We present a functional quantization scheme for a symmetric tensor field de- fined on a differentiable temporal base without metric structure. By replacing the Hilbert space formalism with a functional integral over admissible configurations of the field, we show that quantum phenomena (interference, decoherence, measure- ment) emerge as statistical effects of dynamically stabilized configurations. This framework offers a rigorous alternative to operator-based quantum mechanics, with- out invoking background geometry, and recovers classical structures in appropriate asymptotic limits.

Abstract We present a functional quantization scheme for a symmetric tensor field defined on a differentiable temporal base without metric structure. By replacing the Hilbert space formalism with a functional integral over admissible configurations of the field, we show that quantum phenomena (interference, decoherence, measurement) emerge as statistical effects of dynamically stabilized configurations. This framework offers a rigorous alternative to operator-based quantum mechanics, without invoking background geometry, and recovers classical structures in appropriate asymptotic limits.

The quantization error in a fixed-size Self-Organizing Map (SOM) with unsupervised winner-take-all learning has previously been used successfully to detect, in minimal computation time, highly meaningful changes across images in medical time series and in time series of satellite images. Here, the functional properties of the quantization error in SOM are explored further to show that the metric is capable of reliably discriminating between the finest differences in local contrast intensities and contrast signs. While this capability of the (...) QE is akin to functional characteristics of a specific class of retinal ganglion cells (the so-called Y-cells) in the visual systems of the primate and the cat, the sensitivity of the QE surpasses the capacity limits of human visual detection. Here, the quantization error in the SOM is found to reliably signal changes in contrast or colour when contrast information is removed from or added to the image, but not when the amount and relative weight of contrast information is constant and only the local spatial position of contrast elements in the pattern changes. While the RGB Mean reflects coarser changes in colour or contrast well enough, the SOM-QE is shown to outperform the RGB Mean in the detection of single-pixel changes in images with up to five million pixels. This could have important implications in the context of unsupervised image learning and computational building block approaches to large sets of image data (big data). (shrink)

We introduce a semiclassical quantization method which is based on a stroboscopic description of the classical and the quantum flows. We show that this approach emerges naturally when one is interested in extracting the energy spectrum within a prescribed and finite energy interval. The resulting semiclassical expression involves a finite number of periodic orbits whose energies are in the considered interval. Higher order corrections which reflect the sharp restriction of the spectrum to an interval are explicitly given. The relation (...) to Fourier methods for extracting semiclassical spectra, such as harmonic inversion, is worked out. The constraints due to the finite dimension of the Hilbert space and the unitarity of the restricted quantum evolution operator are important ingredients in this context. (shrink)

Karim Thébault has argued that for ontic structural realism to be a viable ontology it should accommodate two principles: physico-mathematical structures it deploys must be firstly consistent and secondly substantial. He then contends that in geometric quantization, a transitional machinery from classical to quantum mechanics, the two principles are followed, showing that it is a guide to ontic structure. In this article, I will argue that geometric quantization violates the consistency principle. To compensate for this shortcoming, the deformation (...) quantization procedure will be offered. (shrink)

The symplectic quantization scheme proposed for matter scalar fields in the companion paper (Gradenigo and Livi, arXiv:2101.02125, 2021) is generalized here to the case of space–time quantum fluctuations. That is, we present a new formalism to frame the quantum gravity problem. Inspired by the stochastic quantization approach to gravity, symplectic quantization considers an explicit dependence of the metric tensor gμν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{\mu \nu }$$\end{document} on an additional time variable, named intrinsic (...) time at variance with the coordinate time of relativity, from which it is different. The physical meaning of intrinsic time, which is truly a parameter and not a coordinate, is to label the sequence of gμν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{\mu \nu }$$\end{document} quantum fluctuations at a given point of the four-dimensional space–time continuum. For this reason symplectic quantization necessarily incorporates a new degree of freedom, the derivative g˙μν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\dot{g}}_{\mu \nu }$$\end{document} of the metric field with respect to intrinsic time, corresponding to the conjugated momentum πμν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{\mu \nu }$$\end{document}. Our proposal is to describe the quantum fluctuations of gravity by means of a symplectic dynamics generated by a generalized action functional A[gμν,πμν]=K[gμν,πμν]-S[gμν]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}[g_{\mu \nu },\pi _{\mu \nu }] = {\mathcal {K}}[g_{\mu \nu },\pi _{\mu \nu }] - S[g_{\mu \nu }]$$\end{document}, playing formally the role of a Hamilton function, where S[gμν]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S[g_{\mu \nu }]$$\end{document} is the standard Einstein–Hilbert action while K[gμν,πμν]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {K}}[g_{\mu \nu },\pi _{\mu \nu }]$$\end{document} is a new term including the kinetic degrees of freedom of the field. Such an action allows us to define an ensemble for the quantum fluctuations of gμν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{\mu \nu }$$\end{document} analogous to the microcanonical one in statistical mechanics, with the only difference that in the present case one has conservation of the generalized action A[gμν,πμν]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}[g_{\mu \nu },\pi _{\mu \nu }]$$\end{document} and not of energy. Since the Einstein–Hilbert action S[gμν]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S[g_{\mu \nu }]$$\end{document} plays the role of a potential term in the new pseudo-Hamiltonian formalism, it can fluctuate along the symplectic action-preserving dynamics. These fluctuations are the quantum fluctuations of gμν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{\mu \nu }$$\end{document}. Finally, we show how the standard path-integral approach to gravity can be obtained as an approximation of the symplectic quantization approach. By doing so we explain how the integration over the conjugated momentum field πμν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{\mu \nu }$$\end{document} gives rise to a cosmological constant term in the path-integral approach. (shrink)

