An Analogue of the Kac—Wakimoto Formula¶and Black Hole Conditional Entropy |
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Authors: | Roberto Longo |
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Institution: | (1) Dipartimento di Matematica, Università di Roma “Tor Vergata”, via della Ricerca Scientifica, I—00133 Roma, Italy, and Centro Linceo Interdisciplinare, Accademia Nazionale dei Lincei via della Lungara 10, I—00165 Roma, Italy. E-mail: longo@mat.utovrm.it, IT |
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Abstract: | A local formula for the dimension of a superselection sector in Quantum Field Theory is obtained as vacuum expectation value
of the exponential of the proper Hamiltonian. In the particular case of a chiral conformal theory, this provides a local analogue
of a global formula obtained by Kac and Wakimoto within the context of representations of certain affine Lie algebras. Our
formula is model independent and its version in general Quantum Field Theory applies to black hole thermodynamics. The relative
free energy between two thermal equilibrium states associated with a black hole turns out to be proportional to the variation
of the conditional entropy in different superselection sectors, where the conditional entropy is defined as the Connes—St?rmer
entropy associated with the DHR localized endomorphism representing the sector. The constant of proportionality is half of
the Hawking temperature. As a consequence the relative free energy is quantized proportionally to the logarithm of a rational
number, in particular it is equal to a linear function of the logarithm of an integer once the initial state or the final
state is taken fixed.
Received: 31 May 1996 / Accepted: 8 November 1996 |
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