lorentz group representation? – Solidification Nov 09 '19 at 08:48
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    @mithusengupta123 There is no direct correspondence between one $SU(2)$ and $SO(1,3)$. A $\text{Triplet}\times\text{Triplet}$ of $SU(2)\times SU(2)$ would correspond to the traceless symmetric tensor representation of $SO(1,3)$. A $\text{Triplet}\times\text{Singlet}$ of $SU(2)\times SU(2)$ would correspond to the self-dual $2-$form representation of $SO(1,3)$. And so on. –  Nov 09 '19 at 08:53
  • For the Lorentz group, we use the terms tensorial and spinorial representations. For SU(2) (or SU(n), in general) we use the terms singlet, doublet, triplet etc representations. Is this true? I guess I am confused with terminology. – Solidification Nov 10 '19 at 12:30