Abstract
Associated with any density matrix $\rho$ on a Hilbert space, there
is a GAP measure, a natural probability measure $GAP(\rho)$ on the
unit sphere of the Hilbert space, having covariance $\rho$. A system
whose wave function is random with distribution $GAP(\rho)$ is a
system which, according to quantum mechanics, is in the state
$\rho$. For a system in thermal equilibrium with canonical density
matrix $\rho_\exp(-H)$, its wave function should
be regarded as random, with distribution $GAP(\rho_\beta)$ on the
unit sphere of the system's Hilbert space.
The proof of this claim is our main result. Crucial ingredients are:
(i) the general claim that for a system 1 in interaction with a much
larger system 2 and such that the composite is in a pure state
$\psi$ for which the reduced density matrix of system 1 is fixed to
be $\rho_1$, then a typical such $\psi$ yields a wave function
$\psi_1$ for system 1 that is random with distribution
$GAP(\rho_1)$, and (ii) canonical typicality, the fact that when
system 2 is a heat bath, then for a typical pure state $\psi$ of the
composite, the reduced density matrix $\rho_1$ of system 1 is
canonical.
The talk is based on joint work with S. Goldstein, J.L. Lebowitz, R.
Tumulka and N. Zangh\`ı.
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