index_3.html

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SU(2) Cayley-Hamilton rules * all this should be abstracted and automatized
*

${< χ _ + '6 u _ μ '6 u _ μ > -> -1/2 < χ _ + > ^2 + < χ _ + '6 u _ μ > + 1/2 < χ _ + >}$

${-< 鱯 _ μ(鲃)^† '6 鱯 _ μ(鲃) > < 鲃^† '6 χ >, -< 鱯 _ μ(鲃)^† '6 鱯 _ μ(鲃) > < χ^† '6 鲃 >, 2 < 鱯 _ μ(鲃)^† '6 鱯 _ μ(鲃) '6 χ^† '6 鲃 >, -2 < 鲃 '6 鱯 _ μ(鲃)^† '6 χ '6 鱯 _ μ(鲃)^† >}$

$su2chicalhamRule = (-su2chicalham[[1]] /. {μ -> μ_, x -> x_}) -> Plus @@ Drop[su2chicalham, {1}]$

$clr1 = CayleyHamiltonRules[{{HLeft[x], HLeft[x], UChiPlus[x]}}, UDimension -> 2] /. UTrace1[(HLeft | USmall[_])[_]] -> 0$

${< χ _ + '6 H _ L '6 H _ L > -> 1/2 < H _ L '6 H _ L > < χ _ + >}$

${-1/2 < 鲃^† '6 χ > < Q _ L '6 Q _ L >, -1/2 < χ^† '6 鲃 > < Q _ L '6 Q _ L >, -1/2 < 鲃^† '6 χ > < Q _ R '6 Q _ R >, -1/2 < χ^† '6 鲃 > < Q _ R '6 Q _ R >, < 鲃^† '6 χ '6 Q _ L '6 Q _ L >, < 鲃^† '6 χ > < 鲃^† '6 Q _ R '6 鲃 '6 Q _ L >, < χ^† '6 鲃 > < 鲃^† '6 Q _ R '6 鲃 '6 Q _ L >, -< 鲃^† '6 Q _ R '6 χ '6 Q _ L >, < 鲃^† '6 Q _ R '6 Q _ R '6 χ >, < χ^† '6 鲃 '6 Q _ L '6 Q _ L >, -< χ^† '6 Q _ R '6 鲃 '6 Q _ L >, < χ^† '6 Q _ R '6 Q _ R '6 鲃 >, -< 鲃^† '6 Q _ R '6 鲃 '6 χ^† '6 鲃 '6 Q _ L >, -< 鲃 '6 Q _ L '6 鲃^† '6 χ '6 鲃^† '6 Q _ R >}$

$su2chicalhamRule1 = (su2chicalham[[5]] /. {μ -> μ_, x -> x_}) -> -Plus @@ Drop[su2chicalham, {5}]$

$clr1 = CayleyHamiltonRules[{{HRight[x], HRight[x], UChiPlus[x]}}, UDimension -> 2] /. UTrace1[(HLeft | USmall[_])[_]] -> 0$

${< χ _ + '6 H _ R '6 H _ R > -> -1/2 < χ _ + > < H _ R >^2 + < H _ R '6 χ _ + > < H _ R > + 1/2 < H _ R '6 H _ R > < χ _ + >}$

${-1/2 < 鲃^† '6 χ > < Q _ L '6 Q _ L >, -1/2 < χ^† '6 鲃 > < Q _ L '6 Q _ L >, -1/2 < 鲃^† '6 χ > < Q _ R '6 Q _ R >, -1/2 < χ^† '6 鲃 > < Q _ R '6 Q _ R >, < 鲃^† '6 χ '6 Q _ L '6 Q _ L >, -< 鲃^† '6 χ > < 鲃^† '6 Q _ R '6 鲃 '6 Q _ L >, -< χ^† '6 鲃 > < 鲃^† '6 Q _ R '6 鲃 '6 Q _ L >, < 鲃^† '6 Q _ R '6 χ '6 Q _ L >, < 鲃^† '6 Q _ R '6 Q _ R '6 χ >, < χ^† '6 鲃 '6 Q _ L '6 Q _ L >, < χ^† '6 Q _ R '6 鲃 '6 Q _ L >, < χ^† '6 Q _ R '6 Q _ R '6 鲃 >, < 鲃^† '6 Q _ R '6 鲃 '6 χ^† '6 鲃 '6 Q _ L >, < 鲃 '6 Q _ L '6 鲃^† '6 χ '6 鲃^† '6 Q _ R >, -2 < 鲃^† '6 χ '6 Q _ L > < Q >, -2 < 鲃^† '6 Q _ R '6 χ > < Q >, -2 < χ^† '6 鲃 '6 Q _ L > < Q >, -2 < χ^† '6 Q _ R '6 鲃 > < Q >, 2 < 鲃^† '6 χ > < Q >^2, 2 < χ^† '6 鲃 > < Q >^2}$

