We use the notion of polar duality from convex geometry and the theory of Lagrangian planes from symplectic geometry to construct a fiber bundle over ellipsoids that can be viewed as a quantum-mechanical substitute for the classical symplectic phase space. The total space of this fiber bundle consists of geometric quantum states, products of convex bodies carried by Lagrangian planes by their polar duals with respect to a second transversal Lagrangian plane. Using the theory of the John ellipsoid we relate (...) these geometric quantum states to the notion of “quantum blobs” introduced in previous work; quantum blobs are the smallest symplectic invariant regions of the phase space compatible with the uncertainty principle. We show that the set of equivalence classes of unitarily related geometric quantum states is in a one-to-one correspondence with the set of all Gaussian wavepackets. We emphasize that the uncertainty principle appears in this paper as geometric property of the states we define, and is not expressed in terms of variances and covariances, the use of which was criticized by Hilgevoord and Uffink. (shrink)

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 aim of the famous Born and Jordan 1925 paper was to put Heisenberg’s matrix mechanics on a firm mathematical basis. Born and Jordan showed that if one wants to ensure energy conservation in Heisenberg’s theory it is necessary and sufficient to quantize observables following a certain ordering rule. One apparently unnoticed consequence of this fact is that Schrödinger’s wave mechanics cannot be equivalent to Heisenberg’s more physically motivated matrix mechanics unless its observables are quantized using this rule, and not (...) the more symmetric prescription proposed by Weyl in 1926, which has become the standard procedure in quantum mechanics. This observation confirms the superiority of Born–Jordan quantization, as already suggested by Kauffmann. We also show how to explicitly determine the Born–Jordan quantization of arbitrary classical variables, and discuss the conceptual advantages in using this quantization scheme. We finally suggest that it might be possible to determine the correct quantization scheme by using the results of weak measurement experiments. (shrink)

The imprints left by quantum mechanics in classical (Hamiltonian) mechanics are much more numerous than is usually believed. We show that the Schrödinger equation for a nonrelativistic spinless particle is a classical equation which is equivalent to Hamilton’s equations. Our discussion is quite general, and incorporates time-dependent systems. This gives us the opportunity of discussing the group of Hamiltonian canonical transformations which is a non-linear variant of the usual symplectic group.

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)

Quantum blobs are the smallest phase space units of phase space compatible with the uncertainty principle of quantum mechanics and having the symplectic group as group of symmetries. Quantum blobs are in a bijective correspondence with the squeezed coherent states from standard quantum mechanics, of which they are a phase space picture. This allows us to propose a substitute for phase space in quantum mechanics. We study the relationship between quantum blobs with a certain class of level sets defined by (...) Fermi for the purpose of representing geometrically quantum states. (shrink)

We show that the strong form of Heisenberg’s inequalities due to Robertson and Schrödinger can be formally derived using only classical considerations. This is achieved using a statistical tool known as the “minimum volume ellipsoid” together with the notion of symplectic capacity, which we view as a topological measure of uncertainty invariant under Hamiltonian dynamics. This invariant provides a right measurement tool to define what “quantum scale” is. We take the opportunity to discuss the principle of the symplectic camel, which (...) is at the origin of the definition of symplectic capacities, and which provides an interesting link between classical and quantum physics. (shrink)

The necessary appearance of Clifford algebras in the quantum description of fermions has prompted us to re-examine the fundamental role played by the quaternion Clifford algebra, C 0,2 . This algebra is essentially the geometric algebra describing the rotational properties of space. Hidden within this algebra are symplectic structures with Heisenberg algebras at their core. This algebra also enables us to define a Poisson algebra of all homogeneous quadratic polynomials on a two-dimensional sub-space, $\mathbb{F}^{a}$ of the Euclidean three-space. This enables (...) us to construct a Poisson Clifford algebra, ℍ F , of a finite dimensional phase space which will carry the dynamics. The quantum dynamics appears as a realisation of ℍ F in terms of a Clifford algebra consisting of Hermitian operators. (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 hard problem of consciousness is the problem of explaining how and why physical processes give rise to consciousness (Chalmers 1995). Regardless of many attempts to solve the problem, there is still no commonly agreed solution. It is thus very likely that some radically new ideas are required if we are to make any progress. In this paper we turn to quantum theory to find out whether it has anything to offer in our attempts to understand the place of mind (...) and conscious experience in nature. In particular we will be focusing on the ontological interpretation of quantum theory proposed by Bohm and Hiley (1987, 1993), its further development by Hiley (Hiley and Callaghan 2012; Hiley, Dennis and de Gosson 2021), and its philosophical interpretation by Pylkkänen (2007, 2020). The ontological interpretation makes the radical proposal that quantum reality includes a new type of potential energy which contains active information. This proposal, if correct, constitutes a major change in our notion of matter. We are used to having in physics only mechanical concepts, such as position, momentum and force. Our intuition that it is not possible to understand how and why physical processes can give rise to consciousness is partly the result of our assuming that physical processes (including neurophysiological processes) are always mechanical. If, however, we are willing to change our view of physical reality by allowing non-mechanical, organic and holistic concepts such as active information to play a fundamental role, this, we argue, makes it possible to understand the relationship between physical and mental processes in a new way. It might even be a step toward solving the hard problem. (shrink)

