We outline the rationale and preliminary results of using the State Context Property (SCOP) formalism, originally developed as a generalization of quantum mechanics, to describe the contextual manner in which concepts are evoked, used, and combined to generate meaning. The quantum formalism was developed to cope with problems arising in the description of (1) the measurement process, and (2) the generation of new states with new properties when particles become entangled. Similar problems arising with concepts motivated the formal treatment introduced here. Concepts are viewed not as fixed representations, but entities existing in states of potentiality that require interaction with a context--a stimulus or another concept--to 'collapse' to an instantiated form (e.g. exemplar, prototype, or other possibly imaginary instance). The stimulus situation plays the role of the measurement in physics, acting as context that induces a change of the cognitive state from superposition state to collapsed state. The collapsed state is more likely to consist of a conjunction of concepts for associative than analytic thought because more stimulus or concept properties take part in the collapse. We provide two contextual measures of conceptual distance--one using collapse probabilities and the other weighted properties--and show how they can be applied to conjunctions using the pet fish problem
We outline the rationale and preliminary results of using the state context property (SCOP) formalism, originally developed as a generalization of quantum mechanics, to describe the contextual manner in which concepts are evoked, used and combined to generate meaning. The quantum formalism was developed to cope with problems arising in the description of (i) the measurement process, and (ii) the generation of new states with new properties when particles become entangled. Similar problems arising with concepts motivated the formal treatment introduced here. Concepts are viewed not as fixed rep…
We give the mathematical theory of duality computer in the density matrix formalism. This result complements the mathematical theory of duality computer of Gudder in the pure state formalism.
We present a mathematical framework for simulation of optical fields in complex gravitational-wave interferometers. The simulation framework uses the two-photon formalism for optical fields and includes radiation pressure effects, an important addition required for simulating signal and noise fields in next-generation interferometers with high circulating power. We present a comparison of results from the simulation with analytical calculation and show that accurate agreement is achieved.
A scheme for constructing quantum mechanics is given that does not have Hilbert space and linear operators as its basic elements. Instead, a version of algebraic approach is considered. Elements of a noncommutative algebra (observables) and functionals on this algebra (elementary states) associated with results of single measurements are used as primary components of the scheme. On the one hand, it is possible to use within the scheme the formalism of the standard (Kolmogorov) probability theory, and, on the other hand, it is possible to reproduce the mathematical formalism of standard quantum…
We provide here a general mathematical framework to model attitudes towards ambiguity which uses the formalism of quantum theory as a ``purely mathematical formalism, detached from any physical interpretation''. We show that the quantum-theoretic framework enables modelling of the "Ellsberg paradox", but it also successfully applies to more concrete human decision-making (DM) tests involving financial, managerial and medical decisions. In particular, we elaborate a mathematical representation of various empirical studies which reveal that attitudes of managers towards uncertainty shift from "a…
Conditional probabilities in quantum systems which have both initial and final boundary conditions are commonly evaluated using the Aharonov-Bergmann-Lebowitz rule. In this short note we present a seemingly disturbing paradox that appears when applying the rule to systems with slightly broken degeneracies. In these cases we encounter a singular limit - the probability "jumps" when going from perfect degeneracy to negligibly broken one. We trace the origin of the paradox and solve it from both traditional and modern perspectives in order to highlight the physics behind it: the necessity to take…
It is well known that nonrelativistic quantum mechanics presents a clear asymmetry between space and time. Much of this asymmetry is attributed to the lack of Lorentz invariance of the theory. Nonetheless, a recent work [Phys. Rev. A \textbf{95}, 032133 (2017)] showed that even though this is partially true, there is a broader physical scenario in which space and time can be handled in nonrelativistic quantum theory in a more symmetric way. In this space-time-symmetric formalism, an additional Hilbert space is defined so that time is raised to the status of operator and position becomes a para…
