The basic concepts appropriate for anS matrix theory for classical nonlinear physics are formulated here. These concepts are illustrated by a discussion of shock wave diffraction patterns. Other information concerning solutions of non-linear conservation laws is surveyed, so that a coherent picture of this theory can be seen. Within thisS matrix framework, a number of open problems as well as a few solved ones will be discussed.

Recent theories of art that subscribe to the view that art objects are agents enchanting their target audience, have tended to explain the operation of art objects as an agent–patient dynamic, a causal nexus of agency. They face a challenge, however, when they also aspire to embrace the idea – dominant in modernist and contemporary art theory – that the function of art is to unsettle its spectators’ habitual ways of perceiving and understanding, that is, to disenchant them: If artworks (...) are to be understood as agents enchanting their recipients, how can they be forces of disenchantment at the same time? I argue that the shift in perspective from perceiving art objects as enchanting agents to approaching them as possible means of gaining cognitive distance is inadequately addressed by these theories. (shrink)

One of the major philosophical problems in physical sciences is what criteria should determine how scientific theories are selected and justified in practice and whether, in describing observable physical phenomena, such theories are effectively constrained to be unique. This book studies the example of a particular theory, the S-matrix theory. The S-matrix program was initiated by Heisenberg to deal with difficulties encountered in quantum field theories in describing particular phenomena. Since then, each theory has at different times been (...) favored as the explanation of observed phenomena. Certainly the S-matrix theory was adequate, feasible and fertile. However, the quantum field theory interpretation is now widely accepted and the study of alternative theories is all but abandoned. By examining the philosophy which influenced the turns in this story, the author explains how an adequate and viable theory fell out of favor and concludes with a critique of different methodologies in the history of science. (shrink)

In this revolutionary work, the author sets the stage for the science of the 21st Century, pursuing an unprecedented synthesis of fields previously considered unrelated. Beginning with simple classical concepts, he ends with a complex multidisciplinary theory requiring a high level of abstraction. The work progresses across the sciences in several multidisciplinary directions: Mathematical logic, fundamental physics, computer science and the theory of intelligence. Extraordinarily enough, the author breaks new ground in all these fields. In the field of fundamental physics (...) the author reaches the revolutionary conclusion that physics can be viewed and studied as logic in a fundamental sense, as compared with Einstein's view of physics as space-time geometry. This opens new, exciting prospects for the study of fundamental interactions. A formulation of logic in terms of matrix operators and logic vector spaces allows the author to tackle for the first time the intractable problem of cognition in a scientific manner. In the same way as the findings of Heisenberg and Dirac in the 1930s provided a conceptual and mathematical foundation for quantum physics, matrix operator logic supports an important breakthrough in the study of the physics of the mind, which is interpreted as a fractal of quantum mechanics. Introducing a concept of logic quantum numbers, the author concludes that the problem of logic and the intelligence code in general can be effectively formulated as eigenvalue problems similar to those of theoretical physics. With this important leap forward in the study of the mechanism of mind, the author concludes that the latter cannot be fully understood either within classical or quantum notions. A higher-order covariant theory is required to accommodate the fundamental effect of high-level intelligence. The landmark results obtained by the author will have implications and repercussions for the very foundations of science as a whole. Moreover, Stern's Matrix Logic is suitable for a broad spectrum of practical applications in contemporary technologies. (shrink)

This book explores Gilles Deleuze's contribution to film theory. According to Deleuze, we have come to live in a universe that could be described as metacinematic. His conception of images implies a new kind of camera consciousness, one that determines our perceptions and sense of selves: aspects of our subjectivities are formed in, for instance, action-images, affection-images and time-images. We live in a matrix of visual culture that is always moving and changing. Each image is always connected to an (...) assemblage of affects and forces. This book presents a model, as well as many concrete examples, of how to work with Deleuze in film theory. It asks questions about the universe as metacinema, subjectivity, violence, feminism, monstrosity, and music. Among the contemporary films it discusses within a Deleuzian framework are Strange Days, Fight Club, and Dancer in the Dark. (shrink)

