arXiv subjects

Roderich Tumulka

Publications and source records attributed to Roderich Tumulka.

At least 37 records · Page 2Linked to original sources

arXiv2024

On the Problem of Defining Charge Operators for the Dirac Quantum Field

It is well known how to define the operator $Q$ for the total charge (i.e., positron number minus electron number) on the standard Hilbert space of the second-quantized Dirac equation. Here we ask about operators $Q_A$ representing the charge content of a region $A\subseteq \mathbb{R}^3$ in 3d physical space. There is a natural formula for $Q_A$ but, as we explain, there are difficulties about turning it into a mathematically precise definition. First, $Q_A$ can be written as a series but its convergence seems hopeless. Second, we show for some choices of $A$ that if $Q_A$ could be defined then its domain could not contain either the vacuum vector or any vector obtained from the vacuum by applying a polynomial in creation and annihilation operators. Both observations speak against the existence of $Q_A$ for generic $A$.

math-ph↗

arXiv2024

Arrival Times Versus Detection Times

How to compute the probability distribution of a detection time, i.e., of the time which a detector registers as the arrival time of a quantum particle, is a long-debated problem. In this regard, Bohmian mechanics provides in a straightforward way the distribution of the time at which the particle actually does arrive at a given surface in 3-space in the absence of detectors. However, as we discuss here, since the presence of detectors can change the evolution of the wave function and thus the particle trajectories, it cannot be taken for granted that the arrival time of the Bohmian trajectories in the absence of detectors agrees with the one in the presence of detectors, and even less with the detection time. In particular, we explain why certain distributions that Das and Dürr [arXiv:1802.07141] presented as the distribution of the detection time in a case with spin, based on assuming that all three times mentioned coincide, is actually not what Bohmian mechanics predicts.

quant-ph↗

arXiv2024

Boundary Conditions that Remove Certain Ultraviolet Divergences

In quantum field theory, Hamiltonians contain particle creation and annihilation terms that are usually ultraviolet (UV) divergent. It is well known that these divergences can sometimes be removed by adding counter-terms and taking limits in which an UV cut-off tends to infinity. Here, I review a novel way of removing UV divergences: by imposing a kind of boundary condition on the wave function. These conditions, called interior-boundary conditions (IBCs), relate the values of the wave function at two configurations linked by the creation or annihilation of a particle. They allow for a direct definition of the Hamiltonian without renormalization or limiting procedures. In the last section, I review another boundary condition that serves for determining the probability distribution of detection times and places on a timelike 3-surface.

hep-th↗

arXiv2024

Canonical Typicality For Other Ensembles Than Micro-Canonical

We generalize Lévy's lemma, a concentration-of-measure result for the uniform probability distribution on high-dimensional spheres, to a much more general class of measures, so-called GAP measures. For any given density matrix $ρ$ on a separable Hilbert space $\mathcal{H}$, GAP$(ρ)$ is the most spread out probability measure on the unit sphere of $\mathcal{H}$ that has density matrix $ρ$ and thus forms the natural generalization of the uniform distribution. We prove concentration-of-measure whenever the largest eigenvalue $\|ρ\|$ of $ρ$ is small. We use this fact to generalize and improve well-known and important typicality results of quantum statistical mechanics to GAP measures, namely canonical typicality and dynamical typicality. Canonical typicality is the statement that for ``most'' pure states $ψ$ of a given ensemble, the reduced density matrix of a sufficiently small subsystem is very close to a $ψ$-independent matrix. Dynamical typicality is the statement that for any observable and any unitary time-evolution, for ``most'' pure states $ψ$ from a given ensemble the (coarse-grained) Born distribution of that observable in the time-evolved state $ψ_t$ is very close to a $ψ$-independent distribution. So far, canonical typicality and dynamical typicality were known for the uniform distribution on finite-dimensional spheres, corresponding to the micro-canonical ensemble, and for rather special mean-value ensembles. Our result shows that these typicality results hold also for GAP$(ρ)$, provided the density matrix $ρ$ has small eigenvalues. Since certain GAP measures are quantum analogs of the canonical ensemble of classical mechanics, our results can also be regarded as a version of equivalence of ensembles.