La niñez es un tema que aparece a menudo en la filosofía de Vico. En la Sn44 la niñez del mundo es la barbarie de los sentidos y la niñez de la persona singular del siglo de las luces es la barbarie del intelecto. Deseamos en este artículo investigar en los primeros escritos de Vico las consideraciones que fueron hechas respecto a la niñez singular y que, más adelante, sirvieron de soporte para las investigaciones de los tiempos oscuros. La Vita (...) es el texto privilegiado para la realización de nuestro propósito; aunque son pocos los renglones referidos a los primeros años de la vida del filósofo, la narración del accidente –la caída por una escalera– sufrido cuando Vico tenía siete años nos ofrece la comprensión de la concepción filosófica de la niñez de Vico.Childhood is indeed a subject which appears quite often in Vico’s philosophy. In Sn 44 the world’s childhood is sense perception’s barbarism as that of the eighteen century enlightened singular person’s is intellect’s barbarism. We purport to study in this paper the account Vico gives about singular person’s childhood in his early writings which later is used to shed some light on the enquiries about the dark ages. The Vita is –or thus is claimed– a privileged text to achieve our aims and, though few are lines devoted to his own infancy, it gives the key to understanding Vico’s philosophical view of childhood. (shrink)

Ondanks al zijn twijfels staat Nietzsche vrij positief tegenover de waarde van kennis en waarheid. Maar hij is tevens sceptisch over de mogelijkheid om de waarheid te kennen. Daarom ruimte hij een grote plaats voor de kunst in, die zijn creativiteit put uit het ongeloof in de waarheid.

This study addresses philosophy’s traditional rejection of the sense of smell, taking into account the most unpleasant but also the most beneficial aspects of odour, from antiquity to the present day. The topic is investigated following a thread through several texts by Derrida, Nietzsche, and Freud. It explains the reasons– sometimes apparent, sometimes hidden– behind the most palpable effects of the philosophical marginalization of the sense of smell.

Tal y como reza la cita de la Apología platónica que encabeza estas líneas, ya el propio Sócrates habría advertido a sus jueces de que no le iban a tomar en serio al pensar que hablaba de forma irónica, en broma. Pues bien, en este trabajo de tesis doctoral, defenderemos que, hoy por hoy, esa advertencia sigue vigente; que sigue siendo necesario tomar a Sócrates en serio, pensar que no habló irónicamente y todo ello asumiendo expresamente los postulados de la (...) hermenéutica gadameriana. Unos postulados a los que nos referimos en el apartado inicial: “Introducción y cuestiones metodológicas” y según los cuales cabe conocer, desde nuestro propio horizonte de praxis, lo que la tradición nos dice en su sentido propio y diferente, si bien hemos de ser conscientes de la posible aplicación de prejuicios no percibidos, ocultos, que pueden volvernos sordos a eso que la tradición nos dice. (shrink)

Selon Derrida, 'Tout vrai questionnement est sommé par un certain type d'eschatologie' : la philosophie serait tirée de son sommeil dogmatique par une injonction eschatologique. Ainsi, alors que la notion d'eschatologie est originellement théologique et désigne la doctrine de la fin des temps, elle est convoquée par la philosophie contemporaine chez Heidegger, puis chez Levinas et Derrida, justement pour réfuter toute philosophie téléologique de l'histoire. Comment comprendre ce réinvestissement inattendu? Cet ouvrage propose d'abord une généalogie du motif eschatologique dans la (...) philosophie contemporaine, afin de comprendre comment l'eschatologie peut s'émanciper du domaine théologique et acquérir une spécificité philosophique. Ensuite, il s'attelle à décrire le motif eschatologique propre à Levinas et Derrida, qu'ils pensent en dialogue avec le messianisme et la phénoménologie. Enfin, il s'intéresse à l'opérativité spéculative de ce motif, qu'on appelle 'eschatologie du présent' : un schème qui exige d'interrompre le présent et la présence, pour construire une éthique de l'événement. L'eschatologie doit ainsi être pensée comme l'insomnie de la philosophie, 'l'appel à une infinie responsabilité de l'homme, à un inlassable éveil', selon la formule de Levinas. Elle devient l'injonction éthique, le nom de l'énigme de l'altérité - une altérité que je ne pourrai jamais comprendre au sein d'un horizon, mais qui m'assigne comme responsabilité et vigilance."--Page 4 of cover. (shrink)

One of the major contributions of Henri de Lubac to contemporary theology is his reflection on the senses of Holy Scripture. Started in Catholicisme , his reflection on this subject reached its full development in Exégèse médiévale. Les quatre sens de l'Ecriture , and was taken up again by way of a synthesis in L'Ecriture dans la Tradition . His book on Origen expresses well the central idea: God reveals himself in history, but these historical events are to be understood (...) spiritually. (shrink)

Neokantismes: la marque du pluriel indique d'emblee que le mot d'ordre des annees 1865 Retour a Kant! regroupe differentes ecoles. Les auteurs dont les textes inedits en francais se trouvent ici rassembles et traduits, appartiennent en effet soit a l'ecole de Marbourg (H. Cohen, P. Natorp et E. Cassirer) soit a l'ecole de Bade (W. Windelband, H. Rickert, E. Lask, J. Cohn). Le point commun du neokantisme logique des premiers et du neokantisme axiologique des seconds consiste en une volonte d'asseoir (...) le positivisme sur un fondement philosophique respectueux de l'attitude phenomenologique. Kant apparut donc a tous ces penseurs comme science rigoureuse a condition toutefois de ne considerer le sujet transcendantal que comme une fonction logique de la pensee qui elabore la forme et le contenu de la connaissance. A condition egalement de ne pas limiter l'entreprise critique a la physique, mais de l'etendre au fondement de toutes les sciences de l'esprit. (shrink)