In this paper, we introduce the concept of N-dimensional generalized Minkowski space, i.e., a space endowed with a metric tensor, whose coefficients do depend on a set of non-metrical coordinates. This is the first of a series of papers devoted to the investigation of the Killing symmetries of generalized Minkowski spaces. In particular, we discuss here the infinitesimal-algebraic structure of the space-time rotations in such spaces. It is shown that the maximal Killing group of (...) these spaces is the direct product of a generalized Lorentz group and a generalized translation group. We derive the explicit form of the generators of the generalized Lorentz group in the self-representation and their related, generalized Lorentz algebra. The results obtained are specialized to the case of a 4-dimensional, “deformed” Minkowski space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${\widetilde M}$$ \end{document}4, i.e., a pseudoeuclidean space with metric coefficients depending on energy. (shrink)

In this paper, we continue the study of the Killing symmetries of an N-dimensional generalized Minkowski space, i.e., a space endowed with a metric tensor, whose coefficients do depend on a set of non-metrical coordinates. We discuss here the finite structure of the space–time rotations in such spaces, by confining ourselves to the four-dimensional case. In particular, the results obtained are specialized to the case of a “deformed” Minkowski space M_4, for which we derive the explicit (...) general form of the finite rotations and boosts in different parametric bases. (shrink)

In this paper, we continue the study of the Killing symmetries of a N-dimensional generalized Minkowski space, i.e., a space endowed with a metric tensor, whose coefficients do depend on a set of non-metrical coordinates. We discuss here the translations in such spaces, by confining ourselves to the four-dimensional case. In particular, the results obtained are specialized to the case of a “deformed” Minkowski space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\widetilde M_4 (...) $$ \end{document}. (shrink)

A homeomorphism is built between the separable complex Hilbert space (quantum mechanics) and Minkowski space (special relativity) by meditation of quantum information (i.e. qubit by qubit). That homeomorphism can be interpreted physically as the invariance to a reference frame within a system and its unambiguous counterpart out of the system. The same idea can be applied to Poincaré’s conjecture (proved by G. Perelman) hinting at another way for proving it, more concise and meaningful physically. Furthermore, the conjecture can be (...) generalized and interpreted in relation to the pseudo-Riemannian space of general relativity therefore allowing for both mathematical and philosophical interpretations of the force of gravitation due to the mismatch of choice and ordering and resulting into the “curving of information” (e.g. entanglement). Mathematically, that homeomorphism means the invariance to choice, the axiom of choice, well-ordering, and well-ordering “theorem” (or “principle”) and can be defined generally as “information invariance”. Philosophically, the same homeomorphism implies transcendentalism once the philosophical category of the totality is defined formally. The fundamental concepts of “choice”, “ordering” and “information” unify physics, mathematics, and philosophy and should be related to their shared foundations. (shrink)

An isomorphism is built between the separable complex Hilbert space (quantum mechanics) and Minkowski space (special relativity) by meditation of quantum information (i.e. qubit by qubit). That isomorphism can be interpreted physically as the invariance between a reference frame within a system and its unambiguous counterpart out of the system. The same idea can be applied to Poincaré’s conjecture (proved by G. Perelman) hinting another way for proving it, more concise and meaningful physically. Mathematically, the isomorphism means the invariance (...) to choice, the axiom of choice, well-ordering, and well-ordering “theorem” (or “principle”) and can be defined generally as “information invariance”. (shrink)

The semigroup of trajectories in Minkowski space-time and its induced representations are constructed as a generalization of the Galilei case. They describe relativistic pointlike particles and yield the free propagator as a path integral in the space of trajectories parametrized by a fifth parameter. This non physical propagator in a five-dimensional space is integrated over the fifth parameter to yield the physical propagator in Minkowski space. Thereafter, this notion is applied to a model of extended particles with internal (...) Poincaré symmetry and moving in an external Minkowski space. The geometrical structure is of Hilbert bundles and the interaction is introduced as a connection. The propagator is a path integral with respect to either the internal and external trajectories and reduces to a product of an internal and an external propagator when the interaction is ignored, just as has been found in a previous work with representations of the group rather than those of the semigroup. (shrink)

