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Thermodynamics

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Quantum Mathematical Physics

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Abstract

The framework for this discussion will be an algebra A of observables with a strongly continuous time-automorphism and a time-invariant state ρ In the GNS representation the invariant state is made into a vector \(\left| \Omega \right\rangle \), and the timeautomorphism is represented as a unitary group of operators \(U = \left\{ {\exp (iHt)} \right\},U\left| \Omega \right\rangle = \left| \Omega \right\rangle \). The time-evolution then extends to the weak closure A″. If the representation is reducible, then it may occur that U ⊄ A″, even if \(U_t^{ - 1}A{U_t} \subset A.\) The von Neumann algebra

$$ R \equiv {\left\{ {A \cup U} \right\}^{\prime \prime }},R' = A' \cap U', $$

generated by A and U is known as the covariance algebra and will figure prominently in what follows.

Whereas small systems evolve almost periodically in time, large systems appear chaotic and their time-evolution mixes the observables thoroughly.

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  1. Institute for Theoretical Physics, University of Vienna, Boltzmanngasse 5, 1090, Vienna, Austria

    Walter Thirring

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  1. Walter Thirring
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© 2002 Springer-Verlag Berlin Heidelberg

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Thirring, W. (2002). Thermodynamics. In: Quantum Mathematical Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-05008-8_7

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  • DOI: https://doi.org/10.1007/978-3-662-05008-8_7

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  • Print ISBN: 978-3-642-07711-1

  • Online ISBN: 978-3-662-05008-8

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