arXiv subjects

Roderich Tumulka

Publications and source records attributed to Roderich Tumulka.

At least 127 records · Page 7Linked to original sources

arXiv2005

On Spontaneous Wave Function Collapse and Quantum Field Theory

One way of obtaining a version of quantum mechanics without observers, and thus of solving the paradoxes of quantum mechanics, is to modify the Schroedinger evolution by implementing spontaneous collapses of the wave function. An explicit model of this kind was proposed in 1986 by Ghirardi, Rimini, and Weber (GRW), involving a nonlinear, stochastic evolution of the wave function. We point out how, by focussing on the essential mathematical structure of the GRW model and a clear ontology, it can be generalized to (regularized) quantum field theories in a simple and natural way.

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arXiv2005

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.

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arXiv2005

Smoothness of Wave Functions in Thermal Equilibrium

We consider the thermal equilibrium distribution at inverse temperature $β$, or canonical ensemble, of the wave function $Ψ$ of a quantum system. Since $L^2$ spaces contain more nondifferentiable than differentiable functions, and since the thermal equilibrium distribution is very spread-out, one might expect that $Ψ$ has probability zero to be differentiable. However, we show that for relevant Hamiltonians the contrary is the case: with probability one, $Ψ$ is infinitely often differentiable and even analytic. We also show that with probability one, $Ψ$ lies in the domain of the Hamiltonian.

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arXiv2005

The Analogue of Bohm-Bell Processes on a Graph

Bohm-Bell processes, of interest in the foundations of quantum field theory, form a class of Markov processes $Q_t$ generalizing in a natural way both Bohm's dynamical system in configuration space for nonrelativistic quantum mechanics and Bell's jump process for lattice quantum field theories. They are such that at any time $t$ the distribution of $Q_t$ is $|ψ_t|^2$ with $ψ$ the wave function of quantum theory. We extend this class here by introducing the analogous Markov process for quantum mechanics on a graph (also called a network, i.e., a space consisting of line segments glued together at their ends). It is a piecewise deterministic process whose innovations occur only when it passes through a vertex.

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arXiv2005

On the Distribution of the Wave Function for Systems in Thermal Equilibrium

For a quantum system, a density matrix rho that is not pure can arise, via averaging, from a distribution mu of its wave function, a normalized vector belonging to its Hilbert space H. While rho itself does not determine a unique mu, additional facts, such as that the system has come to thermal equilibrium, might. It is thus not unreasonable to ask, which mu, if any, corresponds to a given thermodynamic ensemble? To answer this question we construct, for any given density matrix rho, a natural measure on the unit sphere in H, denoted GAP(rho). We do this using a suitable projection of the Gaussian measure on H with covariance rho. We establish some nice properties of GAP(rho) and show that this measure arises naturally when considering macroscopic systems. In particular, we argue that it is the most appropriate choice for systems in thermal equilibrium, described by the canonical ensemble density matrix rho_beta = (1/Z) exp(- beta H). GAP(rho) may also be relevant to quantum chaos and to the stochastic evolution of open quantum systems, where distributions on H are often used.

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arXiv2005

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.

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arXiv2005

Are All Particles Real?

In Bohmian mechanics elementary particles exist objectively, as point particles moving according to a law determined by a wavefunction. In this context, questions as to whether the particles of a certain species are real--questions such as, Do photons exist? Electrons? Or just the quarks?--have a clear meaning. We explain that, whatever the answer, there is a corresponding Bohm-type theory, and no experiment can ever decide between these theories. Another question that has a clear meaning is whether particles are intrinsically distinguishable, i.e., whether particle world lines have labels indicating the species. We discuss the intriguing possibility that the answer is no, and particles are points--just points.

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arXiv2005

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.

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arXiv2004

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.