The main goal of this paper is to put on solid mathematical grounds the so-called Non-Equilibrium Green's Function (NEGF) transport formalism for open systems. In particular, we derive the Jauho-Meir-Wingreen formula for the time-dependent current through an interacting sample coupled to non-interacting leads. Our proof is non-perturbative and uses neither complex-time Keldysh contours, nor Langreth rules of 'analytic continuation'. We also discuss other technical identities (Langreth, Keldysh) involving various many body Green's functions. Finally, we study the Dyson equation for the advanced…
We combine the ideas of Dirac's orthonormal representation, Everett's relative state, and 't Hooft's ontological basis to define the notion of a world for quantum mechanics. Mathematically, for a quantum system $\mathcal{Q}$ with an associated Hilbert space $\mathbb{H},$ a world of $\mathcal{Q}$ is defined to be an orthonormal basis of $\mathbb{H}.$ The evolution of the system is governed by Schrödinger's equation for the worlds of it. An observable in a certain world is a self-adjoint operator diagonal under the corresponding basis. Moreover, a state is defined in an associated world but can …
We present a mathematical formalism of non-Hermitian quantum mechanics, following the Dirac-von Neumann formalism of quantum mechanics. In this formalism, the state postulate is the same as in the Dirac-von Neumann formalism, but the observable postulate should be changed to include para-Hermitian operators (spectral operators of scalar type with real spectrum) representing observable, as such both the measurement postulate and the evolution postulate must be modified accordingly. This is based on a Stone type theorem as proved here that the dynamics of non-Hermitian quantum systems is governe…
By a series of simple examples, we illustrate how the lack of mathematical concern can readily lead to surprising mathematical contradictions in wave mechanics. The basic mathematical notions allowing for a precise formulation of the theory are then summarized and it is shown how they lead to an elucidation and deeper understanding of the aforementioned problems. After stressing the equivalence between wave mechanics and the other formulations of quantum mechanics, i.e. matrix mechanics and Dirac's abstract Hilbert space formulation, we devote the second part of our paper to the latter approac…
We have presented a consistent electronic transport framework for the two-dimensional extended Holstein's organic molecular-crystal based upon complete quantum-mechanical treatment through the non-equilibrium Green's function (NEGF) formalism and corresponding one-to-one semi-classical framework based upon Boltzmann transport theory for the narrow-bandwidth electronic energy organic semiconductor crystal material and device. In this process, we have formulated electronic self-energy interaction with acoustic and polar optical phonon mode with one phonon and two phonon interactions through Feyn…
We discuss the possibility to modify many-body Hilbert quantum formalism that is necessary for the representation of quantum systems dynamics. The notion of effective classical algorithm and visualization of quantum dynamics play the key role.
The relational interpretation of quantum mechanics (RQM) has received a growing interest since its first formulation in 1996. Usually presented as an interpretational layer over the usual quantum mechanics formalism, it appears as a philosophical perspective without proper mathematical counterparts. This state of affairs has direct consequences on the scientific debate on RQM which still suffers from misunderstandings and imprecise statements. In an attempt to clarify those debates, the present paper proposes a radical reformulation of the mathematical framework of quantum mechanics which is r…
The Coupled-Cluster theory is one of the most successful high precision methods used to solve the stationary Schrödinger equation. In this article, we address the mathematical foundation of this theory with focus on the advances made in the past decade. Rather than solely relying on spectral gap assumptions (non-degeneracy of the ground state), we highlight the importance of coercivity assumptions - Gårding type inequalities - for the local uniqueness of the Coupled-Cluster solution. Based on local strong monotonicity, different sufficient conditions for a local unique solution are suggested. …
