arXiv subjects

Roderich Tumulka

Publications and source records attributed to Roderich Tumulka.

At least 19 recordsLinked to original sources

arXiv2026

GAP Measures and Wave Function Collapse

GAP measures (also known as Scrooge measures) are a natural class of probability distributions on the unit sphere of a Hilbert space that come up in quantum statistical mechanics; for each density matrix $ρ$ there is a unique measure GAP$_ρ$. We describe and prove a property of these measures that was not recognized so far: If a wave function $Ψ$ is GAP$_ρ$ distributed and a collapse occurs, then the collapsed wave function $Ψ'$ is again GAP distributed (relative to the appropriate $ρ'$). This fact applies to collapses due to a quantum measurement carried out by an observer, as well as to spontaneous collapse theories such as CSL or GRW. More precisely, it is the conditional distribution of $Ψ'$, given the measurement outcome (respectively, the noise in CSL or the collapse history in GRW), that is GAP$_{ρ'}$.

quant-ph↗

arXiv2026

On the Increase of Boltzmann Entropy in Closed Macroscopic Quantum Systems

We consider a closed macroscopic quantum system in a pure state $ψ$ evolving unitarily with Hamiltonian $H$ in a finite-dimensional Hilbert space $\mathscr{H}$. Following von Neumann, we assume a decomposition of $\mathscr{H}$ into mutually orthogonal subspaces $\mathscr{H}_ν$, where each $ν$ represents a macro state, and $\mathscr{H}_ν$ contains the pure states compatible with $ν$. We define the quantum Boltzmann entropy for a macro state $ν$ as $S_\mathrm{qB}(ν) = k_{\mathrm{B}} \log \dim \mathscr{H}_ν$, where $k_\mathrm{B}$ is the Boltzmann constant. Goldstein et al. [http://arxiv.org/abs/1903.11870] have suggested, as a version of the second law of thermodynamics, that $S_\mathrm{qB}$ predominantly increases with time for realistic macroscopic systems. Here, we discuss precise versions of this statement and confirm them for several simple models of $H$ and $\mathscr{H}_ν$.

quant-ph↗

arXiv2026

Extended Results on Absorbing Boundary Conditions as Limits of Complex Potentials

In quantum mechanics, among the proposals for how to compute the probability distribution of detection time and place along a given spatial surface or in a given volume, two leading candidates involve absorbing boundary conditions (ABCs) and complex absorbing potentials (CAPs) for the Schrödinger equation. Here, we extend earlier results on how the two are related. It was previously shown that ABCs arise as a limiting case of CAPs, but for that a (reflecting) Neumann boundary condition was assumed at the end of the region with the CAP. Here, we obtain ABCs in a different limit using a Dirichlet boundary condition at the end of the CAP region, which seems more practically relevant and realizable.

quant-ph↗

arXiv2026

Quantum Zeno-like Paradox for Position Measurements: A Particle Precisely Found in Space is Nowhere to be Found in Hilbert Space

On a quantum particle in the unit interval $[0,1]$, perform a position measurement with inaccuracy $1/n$ and then a quantum measurement of the projection $|ϕ\rangle\langleϕ|$ with some arbitrary but fixed normalized $ϕ$. Call the outcomes $X \in[0,1]$ and $Y \in\{0,1\}$. We show that in the limit $n\to\infty$ corresponding to perfect precision for $X$, the probability of $Y=1$ tends to 0 for every $ϕ$. Since there is no density matrix, pure or mixed, which upon measurement of any $|ϕ\rangle\langleϕ|$ yields outcome 1 with probability 0, our result suggests that a novel type of quantum state beyond Hilbert space is necessary to describe a quantum particle after a perfect position measurement.

quant-ph↗

arXiv2026

Dirac Wave Functions of Positive Energy with Arbitrarily Small Position Uncertainty

We consider wave functions in the Hilbert space $\mathcal{H}=L^2(\mathbb{R}^3,\mathbb{C}^4)$ of a single Dirac particle, specifically from the positive-energy subspace $\mathcal{H}_+$ of the free Dirac Hamiltonian. Over the decades, various authors hypothesized that for wave functions from $\mathcal{H}_+$, there is a positive lower bound to the position uncertainty $σ_x$; in other words, that such states cannot be arbitrarily narrow in $x$. Using a sequence of wave functions introduced by Bracken and Melloy, we show that this hypothesis is false. (In fact, they already stated that it is false, but their proof that their sequence is a counter-example had a gap.)

