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EDIT: The vector space for the $(\frac{1}{2},0)$ Representation is $\mathbb{C}^2$ as mentioned by Qmechanic in the comments to his answer below! The vector spaces for the other representations remain unanswered.

The definition of a representation is a map (a homomorphism) 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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2 Answers 2

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I) Representation theory for the double cover $SL(2,\mathbb{C})$ of the restricted$^1$ Lorentz group $SO^+(1,3;\mathbb{R})$ is a fairly broad subject covered in many textbooks, see e.g. Ref. 1 for further information.

An irreducible representation$^2$

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

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.

Examples:

  1. $(j_L,j_R)=(0,0)$. This is the trivial/singlet representation. Then the vector-space is $V\cong\mathbb{C}$. Note that the trivial representation $(0,0)$ is the multiplicative identity for the tensor product $\otimes_{\mathbb{C}}$, i.e.

$$\forall V:~~(0,0)\otimes_{\mathbb{C}}V~\cong~ V~\cong~ V\otimes_{\mathbb{C}}(0,0).\tag{2}$$

  1. $(j_L,j_R)=(\frac{1}{2},0)$. This is known as the left-handed Weyl-spinor representation. Then the vector-space is $V\cong\mathbb{C}^2$. It is the fundamental/defining representation of $SL(2,\mathbb{C})$.

  2. $(j_L,j_R)=(0,\frac{1}{2})$. This is known as the right-handed Weyl-spinor representation. It is the complex conjugate representation of the left-handed Weyl-spinor representation.

  3. $(j_L,j_R)=(\frac{1}{2},\frac{1}{2})$. This is isomorphic to the vector representation of the Lorentz group.

  4. The Dirac-spinor representation $(\frac{1}{2},0) \oplus (0,\frac{1}{2})$ is a direct sum of the left- and right-handed Weyl-spinor representation.

An irreducible representation (1) can be written with the help of the symmetric tensor product $\odot$ of the left-handed and right-handed Weyl-spinor representation

$$(j_L,j_R)~=~(\frac{1}{2},0)^{\odot 2j_L} \otimes (0,\frac{1}{2})^{\odot 2j_R}$$ $$~:=~\underbrace{\left\{(\frac{1}{2},0)\odot\ldots\odot(\frac{1}{2},0)\right\}}_{2j_L\text{ symmetrized factors}} \otimes \underbrace{\left\{(0,\frac{1}{2})\odot\ldots\odot(0,\frac{1}{2})\right\}}_{2j_R\text{ symmetrized factors}} .\tag{3} $$

Here $\otimes$ denotes the standard (un-symmetrized) tensor product.

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

$$ G~=~SL(2,\mathbb{C})_L\times SL(2,\mathbb{C})_R,\tag{4}$$

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

In more detail, the irreducible representation (1) for $SL(2,\mathbb{C})$ lifts to an irreducible representation

$$ \rho~=~\rho_L\otimes \rho_R:G\to GL(V,\mathbb{C})\tag{5}$$

for the product Lie group (4) given as

$$ \rho(g_L,g_R)(\sum_iv^i_L\otimes v^i_R)~=~\sum_i\rho_L(g_L)v^i_L\otimes\rho_R(g_R)v^i_R ,\tag{6}$$

where both

$$ \rho_{L/R}:SL(2,\mathbb{C})\to GL(V_{L/R},\mathbb{C})\tag{7}$$

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

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 $O(1,3;\mathbb{R})$. To allow for spinor representations, we need to go to the double cover $SL(2,\mathbb{C})$.

$^2$ We can w.l.o.g. assume that a representation over a real vector space is complexified to a complex vector space.

$^3$ It turns out that relativistic physical theories often have pertinent complex analytic properties. See also this related Phys.SE.

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  • $\begingroup$ Regarding 3. — by isomorphic to the vector rep, do you mean the complexificiation of the vector rep? Otherwise I don't see how it could be the case that $\mathbb{C}^2\otimes\mathbb{C}^2 \cong \mathbb{R^4}$ $\endgroup$
    – Craig
    Mar 30, 2022 at 6:22
  • $\begingroup$ $\uparrow$ Yes. $\endgroup$
    – Qmechanic
    Mar 30, 2022 at 7:03
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(A)

(0,0) acts on a trivial space $\mathbb{C}. $


(B)

$(\frac{1}{2},0)$ acts on a vector space which is same as a spin space $( \alpha|\uparrow \rangle +\beta | \downarrow\rangle) $ , ignoring the meaning of spin up and down now. This space is just $\mathbb{C}^2 $ up to a normalization constraint $|\alpha|^2+|\beta|^2=1.$


(C)

$(0,\frac{1}{2})$ acts on a vector space, which has the same structure as $(\frac{1}{2},0)$'s space, but may have different meaning, I write it as $( \gamma|\Uparrow \rangle +\delta | \Downarrow\rangle). $


(D)

$(\frac{1}{2},0) \oplus (0,\frac{1}{2}) $ acts on $(\alpha|\uparrow \rangle +\beta | \downarrow\rangle) \oplus (\gamma|\Uparrow\rangle +\delta | \Downarrow\rangle)=( \alpha|\uparrow\rangle +\beta |\downarrow\rangle + \gamma|\Uparrow\rangle +\delta | \Downarrow\rangle). $


(E)

$(\frac{1}{2}, \frac{1}{2}) $ acts on $(\alpha|\uparrow\rangle +\beta | \downarrow\rangle)\otimes (\gamma|\Uparrow\rangle +\delta | \Downarrow\rangle )=(a|A\rangle + b|B\rangle +c|C\rangle +d|D\rangle).$

$|\alpha|^2+|\beta|^2=1$ and $|\gamma|^2+|\delta|^2=1$ may not hold, it becomes one expression for $a \ b \ c \ d.$


(F)

infinite basis, adding extra momentum to (B) for example:

$(\alpha_1|\uparrow,p_1\rangle +\beta_1 | \downarrow,p_1\rangle)\oplus( \alpha_2|\uparrow,p_2\rangle +\beta_2 | \downarrow,p_2\rangle)\oplus( \alpha_3|\uparrow,p_3 \rangle +\beta_3 | \downarrow,p_3\rangle)\oplus...$

I am using $\oplus$, since $\langle s_1,p_i|s_2,p_j\rangle =\delta_{ij} \langle s_1 |s_2\rangle .$

Therefore the space is:

$$(\sum_{s=1,2} \sum_{p} a_{s,p} |s,p\rangle)$$ with normalization constraint $\sum_{s=1,2} \sum_{ p} |a_{s,p}|^2=1.$

similarly, you can add extra momentum to (A) (C) (D) (E), to realize their infinite versions.

for (A)'s infinite version, that vector space is just $ \{ |p\rangle \} $ itself.

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    $\begingroup$ (D) is not an irreducible representation, is it? $\endgroup$
    – gented
    Nov 9, 2016 at 0:27
  • $\begingroup$ @gented . No, the bispinor rep is reducible, cf. Dirac spinors. $\endgroup$ May 2, 2018 at 21:38

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