arXiv subjects

Nino Zanghi

Publications and source records attributed to Nino Zanghi.

At least 19 recordsLinked to original sources

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↗

arXiv2021

Remarks About the Relationship Between Relational Physics and a Large Kantian Component of the Laws of Nature

Relational mechanics is a reformulation of mechanics (classical or quantum) for which space is relational. This means that the configuration of an $N$-particle system is a shape, which is what remains when the effects of rotations, translations, and dilations are quotiented out. This reformulation of mechanics naturally leads to a relational notion of time as well, in which a history of the universe is just a curve in shape space without any reference to a special parametrization of the curve given by an absolute Newtonian time. When relational mechanics (classical or quantum) is regarded as fundamental, the usual descriptions in terms of absolute space and absolute time emerge merely as corresponding to the choice of a gauge. This gauge freedom forces us to recognize that what we have traditionally regarded as fundamental in physics might in fact be imposed by us through our choice of gauge. It thus imparts a somewhat Kantian aspect to physical theory.

quant-ph↗

arXiv2019

Gibbs and Boltzmann Entropy in Classical and Quantum Mechanics

The Gibbs entropy of a macroscopic classical system is a function of a probability distribution over phase space, i.e., of an ensemble. In contrast, the Boltzmann entropy is a function on phase space, and is thus defined for an individual system. Our aim is to discuss and compare these two notions of entropy, along with the associated ensemblist and individualist views of thermal equilibrium. Using the Gibbsian ensembles for the computation of the Gibbs entropy, the two notions yield the same (leading order) values for the entropy of a macroscopic system in thermal equilibrium. The two approaches do not, however, necessarily agree for non-equilibrium systems. For those, we argue that the Boltzmann entropy is the one that corresponds to thermodynamic entropy, in particular in connection with the second law of thermodynamics. Moreover, we describe the quantum analog of the Boltzmann entropy, and we argue that the individualist (Boltzmannian) concept of equilibrium is supported by the recent works on thermalization of closed quantum systems.

cond-mat.stat-mech↗

arXiv2016

Is the Hypothesis About a Low Entropy Initial State of the Universe Necessary for Explaining the Arrow of Time?

According to statistical mechanics, micro-states of an isolated physical system (say, a gas in a box) at time $t_0$ in a given macro-state of less-than-maximal entropy typically evolve in such a way that the entropy at time $t$ increases with $|t-t_0|$ in both time directions. In order to account for the observed entropy increase in only one time direction, the thermodynamic arrow of time, one usually appeals to the hypothesis that the initial state of the universe was one of very low entropy. In certain recent models of cosmology, however, no hypothesis about the initial state of the universe is invoked. We discuss how the emergence of a thermodynamic arrow of time in such models can nevertheless be compatible with the above-mentioned consequence of statistical mechanics, appearances to the contrary notwithstanding.

astro-ph.CO↗

arXiv2015

Universal Probability Distribution for the Wave Function of a Quantum System Entangled with Its Environment

A quantum system (with Hilbert space $\mathscr{H}_1$) entangled with its environment (with Hilbert space $\mathscr{H}_2$) is usually not attributed a wave function but only a reduced density matrix $ρ_1$. Nevertheless, there is a precise way of attributing to it a random wave function $ψ_1$, called its conditional wave function, whose probability distribution $μ_1$ depends on the entangled wave function $ψ\in\mathscr{H}_1\otimes\mathscr{H}_2$ in the Hilbert space of system and environment together. It also depends on a choice of orthonormal basis of $\mathscr{H}_2$ but in relevant cases, as we show, not very much. We prove several universality (or typicality) results about $μ_1$, e.g., that if the environment is sufficiently large then for every orthonormal basis of $\mathscr{H}_2$, most entangled states $ψ$ with given reduced density matrix $ρ_1$ are such that $μ_1$ is close to one of the so-called GAP (Gaussian adjusted projected) measures, $GAP(ρ_1)$. We also show that, for most entangled states $ψ$ from a microcanonical subspace (spanned by the eigenvectors of the Hamiltonian with energies in a narrow interval $[E,E+δE]$) and most orthonormal bases of $\mathscr{H}_2$, $μ_1$ is close to $GAP(\mathrm{tr}_2 ρ_{mc})$ with $ρ_{mc}$ the normalized projection to the microcanonical subspace. In particular, if the coupling between the system and the environment is weak, then $μ_1$ is close to $GAP(ρ_β)$ with $ρ_β$ the canonical density matrix on $\mathscr{H}_1$ at inverse temperature $β=β(E)$. This provides the mathematical justification of our claim in [J. Statist. Phys. 125:1193 (2006), http://arxiv.org/abs/quant-ph/0309021] that $GAP$ measures describe the thermal equilibrium distribution of the wave function.

math-ph↗

arXiv2015

Any Orthonormal Basis in High Dimension is Uniformly Distributed over the Sphere

Let X be a real or complex Hilbert space of finite but large dimension d, let S(X) denote the unit sphere of X, and let u denote the normalized uniform measure on S(X). For a finite subset B of S(X), we may test whether it is approximately uniformly distributed over the sphere by choosing a partition A_1,...,A_m of S(X) and checking whether the fraction of points in B that lie in A_k is close to u(A_k) for each k=1,...,m. We show that if B is any orthonormal basis of X and m is not too large, then, if we randomize the test by applying a random rotation to the sets A_1,...,A_m, B will pass the random test with probability close to 1. This statement is related to, but not entailed by, the law of large numbers. An application of this fact in quantum statistical mechanics is briefly described.

math.PR↗

arXiv2014

Fermionic Wave Functions on Unordered Configurations

Quantum mechanical wave functions of N identical fermions are usually represented as anti-symmetric functions of ordered configurations. Leinaas and Myrheim proposed how a fermionic wave function can be represented as a function of unordered configurations, which is desirable as the ordering is artificial and unphysical. In this approach, the wave function is a cross-section of a particular Hermitian vector bundle over the configuration space, which we call the fermionic line bundle. Here, we provide a justification for Leinaas and Myrheim's proposal, that is, a justification for regarding cross-sections of the fermionic line bundle as equivalent to anti-symmetric functions of ordered configurations. In fact, we propose a general notion of equivalence of two quantum theories on the same configuration space; it is based on specifying a quantum theory as a triple $(\mathscr{H},H,Q)$ (``quantum triple'') consisting of a Hilbert space $\mathscr{H}$, a Hamiltonian $H$, and a family of position operators (technically, a projection-valued measure on configuration space acting on $\mathscr{H}$).

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↗

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↗

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

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

On the Weak Measurement of Velocity in Bohmian Mechanics

In a recent article (New Journal of Physics 9, 165, 2007), Wiseman has proposed the use of so-called weak measurements for the determination of the velocity of a quantum particle at a given position, and has shown that according to quantum mechanics the result of such a procedure is the Bohmian velocity of the particle. Although Bohmian mechanics is empirically equivalent to variants based on velocity formulas different from the Bohmian one, and although it has been proven that the velocity in Bohmian mechanics is not measurable, we argue here for the somewhat paradoxical conclusion that Wiseman's weak measurement procedure indeed constitutes a genuine measurement of velocity in Bohmian mechanics. We reconcile the apparent contradictions and elaborate on some of the different senses of measurement at play here.

quant-ph↗