r_2,s_2)\varphi_{\alpha_2}(r_1,s_1) \right).
$$
The two above expression certainly look to be equivalent. However if we consider the wave functions to be $N$-component vectors I get some strange results. Suppose we want to calculate the normalization of the two particle wave function. We would then write the following
$$
\sum_{s_1} \sum_{s_2} \int \textrm{d}r_1 \textrm{d}r_2 \Psi^{\dagger}(r_1,s_1,r_2,s_2) \Psi(r_1,s_1,r_2,s_2)
$$
Note that I have introduced the $\dagger$ instead of the complex conjugate $*$ as the normalization should give a scalar. If we now replace $\Psi$ with the expression for the "label" permutation case we get,
\begin{align*}
\sum_{s_1} \sum_{s_2} & \int \textrm{d}r_1 \textrm{d}r_2 \Psi^{\dagger}(r_1,s_1,r_2,s_2) \Psi(r_1,s_1,r_2,s_2) \\
&= \frac{1}{2} \sum_{s_1} \sum_{s_2} \int \textrm{d}r_1 \textrm{d}r_2 \left(\varphi_{\alpha_1}(r_1,s_1)\varphi_{\alpha_2}(r_2,s_2) - \varphi_{\alpha_2}(r_1,s_1)\varphi_{\alpha_1}(r_2,s_2) \right)^{\dagger} \\
& \times \left(\varphi_{\alpha_1}(r_1,s_1)\varphi_{\alpha_2}(r_2,s_2) - \varphi_{\alpha_2}(r_1,s_1)\varphi_{\alpha_1}(r_2,s_2) \right) \\
&= \frac{1}{2} \sum_{s_1} \sum_{s_2} \int \textrm{d}r_1 \textrm{d}r_2 \left( \varphi^{\dagger}_{\alpha_2}(
r_2,s_2)\varphi_{\alpha_2}(r_1,s_1) \right). 
 $ $ 
 The two above expression certainly look to be equivalent. However if we consider the wave function to be $ N$-component vector I get some strange results. speculate we want to calculate the standardization of the two particle wave function. We would then write the come 
 $ $ 
 \sum_{s_1 } \sum_{s_2 } \int \textrm{d}r_1 \textrm{d}r_2 \Psi^{\dagger}(r_1,s_1,r_2,s_2) \Psi(r_1,s_1,r_2,s_2) 
 $ $ 
 Note that I have introduced the $ \dagger$ rather of the complex conjugate $ * $ as the normalization should give a scalar. If we nowadays replace $ \Psi$ with the expression for the " label " permutation case we get, 
 \begin{align * } 
 \sum_{s_1 } \sum_{s_2 } & \int \textrm{d}r_1 \textrm{d}r_2 \Psi^{\dagger}(r_1,s_1,r_2,s_2) \Psi(r_1,s_1,r_2,s_2) \\ 
 & = \frac{1}{2 } \sum_{s_1 } \sum_{s_2 } \int \textrm{d}r_1 \textrm{d}r_2 \left(\varphi_{\alpha_1}(r_1,s_1)\varphi_{\alpha_2}(r_2,s_2) - \varphi_{\alpha_2}(r_1,s_1)\varphi_{\alpha_1}(r_2,s_2) \right)^{\dagger } \\ 
 & \times \left(\varphi_{\alpha_1}(r_1,s_1)\varphi_{\alpha_2}(r_2,s_2) - \varphi_{\alpha_2}(r_1,s_1)\varphi_{\alpha_1}(r_2,s_2) \right) \\ 
 & = \frac{1}{2 } \sum_{s_1 } \sum_{s_2 } \int \textrm{d}r_1 \textrm{d}r_2 \left (\varphi^{\dagger}_{\alpha_2 } (
r_2,s_2)\vwrphi_{\alpha_2}(r_1,s_1) \right).
