By MadScientist


2012-05-17 15:11:11 8 Comments

This question is based on problem II.3.1 in Anthony Zee's book Quantum Field Theory in a Nutshell

Show, by explicit calculation, that $(1/2,1/2)$ is the Lorentz Vector.

I see that the generators of SU(2) are the Pauli Matrices and the generators of SO(3,1)is a matrix composed of two Pauli Matrices along the diagonal. Is it always the case that the Direct Product of two groups is formed from the generators like this?

I ask this because I'm trying to write a Lorentz boost as two simultaneous quatertion rotations [unit quaternions rotations are isomorphic to SU(2)] and tranform between the two methods. Is this possible?

In other words, How do I construct the SU(2) representation of the Lorentz Group using the fact that $SU(2)\times SU(2) \sim SO(3,1)$?

Here is some background information:

Zee has shown that the algebra of the Lorentz group is formed from two separate $SU(2)$ algebras [$SO(3,1)$ is isomorphic to $SU(2)\times SU(2)$] because the Lorentz algebra satisfies:

$$\begin{align}[J_{+i},J_{+j}] &= ie_{ijk}J_{k+} & [J_{-i},J_{-j}] &= ie_{ijk} J_{k-} & [J_{+i},J_{-j}] &= 0\end{align}$$

The representations of $SU(2)$ are labeled by $j=0,\frac{1}{2},1,\ldots$ so the $SU(2)\times SU(2)$ rep is labelled by $(j_+,j_-)$ with the $(1/2,1/2)$ being the Lorentz 4-vector because and each representation contains $(2j+1)$ elements so $(1/2,1/2)$ contains 4 elements.

2 comments

@DanielC 2017-09-20 22:10:57

For the problem at hand formulated in a precise manner, „Show that the $\left(\frac{1}{2},\frac{1}{2}\right)$ representation of the $\mbox{SL}(2,\mathbb{C})$ group is* the Lorentz 4-vector", the solution - which is not so apparent from Qmechanic's otherwise good post - should be exhibited by direct / brute-force computation. This is relatively easy, and I quote from my diploma/Batchlor's Degree graduation paper (written in my native Romanian)

PART 1:

Let $\psi_{\alpha}$ be the components of a Weyl spinor wrt the canonical basis in a 2-dimensional vector space in which the fundamental $\left(\frac{1}{2},0\right)$ representation of $\mbox{SL}(2,\mathbb{C})$ "lives". Idem for $\bar{\chi}_{\dot{\alpha}}$ and the contragradient representation of the same group, $\left(0,\frac{1}{2}\right)$. Then, as an application of the Clebsch-Gordan theorem for $\mbox{SL}(2,\mathbb{C})$:

LEMMA:

$\begin{equation} \psi _{\alpha }\otimes \overline{\chi }_{\stackrel{\bullet }{\alpha }}\equiv \psi _{\alpha }\overline{\chi }_{\stackrel{\bullet }{\alpha }}=\left[ \frac{1% }{2}\psi ^{\beta }\left( \sigma ^{\mu }\right) _{\beta \stackrel{\bullet }{% \beta }}\overline{\chi }^{\stackrel{\bullet }{\beta }}\right] \left( \sigma _{\mu }\right) _{\alpha \stackrel{\bullet }{\alpha }}\equiv V^{\mu}\left( \sigma _{\mu }\right) _{\alpha \stackrel{\bullet }{\alpha }}\text{.} \end{equation}$

PROOF:

$\left[ \frac{1}{2}\psi ^{\beta }\left( \sigma _{\mu }\right) _{\beta \stackrel{\bullet }{\beta }}\overline{\chi }^{\stackrel{\bullet }{\beta }% }\right] \left( \sigma ^{\mu }\right) _{\alpha \stackrel{\bullet }{\alpha }}=% \frac{1}{2}\left( \varepsilon ^{\beta \gamma }\psi _{\gamma }\right) \left( \sigma ^{\mu }\right) _{\beta \stackrel{\bullet }{\beta }}\left( \varepsilon ^{\stackrel{\bullet }{\beta }\stackrel{\bullet }{\gamma }}\overline{\chi }_{% \stackrel{\bullet }{\gamma }}\right) \left( \sigma _{\mu }\right) _{\alpha \stackrel{\bullet }{\alpha }} \\ =-\frac{1}{2}\psi _{\gamma }\varepsilon ^{\beta \gamma }\varepsilon ^{% \stackrel{\bullet }{\gamma }\stackrel{\bullet }{\beta }}\left( \sigma ^{\mu }\right) _{\beta \stackrel{\bullet }{\beta }}\overline{\chi }_{\stackrel{% \bullet }{\gamma }}\left( \sigma _{\mu }\right) _{\alpha \stackrel{\bullet }{% \alpha }} \\ =\frac{1}{2}\psi _{\gamma }\left[ \varepsilon ^{\gamma \beta }\varepsilon ^{% \stackrel{\bullet }{\gamma }\stackrel{\bullet }{\beta }}\left( \sigma ^{\mu }\right) _{\beta \stackrel{\bullet }{\beta }}\right] \overline{\chi }_{% \stackrel{\bullet }{\gamma }}\left( \sigma _{\mu }\right) _{\alpha \stackrel{% \bullet }{\alpha }} \\ =\frac{1}{2}\psi _{\gamma }\overline{\chi }_{\stackrel{\bullet }{\gamma }% }\left( \overline{\sigma }^{\mu }\right) ^{\stackrel{\bullet }{\gamma }% \gamma }\left( \sigma _{\mu }\right) _{\alpha \stackrel{\bullet }{\alpha }} \\ =\psi _{\gamma }\overline{\chi }_{\stackrel{\bullet }{\gamma }}\delta _{% \stackrel{\bullet }{\alpha }}^{\stackrel{\bullet }{\gamma }}\delta _{\alpha }^{\gamma }=\psi _{\alpha }\overline{\chi }_{\stackrel{\bullet }{\alpha }} $

This proof makes the Pauli matrices to be seen as Clebsch-Gordan coefficients.

PART 2:

THEOREM:

$V^{\mu}\left(\psi,\chi\right)$ defined above is a Lorentz 4-vector (i.e. they are components of a Lorentz 4-vector seen as a generic member of a vector space carrying the fundamental representation of the restricted Lorentz group $\mathfrak{Lor}(1,3)$).

PROOF:

$V'^{\mu}\equiv \left( \phi ^{\prime }\right) ^{\alpha }\left( \sigma ^{\mu }\right) _{\alpha \stackrel{\bullet }{\beta }}\left( \overline{\chi }^{\prime }\right) ^{\stackrel{\bullet }{\beta }}=-\left( \overline{\chi }^{\prime }\right) _{\stackrel{\bullet }{\alpha }}\left( \overline{\sigma }^{\mu }\right) ^{\stackrel{\bullet }{\alpha }\beta }\left( \phi ^{\prime }\right) _{\beta }=-\left( M^{*}\right) _{\stackrel{\bullet }{\alpha }}{}^{\stackrel{% \bullet }{\beta }}\overline{\chi }_{\stackrel{\bullet }{\beta }}\left( \overline{\sigma }^{\mu }\right) ^{\stackrel{\bullet }{\alpha }\beta }M_{\beta }{}^{\gamma }\phi _{\gamma } \\ =-\overline{\chi }_{\stackrel{\bullet }{\beta }}\left( M^{\dagger }\right) ^{% \stackrel{\bullet }{\beta }}{}_{\stackrel{\bullet }{\alpha }}\left( \overline{\sigma }^{\mu }\right) ^{\stackrel{\bullet }{\alpha }\beta }M_{\beta }{}^{\gamma }\phi _{\gamma } \\ =-\overline{\chi }_{\stackrel{\bullet }{\beta }}\delta _{\stackrel{\bullet }{% \gamma }}^{\stackrel{\bullet }{\beta }}\left( M^{\dagger }\right) ^{% \stackrel{\bullet }{\gamma }}{}_{\stackrel{\bullet }{\alpha }}\left( \overline{\sigma }^{\mu }\right) ^{\stackrel{\bullet }{\alpha }\beta }M_{\beta }{}^{\gamma }\delta _{\gamma }^{\zeta }\phi _{\zeta } \\ =-\frac{1}{2}\overline{\chi }_{\stackrel{\bullet }{\beta }}\left( \overline{% \sigma }^{\nu }\right) ^{\stackrel{\bullet }{\beta }\zeta }\left( \sigma _{\nu }\right) _{\gamma \stackrel{\bullet }{\gamma }}\left( M^{\dagger }\right) ^{\stackrel{\bullet }{\gamma }}{}_{\stackrel{\bullet }{\alpha }% }\left( \overline{\sigma }^{\mu }\right) ^{\stackrel{\bullet }{\alpha }\beta }M_{\beta }{}^{\gamma }\phi _{\zeta } \\ =-\frac{1}{2}\left[ \left( M^{\dagger }\right) ^{\stackrel{\bullet }{\gamma }% }{}_{\stackrel{\bullet }{\alpha }}\left( \overline{\sigma }^{\mu }\right) ^{% \stackrel{\bullet }{\alpha }\beta }M_{\beta }{}^{\gamma }\left( \sigma _{\nu }\right) _{\gamma \stackrel{\bullet }{\gamma }}\right] \left[ \overline{\chi }_{\stackrel{\bullet }{\beta }}\left( \overline{\sigma }^{\nu }\right) ^{% \stackrel{\bullet }{\beta }\zeta }\phi _{\zeta }\right] \\ =-\frac{1}{2}Tr\left( M^{\dagger }\overline{\sigma }^{\mu }M\sigma _{\nu }\right) \left( \overline{\chi }\overline{\sigma }^{\nu }\phi \right) \\ =-\Lambda ^{\mu }{}_{\nu }\left( M\right) \left( \overline{\chi }\overline{% \sigma }^{\nu }\phi \right) \\ =\Lambda ^{\mu }{}_{\nu }\left( M\right) \left( \phi \sigma ^{\nu }\overline{% \chi }\right) \equiv \Lambda ^{\mu }{}_{\nu }\left( M\right) V^{\nu} $

*is = in the sense of group representation theory, it means that the carrier vector spaces of the two representations are isomorphic which is the content of the lemma. Note to the reader: the proof of the theorem uses the fact that these "classical" spinors have Grassmann parity 1. This explains the appearance and disappearance of the "-" sign.

@Qmechanic 2012-05-20 15:01:36

Here is a mathematical derivation. We use the sign convention $(+,-,-,-)$ for the Minkowski metric $\eta_{\mu\nu}$.

I) First recall the fact that

$SL(2,\mathbb{C})$ is (the double cover of) the restricted Lorentz group $SO^+(1,3;\mathbb{R})$.

This follows partly because:

  1. There is a bijective isometry from the Minkowski space $(\mathbb{R}^{1,3},||\cdot||^2)$ to the space of $2\times2 $ Hermitian matrices $(u(2),\det(\cdot))$, $$\mathbb{R}^{1,3} ~\cong ~ u(2) ~:=~\{\sigma\in {\rm Mat}_{2\times 2}(\mathbb{C}) \mid \sigma^{\dagger}=\sigma \} ~=~ {\rm span}_{\mathbb{R}} \{\sigma_{\mu} \mid \mu=0,1,2,3\}, $$ $$\mathbb{R}^{1,3}~\ni~\tilde{x}~=~(x^0,x^1,x^2,x^3) \quad\mapsto \quad\sigma~=~x^{\mu}\sigma_{\mu}~\in~ u(2), $$ $$ ||\tilde{x}||^2 ~=~x^{\mu} \eta_{\mu\nu}x^{\nu} ~=~\det(\sigma), \qquad \sigma_{0}~:=~{\bf 1}_{2 \times 2}.\tag{1}$$

