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A local formula for the dimension of a superselection sector in Quantum FieldTheory is obtained as vacuum expectation value of the exponential of the properHamiltonian. In the particular case of a chiral conformal theory, this providesa local analogue of a global formula obtained by Kac and Wakimoto within thecontext of representations of certain affine Lie algebras. Our formula is modelindependent and its version in general Quantum Field Theory applies to blackhole thermodynamics. The relative free energy between two thermal equilibriumstates associated with a black hole turns out to be proportional to thevariation of the conditional entropy in different superselection sectors, wherethe conditional entropy is defined as the Connes-Stoermer entropy associatedwith the DHR localized endomorphism representing the sector. The constant ofproportionality is half of the Hawking temperature. As a consequence therelative free energy is quantized proportionally to the logarithm of a rationalnumber, in particular it is equal to a linear function the logarithm of aninteger once the initial state or the final state is taken fixed.Comment: AMS-TeX v1.1c, minor grammatical correction

An Analogue of the Kac--Wakimoto Formula¶and Black Hole Conditional Entropy