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arXiv2004

Understanding Bohmian mechanics: A dialogue

This paper is an introduction to the ideas of Bohmian mechanics, an interpretation of quantum mechanics in which the observer plays no fundamental role. Bohmian mechanics describes, instead of probabilities of measurement results, objective microscopic events. In recent years, Bohmian mechanics has attracted increasing attention by researchers. The form of a dialogue allows me to address questions about the Bohmian view that often arise.

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arXiv2004

Quantum Hamiltonians and Stochastic Jumps

With many Hamiltonians one can naturally associate a |Psi|^2-distributed Markov process. For nonrelativistic quantum mechanics, this process is in fact deterministic, and is known as Bohmian mechanics. For the Hamiltonian of a quantum field theory, it is typically a jump process on the configuration space of a variable number of particles. We define these processes for regularized quantum field theories, thereby generalizing previous work of John S. Bell [Phys. Rep. 137, 49 (1986)] and of ourselves [J. Phys. A: Math. Gen. 36, 4143 (2003)]. We introduce a formula expressing the jump rates in terms of the interaction Hamiltonian, and establish a condition for finiteness of the rates.

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arXiv2004

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.

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arXiv2004

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 Schroedinger 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 a simpler and more transparent proof of the known global existence result for spinless Schroedinger particles and extend the result to particles with spin, to the presence of magnetic fields, and to Dirac wavefunctions. Our main new result are conditions on the current vector field on configuration-space-time which are sufficient for almost-sure global existence.

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arXiv2004

Global Existence of Bell's Time-Inhomogeneous Jump Process for Lattice Quantum Field Theory

We consider the time-inhomogeneous Markovian jump process introduced by John S. Bell [Phys.Rep. 137, 49] for a lattice quantum field theory, which runs on the associated configuration space. Its jump rates, tailored to give the process the quantum distribution $|Ψ_t|^2$ at all times $t$, typically exhibit singularities. We establish the existence of a unique such process for all times, under suitable assumptions on the Hamiltonian or the initial state vector $Ψ_0$. The proof of non-explosion takes advantage of the special role of the $|Ψ_t|^2$ distribution.

math.PR↗

arXiv2004

Some Jump Processes in Quantum Field Theory

A jump process for the positions of interacting quantum particles on a lattice, with time-dependent transition rates governed by the state vector, was first considered by J.S. Bell. We review this process and its continuum variants involving ``minimal'' jump rates, describing particles as they get created, move, and get annihilated. In particular, we sketch a recent proof of global existence of Bell's process. As an outlook, we suggest how methods of this proof could be applied to similar global existence questions, and underline the particular usefulness of minimal jump rates on manifolds with boundaries.

math.PR↗

arXiv2003

On the Role of Density Matrices in Bohmian Mechanics

It is well known that density matrices can be used in quantum mechanics to represent the information available to an observer about either a system with a random wave function (``statistical mixture'') or a system that is entangled with another system (``reduced density matrix''). We point out another role, previously unnoticed in the literature, that a density matrix can play: it can be the ``conditional density matrix,'' conditional on the configuration of the environment. A precise definition can be given in the context of Bohmian mechanics, whereas orthodox quantum mechanics is too vague to allow a sharp definition, except perhaps in special cases. In contrast to statistical and reduced density matrices, forming the conditional density matrix involves no averaging. In Bohmian mechanics with spin, the conditional density matrix replaces the notion of conditional wave function, as the object with the same dynamical significance as the wave function of a Bohmian system.

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arXiv2003

John Bell Across Space and Time

This is a review of the book Quantum [Un]speakables: From Bell to Quantum Information. Reinhold A. Bertlmann and Anton Zeilinger (editors).

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arXiv2003

Trajectories and Particle Creation and Annihilation in Quantum Field Theory

We develop a theory based on Bohmian mechanics in which particle world lines can begin and end. Such a theory provides a realist description of creation and annihilation events and thus a further step towards a "beable-based" formulation of quantum field theory, as opposed to the usual "observable-based" formulation which is plagued by the conceptual difficulties--like the measurement problem--of quantum mechanics.

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