math-ph↗

arXiv2024

Some Things I Have Learned From Detlef Dürr

Detlef Dürr (1951-2021) was a theoretical and mathematical physicist who worked particularly on the foundations of quantum mechanics, electromagnetism, and statistical mechanics. This piece is a rather personal look back at him and his science.

physics.hist-ph↗

arXiv2023

On a Derivation of the Absorbing Boundary Rule

Consider detectors waiting for a quantum particle to arrive at a surface $S$ in 3-space. For predicting the probability distribution of the time and place of detection, a rule was proposed in [arXiv:1601.03715], called the absorbing boundary rule (ABR) and involving a 1-particle Schrödinger equation with an absorbing boundary condition on $S$. While plausibility arguments for the ABR were given there, it would be desirable to derive the ABR from a microscopic model of a detector. We outline here such a derivation by putting together known results from the literature. Our derivation is non-rigorous, and it would still be desirable to have a rigorous version of it in the future.

quant-ph↗

arXiv2023

On the Spin Dependence of Detection Times and the Nonmeasurability of Arrival Times

According to a well-known principle of quantum physics, the statistics of the outcomes of any quantum experiment are governed by a Positive Operator-Valued Measure (POVM). In particular, for experiments designed to measure a specific physical quantity, like the time of a particle's first arrival at a surface, this principle establishes that if the probability distribution of that quantity does not arise from a POVM, no such experiment exists. Such is the case with the arrival time distributions proposed by Das and Dürr [arXiv:1802.07141], due to the nature of their spin dependence.

quant-ph↗

arXiv2023

Time Evolution of Typical Pure States from a Macroscopic Hilbert Subspace

We consider a macroscopic quantum system with unitarily evolving pure state $ψ_t\in \mathcal{H}$ and take it for granted that different macro states correspond to mutually orthogonal, high-dimensional subspaces $\mathcal{H}_ν$ (macro spaces) of $\mathcal{H}$. Let $P_ν$ denote the projection to $\mathcal{H}_ν$. We prove two facts about the evolution of the superposition weights $\|P_νψ_t\|^2$: First, given any $T>0$, for most initial states $ψ_0$ from any particular macro space $\mathcal{H}_μ$ (possibly far from thermal equilibrium), the curve $t\mapsto \|P_νψ_t\|^2$ is approximately the same (i.e., nearly independent of $ψ_0$) on the time interval $[0,T]$. And second, for most $ψ_0$ from $\mathcal{H}_μ$ and most $t\in[0,\infty)$, $\|P_νψ_t\|^2$ is close to a value $M_{μν}$ that is independent of both $t$ and $ψ_0$. The first is an instance of the phenomenon of dynamical typicality observed by Bartsch, Gemmer, and Reimann, and the second modifies, extends, and in a way simplifies the concept, introduced by von Neumann, now known as normal typicality.

quant-ph↗

arXiv2022

Creation Rate of Dirac Particles at a Point Source

Only recently has it been possible to construct a self-adjoint Hamiltonian that involves the creation of Dirac particles at a point source in 3d space. Its definition makes use of an interior-boundary condition. Here, we develop for this Hamiltonian a corresponding theory of the Bohmian configuration. That is, we construct a Markov jump process $(Q_t)_{t\in\mathbb{R}}$ in the configuration space of a variable number of particles that is $|ψ_t|^2$-distributed at every time $t$ and follows Bohmian trajectories between the jumps. The jumps correspond to particle creation or annihilation events and occur either to or from a configuration with a particle located at the source. The process is the natural analog of Bell's jump process, and a central piece in its construction is the determination of the rate of particle creation. The construction requires an analysis of the asymptotic behavior of the Bohmian trajectories near the source. We find that the particle reaches the source with radial speed 0, but orbits around the source infinitely many times in finite time before absorption (or after emission).