The two-component spinor theory of van der Waerden is put into a convenient matrix notation. The mathematical relations among various types of matrices and the rule for forming covariant expressions are developed. Relativistic equations of classical mechanics and electricity and magnetism are expressed in this notation. In this formulation the distinction between time and space coordinates in the four-dimensional space-time continuum falls out naturally from the assumption that a four-vector is represented by a Hermitian matrix. The indefinite metric (...) of tensor analysis is a derived result rather than an arbitrary ad hoc assumption. The relation to four-component spinor theory is also discussed. (shrink)

It is shown that the matrix models which give non-perturbative definitions of string and M theory may be interpreted as non-local hidden variables theories in which the quantum observables are the eigenvalues of the matrices while their entries are the non-local hidden variables. This is shown by studying the bosonic matrix model at finite temperature, with T taken to scale as 1/N, with N the rank of the matrices. For large N the eigenvalues of the matrices undergo Brownian (...) motion due to the interaction of the diagonal elements with the off diagonal elements, giving rise to a diffusion constant that remains finite as N goes to infinity. The resulting probability density and current for the eigenvalues are then found to evolve in agreement with the Schroedinger equation, to leading order in 1/N, with hbar proportional to the thermal diffusion constant for the eigenvalues. The quantum uctuations and uncertainties in the eigenvalues are then consequences of ordinary statistical uctuations in the values of the off-diagonal matrix elements. Furthermore, this formulation of the quantum theory is background independent, as the definition of the thermal ensemble makes no use of a particular classical solution. The derivation relies on Nelson's stochastic formulation of quantum theory, which is expressed in terms of a variational principle. (shrink)

Realism about quantum theory naturally leads to realism about the quantum state of the universe. It leaves open whether it is a pure state represented by a wave function, or an impure one represented by a density matrix. I characterize and elaborate on Density Matrix Realism, the thesis that the universal quantum state is objective but can be impure. To clarify the thesis, I compare it with Wave Function Realism, explain the conditions under which they are empirically equivalent, (...) consider two generalizations of Density Matrix Realism, and answer some frequently asked questions. I end by highlighting an implication for scientific realism. (shrink)

During the past few decades, the global food system has confronted new sustainability challenges related not only to public health and the environment but also to ethical concerns over the treatment of farmed animals. However, the traditional threedimensional framework of sustainable development is ill equipped to take ethical concerns related to non-human animals into account. For instance, the interests of farmed animals are often overridden by objectives associated with social, economic or environmental sustainability, despite their vast numbers and influence on (...) contemporary societies. Moreover, sustainability policies necessarily involve an element of ethical evaluation; yet this element is not explicitly incorporated in prevailing frameworks of sustainable development. Our purpose in this article is to address these shortcomings by developing a Sustainability Matrix that recognizes the need to consider food system sustainability from the perspective of all interest groups affected by the issue under consideration, from a plurality of ethical standpoints. Combing sustainability principles with the basic idea of an ethical evaluation tool, the proposed Sustainability Matrix evaluates the sustainability of food-related systems, decisions and policies from the perspectives of three major strands of ethical theory and from the perspectives of human beings, farmed animals and wildlife. In terms of policy implications, the Sustainability Matrix can be applied in deciding on the specific targets of food system sustainability that can then be utilized as a basis for designing policies and measures towards the achievement of these goals. (shrink)

This article endeavors to propose a legal matrix, the horizontal axis of which represents the structure of legal studies, which consists of doctrinal and non-doctrinal legal knowledge. Non-doctrinal knowledge provides empirical demonstration and legitimacy judgment, but only within the framework set by legal doctrine can it be transformed to legal/illegal judgment. The vertical axis of the matrix measures the extent of application of the legal studies. Differentiated from other disciplines, especially the natural science, the various branches of legal (...) knowledge are either weak or strong practical, other than ‘theoretical’ or ‘applied’. (shrink)