We discuss the possible breakdown of Lorentz invariance—at distances greater than the Planck length—from both the theoretical and the phenomenological point of view. The theoretical tool to deal with such a problem is provided by a “deformation” of the Minkowski metric, with parameters dependent on the energy of the physical system considered. Such a deformed metric realizes, for any interaction, the “solidarity principle” between interactions and spacetime geometry (usually assumed for gravitation), according to which the peculiar features of every (...) interaction determine—locally—its own spacetinie structure. The generalized theory of relativity, based on the locally deformed Minkowski spacetime, is called “deformed special relativity” (DSR). In the first part of the paper, we give the foundations and the basic laws of DSR. In the second part, we analyze some experimental data, which admit an interpretation in terms of the DSR formalism and are, therefore, candidates for displaying a breakdown of the Lorentz symmetry. They are (i) the superluminal propagation of evanescent electromagnetic waves in waveguides, (ii) the meanlife of the KS 0, (iii) the Bose-Einstein correlation in pion production and (iv) the comparison of clock rates in the gravitational field of Earth. Such analysis provides us with the explicit forms of the related deformed metrics as functions of the energy, thus putting in evidence, in all four cases (and therefore for all four fundamental interactions), departures from the usual Minkowski metric. This preliminary evidence for a broken Lorentz invariance may be regarded as the signature of possible nonlocal effects involved in the processes examined. Moreover, the corresponding deformed metrics obtained by our analysis provide an effective dynamical description of the interactions (at least in the energy range considered). (shrink)

Relativistic quantum theories are equipped with a background Minkowski spacetime and non-relativistic quantum theories with a Galilean space-time. Traditional investigations have distinguished their distinct space-time structures and have examined ways in which relativistic theories become sufficiently like Galilean theories in a low velocity approximation or limit. A different way to look at their relationship is to see that both kinds of theories are special cases of a certain five-dimensional generalization involving no limiting procedures or approximations. When one compares them, (...) striking features emerge that bear on philosophical questions, including the ontological status of the wave function and time reversal invariance. (shrink)

We here present explicit relational theories of a class of geometrical systems (namely, inner product spaces) which includes Euclidean space and Minkowski spacetime. Using an embedding approach suggested by the theory of measurement, we prove formally that our theories express the entire empirical content of the corresponding geometric theory in terms of empirical relations among a finite set of elements (idealized point-particles or events) thought of as embedded in the space. This result is of interest within the general (...) phenomenalist tradition as well as the theory of space and time, since it seems to be the first example of an explicit phenomenalist reconstruction of a realist theory which is provably equivalent to it in observational consequences. The interesting paper "On the Space-Time Ontology of Physical Theories" by Ken Manders, Philosophy of Science, vol. 49, number 4, December 1982, p. 575-590, has significant affinities to this one. We both, in a sense, try to formally vindicate Leibniz's notion of a relational theory of space, by constructing theories of spatial relations among physical objects which are provably equivalent to the standard absolutist theories. The essential difference between our approaches is that Manders retains Leibniz's explicitly modal framework, whereas I do not. Manders constructs a spacetime theory which explicitly characterizes the totality of possible configurations of physical objects, using a modal language in which the notion of a possible configuration occurs as a primitive. There is no doubt that this is a more accurate realization of Leibniz's own conception of space than the embedding-based approach developed here. However, it also remains open to objections (such as those cited here from Sklar) on account of the special appeal to modal notions. Our approach here, by contrast, aims to avoid the special appeal to modal notions by giving directly a set of laws which are satisfied by a configuration individually, if and only if it is one of the allowable ones. One thus avoids the need for reference to possible but not actual configurations or objects, in the statement of the spacetime laws. We may then take this alternative set of laws as the actual geometric theory, and do away with the hypothetical entity called 'space'. Yet at the same time there is no invocation of modality, except in the ordinary sense in which every physical theory constrains what is possible. So that a relationalist is not forced to utilize a modal language (though Leibniz certainly does.). (shrink)