The ontic structural realist stance is motivated by a desire to do philosophical justice to the success of science, whilst withstanding the metaphysical undermining generated by the various species of ontological underdetermination. We are, however, as yet in want of general principles to provide a scaffold for the explicit construction of structural ontologies. Here we will attempt to bridge this gap by utilizing the formal procedure of quantization as a guide to ontic structure of modern physical theory. The example (...) of non-relativistic particle mechanics will be considered and, for that case, it will be shown that a viable candidate for an ontic structural realism framework can be constituted in terms of the combination of a state-space with Poisson bracket structure, and a set of observables, with Lie algebra structure. 1 Introduction2 Formulation Underdetermination and Structural Ontologies3 Quantization and Structural Ontologies4 The Case of Non-relativistic Particle Mechanics4.1 Formulation underdetermination4.2 The classical ontology4.3 Quantum theory and the generalized structural ontology4.4 Interpretation of results5 Conclusion and Prospects. (shrink)

We propose here a new symplectic quantization scheme, where quantum fluctuations of a scalar field theory stem from two main assumptions: relativistic invariance and equiprobability of the field configurations with identical value of the action. In this approach the fictitious time of stochastic quantization becomes a genuine additional time variable, with respect to the coordinate time of relativity. Thisintrinsic timeis associated to a symplectic evolution in the action space, which allows one to investigate not only asymptotic, i.e. equilibrium, (...) properties of the theory, but also its non-equilibrium transient evolution. In this paper, which is the first one in a series of two, we introduce a formalism which will be applied to general relativity in its companion work (Gradenigo, Symplectic quantization II: dynamics of space-time quantum fluctuations and the cosmological constant, 2021). (shrink)

The existence of a fundamental length in general relativity, the Planck length, may lead to a breakdown of Lorentz invariance of the vacuum. The third quantization introduces renormalization fields of negative energy which do not interact with matter however. This revision leads to a measurable modification of the Casimir effect and can, at least in principle, lead to an eternal universe.

Relativistic quantum mechanics is generalized to account for a universally constant quantum of length a. Its value depends on the total convertible energy content of our universe: Eu = hc/2a. The eigenvalues of all (x,y,z,ct) coordinates are integer or half-integer multiples of a in every particular inertial frame. There are thus several spacetime lattices of lattice-constant a: the “normal lattice” contains the origin of the chosen frame, while “inserted lattices” are displaced by a/2 along one or several reference axes. States (...) of motion are defined by possible variations of ψ-functions on any one of these lattices. Particle states are defined by their relative phases, specified by four new quantum numbers, ux, uy, uz, uct = 0, ±1, ±2,.... They account for all known elementary particles and yield a natural extension of the standard model. Spacetime quantization solves also the EPR paradox and other difficulties that subsisted in the usual continuum theories. It defines inertial frames and is related to cosmology. (shrink)