A recent rethinking of the early history of Quantum Mechanics deemed the late 1920s agreement on the equivalence of Matrix Mechanics and Wave Mechanics, prompted by Schrödinger’s 1926 proof, a myth. Schrödinger supposedly failed to achieve the goal of proving isomorphism of the mathematical structures of the two theories, while only later developments in the early 1930s, especially the work of mathematician John von Neumman (1932) provided sound proof of equivalence. The alleged agreement about the Copenhagen Interpretation, predicated to (...) a large extent on this equivalence, was deemed a myth as well. If such analysis is correct, it provides considerable evidence that, in its critical moments, the foundations of scientific practice might not live up to the minimal standards of rigor, as such standards are established in the practice of logic, mathematics, and mathematical physics, thereby prompting one to question the rationality of the practice of physics. In response, I argue that Schrödinger’s proof concerned primarily a domain-specific ontological equivalence, rather than the isomorphism. It stemmed initially from the agreement of the eigenvalues of Wave Mechanics and energy-states of Bohr’s Model that was discovered and published by Schrödinger in his First and Second Communications of 1926. Schrödinger demonstrated in this proof that the laws of motion arrived at by the method of Matrix Mechanics could be derived successfully from eigenfunctions as well (while he only outlined the reversed derivation of eigenfunctions from Matrix Mechanics, which was necessary for the proof of isomorphism of the two theories). This result was intended to demonstrate the domain-specific ontological equivalence of Matrix Mechanics and Wave Mechanics, with respect to the domain of Bohr’s atom. And although the full-fledged mathematico-logical equivalence of the theories did not seem out of the reach of existing theories and methods, Schrödinger never intended to fully explore such a possibility in his proof paper. In a further development of Quantum Mechanics, Bohr’s complementarity and Copenhagen Interpretation captured a more substantial convergence of the subsequently revised (in light of the experimental results) Wave and Matrix Mechanics. I argue that both the equivalence and Copenhagen Interpretation can be deemed myths if one predicates the philosophical and historical analysis on a narrow model of physical theory which disregards its historical context, and focuses exclusively on its formal aspects and the exploration of the logical models supposedly implicit in it. (shrink)

We derive an expression for the relation between two scattering transition amplitudes which, reflect the same dynamics, but which differ in the description of their initial and final state vectors. In one version, the incident and scattered states are elements of a perturbative Fock space, and solve the eigenvalue problem for the “free” pari of the Hamiltonian—the part that remains after the interactions between particle excitations have been “switched off”. Alternatively, the incident and scattered states may be coherent state that (...) are transforms of these Fock states. In earlier work, we reported on the scattering amplitudes for QED, in which a unitary transformation relates perturbative and nonperturbative sets of incident and scattered states. In this work, we generalize this earlier result to the case of transformations that are not necessarily unitary and that may not have unique inverses. We discuss the implication of this relationship for Abelian and non-Abelian gauge theories in which the “transformed”, nonperturbative states implement constraints, such as Gauss’s law. (shrink)

This article aims to draw a portrait of the influence of cybernetics on soft science. To this end, structuralism, post-structuralism and postmodern philosophy will be successively analyzed in a perspective based on importing concepts stemming from the cybernetic paradigm. By focusing more specifically on the American postwar context, we intend to remind the audience that many soft science specialists were involved in the elaboration of this ‘new science’. We will then retrace the influence of the cybernetic paradigm on structuralism. Starting (...) with the historic meeting between Roman Jakobson and Claude Lévi-Strauss, we will illustrate that structural phonology is directly inspired by discoveries stemming from the informational model. In the same perspective, the conceptual borrowings of psychoanalyst Jacques Lacan from cybernetics will be identified and analyzed. Then, we will address the matter of the relationship between postmodern theories and the cybernetic paradigm. The philosophical movement towards deconstruction, as well as Deleuze and Guattari’s philosophy, will be analyzed based on how they relate to this paradigm. We will also insist on the fact that the philosophy of Jean- François Lyotard’s La Condition postmoderne is fully in line with the epistemological revolution launched by cybernetics. (shrink)