An often repeated account of the genesis of special relativity tells us that relativity theory was to a considerable extent the fruit of an operationalist philosophy of science. Indeed, Einstein’s 1905 paper stresses the importance of rods and clocks for giving concrete physical content to spatial and temporal notions. I argue, however, that it would be a mistake to read too much into this. Einstein’s operationalist remarks should be seen as serving rhetoric purposes rather than as attempts to promulgate a (...) particular philosophical position --- in fact, Einstein never came close to operationalism in any of his philosophical writings. By focussing on what could actually be measured with rods and clocks Einstein shed doubt on the empirical status of a number of pre-relativistic concepts, with the intention to persuade his readers that the applicability of these concepts was not obvious. This rhetoric manoeuvre has not always been rightly appreciated in the philosophy of physics. Thus, the influence of operationalist misinterpretations, according to which associated operations strictly define what a concept means, can still be felt in present-day discussions about the conventionality of simultaneity. The standard story continues by pointing out that Minkowski in 1908 supplanted Einstein’s approach with a realist spacetime account that has no room for a foundational role of rods and clocks: relativity theory became a description of a four-dimensional ‘absolute world’. As it turns out, however, it is not at all clear that Minkowski was proposing a substantivalist position with respect to spacetime. On the contrary, it seems that from a philosophical point of view Minkowski’s general position was not very unlike the one in the back of Einstein’s mind. However, in Minkowski’s formulation of special relativity it becomes more explicit that the content of spatiotemporal concepts relates to considerations about the form of physical laws. If accepted, this position has important consequences for the discussion about the conventionality of simultaneity. (shrink)

This chapter contains sections titled: Non‐Euclidean Geometry The General Theory Superluminal Constraints and the GTR Lorentz Invariance and the GTR Quantum Theories in Non‐Minkowski Space‐times The GTR to the Rescue?

What if gravity satisfied the Klein-Gordon equation? Both particle physics from the 1920s-30s and the 1890s Neumann-Seeliger modification of Newtonian gravity with exponential decay suggest considering a "graviton mass term" for gravity, which is _algebraic_ in the potential. Unlike Nordström's "massless" theory, massive scalar gravity is strictly special relativistic in the sense of being invariant under the Poincaré group but not the 15-parameter Bateman-Cunningham conformal group. It therefore exhibits the whole of Minkowski space-time structure, albeit only indirectly concerning volumes. (...) Massive scalar gravity is plausible in terms of relativistic field theory, while violating most interesting versions of Einstein's principles of general covariance, general relativity, equivalence, and Mach. Geometry is a poor guide to understanding massive scalar gravity: matter sees a conformally flat metric due to universal coupling, but gravity also sees the rest of the flat metric in the mass term. What is the 'true' geometry, one might wonder, in line with Poincaré's modal conventionality argument? Infinitely many theories exhibit this bimetric 'geometry,' all with the total stress-energy's trace as source; thus geometry does not explain the field equations. The irrelevance of the Ehlers-Pirani-Schild construction to a critique of conventionalism becomes evident when multi-geometry theories are contemplated. Much as Seeliger envisaged, the smooth massless limit indicates underdetermination of theories by data between massless and massive scalar gravities---indeed an unconceived alternative. At least one version easily could have been developed before General Relativity; it then would have motivated thinking of Einstein's equations along the lines of Einstein's newly re-appreciated "physical strategy" and particle physics and would have suggested a rivalry from massive spin 2 variants of General Relativity. The Putnam-Grünbaum debate on conventionality is revisited with an emphasis on the broad modal scope of conventionalist views. Massive scalar gravity thus contributes to a historically plausible rational reconstruction of much of 20th-21st century space-time philosophy in the light of particle physics. An appendix reconsiders the Malament-Weatherall-Manchak conformal restriction of conventionality and constructs the 'universal force' influencing the causal structure. Subsequent works will discuss how massive gravity could have provided a template for a more Kant-friendly space-time theory that would have blocked Moritz Schlick's supposed refutation of synthetic _a priori_ knowledge, and how Einstein's false analogy between the Neumann-Seeliger-Einstein modification of Newtonian gravity and the cosmological constant \Lambda generated lasting confusion that obscured massive gravity as a conceptual possibility. (shrink)

