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

Are All Particles Identical?

We consider the possibility that all particles in the world are fundamentally identical, i.e., belong to the same species. Different masses, charges, spins, flavors, or colors then merely correspond to different quantum states of the same particle, just as spin-up and spin-down do. The implications of this viewpoint can be best appreciated within Bohmian mechanics, a precise formulation of quantum mechanics with particle trajectories. The implementation of this viewpoint in such a theory leads to trajectories different from those of the usual formulation, and thus to a version of Bohmian mechanics that is inequivalent to, though arguably empirically indistinguishable from, the usual one. The mathematical core of this viewpoint is however rather independent of the detailed dynamical scheme Bohmian mechanics provides, and it amounts to the assertion that the configuration space for N particles, even N ``distinguishable particles,'' is the set of all N-point subsets of physical 3-space.Comment: 12 pages LaTeX, no figure

Feynman's Path Integrals and Bohm's Particle Paths

Both Bohmian mechanics, a version of quantum mechanics with trajectories, and Feynman's path integral formalism have something to do with particle paths in space and time. The question thus arises how the two ideas relate to each other. In short, the answer is, path integrals provide a re-formulation of Schroedinger's equation, which is half of the defining equations of Bohmian mechanics. I try to give a clear and concise description of the various aspects of the situation.Comment: 4 pages LaTeX, no figures; v2 shortened a bi

On the speed of fluctuations around thermodynamic equilibrium

We study the speed of fluctuation of a quantum system around its thermodynamic equilibrium state, and show that the speed will be extremely small for almost all times in typical thermodynamic cases. The setting considered here is that of a quantum system couples to a bath, both jointly described as a closed system. This setting, is the same as the one considered in [N. Linden et al., Phys. Rev. E 79:061103 (2009)] and the ``thermodynamic equilibrium state'' refers to a situation that includes the usual thermodynamic equilibrium case, as well as far more general situations

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

On the Time-Dependent Analysis of Gamow Decay

Gamow's explanation of the exponential decay law uses complex "eigenvalues" and exponentially growing "eigenfunctions". This raises the question, how Gamow's description fits into the quantum mechanical description of nature, which is based on real eigenvalues and square integrable wave functions. Observing that the time evolution of any wave function is given by its expansion in generalized eigenfunctions, we shall answer this question in the most straightforward manner, which at the same time is accessible to graduate students and specialists. Moreover the presentation can well be used in physics lectures to students.Comment: 10 pages, 4 figures; heuristic argument simplified, different example discussed, calculation of decay rate adde

Parameter Diagrams of the GRW and CSL Theories of Wave Function Collapse

It has been hypothesized that the time evolution of wave functions might include collapses, rather than being governed by the Schroedinger equation. The leading models of such an evolution, GRW and CSL, both have two parameters (or new constants of nature), the collapse width sigma and the collapse rate lambda. We draw a diagram of the sigma-lambda-plane showing the region that is empirically refuted and the region that is philosophically unsatisfactory.Comment: 17 pages LaTeX, 7 figure

Free Will in a Quantum World?

In this paper, I argue that Conway and Kochen’s Free Will Theorem (1,2) to the conclusion that quantum mechanics and relativity entail freedom for the particles, does not change the situation in favor of a libertarian position as they would like. In fact, the theorem more or less implicitly assumes that people are free, and thus it begs the question. Moreover, it does not prove neither that if people are free, so are particles, nor that the property people possess when they are said to be free is the same as the one particles possess when they are claimed to be free. I then analyze the Free State Theorem (2), which generalizes the Free Will Theorem without the assumption that people are free, and I show that it does not prove anything about free will, since the notion of freedom for particles is either inconsistent, or it does not concern our common understanding of freedom. In both cases, the Free Will Theorem and the Free State Theorem do not provide any enlightenment on the constraints physics can pose on free will