On the Distribution of the Wave Function for Systems in Thermal Equilibrium

A density matrix that is not pure can arise, via averaging, from many different distributions of the wave function. This raises the question, which distribution of the wave function, if any, should be regarded as corresponding to systems in thermal equilibrium as represented, for example, by the density matrix $\rho_\beta = (1/Z) \exp(- \beta H)$ of the canonical ensemble. To answer this question, we construct, for any given density matrix $\rho$, a measure on the unit sphere in Hilbert space, denoted GAP($\rho$), using the Gaussian measure on Hilbert space with covariance $\rho$. We argue that GAP($\rho_\beta$) corresponds to the canonical ensemble.