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The thermal time hypothesis (TTH) of Connes and Rovelli might be the sort of thing you're looking for.

By way of background, let $\mathcal{H}$ be a Hamiltonian. The thermal density matrix is $\omega = Z^{-1}\mbox{Tr}(e^{-\beta \mathcal{H}})$, and the time evolution of an observable $A$ is given as usual by $e^{i\mathcal{H}t/\hbar} A e^{-i\mathcal{H}t/\hbar}$. Now the one-parameter modular group of $\omega$ that appears in the Tomita-Takesaki theory of von Neumann algebras can be shown to coincide with the time evolution group: if $s$ is the modular parameter and $t$ is the physical time, then $t = \hbar \beta s/\hbar$s$. In particular,$s$does not depend on$\beta$. The TTH states that physical time is determined by the modular group, which is in turn determined by the state. Besides implying Hamiltonian mechanics, the TTH simultaneously inverts and generalizes the Kubo-Martin-Schwinger condition and hence also the Gibbs relation, with temperature providing the physical link between time evolution and equilibria. A few relevant references: A. Connes and C. Rovelli, Class. Quant. Grav. 11, 2899 (1994) P. Martinetti and C. Rovelli, Class. Quant. Grav. 20, 4919 (2003) Y. Tian, JHEP06, 045 (2005) C. Rovelli and M. Smerlak, arxiv:1005.2985 Post Made Community Wiki by S. Carnahan 2 added 253 characters in body The thermal time hypothesis (TTH) of Connes and Rovelli might be the sort of thing you're looking for. By way of background, let$\mathcal{H}$be a Hamiltonian. The thermal density matrix is$\omega = Z^{-1}\mbox{Tr}(e^{-\beta \mathcal{H}})$, and the time evolution of an observable$A$is given as usual by$e^{i\mathcal{H}t/\hbar} A e^{-i\mathcal{H}t/\hbar}$. Now the one-parameter modular group of$\omega$that appears in the Tomita-Takesaki theory of von Neumann algebras can be shown to coincide with the time evolution group: if$s$is the modular parameter and$t$is the physical time, then$t = \beta s/\hbar$. In particular,$s$does not depend on$\beta\$.

The TTH states that physical time is determined by the modular group, which is in turn determined by the state. Besides implying Hamiltonian mechanics, the TTH simultaneously inverts and generalizes the Kubo-Martin-Schwinger condition and hence also the Gibbs relation, with temperature providing the physical link between time evolution and equilibria.

A few relevant references:

A. Connes and C. Rovelli, Class. Quant. Grav. 11, 2899 (1994)

P. Martinetti and C. Rovelli, Class. Quant. Grav. 20, 4919 (2003)

Y. Tian, JHEP06, 045 (2005)

C. Rovelli and M. Smerlak, arxiv:1005.2985

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