Page - 40 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 whereΘb :G→G∗ is the symplectic1-cocycledefined inCorollary4. Proof. WehaveseeninCorollary4 thatb∈kerΘb. Thefact that theequalitygiven inthestatement abovedefines indeedasymmetricbilinear formonFbdirectly followsfromLemma1.Weonlyhave toprove that this symmetricbilinear formisnegative. LetX∈ Fb andX1∈G suchthatX=[X1,b]. UsingProposition17andCorollary3,wehave Γb(X,X)= 〈 Θb(X1), [X1,b] 〉 = 〈 Θ(X1)−ad∗X1 EJ(b), [X1,b] 〉 = 〈 DEJ(b)[X1,b], [X1,b] 〉 ≤0. Thesymmetricbilinear formΓbonFb is thereforenegative. Remark17. Thesymmetricnegativebilinear formsencountered inTheorem7andCorollary3seemtobe linked with theFishermetric in informationgeometrydiscussed in [31,60,61]. 7.3. ExamplesofGeneralizedGibbsStates 7.3.1.Actionof theGroupofRotationsonaSphere The symplecticmanifold (M,ω) considered here is the two-dimensional sphere of radius R centeredat theoriginOofa three-dimensionalorientedEuclideanvectorspace −→ E , equippedwith its areaelementassymplectic form. ThegroupGof rotationsaroundtheorigin (isomorphic toSO(3)) acts on the sphere M by aHamiltonian action. TheLie algebraG ofG can be identifiedwith−→E , the fundamental vector field on M associated to an element −→ b inG ≡ −→E being the vector field on Mwhose value at a pointm ∈ M is given by the vector product−→b ×−→Om. The dualG∗ ofG willbe too identifiedwith −→ E , thecouplingbydualitybeinggivenbytheEuclideanscalarproduct. Themomentummap J :M→G∗≡−→E isgivenby J(m)=−R−→Om , m∈M . Therefore, forany −→ b ∈G≡−→E ,〈 J(m), −→ b 〉 =−R−→Om ·−→b . Let −→ b be any element in G ≡ −→E . To calculate the partition function P(−→b )we choose an orthonormalbasis (−→ex ,−→ey ,−→ez)of−→E such that−→b = ‖−→b ‖−→ez ,with‖−→b ‖ ∈R+, andweuseangular coordinates (ϕ,θ)onthesphereM. Thecoordinatesofapointm∈Mare x=Rcosθcosϕ , y=Rcosθsinϕ , z=Rsinθ . Wehave P( −→ b )= ∫ 2π 0 (∫ π/2 −π/2 R2exp(R‖−→b ‖sinθdθ ) dϕ= 4πR ‖−→b ‖ sh ( R‖−→b ‖) . The probability density (with respect to the natural area measure on the sphere M) of the generalizedGibbsstateassociatedto −→ b is ρb(m)= 1 P( −→ b ) exp( −→ Om ·−→b ) , m∈M . 40