Page - 132 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 254 Finally: z= 1 2 ln(det(C))− 3 2 lnβ+Cte , (47) where the value of the constant is not relevant in the sequel since it does not depend onW and F (through C). It isworth remarking that, unlike orb(μ), the subsetV0×R3×S2 is not preservedbytheactionanddependsonthearbitrarychoiceofV0.Nevertheless,z—then sand M—dependsonV0 only throughln(V0)which isabsorbedintheconstantandhasnoinfluence onthederivatives (17). Aspointedout byBarbaresco [17], there is apuzzlinganalogybetween the integral occuring in (10)andKoszul–Vinbergcharacteristic function[18,19]: ψΩ(Z)= ∫ Ω∗ e−μZdλ , whereΩ isasharpopenconvexconeandΩ∗ is thesetof linearstrictlypositive formson Ω¯−{0}. ConsideringGalileo’sgroup, it isworthremarkingthat theconeoffuturedirectedtimelikevectors (i.e., suchthatβ>0) [20] ispreservedbylinearGalileantransformations. Themomentumorbits arecontainedinΩ∗but the integraldoesnotconvergeontheorbitsoronΩ∗. • Step5: identification. It isbasedonthe followingresult. Theorem2. The transformation lawof the temperaturevector ˆW is the sameas theoneof affinemapsΘ on theaffine spaceofmomentumtensors through the identification: Z=(−W,0), z=mζ , Proof. Firstofall, letusverifythat theformZ=(−W,0)doesnotdependonthechoiceof the affineframe. Indeed,startingfromZ′=(−W′,0)andapplyingtheadjointrepresentation(5)with dC′=−W′anddP′=0,wefindthatdC=−WanddP=0with: W=PW′ . Besides,usingthenotationsof (30),Equation(9)gives: z= z′−θ(a)Ad(a)Z′= z′+KmPW′ . Ontheotherhand, letWˆbethe5-column(20) representingthe temperaturevector: Wˆ= ( W ζ ) = ⎛⎜⎝ βw ζ ⎞⎟⎠ . Takingintoaccount (12)and(31), it iseasytoverify that its transformationlaw(25)withthe linear Bargmanniantransformation(24)canberecastas:( W ζ ) = ( P 0 F1P 1 )( W′ ζ′ ) , whichis thetransformationlawoftheaffinemapΘprovidedz=mζ, thatachievestheproof. 132