$su2chicalhamRule2 = (-1/2 su2qcalham[[-3]] /. {μ -> μ_, x -> x_}) -> 1/2 Plus @@ Drop[su2qcalham, {-3}]$

$clr1 = CayleyHamiltonRules[{{HLeft[x], USmall[μ][x], NM[USmall[μ][x], HLeft[x]]}}, UDimension -> 2] /. UTrace1[(HLeft | USmall[_])[_]] -> 0$

${< H _ L '6 u _ μ '6 H _ L '6 u _ μ > -> < H _ L '6 u _ μ >^2 - < H _ L '6 H _ L '6 u _ μ '6 u _ μ >}$

${< 鲃^† '6 鱯 _ μ(鲃) '6 Q _ L >^2, -2 < 鲃^† '6 鱯 _ μ(鲃) '6 Q _ L > < 鲃^† '6 Q _ R '6 鱯 _ μ(鲃) >, < 鲃^† '6 Q _ R '6 鱯 _ μ(鲃) >^2, < 鱯 _ μ(鲃)^† '6 鱯 _ μ(鲃) '6 Q _ L '6 Q _ L >, -2 < 鱯 _ μ(鲃)^† '6 Q _ R '6 鱯 _ μ(鲃) '6 Q _ L >, < 鱯 _ μ(鲃)^† '6 Q _ R '6 Q _ R '6 鱯 _ μ(鲃) >, < 鱯 _ μ(鲃)^† '6 鲃 '6 Q _ L '6 鲃^† '6 鱯 _ μ(鲃) '6 Q _ L >, < 鱯 _ μ(鲃)^† '6 Q _ R '6 鱯 _ μ(鲃) '6 鲃^† '6 Q _ R '6 鲃 >, < Q _ L '6 鱯 _ μ(鲃)^† '6 鲃 '6 鱯 _ μ(鲃)^† '6 Q _ R '6 鲃 >, < Q _ L '6 鲃^† '6 Q _ R '6 鱯 _ μ(鲃) '6 鲃^† '6 鱯 _ μ(鲃) >}$

$su2qchalhamrule1 = (su2qchalham[[7]] /. {μ -> μ_, x -> x_}) -> -Plus @@ Drop[su2qchalham, {7}]$

$clr1 = CayleyHamiltonRules[{{HLeft[x], HLeft[x], NM[HRight[x], HRight[x]]}}, UDimension -> 2] /. UTrace1[(HLeft | USmall[_])[_]] -> 0$

${< H _ L '6 H _ L '6 H _ R '6 H _ R > -> 1/2 < H _ L '6 H _ L > < H _ R '6 H _ R >}$

-1/2 < Q _ L '6 Q _ L >^2 - < Q _ R '6 Q _ R > < Q _ L '6 Q _ L > - 1/2 < Q _ R '6 Q _ R >^2 + 2 < 鲃^† '6 Q _ R '6 鲃 '6 Q _ L >^2 + < Q _ L '6 Q _ L '6 Q _ L '6 Q _ L > + < Q _ R '6 Q _ R '6 Q _ R '6 Q _ R > - 2 < 鲃^† '6 Q _ R '6 鲃 '6 Q _ L '6 鲃^† '6 Q _ R '6 鲃 '6 Q _ L >

${-1/2 < Q _ L '6 Q _ L >^2, -< Q _ L '6 Q _ L > < Q _ R '6 Q _ R >, -1/2 < Q _ R '6 Q _ R >^2, 2 < 鲃^† '6 Q _ R '6 鲃 '6 Q _ L >^2, < Q _ L '6 Q _ L '6 Q _ L '6 Q _ L >, < Q _ R '6 Q _ R '6 Q _ R '6 Q _ R >, -2 < 鲃^† '6 Q _ R '6 鲃 '6 Q _ L '6 鲃^† '6 Q _ R '6 鲃 '6 Q _ L >}$