This paper presents a new mathematical formalism that describes the quantization of games. The study of so-called quantum games is quite new, arising from a seminal paper of D. Meyer \cite{Meyer} published in Physics Review Letters in 1999. The ensuing near decade has seen an explosion of contributions and controversy over what exactly a quantized game really is and if there is indeed anything new for game theory. What has clouded many of the issues is the lack of a mathematical formalism for the subject in which these various issues can be clearly and precisely expressed, and which provides a…
The usual mathematical formalism of quantum field theory is non-rigorous because it contains divergences that can only be renormalized by non-rigorous mathematical methods. The purpose of this paper is to present a method of subtraction of this divergences using the formalism of decoherence. This is achieved by replacing the standard renormalization method by a projector on a well defined Hilbert subspace. In this way a list of problems of the standard formalism disappears while the physical results of QFT remains valid. From it own nature, this formalism can be used in non-renormalizable theo…
We show how various mathematical formalisms, specifically the catastrophe formalism and group theory, aid in the study of relevant systems in quantum optics. We describe the phase transition of the Dicke model for a finite number N of atoms, via 3 different methods, which lead to universal parametric curves for the expectation value of the first quadrature of the electromagnetic field and the expectation value of the number operator, as functions of the atomic relative population. These are valid for all values of the matter-field coupling parameter, and valid for both the ground and first-exc…
We develop a kind of quantum formalism (Hilbert space probabilistic calculus) for measurements performed over cognitive (in particular, conscious) systems. By using this formalism we could predict averages of cognitive observables. Reflecting the basic idea of neurophisiological and psychological studies on a hierarchic structure of cognitive processes, we use p-adic hierarchic trees as a mathematical model of a mental space. We also briefly discuss the general problem of the choice of adequate mental geometry.
Existing computer algebra packages do not fully support quantum mechanics calculations in Dirac's notation. I present the foundation for building such support: a mathematical system for the symbolic manipulation of expressions used in the invariant formalism of quantum mechanics. I first describe the essential mathematical features of the Hilbert-space invariant formalism. This is followed by a formal characterisation of all possible algebraic expressions in this formalism. This characterisation is provided in the form of a set of terms. Rewrite rules over this set of terms are then developed …
Gamow vectors have been developed in order to give a mathematical description for quantum decay phenomena. Mainly, they have been applied to radioactive phenomena, scattering and to some decoherence models. They play a crucial role in the description of quantum irreversible processes, and in the formulation of time asymmetry in quantum mechanics. In this paper, we use this formalism to describe a well-known phenomenon of irreversibility: the Loschmidt echo. The standard approach considers that the irreversibility of this phenomenon is the result of an additional term in the backward Hamiltonia…
We propose a unified mathematical framework for classifying phases of matter. The framework is based on different types of combinatorial structures with a notion of locality called lattices. A tensor lattice is a local prescription that associates tensor networks to those lattices. Different lattices are related by local operations called moves. Those local operations define consistency conditions for the tensors of the tensor network, the solutions to which yield exactly solvable models for all kinds of phases. We implement the framework to obtain models for symmetry-breaking and topological …
In this work we present a mathematical description of how one can produce and read a thin hologram. We use different kinds of waves, such as scalar, vector (electromagnetic field, Maxwell-Proca fields, acoustic waves, etc.). For reading of the hologram, we use the Green's function formalism. With the help of computer simulations, we investigate the aberrations created by this procedure for the simplest case: 2d-scalar wave case.
A hybrid formalism is proposed for interacting classical and quantum sytems. This formalism is mathematically consistent and reduces to standard classical and quantum mechanics in the case of no interaction. However, in the presence of interaction, the correspondence principle is violated.