  2. There is a group action $\rho: SL(2,\mathbb{C})\times u(2) \to u(2)$ given by $$g\quad \mapsto\quad\rho(g)\sigma~:= ~g\sigma g^{\dagger}, \qquad g\in SL(2,\mathbb{C}),\qquad\sigma\in u(2), \tag{2}$$ which is length preserving, i.e. $g$ is a pseudo-orthogonal (or Lorentz) transformation. In other words, there is a Lie group homomorphism
    $$\rho: SL(2,\mathbb{C}) \quad\to\quad O(u(2),\mathbb{R})~\cong~ O(1,3;\mathbb{R}) .\tag{3}$$

  3. Since $\rho$ is a continuous map and $SL(2,\mathbb{C})$ is a connected set, the image $\rho(SL(2,\mathbb{C}))$ must again be a connected set. In fact, one may show so there is a surjective Lie group homomorphism$^1$
    $$\rho: SL(2,\mathbb{C}) \quad\to\quad SO^+(u(2),\mathbb{R})~\cong~ SO^+(1,3;\mathbb{R}) , $$ $$\rho(\pm {\bf 1}_{2 \times 2})~=~{\bf 1}_{u(2)}.\tag{4}$$

  4. The Lie group $SL(2,\mathbb{C})=\pm e^{sl(2,\mathbb{C})}$ has Lie algebra $$ sl(2,\mathbb{C}) ~=~ \{\tau\in{\rm Mat}_{2\times 2}(\mathbb{C}) \mid {\rm tr}(\tau)~=~0 \} ~=~{\rm span}_{\mathbb{C}} \{\sigma_{i} \mid i=1,2,3\}.\tag{5}$$

  5. The Lie group homomorphism $\rho: SL(2,\mathbb{C}) \to O(u(2),\mathbb{R})$ induces a Lie algebra homomorphism $$\rho: sl(2,\mathbb{C})\to o(u(2),\mathbb{R})\tag{6}$$ given by $$ \rho(\tau)\sigma ~=~ \tau \sigma +\sigma \tau^{\dagger}, \qquad \tau\in sl(2,\mathbb{C}),\qquad\sigma\in u(2), $$ $$ \rho(\tau) ~=~ L_{\tau} +R_{\tau^{\dagger}},\tag{7}$$ where we have defined left and right multiplication of $2\times 2$ matrices $$L_{\sigma}(\tau)~:=~\sigma \tau~=:~ R_{\tau}(\sigma), \qquad \sigma,\tau ~\in~ {\rm Mat}_{2\times 2}(\mathbb{C}).\tag{8}$$

II) Note that the Lorentz Lie algebra $so(1,3;\mathbb{R}) \cong sl(2,\mathbb{C})$ does not$^2$ contain two perpendicular copies of, say, the real Lie algebra $su(2)$ or $sl(2,\mathbb{R})$. For comparison and completeness, let us mention that for other signatures in $4$ dimensions, one has

$$SO(4;\mathbb{R})~\cong~[SU(2)\times SU(2)]/\mathbb{Z}_2, \qquad\text{(compact form)}\tag{9}$$

$$SO^+(2,2;\mathbb{R})~\cong~[SL(2,\mathbb{R})\times SL(2,\mathbb{R})]/\mathbb{Z}_2.\qquad\text{(split form)}\tag{10}$$