${ -> -1/2 < a >^2 + < a '6 b > < a > + 1/2 < a '6 a > }$

$su2calhamrule2 = (su2calhb[[1, 1]] /. {UMatrix[a] -> a_, UMatrix[b] -> b_}) -> (su2calhb[[1, 2]] /. {UMatrix[a] -> a, UMatrix[b] -> b})$

$su2calhamrule2 = UTrace1[NM[b___, a_, a_, c___]] :> -1/2 UTrace1[a]^2 UTrace1[NM[c, b]] + 1/2 UTrace1[NM[c, b]] UTrace1[NM[a, a]] + UTrace1[a] UTrace1[NM[a, c, b]] /; Length[{b, c}] > 0$

$< b___ '6 a_ '6 a_ '6 c___ > :> -1/2 < c '6 b > < a >^2 + < a '6 c '6 b > < a > + 1/2 < a '6 a > < c '6 b > /; Length[{b, c}] > 0$

${< a '6 a '6 a > -> 3/2 < a '6 a > < a > - < a >^3/2}$

${< a '6 a '6 a '6 a > -> -< a >^4/2 + < a '6 a > < a >^2 + 1/2 < a '6 a >^2}$

$su2qchalhamrule2 = (-1/2 * su2qchalham[[-1]] /. {μ -> μ_, x -> x_}) -> 1/2 (Plus @@ Drop[su2qchalham, {-1}]) /. su2calhamrule4$

< 鲃^† '6 Q _ R '6 鲃 '6 Q _ L '6 鲃^† '6 Q _ R '6 鲃 '6 Q _ L > -> 1/2 (-1/2 < Q _ L >^4 + < Q _ L '6 Q _ L > < Q _ L >^2 - < Q _ R >^4/2 + 2 < 鲃^† '6 Q _ R '6 鲃 '6 Q _ L >^2 + < Q _ R '6 Q _ R > < Q _ R >^2 - < Q _ L '6 Q _ L > < Q _ R '6 Q _ R >)

$clr1 = CayleyHamiltonRules[{{HLeft[x], HRight[x], NM[HLeft[x], HRight[x]]}}, UDimension -> 2] /. UTrace1[(HLeft | USmall[_])[_]] -> 0$

${< H _ L '6 H _ R '6 H _ L '6 H _ R > -> < H _ L '6 H _ R >^2 + < H _ R > < H _ R '6 H _ L '6 H _ L > - < H _ L '6 H _ L '6 H _ R '6 H _ R >}$

${2 < Q _ L '6 Q _ L > < Q _ R '6 Q _ R >, -4 < 鲃^† '6 Q _ R '6 Q _ R '6 鲃 '6 Q _ L '6 Q _ L >, 2 < 鲃^† '6 Q _ R '6 鲃 '6 Q _ L '6 Q _ L > < Q >, 2 < 鲃^† '6 Q _ R '6 Q _ R '6 鲃 '6 Q _ L > < Q >, -< Q _ L '6 Q _ L > < Q >^2, -< Q _ R '6 Q _ R > < Q >^2}$

$su2qchalhamrule3 = (-1/4 su2qchalham[[2]] /. {μ -> μ_, x -> x_}) -> 1/4 (Plus @@ Drop[su2qchalham, {2}])$

< 鲃^† '6 Q _ R '6 Q _ R '6 鲃 '6 Q _ L '6 Q _ L > -> 1/4 (-< Q _ L '6 Q _ L > < Q >^2 - < Q _ R '6 Q _ R > < Q >^2 + 2 < 鲃^† '6 Q _ R '6 鲃 '6 Q _ L '6 Q _ L > < Q > + 2 < 鲃^† '6 Q _ R '6 Q _ R '6 鲃 '6 Q _ L > < Q > + 2 < Q _ L '6 Q _ L > < Q _ R '6 Q _ R >)

$clr1 = CayleyHamiltonRules[{{HLeft[x], HLeft[x], HRight[x]}}, UDimension -> 2] /. UTrace1[(HLeft | USmall[_])[_]] -> 0$

${< H _ R '6 H _ L '6 H _ L > -> 1/2 < H _ R > < H _ L '6 H _ L >}$

${-< 鲃^† '6 Q _ R '6 鲃 '6 Q _ L '6 Q _ L >, -< 鲃^† '6 Q _ R '6 Q _ R '6 鲃 '6 Q _ L >, 1/2 < Q _ L '6 Q _ L > < Q >, 1/2 < Q _ R '6 Q _ R > < Q >, 2 < 鲃^† '6 Q _ R '6 鲃 '6 Q _ L > < Q >, -< Q >^3}$

