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Bruno Galvan

Publications and source records attributed to Bruno Galvan.

17 recordsLinked to original sources
arXiv2024

Cournot's principle for measure-theoretic probability

The problem of relating the mathematics of probability theory to the empirical world of experiments has been debated for centuries. One of the oldest solutions proposed for this problem is a principle that states that an event with probability close to 1 nearly certainly occurs in a single trial of an experiment. This principle is now called $\textit{Cournot' principle}$. Cournot's principle was first formulated in the context of classical probability, in which the probability of any event is given, and the $\textit{product rule}$, i.e., the rule that the probability that two events occur in two separated trials is the product of their probabilities, can be deduced. On the contrary, in the modern measure-theoretic approach to probability, probability measures and experiments are separate entities that must be related in an appropriate way, and the product rule cannot be deduced. In this paper, a version of Cournot's principle suitable for measure-theoretic probability is proposed. Therefore, the principle is reformulated as a criterion for relating probability measures and experiments, and the product rule is explicitly stated. In spite of the vagueness of the notions involved, the new version is formulated in a rigorous manner and an exact result, namely, that at most one probability measure can be related to an experiment, is rigorously proven.

math.PR↗
arXiv2022

Nonprobabilistic typicality with application to quantum mechanics

In this paper two hypotheses are developed. The first hypothesis is the existence of random phenomena/experiments in which the events cannot generally be assigned a definite probability but that nevertheless admit a class of nearly certain events. These experiments are referred to as \textit{typicalistic} (instead of probabilistic) experiments. As probabilistic experiments are represented by probability spaces, typicalistic experiments can be represented by \textit{typicality spaces}, where a typicality space is basically a probability space in which the probability measure has been replaced by a much less structured typicality measure $T$. The condition $T(A) \approx 1$ defines the typical sets, and a typicality space is related to a typicalistic experiment by associating the typical sets of the former with the nearly certain events of the latter. Some elements of a theory of typicality, including the definition of typicality spaces, are developed in the first part of the paper. The second hypothesis is that the evolution of a quantum particle (or of a system of quantum particles) can be considered a typicalistic phenomenon, so that it can be represented by the combination of typicality theory and quantum mechanics. The result is a novel formulation of quantum mechanics that does not present the measurement problem and that could be a viable alternative to Bohmian mechanics. This subject is developed in the second part of the paper.

quant-ph↗
arXiv2017

Asymptotic velocities in quantum and Bohmian mechanics

In this paper the relations between the asymptotic velocity operators of a quantum system and the asymptotic velocities of the associated Bohmian trajectories are studied. In particular it is proved that, under suitable conditions of asymptotic regularity, the probability distribution of the asymptotic velocities of the Bohmian trajectories is equal to the one derived from the asymptotic velocity operators of the associated quantum system. It is also shown that in the relativistic case the distribution of the asymptotic velocities of the Bohmian trajectories is covariant, or equivalently, it does not depend on a preferred foliation (it is well known that this is not the case for the structure of the Bohmian trajectories or for their spatial distribution at a finite time). This result allows us to develop a covariant formulation of relativistic Bohmian mechanics; such a formulation is proposed here merely as a mathematical possibility, while its empirical adequacy will be discussed elsewhere.

quant-ph↗
arXiv2016

Quantum field theory without divergence: the method of the interaction operators

The recently proposed interior boundary conditions approach [S. Teufel and R. Tumulka: Avoiding Ultraviolet Divergence by Means of Interior Boundary Conditions, arXiv:1506.00497] is a method for defining Hamiltonians without UV divergence for quantum field theories. In this approach the interactions between sectors of the Fock space with different number of particles (inter-sector interactions) are obtained by extending the domain of the free Hamiltonian to include functions with singularities. In this paper a similar but alternative strategy is proposed, in which the inter-sector interactions are implemented by specific interaction operators. In its simplest form, an interaction operator is obtained by symmetrizing the asymmetric operator $\|\hat{\textbf{x}} \|^{-1} \Delta \|\hat {\textbf{x}}\|^{-1}$. The inter-sector interactions derive from the singularities generated by the factors $\|\hat{\textbf{x}}\|^{-1}$ enclosing the Laplacian, while the domain of the interaction operator does not include singular functions. As a consequence the interaction operators and the free Hamiltonian have a common dense domain, and they can be added together to form the complete Hamiltonian with interaction.

quant-ph↗
arXiv2015

Relativistic Bohmian mechanics without a preferred foliation

In non-relativistic Bohmian mechanics the universe is represented by a probability space whose sample space is composed of the Bohmian trajectories. In relativistic Bohmian mechanics an entire class of empirically equivalent probability spaces can be defined, one for every foliation of spacetime. In the literature the hypothesis has been advanced that a single preferred foliation is allowed, and that this foliation derives from the universal wave function by means of a covariant law. In the present paper the opposite hypothesis is advanced, i.e., no law exists for the foliations and therefore all the foliations are allowed. The resulting model of the universe is basically the "union" of all the probability spaces associated with the foliations. This hypothesis is mainly motivated by the fact that any law defining a preferred foliation is empirically irrelevant. It is also argued that the absence of a preferred foliation may reduce the well known conflict between Bohmian mechanics and Relativity.

