It is well known that a system $S$ weakly coupled to a heat bath $B$ is described by the canonical ensemble when the composite $S+B$ is described by the microcanonical ensemble corresponding to a suitable energy shell. This is true for both classical distributions on the phase space and quantum density matrices. Here we show that a much stronger statement holds for quantum systems. Even if the state of the composite corresponds to a single wave function rather than a mixture, the reduced density matrix of the system is canonical, for the overwhelming majority of wave functions in the subspace corresponding to the energy interval encompassed by the microcanonical ensemble. This clarifies, expands, and justifies remarks made by Schrödinger in 1952.

We discuss a recently proposed extension of Bohmian mechanics to quantum field theory. For more or less any regularized quantum field theory there is a corresponding theory of particle motion, which, in particular, ascribes trajectories to the electrons or whatever sort of particles the quantum field theory is about. Corresponding to the nonconservation of the particle number operator in the quantum field theory, the theory describes explicit creation and annihilation events: the world lines for the particles can begin and end.

We consider an isolated macroscopic quantum system. Let $\mathcal{H}$ be a microcanonical “energy shell,” i.e., a subspace of the system’s Hilbert space spanned by the (finitely) many energy eigenstates with energies between $E$ and $E+\delta E$. The thermal equilibrium macrostate at energy $E$ corresponds to a subspace ${\mathcal{H}}_{eq}$ of $\mathcal{H}$ such that $\text{dim}{\mathcal{H}}_{eq}/\text{dim}\mathcal{H}$ is close to 1. We say that a system with state vector $\psi ∊\mathcal{H}$ is in thermal equilibrium if $\psi$ is “close” to ${\mathcal{H}}_{eq}$. We show that for “typical” Hamiltonians with given eigenvalues, all initial state vectors ${\psi }_0$ evolve in such a way that ${\psi }_t$ is in thermal equilibrium for most times $t$. This result is closely related to von Neumann’s quantum ergodic theorem of 1929.