quant-ph↗

arXiv2026

Velocity of a Quantum Particle in a Classically Forbidden Region

Recently, Sharoglazova et al. [Nature 643, 67 (2025)] proposed a procedure for determining the speed of a quantum particle in the classically forbidden region of a potential step, and implemented it in a beautiful experiment. The inferred speeds disagree significantly with the Bohmian velocities, which the authors presented as an experimental challenge to Bohmian mechanics. This is puzzling because the speeds are inferred from particle populations in coupled waveguides for which Bohmian mechanics and standard quantum mechanics make identical predictions. We resolve the puzzle by a detailed theoretical analysis of the experimental setup. We show that the speed inference rests on an assumption that fails in the relevant (evanescent) regime according to both Bohmian mechanics and standard quantum mechanics -- namely that the inter-waveguide tunneling time is set by the transverse coupling and is not affected by entanglement with the longitudinal degree of freedom. We also consider a second speed estimate suggested by Sharoglazova et al., which is based on the Büttiker dwell time formula for particles in the forbidden region. We show that the authors applied the formula incorrectly, and that a correct application yields exact agreement with the predictions of Bohmian mechanics. Our analysis includes explicit calculations of Bohmian trajectories, dwell times, and longitudinal speeds in the two-dimensional waveguide model of the experiment.

quant-ph↗

arXiv2026

Detection Time Distribution Predicted Using Absorbing Boundary Conditions and Imaginary Potentials

There are several inequivalent proposals in the literature for how to compute the probability distribution of the time that a detector registers for the arrival of a quantum particle. For three of these proposals, based on two kinds of absorbing boundary conditions and imaginary potentials, we compute the predicted distribution for an experimental setup involving a single non-relativistic quantum particle with spin 0 or 1/2 in a wave guide along the $z$ axis with the detector waiting downstream. We find that the distribution shows signs of partial reflection of the wave function off of the detector; for a spin-1/2 wave function, it is independent of the initial spin orientation for the parameters tested but does depend, for boundary conditions coupling to the spin, on the width of the wave guide. We also compare our predictions with the competing ones of Das and Dürr [arXiv:1802.07141].

quant-ph↗

arXiv2026

Scattering Cross Section Formula Derived From Macroscopic Model of Detectors

We are concerned with the justification of the statement, commonly (explicitly or implicitly) used in quantum scattering theory, that for a free non-relativistic quantum particle with initial wave function $Ψ_0(\boldsymbol{x})$, surrounded by detectors along a sphere of large radius $R$, the probability distribution of the detection time and place has asymptotic density (i.e., scattering cross section) $σ(\boldsymbol{x},t)= m^3 \hbar^{-3} R t^{-4} |\widehatΨ_0(m\boldsymbol{x}/\hbar t)|^2$ with $\widehatΨ_0$ the Fourier transform of $Ψ_0$. We give two derivations of this formula, based on different macroscopic models of the detection process. The first one consists of a negative imaginary potential of strength $λ>0$ in the detector volume (i.e., outside the sphere of radius $R$) in the limit $R\to\infty,λ\to 0, Rλ\to \infty$. The second one consists of repeated nearly-projective measurements of (approximately) the observable $1_{|\boldsymbol{x}|>R}$ at times $\mathscr{T},2\mathscr{T},3\mathscr{T},\ldots$ in the limit $R\to\infty,\mathscr{T}\to\infty,\mathscr{T}/R\to 0$; this setup is similar to that of the quantum Zeno effect, except that there one considers $\mathscr{T}\to 0$ instead of $\mathscr{T}\to\infty$. We also provide a comparison to Bohmian mechanics: while in the absence of detectors, the arrival times and places of the Bohmian trajectories on the sphere of radius $R$ have asymptotic distribution density given by the same formula as $σ$, their deviation from the detection times and places is not necessarily small, although it is small compared to $R$, so the effect of the presence of detectors on the particle can be neglected in the far-field regime. We also cover the generalization to surfaces with non-spherical shape, to the case of $N$ non-interacting particles, to time-dependent surfaces, and to the Dirac equation.