First of all we shortly illustrate how the symplectic quantization scheme (Gradenigo and Livi, Found Phys 51(3):66, 2021) can be applied to a relativistic field theory with self-interaction. Taking inspiration from the stochastic quantization method by Parisi and Wu, this procedure is based on considering explicitly the role of an intrinsic time variable, associated with quantum fluctuations. The major part of this paper is devoted to showing how the symplectic quantization scheme can be extended to the non-relativistic (...) limit for a Schrödinger-like field. Then we also discuss how one can obtain from this non-relativistic theory a linear Schrödinger equation for the single-particle wavefunction. This further passage is based on a suitable coarse-graining procedure, when self-interaction terms can be neglected, with respect to interactions with any external field. In the Appendix we complete our survey on symplectic quantization by discussing how this scheme applies to a non-relativistic particle under the action of a generic external potential. (shrink)

The additional information within a Hamilton–Jacobi representation of quantum mechanics is extra, in general, to the Schrödinger representation. This additional information specifies the microstate of \ that is incorporated into the quantum reduced action, W. Non-physical solutions of the quantum stationary Hamilton–Jacobi equation for energies that are not Hamiltonian eigenvalues are examined to establish Lipschitz continuity of the quantum reduced action and conjugate momentum. Milne quantization renders the eigenvalue J. Eigenvalues J and E mutually imply each other. Jacobi’s theorem (...) generates a microstate-dependent time parametrization \ even where energy, E, and action variable, J, are quantized eigenvalues. Substantiating examples are examined in a Hamilton–Jacobi representation including the linear harmonic oscillator numerically and the square well in closed form. Two byproducts are developed. First, the monotonic behavior of W is shown to ease numerical and analytic computations. Second, a Hamilton–Jacobi representation, quantum trajectories, is shown to develop the standard energy quantization formulas of wave mechanics. (shrink)

This study attempts to spell out more explicitly than has been done previously the connection between two types of formal correspondence that arise in the study of quantum–classical relations: one the one hand, deformation quantization and the associated continuity between quantum and classical algebras of observables in the limit \, and, on the other, a certain generalization of Ehrenfest’s Theorem and the result that expectation values of position and momentum evolve approximately classically for narrow wave packet states. While deformation (...) quantization establishes a direct continuity between the abstract algebras of quantum and classical observables, the latter result makes in-eliminable reference to the quantum and classical state spaces on which these structures act—specifically, via restriction to narrow wave packet states. Here, we describe a certain geometrical re-formulation and extension of the result that expectation values evolve approximately classically for narrow wave packet states, which relies essentially on the postulates of deformation quantization, but describes a relationship between the actions of quantum and classical algebras and groups over their respective state spaces that is non-trivially distinct from deformation quantization. The goals of the discussion are partly pedagogical in that it aims to provide a clear, explicit synthesis of known results; however, the particular synthesis offered aspires to some novelty in its emphasis on a certain general type of mathematical and physical relationship between the state spaces of different models that represent the same physical system, and in the explicitness with which it details the above-mentioned connection between quantum and classical models. (shrink)

This paper develops a formal framework for quantum field quantization without reference to a Hilbert space. Based on a pre-metric tensor field Aμν (x, t∗) defined over a non-metrical time base, we construct a functional quantization procedure relying on dynamically weighted actualization channels. The approach generalizes path integral quantization by grounding it on structural con- straints rather than probabilistic amplitudes, and aims to provide an ontologically viable alternative to Hilbert-based formalisms.

A quantization of classical spinning particle equations is carried out using the Euler angles of the particle. Relativistic corrections are found and compared to the Foldy–Wouthuysen transformation of the Dirac equation. We only consider constant linear electric and magnetic fields, and find agreement up to third order in $$1/{\text{c}}^{2}$$.