$su2qchalhamrule4 = (-su2qchalham[[1]] /. {μ -> μ_, x -> x_}) -> (Plus @@ Drop[su2qchalham, {1}])$

$clr1 = CayleyHamiltonRules[{{HLeft[x], HLeft[x], HRight[x]}}, UDimension -> 2] /. UTrace1[(HLeft | USmall[_])[_]] -> 0$

${< H _ R '6 H _ L '6 H _ L > -> 1/2 < H _ R > < H _ L '6 H _ L >}$

$su2qchalhamrule5 = (-su2qchalham[[1]] /. {μ -> μ_, x -> x_}) -> Plus @@ Drop[su2qchalham, {1}]$

$clr1 = CayleyHamiltonRules[{{HLeft[x], USmall[μ][x], NM[HLeft[x], USmall[μ][x]]}}, UDimension -> 2] /. UTrace1[(HLeft | USmall[_])[_]] -> 0$

${< H _ L '6 u _ μ '6 H _ L '6 u _ μ > -> < H _ L '6 u _ μ >^2 - < H _ L '6 H _ L '6 u _ μ '6 u _ μ >}$

$su2qchalhamrule1 = (su2qchalham[[-4]] /. {μ -> μ_, x -> x_}) -> -Plus @@ Drop[su2qchalham, {-4}]$

$clr1 = CayleyHamiltonRules[{{USmall[μ][x], USmall[μ][x], NM[HLeft[x], HLeft[x]]}}, UDimension -> 2] /. UTrace1[(HLeft | USmall[_])[_]] -> 0$

${< H _ L '6 H _ L '6 u _ μ '6 u _ μ > -> 1/2 < H _ L '6 H _ L > }$

-1/2 < 鱯 _ μ(鲃)^† '6 鱯 _ μ(鲃) > < Q _ L '6 Q _ L > - 1/2 < 鱯 _ μ(鲃)^† '6 鱯 _ μ(鲃) > < Q _ R '6 Q _ R > + < 鱯 _ μ(鲃)^† '6 鱯 _ μ(鲃) '6 Q _ L '6 Q _ L > + < 鱯 _ μ(鲃)^† '6 Q _ R '6 Q _ R '6 鱯 _ μ(鲃) > + < 鱯 _ μ(鲃)^† '6 鱯 _ μ(鲃) > < 鲃^† '6 Q _ R '6 鲃 '6 Q _ L > + < Q _ L '6 鱯 _ μ(鲃)^† '6 鲃 '6 鱯 _ μ(鲃)^† '6 Q _ R '6 鲃 > + < Q _ L '6 鲃^† '6 Q _ R '6 鱯 _ μ(鲃) '6 鲃^† '6 鱯 _ μ(鲃) >

${-1/2 < 鱯 _ μ(鲃)^† '6 鱯 _ μ(鲃) > < Q _ L '6 Q _ L >, -1/2 < 鱯 _ μ(鲃)^† '6 鱯 _ μ(鲃) > < Q _ R '6 Q _ R >, < 鱯 _ μ(鲃)^† '6 鱯 _ μ(鲃) '6 Q _ L '6 Q _ L >, < 鱯 _ μ(鲃)^† '6 Q _ R '6 Q _ R '6 鱯 _ μ(鲃) >, < 鱯 _ μ(鲃)^† '6 鱯 _ μ(鲃) > < 鲃^† '6 Q _ R '6 鲃 '6 Q _ L >, < Q _ L '6 鱯 _ μ(鲃)^† '6 鲃 '6 鱯 _ μ(鲃)^† '6 Q _ R '6 鲃 >, < Q _ L '6 鲃^† '6 Q _ R '6 鱯 _ μ(鲃) '6 鲃^† '6 鱯 _ μ(鲃) >}$

$su2qchalhamrule1a = (su2qchalham[[-2]] /. {μ -> μ_, x -> x_}) -> -Plus @@ Drop[su2qchalham, {-2}]$