quant-ph↗
arXiv2013

Regular Hamiltonians for non-relativistic interacting quantum field theories

In the context of non-relativistic quantum field theory, a method is proposed for multiplying field operators at the same spatial point and obtaining regular (i.e. rigorously defined) interaction terms for the Hamiltonian. The basic idea is to modify the Lebesgue measure of configuration space of many particles by adding singular measures over the subspaces of configuration space in which the positions of two or more particles coincide.

quant-ph↗
arXiv2010

On the preferred-basis problem and its possible solutions

The preferred basis problem is mentioned in the literature in connection with the measurement problem and with the Many World Interpretation. It is argued that this problem actually corresponds to two inequivalent problems: (i) the preferred-decomposition problem, i.e., what singles out a preferred decomposition of a suitable state vector of a system as the sum of a finite or countable set of vectors?, and (ii) the preferred-representation problem, i.e., what singles out a preferred representation for the Hilbert space of a system? In this paper the preferred-decomposition problem is addressed and two processes, namely decoherence and permanent spatial decomposition (PSD), are examined and compared as possible solutions to this problem. It is shown that, perhaps contrary to common belief, in realistic situations decoherence is not sufficient to solve the preferred-decomposition problem. PSD is the (hypothesized) tendency of the wave function of the universe to decompose into permanently non-overlapping wave packets. Three phases can be roughly identified as composing PSD: Microscopic decomposition, amplification, and interaction with the environment. Decoherence theory considers only the interaction with the environment and ignores the first two phases. For this reason PSD is fundamentally different from decoherence and, unlike decoherence, provides a simple and non-elusive solution to the preferred-decomposition problem.

quant-ph↗
arXiv2009

The permanent spatial decomposition of the wave function

Permanent spatial decomposition (PSD) is the (hypothesized) property of the wave function of a macroscopic system of decomposing into localized permanently non-overlapping parts when it spreads over a macroscopic region. The typical example of this phenomenon is the measurement process, in which the wave function of the laboratory (quantum system + apparatus + environment) decomposes into n parts, corresponding to the n outcomes of the measurement: the parts are non-overlapping, because they represent a macroscopic pointer in different positions, and they are permanently non-overlapping due to the irreversible interaction with the environment. PSD is often mentioned in the literature, but until now no formal definition or systematic study of this phenomenon has been undertaken. The aim of this paper is to partially fill this gap by giving a formal definition of PSD and studying its possible connection with scattering theory. The predictive and explanatory powers of this phenomenon are also discussed and compared with those of Bohmian mechanics.

quant-ph↗
arXiv2008

Generalization of the Born rule

A new formulation of quantum mechanics is proposed based on a new principle that can be considered a generalization of the Born rule. The principle is composed of a mathematical expression and an associated interpretation, and establishes a correlation between the positions of a particle at two different times. Under reasonable conditions for the wave function, this correlation implies that the particles follow quasi-classical trajectories. It is also shown that the Born rule is equivalent to a particular case of the new principle. These features allow the principle to provide a unified explanation of the results of the statistical experiments and of the quasi-classical macroscopic evolution. There is a strong analogy between the new quantum principle and a probabilistic principle which is necessary to derive empirical predictions from the mathematical formalism of probability theory. This principle is referred to by some authors as Cournot's principle, while other authors use the equivalent notion of typicality. In this paper probability theory and quantum mechanics are formulated in such a way as to explicitly include the two principles and to emphasize the very similar conceptual structure of the two theories.

quant-ph↗
arXiv2008

Quantum Mechanics and imprecise probability

An extension of the Born rule, the {\it quantum typicality rule}, has recently been proposed [B. Galvan: Found. Phys. 37, 1540-1562 (2007)]. Roughly speaking, this rule states that if the wave function of a particle is split into non-overlapping wave packets, the particle stays approximately inside the support of one of the wave packets, without jumping to the others. In this paper a formal definition of this rule is given in terms of {\it imprecise probability}. An imprecise probability space is a measurable space $(Ω, {\cal A})$ endowed with a {\it set} of probability measures $\cal P$. The quantum formalism and the quantum typicality rule allow us to define a set of probabilities ${\cal P}_Ψ$ on $(X^T, {\cal F})$, where $X$ is the configuration space of a quantum system, $T$ is a time interval and ${\cal F}$ is the $σ$-algebra generated by the cylinder sets. Thus, it is proposed that a quantum system can be represented as the {\it imprecise stochastic process} $(X^T, {\cal F}, {\cal P}_Ψ)$, which is a canonical stochastic process in which the single probability measure is replaced by a set of measures. It is argued that this mathematical model, when used to represent macroscopic systems, has sufficient predictive power to explain both the results of the statistical experiments and the quasi-classical structure of the macroscopic evolution.

