Roderich Tumulka

A Relativistic Version of the Ghirardi–Rimini–Weber Model

Carrying out a research program outlined by John S. Bell in 1987, we arrive at a relativistic version of the Ghirardi-Rimini-Weber (GRW) model of spontaneous wavefunction collapse. The GRW model was proposed as a solution of the measurement problem of quantum mechanics and involves a stochastic and nonlinear modification of the Schrödinger equation. It deviates very little from the Schrödinger equation for microscopic systems but efficiently suppresses, for macroscopic systems, superpositions of macroscopically different states. As suggested by Bell, we take the primitive ontology, or local beables, of our model to be a discrete set of space-time points, at which the collapses are centered. This set is random with distribution determined by the initial wavefunction. Our model is nonlocal and violates Bell’s inequality though it does not make use of a preferred slicing of space-time or any other sort of synchronization of spacelike separated points. Like the GRW model, it reproduces the quantum probabilities in all cases presently testable, though it entails deviations from the quantum formalism that are in principle testable. Our model works in Minkowski space-time as well as in (well-behaved) curved background space-times.

On the Common Structure of Bohmian Mechanics and the Ghirardi–Rimini–Weber Theory

Bohmian mechanics and the Ghirardi–Rimini–Weber theory provide opposite resolutions of the quantum measurement problem: the former postulates additional variables (the particle positions) besides the wave function, whereas the latter implements spontaneous collapses of the wave function by a nonlinear and stochastic modification of Schrödingers equation. Still, both theories, when understood appropriately, share the following structure: They are ultimately not about wave functions but about ‘matter’ moving in space, represented by either particle trajectories, fields on space-time, or a discrete set of space-time points. The role of the wave function then is to govern the motion of the matter. 1. Introduction2. Bohmian Mechanics3. Ghirardi, Rimini, and Weber3.1. GRWm3.2. GRWf3.3. Empirical equivalence between GRWm and GRWf4. Primitive Ontology4.1. Primitive ontology and physical equivalence4.2. Primitive ontology and symmetry4.3. Without primitive ontology4.4. Primitive ontology and quantum state5. Differences between BM and GRW5.1. Primitive ontology and quadratic functionals5.2. Primitive ontology and equivariance6. A Plethora of Theories6.1. Particles, fields, and flashes6.2. Schrödinger wave functions and many-worlds7. The Flexible Wave Function7.1. GRWf without collapse7.2. Bohmian mechanics with collapse7.3. Empirical equivalence and equivariance8. What is a Quantum Theory without Observers? Introduction Bohmian Mechanics Ghirardi, Rimini, and Weber3.1. GRWm3.2. GRWf3.3. Empirical equivalence between GRWm and GRWf GRWm GRWf Empirical equivalence between GRWm and GRWf Primitive Ontology4.1. Primitive ontology and physical equivalence4.2. Primitive ontology and symmetry4.3. Without primitive ontology4.4. Primitive ontology and quantum state Primitive ontology and physical equivalence Primitive ontology and symmetry Without primitive ontology Primitive ontology and quantum state Differences between BM and GRW5.1. Primitive ontology and quadratic functionals5.2. Primitive ontology and equivariance Primitive ontology and quadratic functionals Primitive ontology and equivariance A Plethora of Theories6.1. Particles, fields, and flashes6.2. Schrödinger wave functions and many-worlds Particles, fields, and flashes Schrödinger wave functions and many-worlds The Flexible Wave Function7.1. GRWf without collapse7.2. Bohmian mechanics with collapse7.3. Empirical equivalence and equivariance GRWf without collapse Bohmian mechanics with collapse Empirical equivalence and equivariance What is a Quantum Theory without Observers?

Bohmian mechanics and quantum field theory

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.