The ESR model proposes a new theoretical perspective which incorporates the mathematical formalism of standard (Hilbert space) quantum mechanics (QM) in a noncontextual framework, reinterpreting quantum probabilities as conditional on detection instead of absolute. We have provided in some previous papers mathematical representations of the physical entities introduced by the ESR model, namely observables, properties, pure states, proper and improper mixtures, together with rules for calculating conditional and overall probabilities, and for describing transformations of states induced by meas…
We propose a combined mathematical framework of order statistics and random matrix theory for multicarrier continuous-variable (CV) quantum key distribution (QKD). In a multicarrier CVQKD scheme, the information is granulated into Gaussian subcarrier CVs, and the physical Gaussian link is divided into Gaussian sub-channels. The sub-channels are dedicated to the conveying of the subcarrier CVs. The distribution statistics analysis covers the study of the distribution of the sub-channel transmittance coefficients in the presence of a Gaussian noise and the utilization of the moment generation fu…
We discuss the hypothesis that the debate about the interpretation of the orthodox formalism of quantum mechanics (QM) might have been misguided right from the start by a biased metaphysical interpretation of the formalism and its inner mathematical relations. In particular, we focus on the orthodox interpretation of the congruence relation, '=', which relates equivalent classes of different mathematical representations of a vector in Hilbert space, in terms of metaphysical identity. We will argue that this seemingly "common sense" interpretation, at the semantic level, has severe difficulties…
It is shown that `bipartite' wave functions can present a mathematical formalism of quantum theory for a single particle, in which the associated Schrödinger's wave functions correspond to those `bipartite' wave functions of product forms. This extension of Schrödinger's form establishes a mathematical expression of wave-particle duality and that von Neumann's entropy is a quantitative measure of complementarity between wave-like and particle-like behaviors. In particular, this formalism suggests that collapses of Schrödinger's wave functions can be regarded as the simultaneous transition of t…
A mathematical extension of the weak value formalism to the simultaneous measurement of multiple parameters is presented in the context of an optical focused vector beam scatterometry experiment. In this example, preselection and postselection are achieved via spatially-varying polarization control, which can be tailored to optimize the sensitivity to parameter variations. Initial experiments for the two-parameter case demonstrate that this method can be used to measure physical parameters with resolutions at least 1000 times smaller than the wavelength of illumination.
In a previous preprint (quant-ph/0012122) we introduced a ``contextual objectivity" formulation of quantum mechanics (QM). A central feature of this approach is to define the quantum state in physical rather than in mathematical terms, in such a way that it may be given an "objective reality". Here we use some ideas about the system dimensionality, taken from quant-ph/0101012, to propose a possible axiomatic approach to QM. In this approach the structure of QM appears as a direct consequence of the non-commutative character of the (classical geometrical) group of "knobs transformations", that …
We propose a new formalism of quantum subsystems which allows to unify the existing and new methods of reduced description of quantum systems. The main mathematical ingredients are completely positive maps and correlation functions. In this formalism generalized quantum systems can be composed and there is a notion of generalized entanglement. Models of fermionic and bosonic systems and also quantum systems described by the SU(2) symmetry are studied.
In this paper we shall introduce the mathematical framework for the description of measurements of quantum processes. Using this framework the process estimation problems can be treated in the similar way as the state estimation problems, only replacing the concept of positive operator valued measure (POVM) by the concept of process POVM (PPOVM). In particular, we will show that any measurement of qudit channels can be described by a collection of effects (positive operators) defined on two-qudit system. However, the effects forming a PPOVM are not normalized in the usual sense. We will demons…
In this paper, we investigate the connection between Classical and Quantum Mechanics by dividing Quantum Theory in two parts: - General Quantum Axiomatics (a system is described by a state in a Hilbert space, observables are self-adjoint operators and so on) - Quantum Mechanics properly that specifies the Hilbert space, the Heisenberg rule, the free Hamiltonian... We show that General Quantum Axiomatics (up to a supplementary "axiom of classicity") can be used as a non-standard mathematical ground to formulate all the ideas and equations of ordinary Classical Statistical Mechanics. So the ques…
We present a heuristic derivation of Born's rule and unitary transforms in Quantum Mechanics, from a simple set of axioms built upon a physical phenomenology of quantization. This approach naturally leads to the usual quantum formalism, within a new realistic conceptual framework that is discussed in details. Physically, the structure of Quantum Mechanics appears as a result of the interplay between the quantized number of "modalities" accessible to a quantum system, and the continuum of "contexts" that are required to define these modalities. Mathematically, the Hilbert space structure appear…