The compact form (9) has a nice proof using quaternions

$$(\mathbb{R}^4,||\cdot||^2) ~\cong~ (\mathbb{H},|\cdot|^2)\quad\text{and}\quad SU(2)~\cong~ U(1,\mathbb{H}),\tag{11}$$

see also this Math.SE post and this Phys.SE post. The split form (10) uses a bijective isometry

$$(\mathbb{R}^{2,2},||\cdot||^2) ~\cong~({\rm Mat}_{2\times 2}(\mathbb{R}),\det(\cdot)).\tag{12}$$

To decompose Minkowski space into left- and right-handed Weyl spinor representations, one must go to the complexification, i.e. one must use the fact that

$SL(2,\mathbb{C})\times SL(2,\mathbb{C})$ is (the double cover of) the complexified proper Lorentz group $SO(1,3;\mathbb{C})$.

Note that Refs. 1-2 do not discuss complexification$^2$. One can more or less repeat the construction from section I with the real numbers $\mathbb{R}$ replaced by complex numbers $\mathbb{C}$, however with some important caveats.

  1. There is a bijective isometry from the complexified Minkowski space $(\mathbb{C}^{1,3},||\cdot||^2)$ to the space of $2\times2 $ matrices $({\rm Mat}_{2\times 2}(\mathbb{C}),\det(\cdot))$, $$\mathbb{C}^{1,3} ~\cong ~ {\rm Mat}_{2\times 2}(\mathbb{C}) ~=~ {\rm span}_{\mathbb{C}} \{\sigma_{\mu} \mid \mu=0,1,2,3\}, $$ $$ M(1,3;\mathbb{C})~\ni~\tilde{x}~=~(x^0,x^1,x^2,x^3) \quad\mapsto \quad\sigma~=~x^{\mu}\sigma_{\mu}~\in~ {\rm Mat}_{2\times 2}(\mathbb{C}) , $$ $$ ||\tilde{x}||^2 ~=~x^{\mu} \eta_{\mu\nu}x^{\nu} ~=~\det(\sigma).\tag{13}$$ Note that forms are taken to be bilinear rather than sesquilinear.

  2. There is a surjective Lie group homomorphism$^3$
    $$\rho: SL(2,\mathbb{C}) \times SL(2,\mathbb{C}) \quad\to\quad SO({\rm Mat}_{2\times 2}(\mathbb{C}),\mathbb{C})~\cong~ SO(1,3;\mathbb{C})\tag{14}$$ given by $$(g_L, g_R)\quad \mapsto\quad\rho(g_L, g_R)\sigma~:= ~g_L\sigma g^{\dagger}_R, $$ $$ g_L, g_R\in SL(2,\mathbb{C}),\qquad\sigma~\in~ {\rm Mat}_{2\times 2}(\mathbb{C}).\tag{15} $$

  3. The Lie group $SL(2,\mathbb{C})\times SL(2,\mathbb{C})$ has Lie algebra $sl(2,\mathbb{C})\oplus sl(2,\mathbb{C})$.

  4. The Lie group homomorphism
    $$\rho: SL(2,\mathbb{C})\times SL(2,\mathbb{C}) \quad\to\quad SO({\rm Mat}_{2\times 2}(\mathbb{C}),\mathbb{C})\tag{16}$$ induces a Lie algebra homomorphism $$\rho: sl(2,\mathbb{C})\oplus sl(2,\mathbb{C})\quad\to\quad so({\rm Mat}_{2\times 2}(\mathbb{C}),\mathbb{C})\tag{17}$$ given by $$ \rho(\tau_L\oplus\tau_R)\sigma ~=~ \tau_L \sigma +\sigma \tau^{\dagger}_R, \qquad \tau_L,\tau_R\in sl(2,\mathbb{C}),\qquad \sigma\in {\rm Mat}_{2\times 2}(\mathbb{C}), $$ $$ \rho(\tau_L\oplus\tau_R) ~=~ L_{\tau_L} +R_{\tau^{\dagger}_R}.\tag{18}$$

The left action (acting from left on a two-dimensional complex column vector) yields by definition the (left-handed Weyl) spinor representation $(\frac{1}{2},0)$, while the right action (acting from right on a two-dimensional complex row vector) yields by definition the right-handed Weyl/complex conjugate spinor representation $(0,\frac{1}{2})$. The above shows that

The complexified Minkowski space $\mathbb{C}^{1,3}$ is a $(\frac{1}{2},\frac{1}{2})$ representation of the Lie group $SL(2,\mathbb{C}) \times SL(2,\mathbb{C})$, whose action respects the Minkowski metric.

