Bell's Jump Process in Discrete Time

The jump process introduced by J. S. Bell in 1986, for defining a quantum field theory without observers, presupposes that space is discrete whereas time is continuous. In this letter, our interest is to find an analogous process in discrete time. We argue that a genuine analog does not exist, but provide examples of processes in discrete time that could be used as a replacement.Comment: 7 pages LaTeX, no figure

Universal Probability Distribution for the Wave Function of a Quantum System Entangled with Its Environment

A quantum system (with Hilbert space $\mathscr{H}_1$) entangled with its environment (with Hilbert space $\mathscr{H}_2$) is usually not attributed a wave function but only a reduced density matrix $\rho_1$. Nevertheless, there is a precise way of attributing to it a random wave function $\psi_1$, called its conditional wave function, whose probability distribution $\mu_1$ depends on the entangled wave function $\psi\in\mathscr{H}_1\otimes\mathscr{H}_2$ in the Hilbert space of system and environment together. It also depends on a choice of orthonormal basis of $\mathscr{H}_2$ but in relevant cases, as we show, not very much. We prove several universality (or typicality) results about $\mu_1$, e.g., that if the environment is sufficiently large then for every orthonormal basis of $\mathscr{H}_2$, most entangled states $\psi$ with given reduced density matrix $\rho_1$ are such that $\mu_1$ is close to one of the so-called GAP (Gaussian adjusted projected) measures, $GAP(\rho_1)$. We also show that, for most entangled states $\psi$ from a microcanonical subspace (spanned by the eigenvectors of the Hamiltonian with energies in a narrow interval $[E,E+\delta E]$) and most orthonormal bases of $\mathscr{H}_2$, $\mu_1$ is close to $GAP(\mathrm{tr}_2 \rho_{mc})$ with $\rho_{mc}$ the normalized projection to the microcanonical subspace. In particular, if the coupling between the system and the environment is weak, then $\mu_1$ is close to $GAP(\rho_\beta)$ with $\rho_\beta$ the canonical density matrix on $\mathscr{H}_1$ at inverse temperature $\beta=\beta(E)$. This provides the mathematical justification of our claim in [J. Statist. Phys. 125:1193 (2006), http://arxiv.org/abs/quant-ph/0309021] that $GAP$ measures describe the thermal equilibrium distribution of the wave function.Comment: 27 pages LaTeX, no figures; v2 major revision with simpler proof

The "Unromantic Pictures" of Quantum Theory

I am concerned with two views of quantum mechanics that John S. Bell called ``unromantic'': spontaneous wave function collapse and Bohmian mechanics. I discuss some of their merits and report about recent progress concerning extensions to quantum field theory and relativity. In the last section, I speculate about an extension of Bohmian mechanics to quantum gravity.Comment: 37 pages LaTeX, no figures; written for special volume of J. Phys. A in honor of G.C. Ghirard

Bell-Type Quantum Field Theories

In [Phys. Rep. 137, 49 (1986)] John S. Bell proposed how to associate particle trajectories with a lattice quantum field theory, yielding what can be regarded as a |Psi|^2-distributed Markov process on the appropriate configuration space. A similar process can be defined in the continuum, for more or less any regularized quantum field theory; such processes we call Bell-type quantum field theories. We describe methods for explicitly constructing these processes. These concern, in addition to the definition of the Markov processes, the efficient calculation of jump rates, how to obtain the process from the processes corresponding to the free and interaction Hamiltonian alone, and how to obtain the free process from the free Hamiltonian or, alternatively, from the one-particle process by a construction analogous to "second quantization." As an example, we consider the process for a second quantized Dirac field in an external electromagnetic field.Comment: 53 pages LaTeX, no figure

The Point Processes of the GRW Theory of Wave Function Collapse

The Ghirardi-Rimini-Weber (GRW) theory is a physical theory that, when combined with a suitable ontology, provides an explanation of quantum mechanics. The so-called collapse of the wave function is problematic in conventional quantum theory but not in the GRW theory, in which it is governed by a stochastic law. A possible ontology is the flash ontology, according to which matter consists of random points in space-time, called flashes. The joint distribution of these points, a point process in space-time, is the topic of this work. The mathematical results concern mainly the existence and uniqueness of this distribution for several variants of the theory. Particular attention is paid to the relativistic version of the GRW theory that I developed in 2004.Comment: 72 pages LaTeX, 3 figure