< Q _ L '6 鱯 _ μ_(鲃)^† '6 鲃 '6 鱯 _ μ_(鲃)^† '6 Q _ R '6 鲃 > -> 1/2 < 鱯 _ μ(鲃)^† '6 鱯 _ μ(鲃) > < Q _ L '6 Q _ L > + 1/2 < 鱯 _ μ(鲃)^† '6 鱯 _ μ(鲃) > < Q _ R '6 Q _ R > - < 鱯 _ μ(鲃)^† '6 鱯 _ μ(鲃) '6 Q _ L '6 Q _ L > - < 鱯 _ μ(鲃)^† '6 Q _ R '6 Q _ R '6 鱯 _ μ(鲃) > - < 鱯 _ μ(鲃)^† '6 鱯 _ μ(鲃) > < 鲃^† '6 Q _ R '6 鲃 '6 Q _ L > - < Q _ L '6 鲃^† '6 Q _ R '6 鱯 _ μ(鲃) '6 鲃^† '6 鱯 _ μ(鲃) >

$clr1 = CayleyHamiltonRules[{{HLeft[x], USmall[μ][x], NM[USmall[μ][x], HRight[x]]}}, UDimension -> 2] /. UTrace1[(HLeft | USmall[_])[_]] -> 0$

${< H _ L '6 u _ μ '6 u _ μ '6 H _ R > -> < H _ L '6 u _ μ > < H _ R '6 u _ μ > - < H _ L '6 u _ μ '6 H _ R '6 u _ μ >}$

-< 鲃^† '6 鱯 _ μ(鲃) '6 Q _ L >^2 + < 鲃^† '6 Q _ R '6 鱯 _ μ(鲃) >^2 - < 鱯 _ μ(鲃)^† '6 鱯 _ μ(鲃) '6 Q _ L '6 Q _ L > + < 鱯 _ μ(鲃)^† '6 Q _ R '6 Q _ R '6 鱯 _ μ(鲃) > + < 鲃^† '6 鱯 _ μ(鲃) '6 Q _ L '6 鲃^† '6 鱯 _ μ(鲃) '6 Q _ L > - < 鲃^† '6 Q _ R '6 鱯 _ μ(鲃) '6 鲃^† '6 Q _ R '6 鱯 _ μ(鲃) > + < Q _ L '6 鱯 _ μ(鲃)^† '6 鲃 '6 鱯 _ μ(鲃)^† '6 Q _ R '6 鲃 > - < Q _ L '6 鲃^† '6 Q _ R '6 鱯 _ μ(鲃) '6 鲃^† '6 鱯 _ μ(鲃) >

${-< 鲃^† '6 鱯 _ μ(鲃) '6 Q _ L >^2, < 鲃^† '6 Q _ R '6 鱯 _ μ(鲃) >^2, -< 鱯 _ μ(鲃)^† '6 鱯 _ μ(鲃) '6 Q _ L '6 Q _ L >, < 鱯 _ μ(鲃)^† '6 Q _ R '6 Q _ R '6 鱯 _ μ(鲃) >, < 鲃^† '6 鱯 _ μ(鲃) '6 Q _ L '6 鲃^† '6 鱯 _ μ(鲃) '6 Q _ L >, -< 鲃^† '6 Q _ R '6 鱯 _ μ(鲃) '6 鲃^† '6 Q _ R '6 鱯 _ μ(鲃) >, < Q _ L '6 鱯 _ μ(鲃)^† '6 鲃 '6 鱯 _ μ(鲃)^† '6 Q _ R '6 鲃 >, -< Q _ L '6 鲃^† '6 Q _ R '6 鱯 _ μ(鲃) '6 鲃^† '6 鱯 _ μ(鲃) >}$

$su2qchalhamrule6 = (su2qchalham[[4]] /. {μ -> μ_, x -> x_}) -> -Plus @@ Drop[su2qchalham, {4}]$

< 鱯 _ μ_(鲃)^† '6 Q _ R '6 Q _ R '6 鱯 _ μ_(鲃) > -> < 鲃^† '6 鱯 _ μ(鲃) '6 Q _ L >^2 - < 鲃^† '6 Q _ R '6 鱯 _ μ(鲃) >^2 + < 鱯 _ μ(鲃)^† '6 鱯 _ μ(鲃) '6 Q _ L '6 Q _ L > - < 鲃^† '6 鱯 _ μ(鲃) '6 Q _ L '6 鲃^† '6 鱯 _ μ(鲃) '6 Q _ L > + < 鲃^† '6 Q _ R '6 鱯 _ μ(鲃) '6 鲃^† '6 Q _ R '6 鱯 _ μ(鲃) > - < Q _ L '6 鱯 _ μ(鲃)^† '6 鲃 '6 鱯 _ μ(鲃)^† '6 Q _ R '6 鲃 > + < Q _ L '6 鲃^† '6 Q _ R '6 鱯 _ μ(鲃) '6 鲃^† '6 鱯 _ μ(鲃) >