In today’s era of rapid development in science and technology, the development of digital technology has increasingly higher requirements for data processing functions. The matrix signal commonly used in engineering applications also puts forward higher requirements for processing speed. The eigenvalues of the matrix represent many characteristics of the matrix. Its mathematical meaning represents the expansion of the inherent vector, and its physical meaning represents the spectrum of vibration. The eigenvalue of a matrix is the focus (...) of matrix theory. The problem of matrix eigenvalues is widely used in many research fields such as physics, chemistry, and biology. A neural network is a neuron model constructed by imitating biological neural networks. Since it was proposed, the application research of its typical models, such as recurrent neural networks and cellular neural networks, has become a new hot spot. With the emergence of deep neural network theory, scholars continue to combine deep neural networks to calculate matrix eigenvalues. This article aims to study the estimation and application of matrix eigenvalues based on deep neural networks. This article introduces the related methods of matrix eigenvalue estimation based on deep neural networks, and also designs experiments to compare the time of matrix eigenvalue estimation methods based on deep neural networks and traditional algorithms. It was found that under the serial algorithm, the algorithm based on the deep neural network reduced the calculation time by about 7% compared with the traditional algorithm, and under the parallel algorithm, the calculation time was reduced by about 17%. Experiments are also designed to calculate matrix eigenvalues with Obj and recurrent neural networks (RNNS) models, which proves that the Oja algorithm is only suitable for calculating the maximum eigenvalues of non-negative matrices, while RNNS is commonly used in general models. (shrink)

Two of the most difficult problems in the foundations of physics are (1) what gives rise to the arrow of time and (2) what the ontology of quantum mechanics is. They are difficult because the fundamental dynamical laws of physics do not privilege an arrow of time, and the quantum-mechanical wave function describes a high-dimensional reality that is radically different from our ordinary experiences. -/- In this paper, I characterize and elaborate on the ``Wentaculus” theory, a new approach to time’s (...) arrow in a quantum universe that offers a unified solution to both problems. Central to the Wentaculus are (i) Density Matrix Realism, the idea that the quantum state of the universe is objective but can be impure, and (ii) the Initial Projection Hypothesis, a new law of nature that selects a unique initial quantum state. On the Wentaculus, the quantum state of the universe is sufficiently simple to be a law, and the arrow of time can be traced back to an exact boundary condition. It removes the intrinsic vagueness of the Past Hypothesis, eliminates the Statistical Postulate, provides a higher degree of theoretical unity, and contains a natural realization of “strong determinism." I end by responding to four recent objections. In a companion paper, I elaborate on Density Matrix Realism. (shrink)

Matrix and line develops a social theory using the work of Jacques Derrida as its philosophical basis. In particular, notions of differance, writing, textuality, margin, dissemination, gramme, and others are integrated into the project of formulating a new language of politics. At the same time, the study is focused on the politics in/of language. ;The question of language is at the heart of this study. In the central chapter I argue that, although Habermas is correct to orient social theory (...) toward language, his approach to language is, in my view, not correct. In formulating the alternative, Heidegger, and from analytic philosophy, Donald Davidson, are important contributors in addition to Derrida. ;There are a number of other important engagements in the dissertation, most prominently with Marxist thinkers. However, the study remains focused on what can be developed within the Derridean framework, instead of developing a general response to Marxism. ;The theory addresses postmodern society and culture and is, in that sense, "postmodern." However, postmodernity is theorized as an historical impasse, in which the link between history and memory is severed. Memory is progressively devalued and destroyed. This devaluation is the result of a secularism at the end of its tether. A writerly, participatory approach to society becomes nearly impossible in this context. The notion of "postsecularism" is counterposed to this postmodern hyper-secularism. The prime concern is to develop a new sense of community, one that textually recovers some of the presecular community values , and some of the key achievements of the Enlightenment, and some of the new possibilities that have been generated by the crisis of modernity. (shrink)

'The Matrix Project' comprises two contexts, namely photographic work of architect Thierry Lagrange and the self-reflective trajectory that he developed during his doctorate at KULeuven. Lagrange's practice-based research developed to an autonomous artistic project, in which he studied the signification and nature of observation within a creative process. Through his multifaceted practice as an architect, photographer and researcher, Lagrange developed 'The Matrix Method'; a method that stimulates creativity within different contexts and can be implemented for example in art (...) education. For the Matrix Project, 25 years' worth of photographs was entered into a context of research and presents a new perspective on different ways of seeing in our present visual culture. (shrink)