quant-ph↗

arXiv2026

Grand-Canonical Typicality

We study how the grand-canonical density matrix arises in macroscopic quantum systems. ``Canonical typicality'' is the known statement that for a typical wave function $Ψ$ from a micro-canonical energy shell of a quantum system $S$ weakly coupled to a large but finite quantum system $B$, the reduced density matrix $\hatρ^S_Ψ=\mathrm{tr}^B |Ψ\rangle\langle Ψ|$ is approximately equal to the canonical density matrix $\hatρ_\mathrm{can}=Z^{-1}_\mathrm{can} \exp(-β\hat{H}^S)$. Here, we discuss the analogous statement and related questions for the \emph{grand-canonical} density matrix $\hatρ_\mathrm{gc}=Z^{-1}_\mathrm{gc} \exp(-β(\hat{H}^S-μ_1 \hat{N}_{1}^S-\ldots-μ_r\hat{N}_{r}^S))$ with $\hat{N}_{i}^S$ the number operator for molecules of type $i$ in the system $S$. This includes (i) the case of chemical reactions (which requires some novel considerations) and (ii) that of systems $S$ defined by a spatial region which particles may enter or leave. It includes statements about how $\hatρ_\mathrm{gc}$ arises from the density matrix of the appropriate (generalized micro-canonical) Hilbert subspace $\mathscr{H}_\mathrm{gmc} \subset \mathscr{H}^S \otimes \mathscr{H}^B$ (defined by a micro-canonical interval of total energy and suitable particle number sectors) or from typical $Ψ$ in $\mathscr{H}_\mathrm{gmc}$, as well as statements about the distribution of the (conditional) wave function $ψ^S$ of $S$, which turns out to be a so-called GAP or Scrooge measure. That is, we discuss the foundation and justification of both the density matrix and the distribution of the wave function in the grand-canonical case. To this end (particularly for the chemical reactions), we also need to extend these considerations to the so-called generalized Gibbs ensembles, which apply to systems for which some macroscopic observables are conserved.

quant-ph↗

arXiv2026

Typical Quantum States of the Universe are Observationally Indistinguishable

We establish three impossibility results regarding our knowledge of the quantum state of the universe. Suppose the universal quantum state is a typical unit vector in a high-dimensional subspace $\mathscr{H}_0$ of Hilbert space $\mathscr{H}$, such as the low-entropy subspace defined by the Past Hypothesis. We show that: (1) Any particular observation is incapable of identifying the universal state vector in $\mathscr{H}_0$ or substantially reducing the set of possibilities. In other words, the overwhelming majority of possible state vectors are observationally indistinguishable from each other. (2) For any reasonably probable measurement outcome and for most pairs of vectors in $\mathscr{H}_0$, that outcome will not appreciably favor one vector over the other. (3) Bayesian updating on any measurement result, unless it is extraordinarily improbable, has a negligible effect on the initial uniform probability distribution over the states in $\mathscr{H}_0$. These findings represent the most stringent epistemic constraints known for a quantum universe and are derived from a typicality theorem in quantum statistical mechanics. We close by considering how theoretical considerations beyond empirical evidence might inform our understanding of this fact and our knowledge of the universal quantum state.

quant-ph↗

arXiv2025

Macroscopic Thermalization for Highly Degenerate Hamiltonians After Slight Perturbation

We say of an isolated macroscopic quantum system in a pure state $ψ$ that it is in macroscopic thermal equilibrium (MATE) if $ψ$ lies in or close to a suitable subspace $\mathcal{H}_{eq}$ of Hilbert space. It is known that every initial state $ψ_0$ will eventually reach and stay there most of the time (``thermalize'') if the Hamiltonian is non-degenerate and satisfies the appropriate version of the eigenstate thermalization hypothesis (ETH), i.e., that every eigenvector is in MATE. Tasaki recently proved the ETH for a certain perturbation $H_θ^{fF}$ of the Hamiltonian $H_0^{fF}$ of $N\gg 1$ free fermions on a one-dimensional lattice. The perturbation is needed to remove the high degeneracies of $H_0^{fF}$. Here, we first point out that also for degenerate Hamiltonians all $ψ_0$ thermalize if the ETH holds, i.e., if every eigenbasis lies in MATE, and we prove that this is the case for $H_0^{fF}$. Inspired by the fact that there is one eigenbasis of $H_0^{fF}$ for which MATE can be proved more easily than for the others, with smaller error bounds, and also in higher spatial dimensions, we show for any given $H_0$ that the existence of one eigenbasis in MATE implies quite generally that most eigenbases of $H_0$ lie in MATE. We also show that, as a consequence, after adding a small generic perturbation, $H=H_0+λV$ with $λ\ll 1$, for most perturbations $V$ the perturbed Hamiltonian $H$ satisfies ETH and all states thermalize.

