dextercioby said:
Sam,
sl(2,C) is 3-dimensional only when viewed as a vector space over the field of complex numbers. When viewed as a vector space over the field of real numbers, it's 6 dimensional.
Yes, I am aware of that. Nearly all elements A \in SL( 2 , \mathbb{ C } ) can be parameterized by
A = \exp ( c_{ i } \sigma_{ i } )
where c_{ i } is a complex 3-vector playing the role of local complex coordinates, and the Pauli matrices form a basis in the corresponding Lie algebra. We can also treat SL( 2 , \mathbb{ C } ) as a real six-dimensional Lie group using the following parameterization
A = \exp ( \frac{ 1 }{ 2 } \omega^{ ab } \sigma_{ ab } ),
where
( \omega^{ ab } )^{ * } = \omega^{ ab } = - \omega^{ ba }.
The two parameterizations are related by
<br />
2 c_{ 1 } = \omega^{ 01 } + i \omega^{ 23 }, \ 2 c_{ 2 } = \omega^{ 02 } + i \omega^{ 31 }, \ 2 c_{ 3 } = \omega^{ 03 } + i \omega^{ 12 }.<br />
And [as a basis of the REAL Lie algebra of SL( 2 , \mathbb{ C } )]
<br />
\sigma_{ ab } = - \frac{ 1 }{ 4 } ( \sigma_{ a } \bar{ \sigma }_{ b } - \sigma_{ b } \bar{ \sigma }_{ a } ),<br />
where
\sigma_{ a } \equiv ( I_{ 2 } , \sigma_{ i } ), \ \bar{ \sigma }_{ a } \equiv ( I_{ 2 } , - \sigma_{ i } )
These are related by
<br />
( \bar{ \sigma }_{ a } )^{ \dot{ \alpha } \alpha } = \epsilon^{ \dot{ \alpha } \dot{ \beta } } \epsilon^{ \alpha \beta } ( \sigma_{ a } )_{ \beta \dot{ \beta } }<br />
Notice that Pauli matrices carry mixed indices whereas ( \sigma_{ ab } )_{ \alpha } {}^{ \beta } carry un-dotted indices. There are other generators carrying two dotted indices, I include them here for later use
<br />
( \bar{ \sigma } )^{ \dot{ \alpha }} {}_{ \dot{ \beta } } = - \frac{ 1 }{ 4 } ( \bar{ \sigma }_{ a } \sigma_{ b } - \bar{ \sigma }_{ b } \sigma_{ a } )^{ \dot{ \alpha } } {}_{ \dot{ \beta } } <br />
We will also need the following identities
<br />
\mbox{ Tr } ( \sigma_{ a } \bar{ \sigma }_{ b } ) = - \eta_{ ab }<br />
<br />
( \sigma^{ a } )_{ \alpha \dot{ \alpha } } ( \bar{ \sigma }_{ a } )^{ \dot{ \beta } \beta } = - 2 \delta^{ \beta }_{ \alpha } \delta^{ \dot{ \beta } }_{ \dot{ \alpha } }<br />
Those who have not studied supersymmetry or the representations of SL( 2 , \mathbb{ C } ) and want to follow what I am about to do, they need work through the above and many other relations between the sigmas.