Long-Time Behavior of Macroscopic Quantum Systems: Commentary Accompanying the English Translation of John von Neumann's 1929 Article on the Quantum Ergodic Theorem

AbstractThe renewed interest in the foundations of quantum statistical mechanics in recent years has led us to study John von Neumann’s 1929 article on the quantum ergodic theorem. We have found this almost forgotten article, which until now has been available only in German, to be a treasure chest, and to be much misunderstood. In it, von Neumann studied the long-time behavior of macroscopic quantum systems. While one of the two theorems announced in his title, the one he calls the “quantum H-theorem”, is actually a much weaker statement than Boltzmann’s classical H-theorem, the other theorem, which he calls the “quantum ergodic theorem”, is a beautiful and very non-trivial result. It expresses a fact we call “normal typicality” and can be summarized as follows: for a “typical” finite family of commuting macroscopic observables, every initial wave function ψ0 from a micro-canonical energy shell so evolves that for most times t in the long run, the joint probability distribution of these observables obtained from ψt is close to their micro-canonical distribution.

Many Worlds and Schrödinger’s First Quantum Theory

Schrödinger’s first proposal for the interpretation of quantum mechanics was based on a postulate relating the wave function on configuration space to charge density in physical space. Schrödinger apparently later thought that his proposal was empirically wrong. We argue here that this is not the case, at least for a very similar proposal with charge density replaced by mass density. We argue that when analyzed carefully, this theory is seen to be an empirically adequate many-worlds theory and not an empirically inadequate theory describing a single world. Moreover, this formulation—Schrödinger’s first quantum theory—can be regarded as a formulation of the many-worlds view of quantum mechanics that is ontologically clearer than Everett’s. 1. Monstrosity2. Duality3. Parallelity4. Reality5. Nonlocality6. Relativity7. Probability8. Typicality9. Uncertainty10. Summary Monstrosity Duality Parallelity Reality Nonlocality Relativity Probability Typicality Uncertainty Summary

Normal typicality and von Neumann’s quantum ergodic theorem

We discuss the content and significance of John von Neumann’s quantum ergodic theorem (QET) of 1929, a strong result arising from the mere mathematical structure of quantum mechanics. The QET is a precise formulation of what we call normal typicality, i.e. the statement that, for typical large systems, every initial wave function ψ0 from an energy shell is ‘normal’: it evolves in such a way that |ψt⟩⟨ψt| is, for most t, macroscopically equivalent to the micro-canonical density matrix. The QET has been mostly forgotten after it was criticized as a dynamically vacuous statement in several papers in the 1950s. However, we point out that this criticism does not apply to the actual QET, a correct statement of which does not appear in these papers, but to a different (indeed weaker) statement. Furthermore, we formulate a stronger statement of normal typicality, based on the observation that the bound on the deviations from the average specified by von Neumann is unnecessarily coarse and a much tighter (and more relevant) bound actually follows from his proof.

Matter Density and Relativistic Models of Wave Function Collapse

Mathematical models for the stochastic evolution of wave functions that combine the unitary evolution according to the Schrödinger equation and the collapse postulate of quantum theory are well understood for non-relativistic quantum mechanics. Recently, there has been progress in making these models relativistic. But even with a fully relativistic law for the wave function evolution, a problem with relativity remains: Different Lorentz frames may yield conflicting values for the matter density at a space-time point. We propose here a relativistic law for the matter density function. According to our proposal, the matter density function at a space-time point x is obtained from the wave function ψ on the past light cone of x by setting the i-th particle position in |ψ|2 equal to x, integrating over the other particle positions, and averaging over i. We show that the predictions that follow from this proposal agree with all known experimental facts.

Collapse and Relativity

Ever since we have been in the possession of quantum theories without observers, such as Bohmian mechanics or the Ghirardi‐Rimini‐Weber (GRW) theory of spontaneous wave function collapse, a major challenge in the foundations of quantum mechanics is to devise a relativistic quantum theory without observers. One of the difficulties is to reconcile nonlocality with relativity. I report about recent progress in this direction based on the GRW model: A relativistic version of the model has been devised for the case of N noninteracting (but entangled) particles. A key ingredient was to focus not on the evolution of the wave function but rather on the evolution of the matter in space‐time as determined by the wave function.