This paper addresses the scope behavior of bare nouns (BNs) in Kaingang (Jê, Brazil) and its relevance to the study of semantic variation. I show that Kaingang BNs exhibit variable scope in relation to other clause-mate operators, such as negation, intensional verbs, _if_-clauses, universal quantifiers, and frequency adverbials. Adopting a type-shifting framework (Chierchia in Topics in the syntax and semantics of infinitives and gerunds. PhD diss., University of Massachusetts, 1984 ; Events and grammar, pp. 53–103, 1998a ; Natural Language Semantics (...) 6: 339–405, 1998b ; Partee in Studies in discourse representation theory and the theory of generalized quantifiers, pp. 115–144, 1986 ; Krifka in Proceedings of SALT 13, pp. 180–203, 2003 ; Dayal in Linguistics and Philosophy, 27: 393–450, 2004, among others), I analyze Kaingang indefinite BNs as predicate NPs mapped into arguments via a choice function (CF) that gets existentially closed at any level in the clause, _à la_ English indefinites under Reinhart’s (Linguistics and Philosophy, 20(4): 335–397, 1997 ; Interface strategies: Optimal and costly computations, 2006 ) analysis. I also argue that the difference in scope properties across languages between variable-scope and narrowest-scope BNs is linked to the quantized/cumulative opposition of the predicates they express: the former are quantized predicates, while the latter are cumulative. I then claim that variable-scope and narrowest-scope BNs are created by two distinct type-shifters: the former by CF and the later by Krifka’s ∃ operator (Krifka in Proceedings of SALT 13, pp. 180–203, 2003 ). In this way, I propose that both operators are sensitive to the quantized/cumulative opposition and model this as conditioned by a binary semantic feature, namely [± quantizedness], which checks the (non-)quantized properties of the predicates that undergo type-shifting. The CF type-shifter is [+ quant], and thus defined only for quantized predicates, while Krifka’s ∃ type-shifter is [− quant], selecting only non-quantized ones. This analysis is shown to correctly predict the variation in scope properties of BNs across languages. (shrink)

We consider the quantum mechanical behavior of a driven particle in an infinite ID potential well. We show that the quantum dynamics of the system is induced by the delicate nontrivial properties of the momentum operator in this case, namely, its non-self-adjointness. Using this, we calculate the first order contribution to the cross section and the energy gain, and discuss their classical limit.

This book presents a comprehensive mathematical study of the operators behind the Born-Jordan quantization scheme. The Schrödinger and Heisenberg pictures of quantum mechanics are equivalent only if the Born-Jordan scheme is used. Thus, Born-Jordan quantization provides the only physically consistent quantization scheme, as opposed to the Weyl quantization commonly used by physicists. In this book we develop Born-Jordan quantization from an operator-theoretical point of view, and analyze in depth the conceptual differences between the two schemes. (...) We discuss various physically motivated approaches, in particular the Feynman-integral point of view. One important and intriguing feature of Born-Jordan quantization is that it is not one-to-one: there are infinitely many classical observables whose quantization is zero. (shrink)

The rigorous equivalence of the Schrödinger and Heisenberg pictures requires that one uses Born–Jordan quantization in place of Weyl quantization. We confirm this by showing that the much discussed “ angular momentum dilemma” disappears if one uses Born–Jordan quantization. We argue that the latter is the only physically correct quantization procedure. We also briefly discuss a possible redefinition of phase space quantum mechanics, where the usual Wigner distribution has to be replaced with a new quasi-distribution associated (...) with Born–Jordan quantization, and which has proven to be successful in time-frequency analysis. (shrink)

In the covariant canonical approach to classical physics, each point in phase space represents an entire classical trajectory. Initial data at a fixed time serve as coordinates for this “timeless” phase space, and time evolution can be viewed as a coordinate change. We argue for a similar view in quantum theory. As in the Heisenberg picture, the wave function is fundamentally time-independent. On any given time slice, however, we can diagonalize a complete set of position operators to form a basis, (...) in which the projected wave function depends on the choice of time. In this picture, time evolution can be viewed as a basis change in what is otherwise a block universe. We argue that this may help solve the “problem of time” in quantum gravity, and illustrate the idea with an example from three-dimensional quantum gravity. (shrink)

Our object in this paper is to study the antecedents, contents, implications, and impact of a not well-known or appreciated paper by EHRENFEST in 1911 on the essential nature of the different quantum hypotheses in radiation theory. After a careful analysis of EHRENFEST’s notebooks, correspondence, and publications, we conclude that the essential points of EHRENFEST’s paper were not perceived to a large extent, and hence that its implications were not considered thoroughly. Specifically, we show that EHRENFEST contributed significantly to the (...) clarifications of the differences between PLANCK’s and EINSTEIN’s respective quantum hypotheses, as well as to those of the concepts ‘‘discontinuity,’’ ‘‘quantization,’’ and ‘‘corpuscularity.’’. (shrink)

This paper addresses the robust output feedback model predictive control schemes for networked control systems with input quantization. The logarithmic quantizer is considered in this paper, and the sector bound approach is applied, which appropriately treats the quantization error as a sector-bounded uncertainty. The presented method involves an offline designed state observer using linear matrix inequality and online robust output feedback MPC algorithms which optimize one free control move followed by the output feedback using the estimated state. Moreover, (...) due to the uncertainty of estimation error, a technique of refreshing the bound of estimation error which involves the quantization error is provided so as to guarantee the recursive feasibility of the optimization problem. The proposed MPC schemes inherit the characteristics of the synthesis approach of MPC, guaranteeing the recursive feasibility of the optimization problem and the stability of a closed-loop system, and explicitly account for quantization error. Two simulation examples are given to illustrate the effectiveness of the proposed methods. (shrink)