quant-ph↗

arXiv2022

Limitations to Genuine Measurements in Ontological Models of Quantum Mechanics

Given an ontological model of a quantum system, a "genuine measurement," as opposed to a quantum measurement, means an experiment that determines the value of a beable, i.e., of a variable that, according to the model, has an actual value in nature before the experiment. We prove a theorem showing that in every ontological model, it is impossible to measure all beables. Put differently, there is no experiment that would reliably determine the ontic state. This result shows that the positivistic idea that a physical theory should only involve observable quantities is too optimistic.

quant-ph↗

arXiv2022

Energy-Time Uncertainty Relation for Absorbing Boundaries

We prove the uncertainty relation $σ_T \, σ_E \geq \hbar/2$ between the time $T$ of detection of a quantum particle on the surface $\partial Ω$ of a region $Ω\subset \mathbb{R}^3$ containing the particle's initial wave function, using the "absorbing boundary rule" for detection time, and the energy $E$ of the initial wave function. Here, $σ$ denotes the standard deviation of the probability distribution associated with a quantum observable and a wave function. Since $T$ is associated with a POVM rather than a self-adjoint operator, the relation is not an instance of the standard version of the uncertainty relation due to Robertson and Schrödinger. We also prove that if there is nonzero probability that the particle never reaches $\partial Ω$ (in which case we write $T=\infty$), and if $σ_T$ denotes the standard deviation conditional on the event $T<\infty$, then $σ_T \, σ_E \geq (\hbar/2) \sqrt{\mathrm{Prob}(T<\infty)}$.

quant-ph↗

arXiv2022

Interior-Boundary Conditions for the Dirac Equation at Point Sources in 3 Dimensions

A recently proposed approach for avoiding the ultraviolet divergence of Hamiltonians with particle creation is based on interior-boundary conditions (IBCs). The approach works well in the non-relativistic case, that is, for the Laplacian operator. Here, we study how the approach can be applied to Dirac operators. While this has been done successfully already in 1 space dimension, and more generally for codimension-1 boundaries, the situation of point sources in 3 dimensions corresponds to a codimension-3 boundary. One would expect that, for such a boundary, Dirac operators do not allow for boundary conditions because they are known not to allow for point interactions in 3d, which also correspond to a boundary condition. And indeed, we confirm this expectation here by proving that there is no self-adjoint operator on (a truncated) Fock space that would correspond to a Dirac operator with an IBC at configurations with a particle at the origin. However, we also present a positive result showing that there are self-adjoint operators with IBC (on the boundary consisting of configurations with a particle at the origin) that are, away from those configurations, given by a Dirac operator plus a sufficiently strong Coulomb potential.

math-ph↗

arXiv2022

Empirically Equivalent Distributions in Ontological Models of Quantum Mechanics

We consider ontological models of a quantum system, assuming that not all probability distributions over the space $Λ$ of ontic states are preparable, only those belonging to a certain set C. We assume further that every POVM with a finite value space can be measured and that for every density matrix there exists a distribution in C whose outcome statistics is given by the density matrix. We show that this mapping from C to the set of density matrices must be many-to-one, that is, that there must be empirically indistinguishable distributions in C. This shows that there must be limitations to knowledge in the sense of facts in nature that cannot be discovered empirically.

quant-ph↗

arXiv2022

Distribution of the Time at Which an Ideal Detector Clicks

We consider the problem of computing, for a detector surface waiting for a quantum particle to arrive, the probability distribution of the time and place at which the particle gets detected, from the initial wave function of the particle in the non-relativistic regime. Although the standard rules of quantum mechanics offer no operator for the time of arrival, quantum mechanics makes an unambiguous prediction for this distribution, defined by first solving the Schrödinger equation for the big quantum system formed by the particle of interest, the detector, a clock, and a device that records the time and place of detection, then making a quantum measurement of the record at a very late time, and finally using the distribution of the recorded time and place. This leads to the question whether there is also a practical, simple rule for computing this distribution, at least approximately (i.e., for an idealized detector). We argue here in favor of a rule based on a 1-particle Schrödinger equation with a certain (absorbing) boundary condition at the ideal detecting surface, first considered by Werner in 1987. We present a novel derivation of this rule and describe how it arises as a limit of a "soft" detector represented by an imaginary potential.