A new theory is considered according to which extended objects in n-dimensional space are described in terms of multivector coordinates which are interpreted as generalizing the concept of center of mass coordinates. While the usual center of mass is a point, by generalizing the latter concept, we associate with every extended object a set of r-loops, r=0,1,...,n−1, enclosing oriented (r+1)-dimensional surfaces represented by Clifford numbers called (r+1)-vectors or multivectors. Superpositions of multivectors are called polyvectors or Clifford aggregates and they are (...) elements of Clifford algebra. The set of all possible polyvectors forms a manifold, called C-space. We assume that the arena in which physics takes place is in fact not Minkowski space, but C-space. This has many far reaching physical implications, some of which are discussed in this paper. The most notable is the finding that although we start from the constrained relativity in C-space we arrive at the unconstrained Stueckelberg relativistic dynamics in Minkowski space which is a subspace of C-space. (shrink)

Physical Relativity explores the nature of the distinction at the heart of Einstein's 1905 formulation of his special theory of relativity: that between kinematics and dynamics. Einstein himself became increasingly uncomfortable with this distinction, and with the limitations of what he called the 'principle theory' approach inspired by the logic of thermodynamics. A handful of physicists and philosophers have over the last century likewise expressed doubts about Einstein's treatment of the relativistic behaviour of rigid bodies and clocks in motion in (...) the kinematical part of his great paper, and suggested that the dynamical understanding of length contraction and time dilation intimated by the immediate precursors of Einstein is more fundamental. Harvey Brown both examines and extends these arguments, after giving a careful analysis of key features of the pre-history of relativity theory. He argues furthermore that the geometrization of the theory by Minkowski in 1908 brought illumination, but not a causal explanation of relativistic effects. Finally, Brown tries to show that the dynamical interpretation of special relativity defended in the book is consistent with the role this theory must play as a limiting case of Einstein's 1915 theory of gravity: the general theory of relativity.Appearing in the centennial year of Einstein's celebrated paper on special relativity, Physical Relativity is an unusual, critical examination of the way Einstein formulated his theory. It also examines in detail certain specific historical and conceptual issues that have long given rise to debate in both special and general relativity theory, such as the conventionality of simultaneity, the principle of general covariance, and the consistency or otherwise of the special theory with quantum mechanics. Harvey Brown' s new interpretation of relativity theory will interest anyone working on these central topics in modern physics. (shrink)

_ Source: _Volume 5, Issue 2, pp 290 - 322 Einstein's special theory, as interpreted by Herman Minkowski, suggests that an understanding of space and time requires the replacement of three-dimensional space and one dimensional time with a four-dimensional spacetime continuum, as a natural kind of thing with a characteristic, geometrical, structure. Issues of space and time in general, and of special relativity in particular, are not addressed in Bhaskar's _A Realist Theory of Science_, and their treatment in subsequent (...) realist writings has been patchy and indecisive. Some of Bhaskar's observations in later writings, including his defence of fundamental ontological asymmetries between space and time, and between present, past and future, appear incompatible with a relativistic perspective. Equally problematic, from a critical realist perspective, is the apparent ontological prioritisation of events, and causation as action-by-contact, within the Minkowski interpretation of special relativity. This paper argues that the four-dimensional spacetime continuum of special relativity occupies its own level at the base of the ontological hierarchy of natural kinds, providing lower-order components and laws for higher-order physical, chemical, biological and social structures. While providing no definitive reconciliation of relativistic and realist perspectives, this paper does suggest that A. N. Whitehead's idea of atomic events, or ‘actual occasions’, could pave the way for some possible future reconciliation. (shrink)

