arXiv subjects

Roderich Tumulka

Publications and source records attributed to Roderich Tumulka.

At least 109 records · Page 6Linked to original sources

arXiv2009

What Does the Free Will Theorem Actually Prove?

Conway and Kochen have presented a "free will theorem" (Notices of the AMS 56, pgs. 226-232 (2009)) which they claim shows that "if indeed we humans have free will, then [so do] elementary particles." In a more precise fashion, they claim it shows that for certain quantum experiments in which the experimenters can choose between several options, no deterministic or stochastic model can account for the observed outcomes without violating a condition "MIN" motivated by relativistic symmetry. We point out that for stochastic models this conclusion is not correct, while for deterministic models it is not new.

quant-ph↗

arXiv2009

Bohmian Mechanics at Space-Time Singularities. II. Spacelike Singularities

We develop an extension of Bohmian mechanics by defining Bohm-like trajectories for quantum particles in a curved background space-time containing a spacelike singularity. As an example of such a metric we use the Schwarzschild metric, which contains two spacelike singularities, one in the past and one in the future. Since the particle world lines are everywhere timelike or lightlike, particles can be annihilated but not created at a future spacelike singularity, and created but not annihilated at a past spacelike singularity. It is argued that in the presence of future (past) spacelike singularities, there is a unique natural Bohm-like evolution law directed to the future (past). This law differs from the one in non-singular space-times mainly in two ways: it involves Fock space since the particle number is not conserved, and the wave function is replaced by a density matrix. In particular, we determine the evolution equation for the density matrix, a pure-to-mixed evolution equation of a quasi-Lindblad form. We have to leave open whether a curvature cut-off needs to be introduced for this equation to be well defined.

quant-ph↗

arXiv2009

Bohmian Mechanics

Bohmian mechanics is a theory about point particles moving along trajectories. It has the property that in a world governed by Bohmian mechanics, observers see the same statistics for experimental results as predicted by quantum mechanics. Bohmian mechanics thus provides an explanation of quantum mechanics. Moreover, the Bohmian trajectories are defined in a non-conspiratorial way by a few simple laws.

quant-ph↗

arXiv2008

Two Arrows of Time in Nonlocal Particle Dynamics

Considering what the world would be like if backwards causation were possible is usually mind-bending. Here I discuss something that is easier to study: a toy model that incorporates a very restricted sort of backwards causation. It defines particle world lines by means of a kind of differential delay equation with negative delay. The model presumably prohibits signalling to the past and superluminal signalling, but allows nonlocality while being fully covariant. And that is what constitutes the model's value: it is an explicit example of the possibility of Lorentz invariant nonlocality. That is surprising in so far as many authors thought that nonlocality, in particular nonlocal laws for particle world lines, must conflict with relativity. The development of this model was inspired by the search for a fully covariant version of Bohmian mechanics.

quant-ph↗

arXiv2007

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.

math-ph↗

arXiv2007

A Kolmogorov Extension Theorem for POVMs

We prove a theorem about positive-operator-valued measures (POVMs) that is an analog of the Kolmogorov extension theorem, a standard theorem of probability theory. According to our theorem, if a sequence of POVMs G_n on $\mathbb{R}^n$ satisfies the consistency (or projectivity) condition $G_{n+1}(A\times \mathbb{R}) = G_n(A)$ then there is a POVM G on the space $\mathbb{R}^\mathbb{N}$ of infinite sequences that has G_n as its marginal for the first n entries of the sequence. We also describe an application in quantum theory.

math-ph↗

arXiv2007

Elementary Proof for Asymptotics of Large Haar-Distributed Unitary Matrices

We provide an elementary proof for a theorem due to Petz and Réffy which states that for a random $n\times n$ unitary matrix with distribution given by the Haar measure on the unitary group U(n), the upper left (or any other) $k\times k$ submatrix converges in distribution, after multiplying by a normalization factor $\sqrt{n}$ and as $n\to\infty$, to a matrix of independent complex Gaussian random variables with mean 0 and variance 1.