quant-ph↗

arXiv2022

A Relativistic GRW Flash Process With Interaction

In 2004, I described a relativistic version of the Ghirardi-Rimini-Weber (GRW) model of spontaneous wave function collapse for N non-interacting distinguishable particles. Here I present a generalized version for N interacting distinguishable particles. Presently, I do not know how to set up a similar model for indistinguishable particles or a variable number of particles. The present interacting model is constructed from a given interacting unitary Tomonaga-Schwinger type evolution between spacelike hypersurfaces, into which discrete collapses are inserted. I assume that this unitary evolution is interaction-local (i.e., no interaction at spacelike separation). The model is formulated in terms of Bell's flash ontology but is also compatible with Ghirardi's matter density ontology. It is non-local and satisfies microscopic parameter independence and no-signaling; it also works in curved space-time; in the non-relativistic limit, it reduces to the known non-relativistic GRW model.

quant-ph↗

arXiv2021

Positron Position Operators. I. A Natural Option

By ``position operators,'' I mean here a POVM (positive-operator-valued measure) on a suitable configuration space acting on a suitable Hilbert space that serves as defining the position observable of a quantum theory, and by ``positron position operators,'' I mean a joint treatment of positrons and electrons. I consider the standard free second-quantized Dirac field in Minkowski space-time or in a box. On the associated Fock space (i.e., the tensor product of the positron Fock space and the electron Fock space), there acts an obvious POVM P_obv, but I propose a different one that I call the natural POVM, P_nat. In fact, it is a PVM (projection-valued measure); it captures the sense of locality corresponding to the field operators Psi_s(x) and to the algebra of local observables. The existence of P_nat depends on a mathematical conjecture which at present I can neither prove nor disprove; here I explore consequences of the conjecture. I put up for consideration the possibility that P_nat, and not P_obv, is the physically correct position observable and defines the Born rule for the joint distribution of electron and positron positions. I describe properties of P_nat, including a strict no-superluminal-signaling property, and how it avoids the Hegerfeldt-Malament no-go theorem. I also point out how to define Bohmian trajectories that fit together with P_nat, and how to generalize P_nat to curved space-time.

quant-ph↗

arXiv2021

On the Question Why There Exists Something Rather Than Nothing

In my opinion, nothing useful has ever been written on the question in the title, and small is the contribution that I have to offer. I outline an explanation for why there is something rather than nothing, an explanation which, however, I believe is incorrect because it makes a certain empirical prediction (absence of qualia) that is incorrect. Nevertheless, it may be interesting to discuss this reasoning. It allows, in principle though not in practice, to derive the laws of nature and all physical facts about the universe. Then I elucidate which objections to this explanation are, in my opinion, valid and which are not.

physics.hist-ph↗

arXiv2021

Another Proof of Born's Rule on Arbitrary Cauchy Surfaces

In 2017, Lienert and Tumulka proved Born's rule on arbitrary Cauchy surfaces in Minkowski space-time assuming Born's rule and a corresponding collapse rule on horizontal surfaces relative to a fixed Lorentz frame, as well as a given unitary time evolution between any two Cauchy surfaces, satisfying that there is no interaction faster than light and no propagation faster than light. Here, we prove Born's rule on arbitrary Cauchy surfaces from a different, but equally reasonable, set of assumptions. The conclusion is that if detectors are placed along any Cauchy surface $Σ$, then the observed particle configuration on $Σ$ is a random variable with distribution density $|Ψ_Σ|^2$, suitably understood. The main different assumption is that the Born and collapse rules hold on any spacelike hyperplane, i.e., at any time coordinate in any Lorentz frame. Heuristically, this follows if the dynamics of the detectors is Lorentz invariant.

math-ph↗