Simple Proof for Global Existence of Bohmian Trajectories

We address the question whether Bohmian trajectories exist for all times. Bohmian trajectories are solutions of an ordinary differential equation involving a wavefunction obeying either the Schrödinger or the Dirac equation. Some trajectories may end in finite time, for example by running into a node of the wavefunction, where the law of motion is ill-defined. The aim is to show, under suitable assumptions on the initial wavefunction and the potential, global existence of almost all solutions. We provide an alternative proof of the known global existence result for spinless Schrödinger particles and extend the result to particles with spin, to the presence of magnetic fields, and to Dirac wavefunctions. Our main new result is conditions on the current vector field on configuration-space-time which are sufficient for almost-sure global existence.

Predictions and Primitive Ontology in Quantum Foundations: A Study of Examples

A major disagreement between different views about the foundations of quantum mechanics concerns whether for a theory to be intelligible as a fundamental physical theory it must involve a ‘primitive ontology’ (PO), i.e. variables describing the distribution of matter in four-dimensional space–time. In this article, we illustrate the value of having a PO. We do so by focussing on the role that the PO plays for extracting predictions from a given theory and discuss valid and invalid derivations of predictions. To this end, we investigate a number of examples based on toy models built from the elements of familiar interpretations of quantum theory.1 1 Introduction 2 The GRWm and GRWf Theories   2.1 The GRW process   2.2 GRWm   2.3 GRWf 3 Predictions and Primitive Ontology   3.1 Calibration functions   3.2 Taking the PO seriously   3.3 Examples from the literature   3.4 The main theorem about operators in the GRW formalism   3.5 The GRW formalism 4 A Set of Examples   4.1 Bohmian mechanics   4.2 Bohmian trajectories and GRW collapses     4.2.1 Bohm’s law and GRW’s law     4.2.2 Bohm’s law and a modified GRW law     4.2.3 Trajectories from the GRW wave function     4.2.4 Configuration jumps and GRW law     4.2.5 Another way of configuration jumps and GRW law   4.3 MBM: Bohm-like trajectories from the master equation     4.3.1 Empirical equivalence of MBM with GRWm and GRWf 4.4 Master equation and matter density 4.5 Master equation and flashes 5 Conclusions 1 Introduction 2 The GRWm and GRWf Theories   2.1 The GRW process   2.2 GRWm   2.3 GRWf   2.1 The GRW process   2.2 GRWm   2.3 GRWf 3 Predictions and Primitive Ontology   3.1 Calibration functions   3.2 Taking the PO seriously   3.3 Examples from the literature   3.4 The main theorem about operators in the GRW formalism   3.5 The GRW formalism   3.1 Calibration functions   3.2 Taking the PO seriously   3.3 Examples from the literature   3.4 The main theorem about operators in the GRW formalism   3.5 The GRW formalism 4 A Set of Examples   4.1 Bohmian mechanics   4.2 Bohmian trajectories and GRW collapses     4.2.1 Bohm’s law and GRW’s law     4.2.2 Bohm’s law and a modified GRW law     4.2.3 Trajectories from the GRW wave function     4.2.4 Configuration jumps and GRW law     4.2.5 Another way of configuration jumps and GRW law   4.3 MBM: Bohm-like trajectories from the master equation     4.3.1 Empirical equivalence of MBM with GRWm and GRWf 4.4 Master equation and matter density 4.5 Master equation and flashes   4.1 Bohmian mechanics   4.2 Bohmian trajectories and GRW collapses     4.2.1 Bohm’s law and GRW’s law     4.2.2 Bohm’s law and a modified GRW law     4.2.3 Trajectories from the GRW wave function     4.2.4 Configuration jumps and GRW law     4.2.5 Another way of configuration jumps and GRW law     4.2.1 Bohm’s law and GRW’s law     4.2.2 Bohm’s law and a modified GRW law     4.2.3 Trajectories from the GRW wave function     4.2.4 Configuration jumps and GRW law     4.2.5 Another way of configuration jumps and GRW law   4.3 MBM: Bohm-like trajectories from the master equation     4.3.1 Empirical equivalence of MBM with GRWm and GRWf     4.3.1 Empirical equivalence of MBM with GRWm and GRWf 4.4 Master equation and matter density 4.5 Master equation and flashes 5 Conclusions