Drawing on 75 semi-structured qualitative interviews with Arab, South Asian, and Black Muslim women social justice activists, ages 18–30 years, organizing in the United States and the United Kingdom, I theorize their experiences as the basis of the matrix of gendered Islamophobia. Building upon Jasmine Zine’s concept of gendered Islamophobia, I synthesize this concept with Patricia Hill Collins’s theory of the matrix of domination to give a more in-depth and nuanced structure of how gendered Islamophobia operates and is (...) resisted by Muslim women activists. This article identifies the overlapping configurations of power that affect Muslim women’s lives through structural, disciplinary, hegemonic, and interpersonal domains, countering reductionist accounts of Islamophobia as a universalized, unvariegated social force impacting all Muslims in similar ways. Instead, the matrix of gendered Islamophobia locates Islamophobia within shifting axes of oppression that are simultaneously structured along the lines of gender, race, class, sexuality, and citizenship. The findings of this research reveal a dialectical relationship between Muslim women’s oppression and simultaneous contestation of gendered Islamophobia via their collective remaking of alternative ideas, politics, discourses, and organizing practices. (shrink)

In this pioneering work, the author develops a fundamental formulation of logic in terms of theory of matrices and vector spaces. The discovery of matrix logic represents a landmark in the further formalization of logic. For the first time the power of direct mathematical computation is applied to the whole set of logic operations, allowing the derivation of both the classical and modal logics from the same formal base. The new formalism allows the author to enlarge the alphabet of (...) the truth-values with negative logic antivalues and to link matrix logic descriptions with the Dirac formulation of quantum theory - a result having fundamental implications and repercussions for science as a whole. As a unified language which permits a logical examination of the underlying phenomena of quantum field theory and vice versa, matrix logic opens new avenues for the study of fundamental interactions and gives rise to a revolutionary conclusion that physics as such can be viewed and studied as a logic in the fundamental sense. Finally, modelling itself on exact sciences, matrix logic does not refute the classical logic but instead incorporates it as a special deterministic limit. The book requires multidisciplinary knowledge and will be of interest to physicists, mathematicians, computer scientists and engineers. (shrink)

If the density matrix is treated as an objective description of individual systems, it may become possible to attribute the same objective significance to statistical mechanical properties, such as entropy or temperature, as to properties such as mass or energy. It is shown that the de Broglie--Bohm interpretation of quantum theory can be consistently applied to density matrices as a description of individual systems. The resultant trajectories are examined for the case of the delayed choice interferometer, for which Bell (...) [Int. J. Quantum chem. 155--159, (1980)] appears to suggest that such an interpretation is not possible. Bell’s argument is shown to be based upon a different understanding of the density matrix to that proposed here. (shrink)

In this multidisciplinary book, mathematician Matthew He provides integrative perspectives of algebraic biology, cognitive informatics, and poetic expressions of the human mind. Using classical Pythagorean Theorem and contemporary Category Theory, the proposed matrix models of the human mind connect three domains of the physical space of objective matters, mental space of subjective meanings, and emotional space of bijective modes; draws the connections between neural sparks and idea points, between synapses and idea lines, and between action potentials and frequency curves.

What is the quantum state of the universe? Although there have been several interesting suggestions, the question remains open. In this paper, I consider a natural choice for the universal quantum state arising from the Past Hypothesis, a boundary condition that accounts for the time-asymmetry of the universe. The natural choice is given not by a wave function but by a density matrix. I begin by classifying quantum theories into two types: theories with a fundamental wave function and theories (...) with a fundamental density matrix. The Past Hypothesis is compatible with infinitely many initial wave functions, none of which seems to be particularly natural. However, once we turn to density matrices, the Past Hypothesis provides a natural choice---the normalized projection onto the Past Hypothesis subspace in the Hilbert space. Nevertheless, the two types of theories can be empirically equivalent. To provide a concrete understanding of the empirical equivalence, I provide a novel subsystem analysis in the context of Bohmian theories. Given the empirical equivalence, it seems empirically underdetermined whether the universe is in a pure state or a mixed state. Finally, I discuss some theoretical payoffs of the density-matrix theories and present some open problems for future research. (Bibliographic note: the thesis was submitted for the Master of Science in mathematics at Rutgers University.). (shrink)