References:

  1. Anthony Zee, Quantum Field Theory in a Nutshell, 1st edition, 2003.

  2. Anthony Zee, Quantum Field Theory in a Nutshell, 2nd edition, 2010.


$^1$ It is easy to check that it is not possible to describe discrete Lorentz transformations, such as, e.g. parity $P$, time-reversal $T$, or $PT$ with a group element $g\in GL(2,\mathbb{C})$ and formula (2).

$^2$ For a laugh, check out the (in several ways) wrong second sentence on p.113 in Ref. 1: "The mathematically sophisticated say that the algebra $SO(3,1)$ is isomorphic to $SU(2)\otimes SU(2)$." The corrected statement would e.g. be "The mathematically sophisticated say that the group $SO(3,1;\mathbb{C})$ is locally isomorphic to $SL(2,\mathbb{C})\times SL(2,\mathbb{C})$." Nevertheless, let me rush to add that Zee's book is overall a very nice book. In Ref. 2, the above sentence is removed, and a subsection called "More on $SO(4)$, $SO(3,1)$, and $SO(2,2)$" is added on page 531-532.

$^3$ It is not possible to mimic an improper Lorentz transformations $\Lambda\in O(1,3;\mathbb{C})$ [i.e. with negative determinant $\det (\Lambda)=-1$] with the help of two matrices $g_L, g_R\in GL(2,\mathbb{C})$ in formula (15); such as, e.g., the spatial parity transformation $$P:~~(x^0,x^1,x^2,x^3) ~\mapsto~ (x^0,-x^1,-x^2,-x^3).\tag{19}$$ Similarly, the Weyl spinor representations are representations of (the double cover of) $SO(1,3;\mathbb{C})$ but not of (the double cover of) $O(1,3;\mathbb{C})$. E.g. the spatial parity transformation (19) intertwine between left-handed and right-handed Weyl spinor representations.

@Qmechanic 2012-05-24 15:04:22

Note for later: If $\tilde{\rho}:G\to O(u(2),\mathbb{R})$ is an extension $\tilde{\rho}(g)\sigma:= g\sigma g^{\dagger}$ with $SL(2,\mathbb{C})\subseteq G \subseteq GL(2,\mathbb{C})$, there does not exist a $g\in G$ such that it reproduces the discrete Lorentz transformations, such as, e.g. $\tilde{\rho}(g)=PT:\sigma\mapsto -\sigma$, $\tilde{\rho}(g)=T$, or $\tilde{\rho}(g)=P$.

@Qmechanic 2014-04-05 18:59:38

Note for later: If $g=\begin{pmatrix} a & b \\ c &d\end{pmatrix}$ and $\sigma=\begin{pmatrix} z^+ & z^{\ast} \\ z & z^- \end{pmatrix}$ with $z^{\pm}=x^0\pm x^3$ and $z=x^1+ix^2$, then $g\sigma g^{\dagger} = \begin{pmatrix}az^+a^{\ast} + bza^{\ast} + az^{\ast}b^{\ast} + bz^-b^{\ast} & az^+c^{\ast} + bzc^{\ast} + az^{\ast}d^{\ast} + bz^-d^{\ast} \\ cz^+a^{\ast} + dza^{\ast} + cz^{\ast}b^{\ast} + dz^-b^{\ast} & cz^+c^{\ast} + dzc^{\ast} + cz^{\ast}d^{\ast} + dz^-d^{\ast} \end{pmatrix}$; $\quad T: z^{\pm}\mapsto -z^{\mp}$;

@Issam Ibnouhsein 2014-09-23 22:37:27

How do you go from a representation of the complexified algebra to the original one?

