Seite - 99 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 where β= ⎡⎣ 12R−1 −R−1m 0 0 ⎤⎦ and ξˆ=[ R+mmT m 0 0 ] (259) IfwesetZ1= ⎡⎣ 12Ω−11 −Ω−11 n1 0 0 ⎤⎦ andZ2= ⎡⎣ 12Ω−12 −Ω−12 n2 0 0 ⎤⎦ (260) With 〈..., ...〉 the innerproductgivenby 〈ξ,β〉=Tr [ baT+HTL ] withξ= [ L b 0 0 ] ,β= [ H a 0 0 ] (261) [β,Z2]= βZ2−Z2β= ⎡⎣ 12R−1 −R−1m 0 0 ⎤⎦⎡⎣ 12Ω−12 −Ω−12 n2 0 0 ⎤⎦− ⎡⎣ 12Ω−12 −Ω−12 n2 0 0 ⎤⎦⎡⎣ 12R−1 −R−1m 0 0 ⎤⎦ [β,Z2]= ⎡⎣ 14(R−1Ω−12 −Ω−12 R−1) −12(R−1Ω−12 n2−Ω−12 R−1m) 0 0 ⎤⎦ (262) [ Z1, [ β,Z2 ]] = ⎡⎣ 12Ω−11 −Ω−11 n1 0 0 ⎤⎦⎡⎣ 14 ( R−1Ω−12 −Ω −1 2 R −1) −1 2 ( R−1Ω−12 n2−Ω −1 2 R −1m ) 0 0 ⎤⎦ − ⎡⎢⎣ 14 ( R−1Ω−12 −Ω −1 2 R −1) −1 2 ( R−1Ω−12 n2−Ω −1 2 R −1m ) 0 0 ⎤⎥⎦ ⎡⎢⎣ 12Ω−11 −Ω−11 n1 0 0 ⎤⎥⎦ = ⎡⎢⎣ 18 ( Ω−11 ( R−1Ω−12 −Ω−12 R−1 ) − ( R−1Ω−12 −Ω−12 R−1 ) Ω−11 ) −1 4 ( Ω−11 ( R−1Ω−12 n2−Ω−12 R−1m ) − ( R−1Ω−12 −Ω−12 R−1 ) Ω−11 n1 ) 0 0 ⎤⎥⎦ (263) Wecanthencompute: 〈 ξˆ, [Z1, [β,Z2]] 〉 =Tr [ 1 4 m (( R−1Ω−12 −Ω−12 R−1 ) Ω−11 n1−Ω−11 ( R−1Ω−12 n2−Ω−12 R−1m ))T] +Tr [( 1 8 ( Ω−11 ( R−1Ω−12 −Ω−12 R−1 ) − ( R−1Ω−12 −Ω−12 R−1 ) Ω−11 ))( R+mmT )] (264) TheSouriau-Fishermetric isdefinedinLiealgebragβ([β,Z1] , [β,Z2])where: [β,Z1]= ⎡⎣ 14(R−1Ω−11 −Ω−11 R−1) −12(R−1Ω−11 n1−Ω−11 R−1m) 0 0 ⎤⎦= ⎡⎣ 12G−11 −G−11 g1 0 0 ⎤⎦ withG1=2(Ω1R−RΩ1) andg1=(I−RΩ1R−1Ω−11 )n1+(Ω1RΩ−11 R−1− I)m [β,Z2]= ⎡⎣ 14(R−1Ω−12 −Ω−12 R−1) −12(R−1Ω−12 n2−Ω−12 R−1m) 0 0 ⎤⎦= ⎡⎣ 12G−12 −G−12 g2 0 0 ⎤⎦ withG2=2(Ω2R−RΩ2) andg2=(I−RΩ2R−1Ω−12 )n2+(Ω2RΩ−12 R−1− I)m (265) and β= ⎡⎣ 12R−1 −R−1m 0 0 ⎤⎦ (266) Another approach todevelop the Souriau-Fishermetric gβ([β,Z1] , [β,Z2]) is to compute the tensor Θ˜(X,Y) fromthemomentmap J: Θ˜(X,Y)= J[X,Y]−{JX, JY}with {., .} PoissonBracketand J theMomentMap (267) Θ˜(X,Y) : g×g→ (268) 99