Searcharxiv⌕ Search

arXiv subjects

Roderich Tumulka

Publications and source records attributed to Roderich Tumulka.

At least 91 records · Page 5Linked to original sources
arXiv2014

Multi-Time Schrödinger Equations Cannot Contain Interaction Potentials

Multi-time wave functions are wave functions that have a time variable for every particle, such as $ϕ(t_1,x_1,\ldots,t_N,x_N)$. They arise as a relativistic analog of the wave functions of quantum mechanics but can be applied also in quantum field theory. The evolution of a wave function with N time variables is governed by N Schrödinger equations, one for each time variable. These Schrödinger equations can be inconsistent with each other, i.e., they can fail to possess a joint solution for every initial condition; in fact, the N Hamiltonians need to satisfy a certain commutator condition in order to be consistent. While this condition is automatically satisfied for non-interacting particles, it is a challenge to set up consistent multi-time equations with interaction. We prove for a wide class of multi-time Schrödinger equations that the presence of interaction potentials (given by multiplication operators) leads to inconsistency. We conclude that interaction has to be implemented instead by creation and annihilation of particles, which, in fact, can be done consistently, as we show elsewhere [17]. We also prove the following result: When a cut-off length $δ>0$ is introduced (in the sense that the multi-time wave function is defined only on a certain set of spacelike configurations, thereby breaking Lorentz invariance), then the multi-time Schrödinger equations with interaction potentials of range $δ$ are consistent; however, in the desired limit $δ\to 0$ of removing the cut-off, the resulting multi-time equations are interaction-free, which supports the conclusion expressed in the title.

quant-ph↗
arXiv2014

Multi-Time Formulation of Pair Creation

In a recent work [8], we have described a formulation of a model quantum field theory in terms of a multi-time wave function and proposed a suitable system of multi-time Schroedinger equations governing the evolution of that wave function. Here, we provide further evidence that multi-time wave functions provide a viable formulation of relevant quantum field theories by describing a multi-time formulation, analogous to the one in [8], of another model quantum field theory. This model involves three species of particles, say x-particles, anti-x-particles, and y-particles, and postulates that a y-particle can decay into a pair consisting of an x and an anti-x particle, and that an x-anti-x pair, when they meet, annihilate each other creating a y-particle. (Alternatively, the model can also be interpreted as representing beta decay.) The wave function is a multi-time version of a time-dependent state vector in Fock space (or rather, the appropriate product of Fock spaces) in the particle-position representation. We write down multi-time Schroedinger equations and verify that they are consistent, provided that an even number of the three particle species involved are fermionic.

quant-ph↗
arXiv2014

Multi-Time Equations, Classical and Quantum

Multi-time equations are evolution equations involving several time variables, one for each particle. Such equations have been considered for the purpose of making theories manifestly Lorentz invariant. We compare their status and significance in classical and quantum physics.

quant-ph↗
arXiv2014

Can One Detect Whether a Wave Function Has Collapsed?

Consider a quantum system prepared in state $ψ$, a unit vector in a $d$-dimensional Hilbert space. Let $b_1,...,b_d$ be an orthonormal basis and suppose that, with some probability $0<p<1$, $ψ$ ``collapses,'' i.e., gets replaced by $b_k$ (possibly times a phase factor) with Born's probability $|\langle b_k|ψ\rangle|^2$. The question we investigate is: How well can any quantum experiment on the system determine afterwards whether a collapse has occurred? The answer depends on how much is known about the initial vector $ψ$. We provide a number of different results addressing several variants of the question. In each case, no experiment can provide more than rather limited probabilistic information. In case $ψ$ is drawn randomly with uniform distribution over the unit sphere in Hilbert space, no experiment performs better than a blind guess without measurement; that is, no experiment provides any useful information.

quant-ph↗
arXiv2013

Bohmian Trajectories For a Time Foliation with Kinks

This paper concerns the hypersurface Bohm-Dirac model, i.e., the version of Bohmian mechanics in a relativistic space-time proposed by Dürr et al. [1], which assumes a preferred foliation of space-time into spacelike hypersurfaces (called the time foliation) as given. We show that the leaves of the time foliation do not have to be smooth manifolds but can be allowed to have kinks. More precisely, we show that, also for leaves with kinks, the trajectories are still well defined and the appropriate $|ψ|^2$ distribution is still equivariant, so that the theory is still empirically equivalent to standard quantum mechanics. This result applies to the case where the time foliation is determined by the previously proposed law $dn=0$, since such a foliation generically has kinks.

quant-ph↗
arXiv2013

Spin and the Thermal Equilibrium Distribution of Wave Functions

Consider a quantum system $S$ weakly interacting with a very large but finite system $B$ called the heat bath, and suppose that the composite $S\cup B$ is in a pure state $Ψ$ with participating energies between $E$ and $E+δ$ with small $δ$. Then, it is known that for most $Ψ$ the reduced density matrix of $S$ is (approximately) equal to the canonical density matrix. That is, the reduced density matrix is universal in the sense that it depends only on $S$'s Hamiltonian and the temperature but not on $B$'s Hamiltonian, on the interaction Hamiltonian, or on the details of $Ψ$. It has also been pointed out that $S$ can also be attributed a random wave function $ψ$ whose probability distribution is universal in the same sense. This distribution is known as the "Scrooge measure" or "Gaussian adjusted projected (GAP) measure"; we regard it as the thermal equilibrium distribution of wave functions. The relevant concept of the wave function of a subsystem is known as the "conditional wave function". In this paper, we develop analogous considerations for particles with spin. One can either use some kind of conditional wave function or, more naturally, the "conditional density matrix", which is in general different from the reduced density matrix. We ask what the thermal equilibrium distribution of the conditional density matrix is, and find the answer that for most $Ψ$ the conditional density matrix is (approximately) deterministic, in fact (approximately) equal to the canonical density matrix.