It is argued that Minkowski space-time cannot serve as the deep structure within a ``constructive'' version of the special theory of relativity, contrary to widespread opinion in the philosophical community.

If Einstein's equations are to describe a field theory of gravity in Minkowski spacetime, then causality requires that the effective curved metric must respect the flat background metric's null cone. The kinematical problem is solved using a generalized eigenvector formalism based on the Segré classification of symmetric rank 2 tensors with respect to a Lorentzian metric. Securing the correct relationship between the two null cones dynamically plausibly is achieved using the naive gauge freedom. New variables tied to the (...) generalized eigenvector formalism reduce the configuration space to the causality-respecting part. In this smaller space, gauge transformations do not form a group, but only a groupoid. The flat metric removes the difficulty of defining equal-time commutation relations in quantum gravity and guarantees global hyperbolicity. (shrink)

An axiomatic framework for describing general space-time models is presented. Space-time models to which irreducible propositional systems belong as causal logics are quantum (q) theoretically interpretable and their event spaces are Hilbert spaces. Such aq space-time is proposed via a “canonical” quantization. As a basic assumption, the time t and the radial coordinate r of aq particle satisfy the canonical commutation relation [t,r]=±i $h =$ . The two cases will be considered simultaneously. In that case the event space (...) is the Hilbert space L2(ℝ3). Unitary symmetries consist of Poincaré-like symmetries (translations, rotations, and inversion) and of gauge-like symmetries. Space inversion implies time inversion. Thisq space-time reveals a confinement phenomenon: Theq particle is “confined” in an $h =$ size region of Minkowski space $\mathbb{M}^4$ at any time. One particle mechanics overq space-time provides mass eigenvalue equations for elementary particles. Prugovečki's stochasticq mechanics andq space-time offer a natural way for introducing and interpreting consistently such aq space-time andq particles existing in it. The mass eigenstates ofq particles generate Prugovečki's extended elementary particles. When $h =$ →0, these particles shrink to point particles and $\mathbb{M}^4$ is recovered as the classical (c) limit ofq space-time. Conceptual considerations favor the case [t,r]=+i $h =$ , and applications in hadron physics give the fit $h =$ ⋍2/5 fermi/GeV. (shrink)

This essay explores Kaila's interpretation of the special theory of relativity. Although the relevance of his work to logical empiricism is well-known, not much has been written on what Kaila calls the ‘Einstein-Minkowski invariance theory’. Kaila's interpretation focuses on two salient features. First, he emphasizes the importance of the invariance of the spacetime interval. The general point about spacetime invariance has been known at least since Minkowski, yet Kaila applies his overall tripartite theory of invariances to space, time (...) and spacetime in an original way. Second, Kaila provides a non-conventionalist argument for the isotropic speed of electromagnetic signals. The standard Einstein synchrony is not a mere convention but a part of a larger empirical theory. According to Kaila's holistic principle of testability, which stands in contrast to the theses of translatability and verification, different items in the theory cannot be sharply divided into conventional and empirical. Kaila's invariantism/non-conventionalism about relativity reflects an interesting case in the gradual transition from positivism to realism within the philosophy of science. (shrink)