Following the recent development of a mechanical atomic model based on Louis de Broglie’s double solution theory, we show how it is possible to obtain the famous Landau quantization rules for a purely classical electron coupled to a wave. We present both a qualitative but intuitive explanation of our model of wave-particle duality, as well as a more rigorous derivation of the solutions. This work completes a set of three articles, each using a different combination of external electric and (...) magnetic forces on the same base model. Despite this similarity with previous approaches, it also uncovers some unique properties of this system. We will see that, in order to correctly recover some useful non-relativistic limits, it is necessary to use a scalar field whose mass is the same as an effective mass for the particle that naturally appears in our model. This could be a crucial step to better characterize the necessary ingredients of a double solution theory. (shrink)

The Doubochinski pendulum is a simply-constructed nonlinear oscillator having a discrete series of stable amplitudes, which can be seen to mimic the quantum behavior of microphysical objects in several respects. Apart from this system’s intrinsic interest, the specific form of phase-modulated interaction (so-called argumental interaction) which gives rise to amplitude “quantization” in the Doubochinski pendulum, might prove relevant to developing a more realistic and more intelligible form of quantum physics in the future. In this paper I describe the experimentally-observed (...) behavior of the Doubochinski pendulum in some detail, and sketch some elements of the theory of the pendulum, focusing on insights into the mechanism of amplitude quantization and related phenomena in this system. I shall also briefly touch on a related model system, consisting of a low-frequency mechanical oscillator interacting with a high-frequency wave, which likewise displays a discrete set of stable amplitudes. Finally, I shall discuss some results of A. I. Shumaev and Z. A. Maizelis concerning the statistical distribution of quantized amplitudes of the Doubochinski pendulum as a function of initial conditions, which bears some analogy to probability densities in quantum mechanics. (shrink)

Planck’s and Einstein’s steps toward quantization are discussed, including a historical comparison of these two very different thinkers, their motives and heuristics. Sect. 2.2 studies Albert Einstein’s arguments up to the 1905 paper and how the many important publications from this annus mirabilis and shortly afterwards until 1909 are interconnected. Max Planck’s second quantum theory 1909–13 serves as a contrast: with it Planck attempts to retract his hesitantly introduced quantization of energy from 1900 by blaming it on the (...) material resonators inside an idealized radiator at equilibrium (a ‘black body’) and thereby uphold the theory of a continuous radiation field satisfying Maxwell’s equations. Other papers by Einstein banished this interpretative option by showing that Planck’s formula for the distribution of radiation already presupposes quantized energy in the radiation field itself. Sect. 2.5 discusses the variety of terms used by Einstein and his contemporaries during this work-in-progress. Most of the alternatives considered then, such as ‘light corpuscles’ or ‘energy projectiles’, have been forgotten since. Sect. 2.6 statistically retraces the slow rise of the term ’photon’ in common usage generally attributed to the physical chemist Gilbert N. Lewis (1926). (shrink)

This paper investigates the fixed-time coordinated control problem of six-degree-of-freedom dynamic model for multiple spacecraft formation flying with input quantization, where the communication topology is assumed directed. Firstly, a new multispacecraft nonsingular fixed-time terminal sliding mode vector is derived by using neighborhood state information. Secondly, a hysteretic quantizer is utilized to quantify control force and torque. Utilizing such a quantizer not only can reduce the required communication rate but also can eliminate the control chattering phenomenon induced by the logarithmic (...) quantizer. Thirdly, a 6-DOF fixed-time coordinated control strategy with adaptive tuning laws is proposed, such that the practical fixed-time stability of the controlled system is ensured in the presence of both upper bounds of unknown external disturbances. It theoretically proves that the relative tracking errors of attitude and position can converge into the regions in a fixed time. Finally, a numerical example is exploited to show the usefulness of the theoretical results. (shrink)

In this article, I examine the relationship between physical quantities and physical states in quantum theories. I argue against the claim made by Arageorgis that the approach to interpreting quantum theories known as Algebraic Imperialism allows for “too many states.” I prove a result establishing that the Algebraic Imperialist has very general resources that she can employ to change her abstract algebra of quantities in order to rule out unphysical states.

