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The definition of a representation is a map to the space of linear operators over a vector space. My question is: What are the corresponding vector spaces for the

  • $(0,0)$ Representation
  • $(\frac{1}{2},0)$ Representation
  • $(0,\frac{1}{2})$ Representation

  • $(\frac{1}{2},0) \oplus (0,\frac{1}{2}) $ Representation

  • $(\frac{1}{2},\frac{1}{2})$ Representation

  • infinite dimensional Representation?
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Related: physics.stackexchange.com/q/28505/2451 and links therein. –  Qmechanic Nov 29 at 12:54
    
@Qmechanic After reading through your answer, I'm still not sure. Is the corresponding vectorspace for the $(\frac{1}{2},0)$ representation the space of complex $2\times 2$ matrices ${\rm Mat}_{2\times 2}$? In what sense is the vector space for the $(0,\frac{1}{2})$ representation different? For the $(\frac{1}{2},\frac{1}{2})$ Representation we take the product space and for $(\frac{1}{2},0) \oplus (0,\frac{1}{2})$ the direct sum of those two vector spaces?! –  Tim Dec 1 at 9:31

1 Answer 1

I) Representation theory of the Lorentz$^1$ group is a fairly broad subject covered in many textbooks, see e.g. Ref. 1 for further information.

II) The irreducible representation

$$\tag{1} (j_L,j_R)~=~j_L\otimes_{\mathbb{C}} j_R, \qquad j_L, j_R~\in~ \frac{1}{2}\mathbb{N}_0,$$

is a tensor product of $V=V_L\otimes_{\mathbb{C}} V_R$ of two complex vector spaces $V_L$ and $V_R$, of complex dimension $2j_L+1$ and $2j_R+1$, respectively. The tensor product $V$ is again a complex vector space and has complex dimension $(2j_L+1)(2j_R+1)$. See also this Phys.SE post.

III) The restricted Lorentz group $SO^+(1,3;\mathbb{R})$ is obviously a subgroup of the complexified$^2$ Lorentz group $SO(3,1;\mathbb{C})$. One can show that the double cover of the complexified Lorentz group $SO(3,1;\mathbb{C})$ is isomorphic to the direct or Cartesian product group

$$\tag{2} G~=~SL(2,\mathbb{C})\times SL(2,\mathbb{C}),$$

cf. e.g. Ref. 1 and this Phys.SE post.

IV) In more detail, the irreducible representation

$$\tag{3} \rho~=~\rho_L\otimes \rho_R:G\to GL(V,\mathbb{C})$$

in (1) of the product Lie group (2) is given as

$$\tag{4} \rho(g_L,g_R)(\sum_iv^i_L\otimes v^i_R)~=~\sum_i\rho_L(g_L)v^i_L\otimes\rho_R(g_R)v^i_R ,$$

where

$$\tag{5} \rho_{L/R}:SL(2,\mathbb{C})\to GL(V_{L/R},\mathbb{C})$$

are irreducible representations of $SL(2,\mathbb{C})$ of complex dimensions $2j_{L/R}+1$.

V) In particular, the irreducible representation (1) of the product Lie group (2) is a representation of the restricted Lorentz group $SO^+(1,3;\mathbb{R})$.

References:

  1. I.L. Buchbinder and S.M. Kuzenko, Ideas and Methods of Supersymmetry and Supergravity - Or a Walk Through Superspace, 1998; Chapter 1.

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$^1$ Let us here for simplicity consider the restricted Lorentz group $SO^+(1,3;\mathbb{R})$ rather than the Lorentz group $O(3,1;\mathbb{R})$.

$^2$ It turns out that relativistic physical theories often have pertinent complex analytic properties.

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Thanks for your answer. Your write: $\tag{5} \rho_{L/R}:SL(2,\mathbb{C})\to GL(V_{L/R},\mathbb{C})$, which I do agree with. Unfortunately I still can't see the answer to my original question: What is the vector space $V=V_L\otimes V_R$ here, for the different irreducible reprsentations recited above? –  Tim Dec 3 at 8:41
    
I updated the answer. –  Qmechanic Dec 3 at 9:34

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