math.PR↗

arXiv2007

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 Schrödinger's 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.

quant-ph↗

arXiv2006

Comment on "The Free Will Theorem"

In a recent paper [quant-ph/0604079], Conway and Kochen claim to have established that theories of the GRW type, i.e., of spontaneous wave function collapse, cannot be made relativistic. On the other hand, relativistic GRW-type theories have already been presented, in my recent paper [quant-ph/0406094] and by Dowker and Henson [J. Statist. Phys. 115: 1327 (2004), quant-ph/0209051]. Here, I elucidate why these are not excluded by the arguments of Conway and Kochen.

quant-ph↗

arXiv2006

On Superselection Rules in Bohm-Bell Theories

The meaning of superselection rules in Bohm-Bell theories (i.e., quantum theories with particle trajectories) is different from that in orthodox quantum theory. More precisely, there are two concepts of superselection rule, a weak and a strong one. Weak superselection rules exist both in orthodox quantum theory and in Bohm-Bell theories and represent the conventional understanding of superselection rules. We introduce the concept of strong superselection rule, which does not exist in orthodox quantum theory. It relies on the clear ontology of Bohm-Bell theories and is a sharper and, in the Bohm-Bell context, more fundamental notion. A strong superselection rule for the observable G asserts that one can replace every state vector by a suitable statistical mixture of eigenvectors of G without changing the particle trajectories or their probabilities. A weak superselection rule asserts that every state vector is empirically indistinguishable from a suitable statistical mixture of eigenvectors of G. We establish conditions on G for both kinds of superselection. For comparison, we also consider both kinds of superselection in theories of spontaneous wave function collapse.

quant-ph↗

arXiv2006

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.

quant-ph↗

arXiv2006

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.

quant-ph↗

arXiv2006

Quantum Mechanics in Multiply-Connected Spaces

We explain why, in a configuration space that is multiply connected, i.e., whose fundamental group is nontrivial, there are several quantum theories, corresponding to different choices of topological factors. We do this in the context of Bohmian mechanics, a quantum theory without observers from which the quantum formalism can be derived. What we do can be regarded as generalizing the Bohmian dynamics on $\mathbb{R}^{3N}$ to arbitrary Riemannian manifolds, and classifying the possible dynamics that arise. This approach provides a new understanding of the topological features of quantum theory, such as the symmetrization postulate for identical particles. For our analysis we employ wave functions on the universal covering space of the configuration space.

quant-ph↗

arXiv2006

Determinate Values for Quantum Observables

This is a comment on J. A. Barrett's article ``The Preferred-Basis Problem and the Quantum Mechanics of Everything'' in Brit. J. Phil. Sci. 56 (2005), which concerns theories postulating that certain quantum observables have determinate values, corresponding to additional (often called ``hidden'') variables. I point out that it is far from clear, for most observables, what such a postulate is supposed to mean, unless the postulated additional variable is related to a clear ontology in space-time, such as particle world lines, string world sheets, or fields.

quant-ph↗

arXiv2006

The Message of the Quantum?

We criticize speculations to the effect that quantum mechanics is fundamentally about information. We do this by pointing out how unfounded such speculations in fact are. Our analysis focuses on the dubious claims of this kind recently made by Anton Zeilinger.

quant-ph↗

arXiv2006

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.

quant-ph↗

arXiv2006

Topological Factors Derived From Bohmian Mechanics

We derive for Bohmian mechanics topological factors for quantum systems with a multiply-connected configuration space Q. These include nonabelian factors corresponding to what we call holonomy-twisted representations of the fundamental group of Q. We employ wave functions on the universal covering space of Q. As a byproduct of our analysis, we obtain an explanation, within the framework of Bohmian mechanics, of the fact that the wave function of a system of identical particles is either symmetric or anti-symmetric.

quant-ph↗

arXiv2006

Canonical Typicality

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 both for classical distributions on the phase space and for 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.

cond-mat.stat-mech↗