Experiments witnessing the entanglement between two particles interacting only via the gravitational field have been proposed as a test whether gravity must be quantized. In the language of quantum information, a non-quantum gravitational force would be modeled by local operations with classical communication (LOCC), which cannot generate entanglement in an initially unentangled state. This idea is criticized as too constraining on possible alternatives to quantum gravity. We present a parametrized model for the gravitational interaction of quantum matter on a classical (...) spacetime, inspired by the de Broglie–Bohm formulation of quantum mechanics, which results in entanglement and thereby provides an explicit counterexample to the claim that only a quantized gravitational field possesses this capability. (shrink)

We define and study the notion of quantum polarity, which is a kind of geometric Fourier transform between sets of positions and sets of momenta. Extending previous work of ours, we show that the orthogonal projections of the covariance ellipsoid of a quantum state on the configuration and momentum spaces form what we call a dual quantum pair. We thereafter show that quantum polarity allows solving the Pauli reconstruction problem for Gaussian wavefunctions. The notion of quantum polarity exhibits a strong (...) interplay between the uncertainty principle and symplectic and convex geometry and our approach could therefore pave the way for a geometric and topological version of quantum indeterminacy. We relate our results to the Blaschke–Santaló inequality and to the Mahler conjecture. We also discuss the Hardy uncertainty principle and the less-known Donoho–Stark principle from the point of view of quantum polarity. (shrink)

The gravitational equations were derived in general relativity using the assumption of their covariance relative to arbitrary transformations of coordinates. It has been repeatedly expressed an opinion over the past century that such equality of all coordinate systems may not correspond to reality. Nevertheless, no actual verification of the necessity of this assumption has been made to date. The paper proposes a theory of gravity with a constraint, the degenerate variants of which are general relativity and the unimodular theory of (...) gravity. This constraint is interpreted from a physical point of view as a sufficient condition for the adiabaticity of the process of the evolution of the space–time metric. The original equations of the theory of gravity with the constraint are formulated. On this basis, a unified model of the evolution of the modern, early, and very early Universe is constructed that is consistent with the observational astronomical data but does not require the hypotheses of the existence of dark energy, dark matter or inflatons. It is claimed that: physical time is anisotropic, the gravitational field is the main source of energy of the Universe, the maximum global energy density in the Universe was 64 orders of magnitude smaller the Planckian one, and the entropy density is 18 orders of magnitude higher the value predicted by GR. The value of the relative density of neutrinos at the present time and the maximum temperature of matter in the early Universe are calculated. The wave equation of the gravitational field is formulated, its solution is found, and the nonstationary wave function of the very early Universe is constructed. It is shown that the birth of the Universe was random. (shrink)

We show that there is a hidden freedom in quantum many-body theory associated with overcompleteness of the time evolution through the single-particle subspace of a many-body system. To fix the freedom, an additional constraint is necessary. We argue that the appropriate constraint on the time evolution through the subspace is to quantize the propagation of entangled pairs of particles, represented by the single-particle spectral function, instead of individual particles. This solution method creates a surface that indicates the multiplicity of every (...) solution to the inverse problem defined by matching the freedom to the constraint. Upon measurement, the system collapses nonlocally onto a single quantized solution. In addition to a combinatoric multiplicity, each solution acquires a multiplicity due to its stability when subject to a small variation in the microscopic degrees of freedom. Numerical calculations for a two-level system show that our theory improves upon standard theory in the description of non-quasiparticle spectral features. Our reinterpretation of quantum many-body theory is not based on the Born rule and offers a more faithful representation of experiments than current theory by modeling individual, quantized events with an explicit collapse model. (shrink)

This chapter provides a detailed account of how John Nicholson’s atomic theory of the early 20th century was spectacularly successful in accommodating as well as predicting some spectral lines in the solar corona and in the nebula in Orion’s Belt. The theory was soon shown to be completely mistaken in several respects, and yet it served to introduce the notion of the quantization of angular momentum of electrons in any atom. Armed with this key idea, Bohr was able to (...) develop the first successful model of the hydrogen atom and to initiate the quantum theory of atomic physics. This example is used to argue for an evolutionary and “organic” view of the manner in which scientific theories develop and to dispute Kuhn’s notion of scientific revolutions, while agreeing, nevertheless, with the later Kuhn’s insistence on the evolutionary development of scientific concepts and his denial that science approaches a pre-existing and exterior “truth.”. (shrink)