The correct Lie algebra isomorphisms are
\mbox{so(1,3)} \simeq \mbox{sl}(2,\mathbb{C}) \simeq \mbox{su(2)} \oplus \mbox{su(2)}
\mbox{so(1,3)}_{C} \simeq \mbox{su(2)} \oplus \mbox{su(2)}
DEFINITION (to avoid pages of mathematical gibberish here, I will give a physicist’s definition):
A Lie algebra \mathcal{ L } is the direct sum of two Lie algebras \mathcal{ L }_{ 1 } and \mathcal{ L }_{ 2 } if it is the vector sum and all the elements of \mathcal{ L }_{ 1 } commute with all the elements of \mathcal{ L }_{2}. Symbolically, we represent this by:
<br />
\mathcal{ L } = \mathcal{ L }_{ 1 } \oplus \mathcal{ L }_{ 2 },<br />
if
<br />
[ \mathcal{ L }_{ 1 } , \mathcal{ L }_{ 2 } ] \subset \mathcal{ L }_{ 1 } \cap \mathcal{ L }_{ 2 } = \varnothing<br />
CLAIM 1:
<br />
\mathcal{ so }( 1 , 3 ) \cong \mathcal{ sl }( 2 , \mathbb{ C } ) \oplus \mathcal{ sl }( 2 , \mathbb{ C } ).<br />
PROOF:
We start with the Lorentz algebra which we all know (I hope)
<br />
[ M_{ ab } , M_{ cd} ] = \eta_{ ad } M_{ bc } - \eta_{ ac } M_{ bd } + \eta_{ bc } M_{ ad } - \eta_{ bd } M_{ ac } .<br />
Now, define the following three 2 by 2 matrices ( we met them in my previous post)
<br />
M_{ \alpha \beta } = \frac{ 1 }{ 2 } ( \sigma^{ ab } )_{ \alpha \beta } M_{ ab } = M_{ \beta \alpha } ,<br />
and another 3 by
<br />
\bar{ M }_{ \dot{ \alpha } \dot{ \beta } } = - \frac{ 1 }{ 2 } ( \bar{ \sigma }^{ ab } )_{ \dot{ \alpha } \dot{ \beta } } M_{ ab }<br />
Using these together with the properties of the sigmas, we can split the Lorentz algebra into two commuting algebras:
<br />
2 [ M_{ \alpha \beta } , M_{ \gamma \delta } ] = \epsilon_{ \alpha \gamma } M_{ \beta \delta } + \epsilon_{ \alpha \delta } M_{ \beta \gamma } + \epsilon_{ \beta \gamma } M_{ \alpha \delta } + \epsilon_{ \beta \delta } M_{ \alpha \gamma } \ \ (1)<br />
similar one with the bared M and dotted indices [ \bar{ M }_{ \dot{ \alpha } \dot{ \beta } } , \bar{ M }_{ \dot{ \gamma } \dot{ \delta } } ] and
[ M_{ \alpha \beta } , \bar{ M }_{ \dot{ \gamma } \dot{ \delta } } ] = 0
Ok, if we call M_{ 11 } = E (for Elie Cartan), M_{ 22 } = F (for Felix Klein) and M_{ 12 } = M_{ 21 } = H/2 (for Hermann Weyl), then eq(1) becomes
<br />
[ H , E ] = 2 E, \ \ [ E , F ] = H , \mbox{ and } \ [ F , H ] = 2 F ,<br />
which (I hope) every body recognise as the Lie algebra \mathcal{ sl } ( 2 , \mathbb{ C } ). The bared M’s commutation relations lead to another \mathcal{ sl } ( 2 , \mathbb{ C } ) algebra. Thus as I claimed, the Lorentz algebra is isomorphic to a direct sum of two mutually conjugate \mathcal{ sl } ( 2 , \mathbb{ C } ) algebras.
CLAIM 2:
COMPLEX representation of the REAL Lie algebra \mathcal{ su } ( 2 ) is EQUIVALENT to representation of the COMPLEX Lie algebra
<br />
\mathcal{ su } ( 2 ) \otimes_{ \mathbb{ R } } \mathbb{ C } \left( \equiv \mathcal{ su } ( 2 ) \oplus i \ \mathcal{ su } ( 2 ) \right) = \mathcal{ sl } ( 2 , \mathbb{ C } )<br />
PROOF:
<br />
\mathcal{ su } ( 2 ) = \left \{ H \in M_{ 2 } \mathbb{ C } : \mbox{ Tr } ( H ) = 0 , \ H^{ \dagger } = - H \right \}<br />
<br />
i \ \mathcal{ su } ( 2 ) = \left \{ H \in M_{ 2 } \mathbb{ C } : \mbox{ Tr } ( H ) = 0 , \ H^{ \dagger } = H \right \}<br />
Thus
<br />
\mathcal{ su } ( 2 ) \oplus i \ \mathcal{ su } ( 2 ) = \left \{ H \in M_{ 2 } \mathbb { C } : \mbox{ Tr } ( H ) = 0 \right \} \equiv \mathcal{ sl } ( 2 , \mathbb{ C } ) .<br />
This is a special case of the general theorem which states that the complex representation of a real Lie algebra \mathcal{ L } is equivalent to representation of the complex Lie algebra \mathcal{ L } \oplus i \mathcal{ L } = \mathcal{ gl } ( n , \mathbb{ C } )
Sam
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