We analyze the logical foundations of quantum mechanics (QM) by stressing non-objectivity of quantum observables which is a consequence of the absence of logical atoms in QM. We argue that the matter of quantum non-objectivity is that, on the one hand, the formalism of QM constructed as a mathematical theory is self-consistent, but, on the other hand, quantum phenomena as results of experimenter's performances are not self-consistent. This self-inconsistency is an effect of that the language of QM differs much from the language of human performances. The first is the language of a mathematical…
Electromagnetic phenomena can be described by Maxwell equations written for the vectors of electric and magnetic field. Equivalently, electrodynamics can be reformulated in terms of an electromagnetic vector potential. We demonstrate that the Schrödinger equation admits an analogous treatment. We present a Lagrangian theory of a real scalar field $ϕ$ whose equation of motion turns out to be equivalent to the Schrödinger equation with time independent potential. After introduction the field into the formalism, its mathematical structure becomes analogous to those of electrodynamics. The field $…
The aim of this article is to represent the general description of an entity by means of its states, contexts and properties. The entity that we want to describe does not necessarily have to be a physical entity, but can also be an entity of a more abstract nature, for example a concept, or a cultural artifact, or the mind of a person, etc..., which means that we aim at very general description. The effect that a context has on the state of the entity plays a fundamental role, which means that our approach is intrinsically contextual. The approach is inspired by the mathematical formalisms tha…
In this paper, I consider the issue of how two mathematical models of modern physics, the variational principles and the quantum path integral formalism, relate to reality. I assume that the observed phenomena are consistent with the calculations because both of these models have some common ontological foundations. According to the hypothesis of the summation of coexisting alternative possibilities, at the quantum level, the system at once moves along all histories that possible in given boundary conditions. The actual history is a mere result of the summation of all these alternative possibl…
Construal of observable facts or events, that is, the manner in which we understand reality, is based not only on mathematical formulas of a theory suggested as a reasonable explanation for physical phenomena (like general relativity or quantum mechanics), but also on a mathematical model of reasoning used to analyze and appraise statements regarding the objective world (for example, logic of one type or the other). Hence, every time that a certain construal of reality encounters a problem, there is a choice between a modification to the mathematical formalism of the physical theory and a chan…
We introduce Quantum Register Algebra (QRA) as an efficient tool for quantum computing. We show the direct link between QRA and Dirac formalism. We present GAALOP (Geometric Algebra Algorithms Optimizer) implementation of our approach. Using the QRA basis vectors definitions given in Section 4 and the framework based on the de Witt basis presented in Section 5, we are able to fully describe and compute with QRA in GAALOP using the geometric product. We illustrate the intuitiveness of this computation by presenting the QRA form for the well known SWAP operation on a two qubit register.
Phenomenological quantum mechanics: II. Deducing the formalism from experimental observations
Institute of Physics, Polish Academy of Sciences（波兰科学院物理研究所）；Department Physik, Friedrich-Alexander-Universität Erlangen-Nürnberg（埃尔朗根-纽伦堡大学物理系）；Wilczek Quantum Center, School of Physics and Astronomy, Shanghai Jiao Tong University（上海交通大学物理与天文学院费曼量子中心）；The International Centre for Theory of Quantum Technologies, University of Gdańsk（格但斯克大学量子技术理论国际中心）；Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University（尼古拉·哥白尼大学物理、天文与信息学学院物理研究所）
The motivation for this thesis was to recast quantum self-testing [MY98,MY04] in operational terms. The result is a category-theoretic framework for discussing the following general question: How do different implementations of the same input-output process compare to each other? In the proposed framework, an input-output process is modelled by a causally structured channel in some fixed theory, and its implementations are modelled by causally structured dilations formalising hidden side-computations. These dilations compare through a pre-order formalising relative strength of side-computation…
In this paper we consider general probabilistic theories that pertain to circuits which satisfy two very natural assumptions. We provide a formalism that is local in the following very specific sense: calculations pertaining to any region of spacetime employ only mathematical objects associated with that region. We call this "formalism locality". It incorporates the idea that space and time should be treated on an equal footing. Formulations that use a foliation of spacetime to evolve a state do not have this property nor do histories-based approaches. An operation (see figure on left) has inp…