cond-mat.stat-mech↗

arXiv2025

Existence of Schrodinger Evolution with Absorbing Boundary Condition

Consider a non-relativistic quantum particle with wave function inside a region $Ω\subset \mathbb{R}^3$, and suppose that detectors are placed along the boundary $\partial Ω$. The question how to compute the probability distribution of the time at which the detector surface registers the particle boils down to finding a reasonable mathematical definition of an ideal detecting surface; a particularly convincing definition, called the \emph{absorbing boundary rule}, involves a time evolution for the particle's wave function $ψ$ expressed by a Schrödinger equation in $Ω$ together with an ``absorbing'' boundary condition on $\partial Ω$ first considered by Werner in 1987, viz., $\partial ψ/\partial n=iκψ$ with $κ>0$ and $\partial/\partial n$ the normal derivative. We provide here a discussion of the rigorous mathematical foundation of this rule. First, for the viability of the rule it plays a crucial role that these two equations together uniquely define the time evolution of $ψ$; we point out here how, under some technical assumptions on the regularity (i.e., smoothness) of the detecting surface, the Lumer-Phillips theorem implies that the time evolution is well defined and given by a contraction semigroup. Second, we show that the collapse required for the $N$-particle version of the problem is well defined. We also prove that the joint distribution of the detection times and places, according to the absorbing boundary rule, is governed by a positive-operator-valued measure.

math-ph↗

arXiv2025

How a Space-Time Singularity Helps Remove the Ultraviolet Divergence Problem

Particle creation terms in quantum Hamiltonians are usually ultraviolet divergent and thus mathematically ill defined. A rather novel way of solving this problem is based on imposing so-called interior-boundary conditions on the wave function. Previous papers showed that this approach works in the non-relativistic regime, but particle creation is mostly relevant in the relativistic case after all. In flat relativistic space-time (that is, neglecting gravity), the approach was previously found to work only for certain somewhat artificial cases. Here, as a way of taking gravity into account, we consider curved space-time, specifically the super-critical Reissner-Nordström space-time, which features a naked timelike singularity. We find that the interior-boundary approach works fully in this setting; in particular, we prove rigorously the existence of well-defined, self-adjoint Hamiltonians with particle creation at the singularity, based on interior-boundary conditions. We also non-rigorously analyze the asymptotic behavior of the Bohmian trajectories and construct the corresponding Bohm-Bell process of particle creation, motion, and annihilation. The upshot is that in quantum physics, a naked space-time singularity need not lead to a breakdown of physical laws, but on the contrary allows for boundary conditions governing what comes out of the singularity and thereby removing the ultraviolet divergence.

math-ph↗

arXiv2024

Physical Meaning of Neumann and Robin Boundary Conditions for the Schrödinger Equation

The non-relativistic Schrödinger equation on a domain $Ω\subset \mathbb{R}^d$ with boundary is often considered with homogeneous Dirichlet boundary conditions ($ψ(x)=0$ for $x$ on the boundary), homogeneous Neumann boundary conditions ($\partial_n ψ(x)=0$ for $x$ on the boundary and $\partial_n$ the normal derivative), or Robin boundary conditions ($\partial_nψ(x)=αψ(x)$ for $x$ on the boundary and $α$ a real parameter). Physically, the Dirichlet condition applies if the potential is much higher outside than inside the domain (``potential well''). We ask, when does the Neumann or Robin condition apply physically? Our answer is, when the potential is much lower (at the appropriate level) in a thin layer along the surface of a potential well, or when a negative delta potential of the appropriate strength is added at a surface close to the surface of the potential well.