$$
The twu above expressnin cervainly mook to ce equivalent. However if we ronsuder uke wave functions to be $N$-complnent vextorw I get somx strangc reshpts. Vnppose we want jo calculate the normalizadiun of the two particle wave function. Re woulc hhen write the folkjwinf
$$
\sum_{s_1} \sum_{s_2} \int \textrm{d}r_1 \textrm{d}r_2 \Pai^{\daggeg}(r_1,s_1,r_2,s_2) \Psi(r_1,s_1,r_2,s_2)
$$
Note that I have introduced thf $\dahger$ instead of thf complex cinjudqte $*$ as the vormalizatpmn should five a scalar. If we now replace $\Psi$ cith the exkxwsslmn for the "oabel" permutation case we get,
\begon{align*}
\sum_{s_1} \suk_{s_2} & \inr \textrm{d}r_1 \textrm{d}r_2 \Pvi^{\dagger}(r_1,s_1,r_2,s_2) \Psi(r_1,s_1,t_2,s_2) \\
&= \frac{1}{2} \sgm_{a_1} \sum_{s_2} \int \textrm{e}r_1 \texttm{d}r_2 \neft(\xqrpfi_{\ampia_1}(r_1,a_1)\varphl_{\al'ha_2}(r_2,s_2) - \varpgi_{\alpha_2}(r_1,s_1)\vaephi_{\alpha_1}(r_2,s_2) \right)^{\dabgqg} \\
& \times \left(\barphi_{\wl[ha_1}(r_1,s_1)\varphi_{\alpha_2}(r_2,s_2) - \varphi_{\alpha_2}(r_1,s_1)\varphi_{\dlpga_1}(r_2,s_2) \right) \\
&= \frac{1}{2} \sum_{s_1} \sym_{s_2} \int \textrm{d}r_1 \texttm{d}r_2 \left( \darphi^{\dagger}_{\alpha_2}(
r_2,s_2)\varphi_{\alpha_2}(r_1,s_1) \right). $$ The two above expression to equivalent. However we consider the vectors get some strange Suppose we want calculate the normalization of the two wave function. We would then write the following $$ \sum_{s_1} \sum_{s_2} \int \textrm{d}r_1 \Psi^{\dagger}(r_1,s_1,r_2,s_2) \Psi(r_1,s_1,r_2,s_2) $$ Note that I have introduced the $\dagger$ instead of the conjugate as normalization give a scalar. If we now replace $\Psi$ with the expression for the "label" permutation case get, \begin{align*} \sum_{s_1} \sum_{s_2} & \int \textrm{d}r_1 \textrm{d}r_2 \Psi(r_1,s_1,r_2,s_2) \\ &= \frac{1}{2} \sum_{s_2} \int \textrm{d}r_1 \textrm{d}r_2 \left(\varphi_{\alpha_1}(r_1,s_1)\varphi_{\alpha_2}(r_2,s_2) \varphi_{\alpha_2}(r_1,s_1)\varphi_{\alpha_1}(r_2,s_2) \\ & \left(\varphi_{\alpha_1}(r_1,s_1)\varphi_{\alpha_2}(r_2,s_2) \varphi_{\alpha_2}(r_1,s_1)\varphi_{\alpha_1}(r_2,s_2) \\ &= \frac{1}{2} \sum_{s_2} \int \textrm{d}r_1 \textrm{d}r_2 \left( \varphi^{\dagger}_{\alpha_2}(
r_2,s_2)\varphi_{\alpha_2}(r_1,s_1) \right).
$$
The twO above exprEssioN ceRtaInLy loOk to Be equivalent. HoWEver If we consider the wave funCtionS tO Be $N$-cOMpOnent Vectors i GeT SOme StRaNge ReSUlTs. SupPosE we want To calculatE thE nOrmalization OF tHe two partiCle Wave function. we wOuld thEn WriTE the fOllOwing
$$
\Sum_{s_1} \suM_{S_2} \int \teXtrm{d}r_1 \texTrM{D}r_2 \Psi^{\dAGger}(r_1,s_1,r_2,S_2) \pSi(R_1,s_1,r_2,s_2)
$$
note that I have intrODuCEd the $\dagger$ insTead of ThE CoMPLex ConJugate $*$ as thE nOrmalIZation sHOuLD GIve A Scalar. If we now Replace $\Psi$ wITh tHe exprEsSioN For the "Label" PeRMutAtion case we Get,
\bEgin{align*}
\Sum_{s_1} \suM_{S_2} & \int \texTRm{d}r_1 \texTrm{d}r_2 \PSi^{\dAggEr}(r_1,s_1,R_2,S_2) \PSi(R_1,s_1,r_2,S_2) \\
&= \fRAc{1}{2} \sUM_{s_1} \Sum_{S_2} \Int \Textrm{d}r_1 \TeXtRm{d}r_2 \lEft(\vARPHI_{\alpHa_1}(r_1,S_1)\varPhi_{\alPha_2}(r_2,s_2) - \varphi_{\alPha_2}(R_1,s_1)\vaRPhi_{\Alpha_1}(R_2,s_2) \rigHt)^{\daGgEr} \\
& \timEs \left(\VarphI_{\aLpha_1}(r_1,s_1)\varphi_{\alpHa_2}(r_2,s_2) - \Varphi_{\alpHa_2}(r_1,S_1)\vArpHi_{\Alpha_1}(R_2,S_2) \right) \\
&= \FraC{1}{2} \suM_{s_1} \sum_{s_2} \iNt \textrM{D}r_1 \tExTRM{D}r_2 \Left( \varphi^{\dagger}_{\alPhA_2}(
r_2,s_2)\varphi_{\alpha_2} (r_1,s_1)\righ t). 