quant-ph↗
arXiv2013

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 Schroedinger 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 psi on the past light cone of x by setting the i-th particle position in |psi|^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.

quant-ph↗
arXiv2012

The Quantum Formalism and the GRW Formalism

The Ghirardi-Rimini-Weber (GRW) theory of spontaneous wave function collapse is known to provide a quantum theory without observers, in fact two different ones by using either the matter density ontology (GRWm) or the flash ontology (GRWf). Both theories are known to make predictions different from those of quantum mechanics, but the difference is so small that no decisive experiment can as yet be performed. While some testable deviations from quantum mechanics have long been known, we provide here something that has until now been missing: a formalism that succinctly summarizes the empirical predictions of GRWm and GRWf. We call it the GRW formalism. Its structure is similar to that of the quantum formalism but involves different operators. In other words, we establish the validity of a general algorithm for directly computing the testable predictions of GRWm and GRWf. We further show that some well-defined quantities cannot be measured in a GRWm or GRWf world.

quant-ph↗
arXiv2012

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 4-dimensional space-time. In this paper, 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.

quant-ph↗
arXiv2011

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.

quant-ph↗
arXiv2011

Paradoxical Reflection in Quantum Mechanics

This article concerns a phenomenon of elementary quantum mechanics that is quite counter-intuitive, very non-classical, and apparently not widely known: a quantum particle can get reflected at a downward potential step. In contrast, classical particles get reflected only at upward steps. The conditions for this effect are that the wave length is much greater than the width of the potential step and the kinetic energy of the particle is much smaller than the depth of the potential step. This phenomenon is suggested by non-normalizable solutions to the time-independent Schroedinger equation, and we present evidence, numerical and mathematical, that it is also indeed predicted by the time-dependent Schroedinger equation. Furthermore, this paradoxical reflection effect suggests, and we confirm mathematically, that a quantum particle can be trapped for a long time (though not forever) in a region surrounded by downward potential steps, that is, on a plateau.

quant-ph↗
arXiv2011

Can We Make a Bohmian Electron Reach the Speed of Light, at Least for One Instant?

In Bohmian mechanics, a version of quantum mechanics that ascribes world lines to electrons, we can meaningfully ask about an electron's instantaneous speed relative to a given inertial frame. Interestingly, according to the relativistic version of Bohmian mechanics using the Dirac equation, a massive particle's speed is less than or equal to the speed of light, but not necessarily less. That is, there are situations in which the particle actually reaches the speed of light---a very non-classical behavior. That leads us to the question of whether such situations can be arranged experimentally. We prove a theorem, Theorem 5, implying that for generic initial wave functions the probability that the particle ever reaches the speed of light, even if at only one point in time, is zero. We conclude that the answer to the question is no. Since a trajectory reaches the speed of light whenever the quantum probability current psi-bar gamma^mu psi is a lightlike 4-vector, our analysis concerns the current vector field of a generic wave function and may thus be of interest also independently of Bohmian mechanics. The fact that the current is never spacelike has been used to argue against the possibility of faster-than-light tunnelling through a barrier, a somewhat similar question. Theorem 5, as well as a more general version provided by Theorem 6, are also interesting in their own right. They concern a certain property of a function psi: R^4 --> C^4 that is crucial to the question of reaching the speed of light, namely being transverse to a certain submanifold of C^4 along a given compact subset of space-time. While it follows from the known transversality theorem of differential topology that this property is generic among smooth functions psi: R^4 --> C^4, Theorem 5 asserts that it is also generic among smooth solutions of the Dirac equation.

quant-ph↗
arXiv2010

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

The 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 $ψ_0$ 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 $ψ_t$ is close to their micro-canonical distribution.

quant-ph↗
arXiv2010

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 $ψ_0$ from an energy shell is "normal": it evolves in such a way that $|ψ_t> <ψ_t|$ 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.

quant-ph↗
arXiv2009

Bohmian Trajectories as the Foundation of Quantum Mechanics

Bohmian trajectories have been used for various purposes, including the numerical simulation of the time-dependent Schroedinger equation and the visualization of time-dependent wave functions. We review the purpose they were invented for: to serve as the foundation of quantum mechanics, i.e., to explain quantum mechanics in terms of a theory that is free of paradoxes and allows an understanding that is as clear as that of classical mechanics. Indeed, they succeed in serving that purpose in the context of a theory known as Bohmian mechanics, to which this article is an introduction.

quant-ph↗
arXiv2009

On the Approach to Thermal Equilibrium of Macroscopic Quantum Systems

We consider an isolated, macroscopic quantum system. Let H be a micro-canonical "energy shell," i.e., a subspace of the system's Hilbert space spanned by the (finitely) many energy eigenstates with energies between E and E + delta E. The thermal equilibrium macro-state at energy E corresponds to a subspace H_{eq} of H such that dim H_{eq}/dim H is close to 1. We say that a system with state vector psi in H is in thermal equilibrium if psi is "close" to H_{eq}. We show that for "typical" Hamiltonians with given eigenvalues, all initial state vectors psi_0 evolve in such a way that psi_t is in thermal equilibrium for most times t. This result is closely related to von Neumann's quantum ergodic theorem of 1929.

quant-ph↗
arXiv2009

Many-Worlds and Schroedinger's First Quantum Theory

Schroedinger'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. Schroedinger 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--Schroedinger'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.

quant-ph↗