Mixture item response theory models include a mixture of latent subpopulations such that there are qualitative differences between subgroups but within each subpopulation the measure model based on a continuous latent variable holds. Under this modeling framework, students can be characterized by both their location on a continuous latent variable and by their latent class membership according to Students’ responses. It is important to identify anchor items for constructing a common scale between latent classes beforehand under the mixture IRT framework. (...) Then, all model parameters across latent classes can be estimated on the common scale. In the study, we proposed Q-matrix anchored mixture Rasch model, including a Q-matrix and the traditional mixture Rasch model. The Q-matrix in QAMRM can use class invariant items to place all model parameter estimates from different latent classes on a common scale regardless of the ability distribution. A simulation study was conducted, and it was found that the estimated parameters of the QAMRM recovered fairly well. A real dataset from the Certificate of Proficiency in English was analyzed with the QAMRM, LCDM. It was found the QAMRM outperformed the LCDM in terms of model fit indices. (shrink)

While Judith Butler’s concept of the heterosexual matrix is dominant in gender and sexuality studies, it is a curiously aspatial and atemporal concept. This paper seeks to re-embed it within space and time by situating its emergence within colonial and imperial histories. Based on this discussion, it ends with three lessons for contemporary work on gender and sexuality and a broader theorization of sex-gender-sexuality regimes beyond the heterosexual matrix.

I first argue that the Matrix trilogy is a Gnostic cyber-epic; I then use this interpretive lens to review the films' treatment of fundamental questions in epistemology, metaphysics, and value theory.

This book studies the foundations of quantum theory through its relationship to classical physics. This idea goes back to the Copenhagen Interpretation (in the original version due to Bohr and Heisenberg), which the author relates to the mathematical formalism of operator algebras originally created by von Neumann. The book therefore includes comprehensive appendices on functional analysis and C*-algebras, as well as a briefer one on logic, category theory, and topos theory. Matters of foundational as well as mathematical interest that are (...) covered in detail include symmetry (and its "spontaneous" breaking), the measurement problem, the Kochen-Specker, Free Will, and Bell Theorems, the Kadison-Singer conjecture, quantization, indistinguishable particles, the quantum theory of large systems, and quantum logic, the latter in connection with the topos approach to quantum theory. This book is Open Access under a CC BY licence. (shrink)

The modern concept of ideology was established by the liberal politician and philosopher Destutt de Tracy, with the objective of creating an all-embracing and general science of ideas, which followed the sensualist and empiricist trend initiated by Locke that culminated in the positivism of Comte. Natural selection and immunity are two key concepts in the history of biology that were strongly based on the Malthusian concept of struggle for existence. This concept wrongly assumed that population grew faster than the means (...) of existence. This “natural” law contained implicitly the idea that the poor and least gifted would not survive. This idea led to the progressive development of the concept of natural selection, whose definitive version was given by Darwin. Mechnikov took the concepts of struggle for existence and natural selection and conceived infectious diseases as a struggle between a host and its invader, the so-called phagocytosis theory. This theory created the necessity to possess mechanisms to discriminate between the own and the foreign, and led to the conception of the immune self. These concepts were not developed from ideas coming from perceptions or sensations, but from ideas coming from their values: individual interest, inevitable inequality, property, utility and profit. Values are ideals that constitute an ideological matrix which exerts a numinous activity and influence the development of our future actions. In consequence, science and its practice cannot avoid and ignore the values that drive them and impulse them towards certain directions. (shrink)