$clr1 = CayleyHamiltonRules[{{HLeft[x], USmall[μ][x], NM[HLeft[x], USmall[μ][x]]}}, UDimension -> 2] /. UTrace1[(HLeft | USmall[_])[_]] -> 0$

${< H _ L '6 u _ μ '6 H _ L '6 u _ μ > -> < H _ L '6 u _ μ >^2 - < H _ L '6 H _ L '6 u _ μ '6 u _ μ >}$

$clr2 = CayleyHamiltonRules[{{HLeft[x], HLeft[x], NM[USmall[μ][x], USmall[μ][x]]}}, UDimension -> 2] /. UTrace1[(HLeft | USmall[_])[_]] -> 0$

${< H _ L '6 H _ L '6 u _ μ '6 u _ μ > -> 1/2 < H _ L '6 H _ L > }$

$clr3 = {clr1[[1, 1]] -> clr1[[1, 2]] /. clr2} /. UTrace1[(HLeft | USmall[_])[_]] -> 0$

${< H _ L '6 u _ μ '6 H _ L '6 u _ μ > -> < H _ L '6 u _ μ >^2 - 1/2 < H _ L '6 H _ L > }$

${1/2 < 鱯 _ μ(鲃)^† '6 鱯 _ μ(鲃) > < Q _ L '6 Q _ L >, 1/2 < 鱯 _ μ(鲃)^† '6 鱯 _ μ(鲃) > < Q _ R '6 Q _ R >, < 鱯 _ μ(鲃)^† '6 鲃 '6 Q _ L >^2, -2 < 鱯 _ μ(鲃)^† '6 鲃 '6 Q _ L > < 鱯 _ μ(鲃)^† '6 Q _ R '6 鲃 >, < 鱯 _ μ(鲃)^† '6 Q _ R '6 鲃 >^2, -2 < 鱯 _ μ(鲃)^† '6 Q _ R '6 鱯 _ μ(鲃) '6 Q _ L >, -< 鱯 _ μ(鲃)^† '6 鱯 _ μ(鲃) > < 鲃^† '6 Q _ R '6 鲃 '6 Q _ L >, < 鱯 _ μ(鲃)^† '6 鲃 '6 Q _ L '6 鲃^† '6 鱯 _ μ(鲃) '6 Q _ L >, < 鱯 _ μ(鲃)^† '6 Q _ R '6 鱯 _ μ(鲃) '6 鲃^† '6 Q _ R '6 鲃 >}$

$su2qchalhamrule7 = (su2qchalham[[-2]] /. {μ -> μ_, x -> x_}) -> -Plus @@ Drop[su2qchalham, {-2}]$

$clr1 = CayleyHamiltonRules[{{NM[HLeft[x], HLeft[x]] + NM[HRight[x], HRight[x]], USmall[μ][x], USmall[μ][x]}}, UDimension -> 2] /. UTrace1[(HLeft | USmall[_])[_]] -> 0$

${< H _ R '6 H _ R '6 u _ μ '6 u _ μ > -> 1/2 < H _ L '6 H _ L > + 1/2 < H _ R '6 H _ R > - < H _ L '6 H _ L '6 u _ μ '6 u _ μ >}$

${-< 鱯 _ μ(鲃)^† '6 鱯 _ μ(鲃) > < Q _ L '6 Q _ L >, -< 鱯 _ μ(鲃)^† '6 鱯 _ μ(鲃) > < Q _ R '6 Q _ R >, 2 < 鱯 _ μ(鲃)^† '6 鱯 _ μ(鲃) '6 Q _ L '6 Q _ L >, 2 < 鱯 _ μ(鲃)^† '6 Q _ R '6 Q _ R '6 鱯 _ μ(鲃) >}$

$su2qchalhamrule8 = (1/2 su2qchalham[[-1]] /. {μ -> μ_, x -> x_}) -> -1/2 Plus @@ Drop[su2qchalham, {-1}]$