quant-ph↗

arXiv2024

Bohmian Mechanics at Space-Time Singularities. I. Timelike Singularities

We develop an extension of Bohmian mechanics to a curved background space-time containing a singularity. The present paper focuses on timelike singularities. We use the naked timelike singularity of the super-critical Reissner-Nordstrom geometry as an example. While one could impose boundary conditions at the singularity that would prevent the particles from falling into the singularity, we are interested here in the case in which particles have positive probability to hit the singularity and get annihilated. The wish for reversibility, equivariance, and the Markov property then dictates that particles must also be created by the singularity, and indeed dictates the rate at which this must occur. That is, a stochastic law prescribes what comes out of the singularity. We specify explicit equations of a model involving an interior-boundary condition on the wave function at the singularity, which can be used also in other versions of quantum theory besides Bohmian mechanics. As the resulting theory involves particle creation and annihilation, it can be regarded as a quantum field theory, and the stochastic process for the Bohmian particles is analogous to Bell-type quantum field theories.

quant-ph↗

arXiv2024

Fixed Points of Completely Positive Trace-Preserving Maps in Infinite Dimension

Completely positive trace-preserving maps $S$, also known as quantum channels, arise in quantum physics as a description of how the density operator $ρ$ of a system changes in a given time interval, allowing not only for unitary evolution but arbitrary operations including measurements or other interaction with an environment. It is known that if the Hilbert space $\mathscr{H}$ that $ρ$ acts on is finite-dimensional, then every $S$ must have a fixed point, i.e., a density operator $ρ_0$ with $S(ρ_0)=ρ_0$. In infinite dimension, $S$ need not have a fixed point in general. However, we prove here the existence of a fixed point under a certain additional assumption which is, roughly speaking, that $S$ leaves invariant a certain set of density operators with bounded ``cost'' of preparation. The proof is an application of the Schauder-Tychonoff fixed point theorem. Our motivation for this question comes from a proposal of Deutsch for how to define quantum theory in a space-time with closed timelike curves; our result supports the viability of Deutsch's proposal.

math-ph↗

arXiv2024

Detection Time Distribution for Several Quantum Particles

We address the question of how to compute the probability distribution of the time at which a detector clicks, in the situation of $n$ non-relativistic quantum particles in a volume $Ω\subset \mathbb{R}^3$ in physical space and detectors placed along the boundary $\partial Ω$ of $Ω$. We have previously [arXiv:1601.03715] argued in favor of a rule for the 1-particle case that involves a Schrödinger equation with an absorbing boundary condition on $\partial Ω$ introduced by Werner; we call this rule the "absorbing boundary rule." Here, we describe the natural extension of the absorbing boundary rule to the $n$-particle case. A key element of this extension is that, upon a detection event, the wave function gets collapsed by inserting the detected position, at the time of detection, into the wave function, thus yielding a wave function of $n-1$ particles. We also describe an extension of the absorbing boundary rule to the case of moving detectors.

quant-ph↗

arXiv2024

Typical Macroscopic Long-Time Behavior for Random Hamiltonians

We consider a closed macroscopic quantum system in a pure state $ψ_t$ evolving unitarily and take for granted that different macro states correspond to mutually orthogonal subspaces $\mathcal{H}_ν$ (macro spaces) of Hilbert space, each of which has large dimension. We extend previous work on the question what the evolution of $ψ_t$ looks like macroscopically, specifically on how much of $ψ_t$ lies in each $\mathcal{H}_ν$. Previous bounds concerned the \emph{absolute} error for typical $ψ_0$ and/or $t$ and are valid for arbitrary Hamiltonians $H$; now, we provide bounds on the \emph{relative} error, which means much tighter bounds, with probability close to 1 by modeling $H$ as a random matrix, more precisely as a random band matrix (i.e., where only entries near the main diagonal are significantly nonzero) in a basis aligned with the macro spaces. We exploit particularly that the eigenvectors of $H$ are delocalized in this basis. Our main mathematical results confirm the two phenomena of generalized normal typicality (a type of long-time behavior) and dynamical typicality (a type of similarity within the ensemble of $ψ_0$ from an initial macro space). They are based on an extension we prove of a no-gaps delocalization result for random matrices by Rudelson and Vershynin.

math-ph↗