Referential semantics importantly subscribes to the programme of theory of logical calculi. Defined by Wójcicki in [8], it has been subsequently studied in a series of papers of the author, till the full exposition of the framework in [9] and its intuitive characterisation in [10]. The aim of the article is to present several generalizations of referential semantics as compared and related to the matrix semantics for propositional logics. We show, in a uniform way, some own generalizations of referentiality: (...) the first, directed to unrestricted cluster referential semantics, [4], its "discrete" version, a counterpart of algebraic semantics and a many-valued referentiality based on matrices, whose elements are functions from the set of indices to a finite n-element set of values, n≥2, [3]. Next to this we outline pragmatic matrices introduced by Tokarz in [6] as an alternative for cluster referential approach and discuss together all presented versions of referential semantics. (shrink)

A recent rethinking of the early history of Quantum Mechanics deemed the late 1920s agreement on the equivalence of Matrix Mechanics and Wave Mechanics, prompted by Schrödinger's 1926 proof, a myth. Schrödinger supposedly failed to prove isomorphism, or even a weaker equivalence (“Schrödinger-equivalence”) of the mathematical structures of the two theories; developments in the early 1930s, especially the work of mathematician von Neumann provided sound proof of mathematical equivalence. The alleged agreement about the Copenhagen Interpretation, predicated to a large (...) extent on this equivalence, was deemed a myth as well. In response, I argue that Schrödinger's proof concerned primarily a domain-specific ontological equivalence, rather than the isomorphism or a weaker mathematical equivalence. It stemmed initially from the agreement of the eigenvalues of Wave Mechanics and energy-states of Bohr's Model that was discovered and published by Schrödinger in his first and second communications of 1926. Schrödinger demonstrated in this proof that the laws of motion arrived at by the method of Matrix Mechanics are satisfied by assigning the auxiliary role to eigenfunctions in the derivation of matrices (while he only outlined the reversed derivation of eigenfunctions from Matrix Mechanics, which was necessary for the proof of both isomorphism and Schrödinger-equivalence of the two theories). This result was intended to demonstrate the domain-specific ontological equivalence of Matrix Mechanics and Wave Mechanics, with respect to the domain of Bohr's atom. And although the mathematical equivalence of the theories did not seem out of the reach of existing theories and methods, Schrödinger never intended to fully explore such a possibility in his proof paper. In a further development of Quantum Mechanics, Bohr's complementarity and Copenhagen Interpretation captured a more substantial convergence of the subsequently revised (in light of the experimental results) Wave and Matrix Mechanics. -/- I argue that both the equivalence and Copenhagen Interpretation can be deemed myths if one predicates the philosophical and historical analysis on a narrow model of physical theory which disregards its historical context, and focuses exclusively on its formal aspects and the exploration of the logical models supposedly implicit in it. -/- . (shrink)

Character structures underlie everyone’s personality. When rigidly defended, they limit us; yet as they become more flexible, they can reveal sources of animation, renewal and authenticity. _The Matrix and Meaning of Character_ guides the reader into an awareness of the archetypal depths that underlie character structures, presenting an original developmental model in which current analytic theories are synthesised. The authors examine nine character structures, animating them with fairy tales, mythic images and case material, creating a bridge between the traditional (...) language of psychopathology and the universal realm of image and symbol. This book will appeal to all analytical psychologists, psychoanalysts and psychotherapists who want to strengthen their clinical expertise. It will help clinicians to extend their clinical insights beyond a strictly behavioural, medical or cognitive approach, revealing the potential of the human spirit. (shrink)

The adaptation of the Kramers-Kronig dispersion relations to the causal localization structure of QFT led to an important project in particle physics, the only one with a successful closure. The same cannot be said about the subsequent attempts to formulate particle physics as a pure S-matrix project.The feasibility of a pure S-matrix approach are critically analyzed and their serious shortcomings are highlighted. Whereas the conceptual/mathematical demands of renormalized perturbation theory are modest and misunderstandings could easily be corrected, the (...) correct understanding about the origin of the crossing property requires the use of the mathematical theory of modular localization and its relation to the thermal KMS condition. These new concepts, which combine localization, vacuum polarization and thermal properties under the roof of modular theory, will be explained and their potential use in a new constructive (nonperturbative) approach to QFT will be indicated. The S-matrix still plays a predominant role but, different from Heisenberg’s and Mandelstam’s proposals, the new project is not a pure S-matrix approach. The S-matrix plays a new role as a “relative modular invariant”. (shrink)