$clr1 = CayleyHamiltonRules[{{NM[HRight[x], HRight[x]], USmall[μ][x], USmall[μ][x]}}, UDimension -> 2] /. UTrace1[(HLeft | USmall[_])[_]] -> 0$

${< H _ R '6 H _ R '6 u _ μ '6 u _ μ > -> 1/2 < H _ R '6 H _ R > }$

$clr2 = CayleyHamiltonRules[{{NM[HRight[x], USmall[μ][x]], HRight[x], USmall[μ][x]}}, UDimension -> 2] /. UTrace1[(HLeft | USmall[_])[_]] -> 0 /. clr1$

${< H _ R '6 u _ μ '6 H _ R '6 u _ μ > -> < H _ R '6 u _ μ >^2 - 1/2 < H _ R '6 H _ R > + < H _ R > < H _ R '6 u _ μ '6 u _ μ >}$

$clr1a = CayleyHamiltonRules[{{NM[HLeft[x], HLeft[x]], USmall[μ][x], USmall[μ][x]}}, UDimension -> 2] /. UTrace1[(HLeft | USmall[_])[_]] -> 0$

${< H _ L '6 H _ L '6 u _ μ '6 u _ μ > -> 1/2 < H _ L '6 H _ L > }$

$clr2a = CayleyHamiltonRules[{{NM[HLeft[x], USmall[μ][x]], HLeft[x], USmall[μ][x]}}, UDimension -> 2] /. UTrace1[(HLeft | USmall[_])[_]] -> 0 /. clr1a$

${< H _ L '6 u _ μ '6 H _ L '6 u _ μ > -> < H _ L '6 u _ μ >^2 - 1/2 < H _ L '6 H _ L > }$

${4 < 鲃^† '6 鱯 _ μ(鲃) '6 Q _ L > < 鲃^† '6 Q _ R '6 鱯 _ μ(鲃) >, 4 < 鱯 _ μ(鲃)^† '6 Q _ R '6 鱯 _ μ(鲃) '6 Q _ L >, 2 < 鱯 _ μ(鲃)^† '6 鱯 _ μ(鲃) > < 鲃^† '6 Q _ R '6 鲃 '6 Q _ L >, -2 < 鱯 _ μ(鲃)^† '6 鱯 _ μ(鲃) '6 Q _ L > < Q >, -2 < 鱯 _ μ(鲃)^† '6 Q _ R '6 鱯 _ μ(鲃) > < Q >}$

$su2qchalhamrule9 = (1/4 * su2qchalham[[2]] /. {μ -> μ_, x -> x_}) -> -1/4 * Plus @@ Drop[su2qchalham, {2}]$

< 鱯 _ μ_(鲃)^† '6 Q _ R '6 鱯 _ μ_(鲃) '6 Q _ L > -> 1/4 (-4 < 鲃^† '6 鱯 _ μ(鲃) '6 Q _ L > < 鲃^† '6 Q _ R '6 鱯 _ μ(鲃) > - 2 < 鱯 _ μ(鲃)^† '6 鱯 _ μ(鲃) > < 鲃^† '6 Q _ R '6 鲃 '6 Q _ L > + 2 < 鱯 _ μ(鲃)^† '6 鱯 _ μ(鲃) '6 Q _ L > < Q > + 2 < 鱯 _ μ(鲃)^† '6 Q _ R '6 鱯 _ μ(鲃) > < Q >)

$clr1 = CayleyHamiltonRules[{{HRight[x], USmall[μ][x], USmall[μ][x]}}, UDimension -> 2] /. UTrace1[(HLeft | USmall[_])[_]] -> 0$

${< H _ R '6 u _ μ '6 u _ μ > -> 1/2 < H _ R > }$

${< 鱯 _ μ(鲃)^† '6 鱯 _ μ(鲃) '6 Q _ L >, < 鱯 _ μ(鲃)^† '6 Q _ R '6 鱯 _ μ(鲃) >, -< 鱯 _ μ(鲃)^† '6 鱯 _ μ(鲃) > < Q >}$

$su2qchalhamrule10 = (su2qchalham[[1]] /. {μ -> μ_, x -> x_}) -> -Plus @@ Drop[su2qchalham, {1}]$

Converted by Mathematica (July 10, 2003)

•SU(2) Cayley-Hamilton rules *** all this should be abstracted and automatized ***

•
SU(2) Cayley-Hamilton rules * all this should be abstracted and automatized
*