We argue against claims that the classical ℏ → 0 limit is “singular” in a way that frustrates an eliminative reduction of classical to quantum physics. We show one precise sense in which quantum mechanics and scaling behavior can be used to recover classical mechanics exactly, without making prior reference to the classical theory. To do so, we use the tools of strict deformation quantization, which provides a rigorous way to capture the ℏ → 0 limit. We then use (...) the tools of category theory to demonstrate one way that this reduction is explanatory: it illustrates a sense in which the structure of quantum mechanics determines that of classical mechanics. (shrink)

This paper criticizes the traditional philosophical account of the quantization of gauge theories and offers an alternative. On the received view, gauge theories resist quantization because they feature distinct mathematical representatives of the same physical state of affairs. This resistance is overcome by a sequence of ad hoc modifications, justified in part by reference to semiclassical electrodynamics. Among other things, these modifications introduce "ghosts": particles with unphysical properties which do not appear in asymptotic states and which are said (...) to be purely a notational convenience. I argue that this sequence of modifications is unjustified and inadequate, making it a poor basis for the interpretation of ghosts. I then argue that gauge theories can be quantized by the same method as any other theory. On this account, ghosts are not purely notation: they are coordinates on the classical configuration space of the theory—specifically, on its gauge structure. This interpretation does not fall prey to the standard philosophical arguments against the significance of ghosts, due to Weingard. Weingard’s argumentative strategy, properly applied, in fact tells in favor of ghosts’ physical significance. (shrink)

Bundles of C*-algebras can be used to represent limits of physical theories whose algebraic structure depends on the value of a parameter. The primary example is the ℏ→0 limit of the C*-algebras of physical quantities in quantum theories, represented in the framework of strict deformation quantization. In this paper, we understand such limiting procedures in terms of the extension of a bundle of C*-algebras to some limiting value of a parameter. We prove existence and uniqueness results for such extensions. (...) Moreover, we show that such extensions are functorial for the C*-product, dynamical automorphisms, and the Lie bracket (in the ℏ→0 case) on the fiber C*-algebras. (shrink)

The correspondence principle made of unitarity, locality and renormalizability has been very successful in quantum field theory. Among the other things, it helped us build the standard model. However, it also showed important limitations. For example, it failed to restrict the gauge group and the matter sector in a powerful way. After discussing its effectiveness, we upgrade it to make room for quantum gravity. The unitarity assumption is better understood, since it allows for the presence of physical particles as well (...) as fake particles (fakeons). The locality assumption is applied to an interim classical action, since the true classical action is nonlocal and emerges from the quantization and a later process of classicization. The renormalizability assumption is refined to single out the special role of the gauge couplings. We show that the upgraded principle leads to an essentially unique theory of quantum gravity. In particular, in four dimensions, a fakeon of spin 2, together with a scalar field, is able to make the theory renormalizable while preserving unitarity. We offer an overview of quantum field theories of particles and fakeons in various dimensions, with and without gravity. (shrink)

The Caldeira-Leggett model of quantum Brownian motion is generalized using a generic velocity-dependent coupling. That leads to the description of a set of models able to capture Markovian and non-Markovian versions of Brownian and Lévy statistics, depending on the functional form of the coupling and on the spectral function of the reservoir. One specific coupling force is found that establishes a connection with Lévy statistics of cold atoms in Sisyphus laser cooling. In the low-velocity limit, this also gives rise to (...) additional inertia of the Brownian particle, reproducing the Abraham-Lorentz equation from first principles for a superohmic bath. Through path-integral quantization in Euclidean time, the environment is integrated out, leaving a set of nonlocal effective actions. These results further serve as starting points for several numerical calculations, particularly decoherence properties of nonohmic baths. (shrink)

We present an innovative and robust solution to both biometric and emotion identification using the electrocardiogram (ECG). The ECG represents the electrical signal that comes from the contraction of the heart muscles, indirectly representing the flow of blood inside the heart, it is known to convey a key that allows biometric identification. Moreover, due to its relationship with the nervous system, it also varies as a function of the emotional state. The use of information-theoretic data models, associated with data compression (...) algorithms, allowed to effectively compare ECG records and infer the person identity, as well as emotional state at the time of data collection. The proposed method does not require ECG wave delineation or alignment, which reduces preprocessing error. The method is divided into three steps: (1) conversion of the real-valued ECG record into a symbolic time-series, using a quantization process; (2) conditional compression of the symbolic representation of the ECG, using the symbolic ECG records stored in the database as reference; (3) identification of the ECG record class, using a 1-NN (nearest neighbor) classifier. We obtained over 98% of accuracy in biometric identification, whereas in emotion recognition we attained over 90%. Therefore, the method adequately identify the person, and his/her emotion. Also, the proposed method is flexible and may be adapted to different problems, by the alteration of the templates for training the model. (shrink)