quant-ph↗
arXiv2007

Typicality vs. probability in trajectory-based formulations of quantum mechanics

Bohmian mechanics represents the universe as a set of paths with a probability measure defined on it. The way in which a mathematical model of this kind can explain the observed phenomena of the universe is examined in general. It is shown that the explanation does not make use of the full probability measure, but rather of a suitable set function deriving from it, which defines relative typicality between single-time cylinder sets. Such a set function can also be derived directly from the standard quantum formalism, without the need of an underlying probability measure. The key concept for this derivation is the {\it quantum typicality rule}, which can be considered as a generalization of the Born rule. The result is a new formulation of quantum mechanics, in which particles follow definite trajectories, but which is only based on the standard formalism of quantum mechanics.

quant-ph↗
arXiv2007

Origin of which-way information and generalization of the Born rule

The possibility to recover the which-way information, for example in the two slit experiment, is based on a natural but implicit assumption about the position of a particle {\it before} a position measurement is performed on it. This assumption cannot be deduced from the standard postulates of quantum mechanics. In the present paper this assumption is made explicit and formally postulated as a new rule, the {\it quantum typicality rule}. This rule correlates the positions of the particles at two different times, thus defining their trajectories. Unexpectedly, this rule is also equivalent to the Born rule with regard to the explanation of the results of statistical experiments. For this reason it can be considered a generalization of the Born rule. The existence of the quantum typicality rule strongly suggests the possibility of a new trajectory-based formulation of quantum mechanics. According to this new formulation, a closed quantum system is represented as a {\it quantum process}, which corresponds to a canonical stochastic process in which the probability measure is replaced by the wave function and the usual frequentist interpretation of probability is replaced by the quantum typicality rule.

quant-ph↗
arXiv2004

A Newtonian Hidden Variable Theory

A new hidden variable theory is proposed, according to which particles follows definite trajectories, as in Bohmian Mechanics or Nelson's stochastic mechanics; in the new theory, however, the trajectories are classical, i.e. Newtonian. This result is obtained by developing the following concepts: (i) the essential elements of a hidden variable theory are a set of trajectories and a measure defined on it; the Newtonian HCT will be defined by giving these two elements. (ii) The universal wave function has a tree structure, whose branches are generated by the measurement processes and are spatially disjoined. (iii) The branches have a classical structure, i.e. classical paths go along them; this property derives from the fact that the paths close to the classical ones give the main contribution to the Feynman propagator. (iv) Classical trajectories can give rise to quantum phenomena, like for instance the interference phenomena of the two-slit experiment, by violating the so called Independence Assumption, which is always implicitely made in the conceptual analysis of these phenomena.

quant-ph↗
arXiv2004

On probability, indeterminism and quantum paradoxes

If the quantum mechanical description of reality is not complete and a hidden variable theory is possible, what arises is the problem to explain where the rates of the outcomes of statistical experiments come from, as already noticed by Landé and Popper. In this paper this problem is investigated, and a new "paradigm" about the nature of dynamical and statistical laws is proposed. This paradigm proposes some concepts which contrast with the usual intuitive view of evolution and of physical law, such as: initial conditions could play no privileged role in determining the evolution of the universe; the statistical distribution of the particles emitted by a source could depend on the future interactions of the particles; indeterministic trajectories can be defined by the least action principle. This paradigm is applied to the analysis of the two-slit experiment and of the EPR paradox, and a coherent picture for these phenomena is proposed; this new picture shows how the well known difficulties in completing the quantum mechanical description of reality could be overcome.

quant-ph↗
arXiv1998

Could quantum statistical regularities derive from a measure on the boundary conditions of a classical universe?

The problem of defining the boundary conditions for the universe is considered here in the framework of a classical dynamical theory, pointing out that a measure on boundary conditions must be included in the theory in order to explain the statistical regularities of evolution. It is then suggested that quantum statistical regularities also could derive from this measure. An explicit definition of such a measure is proposed, using both a simplified model of the universe based on classical mechanics and the non-relativistic quantum mechanics formalism. The peculiarity of such a measure is that it does not apply to the initial conditions of the universe, i.e. to the initial positions and momenta of particles, but to their initial and final positions, from which the path is derived by means of the least action principle. This formulation of the problem is crucial and it is supported by the observation that it is incorrect to liken the determination of the boundary conditions of the universe to the preparation of a laboratory system, in which the initial conditions of the system are obviously determined. Some possible objections to this theory are then discussed. Specifically, the EPR paradox is discussed, and it is explained by showing that, in general, a measure on the boundary conditions of the universe generates preinteractive correlations, and that in the presence of such correlations Bell's inequality can no longer be proven true. Finally, it is shown that if one broadens the dynamical scheme of the theory to encompass phenomena such as particle decay and annihilation, the least action principle allows for an indeterministic evolution of the system.

quant-ph↗