@Qmechanic 2014-12-05 13:17:12

A representation of a Lie group (algebra) is also a representation of any subgroup (subalgebra), respectively.

@Qmechanic 2014-12-06 22:46:02

Note for later: If $\tilde{\rho}:O(1,3;\mathbb{C})\to GL(2,\mathbb{C})/\mathbb{Z}_2$ is an extension of the left-handed Weyl rep $\rho:SO(1,3;\mathbb{C})\to SL(2,\mathbb{C})/\mathbb{Z}_2$, and $T:t\mapsto -t$ is the time reversal transformation, it is not hard to show that $\tilde{\rho}(\Lambda_T)$ cannot exists. It has to be diagonal to commute with $xy$ rotations. And traceless to behave correctly under $z$ boosts. While $\tilde{\rho}(\Lambda_T)$ should be in the center to commute with $xz$ rotations. Contradiction. So the extension $\tilde{\rho}$ does not exist.

@Qmechanic 2015-06-04 15:34:13

Note for later: $\sigma_{\mu}$ in this answer is usually called $\bar{\sigma}_{\mu}=\eta_{\mu\nu}\bar{\sigma}^{\nu}$ elsewhere. Instead elsewhere $\sigma_{\mu}=\bar{\sigma}^{\mu}$. Then $\bar{\sigma}=x^{\mu}\bar{\sigma}_{\mu}$ and $x^{\mu}=\frac{1}{2}{\rm tr}(\sigma^{\mu}\bar{\sigma})$, $\frac{1}{2}{\rm tr}(\sigma^{\mu}\bar{\sigma}_{\nu})=\delta^{\mu}_{\nu}$ and $\Lambda^{\mu}{}_{\nu}=\frac{1}{2}{\rm tr}(\sigma^{\mu}g_L\bar{\sigma}_{\nu}g^{\dagger}_R)$.

@Qmechanic 2015-10-04 15:37:39

Note for later: There is a star algebra monomorphism $\Phi:\mathbb{H}\to {\rm Mat}_{2\times 2}(\mathbb{C})$ given by $$\mathbb{H}\ni x=x^0+ix^3 +jx^2+kx^1=\alpha +\beta j $$ $$\mapsto \begin{pmatrix} \alpha & \beta \cr -\bar{\beta} & \bar{\alpha} \end{pmatrix} = x^0{\bf 1}_{2\times 2}+i\sum_{a=1}^3 x^a \sigma_a\in {\rm Mat}_{2\times 2}(\mathbb{C})$$ with $\alpha=x^0+ix^3$, $\beta=x^2+ix^1$; $\Phi(x)^{\dagger}=\Phi(\bar{x})$, and $\Phi(x)^t=\Phi(x|_{x^2\to-x^2})$. The restriction $\Phi:U(1,\mathbb{H}):=\{x\in\mathbb{H}\mid |x|=1 \}\stackrel{\cong}{\to} SU(2)$ is an isomorphism.

@Qmechanic 2015-11-03 10:38:51

Note that $\forall g\in SL(2,\mathbb{C}):~~\sigma_2 g\sigma_2~=~(g^{-1})^t$.

@balu 2018-02-19 12:46:03

I was pondering a question related to Issam's farther above: Namely, why are we considering the complexification of $so(1, 3)$ in the first place (in QFT, that is)? Just because any complex representation will also give us a real representation and considering complex instead of real representations makes things easier (as the representation theory of complex semisimple Lie algebras is well understood)?

@Qmechanic 2018-02-19 13:15:48

The short and pragmatic answer is that the complexified Lorentz group works, is useful, often gets the job done, and is tied to the analytic properties of QFT. When it doesn't work, we have to roll up our sleeves!