This chapter explains why physics could not rest content with classical Yang-Mills theories, and shows how a classical Yang-Mills field theory may be transformed into a quantum field theory. Quantum Yang-Mills theories provide the foundation for the Standard Model of elementary particles — currently our most successful representation of the fundamental forces of nature at high energies and extremely short distances. After pointing out the difficulties for quantization posed by the gauge structure of these theories, the rest of the (...) chapter describes several different methods for overcoming them. By doing so, it lays a technical foundation for the analysis of the rest of the book. (shrink)

Preparing general relativity for quantization in the Hamiltonian approach leads to the `problem of time,' rendering the world fundamentally timeless. One proposed solution is the `thermal time hypothesis,' which defines time in terms of states representing systems in thermal equilibrium. On this view, time is supposed to emerge thermodynamically even in a fundamentally timeless context. Here, I develop the worry that the thermal time hypothesis requires dynamics -- and hence time -- to get off the ground, thereby running into (...) worries of circularity. (shrink)

A novel approach to quantization is shown to allow for superpositions of the cosmological constant in isotropic and homogeneous mini-superspace models. Generic solutions featuring such superpositions display: i) a unitary evolution equation; ii) singularity resolution; iii) a cosmic bounce. Explicit cosmological solutions are constructed. These exhibit characteristic bounce features including a ‘super-inflation’ regime with universal phenomenology that can naturally be made to be insensitive to Planck-scale physics.

In this chapter, I consider whether digital images are digital in the strongest sense; namely, qua images. Assuming that a digital image is one that is made and screened digitally, there is a further question as to whether the representational scheme to which the image belongs has a fundamentally digital structure. Answering this question requires close analysis of Nelson Goodman’s classical account of the analog/digital distinction. It also requires a response to Goodman’s insistence on the essential analogicity of the pictorial. (...) Such a response points to the uses of digital sampling and quantization technology to impose digital structure on encoded, replicable images. (shrink)

In some decision problems, even after reflection, relevant probabilities and utilities are indeterminate. Then a rational decision maximizes utility with respect to a quantization of beliefs and desires. It is robust with respect to a sensitivity analysis. Rationality does not require meeting the maximin rule, which in its standard form expresses excessive aversion to risk. Multiple decisions must form a coherent set, however.

We demonstrate that quantization entails randomness in microscopic processes. Unlike classical random processes that arise from incomplete human knowledge, quantum randomness arises from nature and is perfect. Radioactive decay is a prominent example. Heisenberg’s uncertainty principle quantifies quantum randomness by putting limits on the degree of uncertainty in particular pairs of observable quantities. The diffraction of an electron passing through a 1-slit experiment offers an instructive example of this principle. Quantum uncertainties make it difficult to assign a definite “size” (...) to microscopic objects. Randomness is inherent in the wave nature of all quanta. As we will see in more detail in Chap. 9, quantum randomness is essential to meet the relativistic requirement that information cannot be transferred superluminally. (shrink)

On one popular view, the general covariance of gravity implies that change is relational in a strong sense, such that all it is for a physical degree of freedom to change is for it to vary with regard to a second physical degree of freedom. At a quantum level, this view of change as relative variation leads to a fundamentally timeless formalism for quantum gravity. Here, we will show how one may avoid this acute ‘problem of time’. Under our view, (...) duration is still regarded as relative, but temporal succession is taken to be absolute. Following our approach, which is presented in more formal terms in, it is possible to conceive of a genuinely dynamical theory of quantum gravity within which time, in a substantive sense, remains. 1 Introduction1.1 The problem of time1.2 Our solution2 Understanding Symmetry2.1 Mechanics and representation2.2 Freedom by degrees2.3 Voluntary redundancy3 Understanding Time3.1 Change and order3.2 Quantization and succession4 Time and Gravitation4.1 The two faces of classical gravity4.2 Retaining succession in quantum gravity5 Discussion5.1 Related arguments5.2 Concluding remarks. (shrink)