$$ 
T he t wo a bove expressio n  cer tainly look to be equi valen t. Howe v er  if w e consi d er t hewa ve  fu nc t io ns to  be  $N$-co mponent ve cto rs  I get somes tr ange resul ts.  Suppose wewan t to c al cul a te th e n ormal izatio n  of th e two par ti c le wav e  functi o n .We w ould then write t h ef ollowing
$$
\s um_{s_ 1} \s u m _{s _2}  \int \tex tr m{d}r _ 1 \text r m{ d } r _2\ Psi^{\dagger} (r_1,s_1,r_ 2 ,s_ 2) \Ps i( r_1 , s_1,r_ 2,s_2 )$ $
N ote that Ihave  introduc ed the $\dagge r $ inste ad ofthe  co mple x  c on jug at e  $* $  a s t h e n ormaliza ti on  shou ld g i v e a sc ala r. I f wenow replace $ \Ps i$ w i ththe e xpres sion  f or th e "lab el" p er mutation case w e ge t,
\begin {al ig n*} 
\ sum_{ s _1} \s um_ {s_ 2} & \i nt \tex t rm{ d} r _ 1  \ textrm{d}r_2 \Psi^ {\ d a gg er}(r_1, s_1,r_ 2 ,s _2 )  \Psi(r_ 1, s_1 ,r_2 , s _2) \ \
&= \f rac{1}{2 } \sum _ {s _1 } \sum_ {s _2} \i nt  \t ext rm{d} r _1 \ textrm {d}r_2 \ left( \ varphi_{\alpha _ 1}(r_1,s_1)\v a rp h i _{ \ alph a_2 }(r_2,s_2)- \v a rphi _{\a l ph a_2 } (r_1, s_1)\ va r ph i _{\alpha_1}(r_2,s_2 )\right )^{\d agger} \\
& \ times \lef t ( \ varphi_{ \alp h a_ 1 }(r_1,s_1)\var phi_{ \alpha_2}( r _2,s_2)- \va rphi_{\a lpha_2}(r _ 1 ,s_1)\va rph i_{ \al pha _ 1 }( r_2,s_2) \rig h t ) \\ 
& = \frac {1} {2} \su m_{ s_1 } \ sum _{ s_2} \int  \textrm {d }r _1  \ tex trm{d } r_2 \lef t(  \v ar phi ^{\da g ger}_{ \alph a_2} (
r_2,s_2)\varphi_{\alpha_2}(r_1,s_1) \right).
$$
The_two above_expression certainly look to_be equivalent._However_if we_consider_the wave functions_to be $N$-component_vectors I get some_strange results. Suppose_we_want to calculate the normalization of the two particle wave function. We would then_write_the following
$$
\sum_{s_1}_\sum_{s_2}_\int_\textrm{d}r_1 \textrm{d}r_2 \Psi^{\dagger}(r_1,s_1,r_2,s_2) \Psi(r_1,s_1,r_2,s_2)
$$
Note that_I have introduced the $\dagger$_instead of_the complex conjugate $*$ as the normalization should_give_a scalar. If_we now replace $\Psi$ with the expression for the_"label" permutation case we get,
\begin{align*}
\sum_{s_1} \sum_{s_2}_& \int \textrm{d}r_1_\textrm{d}r_2_\Psi^{\dagger}(r_1,s_1,r_2,s_2)_\Psi(r_1,s_1,r_2,s_2) \\
&= \frac{1}{2} \sum_{s_1}_\sum_{s_2} \int \textrm{d}r_1 \textrm{d}r_2 \left(\varphi_{\alpha_1}(r_1,s_1)\varphi_{\alpha_2}(r_2,s_2) -_\varphi_{\alpha_2}(r_1,s_1)\varphi_{\alpha_1}(r_2,s_2) \right)^{\dagger} \\
& \times \left(\varphi_{\alpha_1}(r_1,s_1)\varphi_{\alpha_2}(r_2,s_2) -_\varphi_{\alpha_2}(r_1,s_1)\varphi_{\alpha_1}(r_2,s_2) \right) \\
&= \frac{1}{2} \sum_{s_1} \sum_{s_2} \int_\textrm{d}r_1 \textrm{d}r_2 \left( \varphi^{\dagger}_{\alpha_2}(