Using a framework of Dirac algebra, the Clifford algebra appropriate for Minkowski space-time, the formulation of classical electromagnetism including both electric and magnetic charge is explored. Employing the two-potential approach of Cabibbo and Ferrari, a Lagrangian is obtained that is dyality invariant and from which it is possible to derive by Hamilton's principle both the symmetrized Maxwell's equations and the equations of motion for both electrically and magnetically charged particles. This latter result is achieved by defining the variation of (...) the action associated with the cross terms of the interaction Lagrangian in terms of a surface integral. The surface integral has an equivalent path-integral form, showing that the contribution of the cross terms is local in nature. The form of these cross terms derives in a natural way from a Dirac algebraic formulation, and, in fact, the use of the geometric product of Dirac algebra is an essential aspect of this derivation. No kinematic restrictions are associated with the derivation, and no relationship between magnetic and electric charge evolves from the (classical) formulation. However, it is indicated that in bound states quantum mechanical considerations will lead to a version of Dirac's quantization condition. A discussion of parity violation of the generalized electromagnetic theory is given, and a new approach to the incorporation of this violation into the formalism is suggested. Possibilities for extensions are mentioned. (shrink)

The massless Thirring model of a self-interacting ferinion field in a curved two-dimensional background spacetime is considered. The exact operator solution for the fields and the equation for the two-point function are given and used to examine the radiation emitted by a two-dimensional black hole. The radiation is found to be thermal in nature, confirming general predictions to this effect. We compute the particle spectrum of the Thirring fermions at finite temperature in Minkowski space and point out errors in (...) a previous attempt at this calculation. Finally we calculate the vacuum expectation value of the stress tensor in an arbitrary two-dimensional spacetime by exploiting the connection between the thermal Hawking radiation from a black hole and the value of the so-called conformal trace anomaly. The latter is also computed quite independently, using dimensional regularization, from general considerations of the renormalization group. (shrink)

We consider the relativistic statistical mechanics of an ensemble of N events with motion in space-time parametrized by an invariant “historical time” τ. We generalize the approach of Yang and Yao, based on the Wigner distribution functions and the Bogoliubov hypotheses to find approximate dynamical equations for the kinetic state of any nonequilibrium system, to the relativistic case, and obtain a manifestly covariant Boltzmann- type equation which is a relativistic generalization of the Boltzmann-Uehling-Uhlenbeck (BUU) equation for indistinguishable particles. This equation (...) is then used to prove the H-theorem for evolution in τ. In the equilibrium limit, the covariant forms of the standard statistical mechanical distributions are obtained. We introduce two-body interactions by means of the direct action potential V(q), where q is an invariant distance in the Minkowski space-time. The two- body correlations are taken to have the support in a relative O(2, 1)-invariant subregion of the full spacelike region. The expressions for the energy density and pressure are obtained and shown to have the same forms (in terms of an invariant distance parameter) as those of the nonrelativistic theory and to provide the correct nonrelativistic limit. (shrink)

Hardly any other problem has been discussed more than that of the status of time in modem physics. This is only natural since there are not many other more important problems in philosophy of science and in philosophy in general. There are also few other areas where controversies as well as confusion were more frequent. This is true not only of popular and semi-popular expositions of the Minkowski concept of space-time but also of a number of its philosophical interpretations. (...) Generally we do not find anything of this kind in the writings of physicists, at least as long as they confine themselves to strictly mathematical and physical expositions; but when they sometimes venture beyond a strictly mathematical approach, they often do not escape certain unconscious or semi-conscious prejudices which are contrary not only to the spirit but sometimes even to the letter of relativity. The true significance of the relativistic fusion of space and time can be understood only when we contrast it with its classical counterpart, i.e., with what may be called the Newtonian space-time. Only on such a contrasting background will the revolutionary meaning of the new concept clearly stand out. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Affective Spaces: Architecture and the Living Body by Federico de MatteisJasna SersicAffective Spaces: Architecture and the Living Body BY FEDERICO DE MATTEIS New York, NY: Routledge, 2021What is architectural space? For architects, urban planners, and all involved in the design and transformation of the environment, space is a central subject. However, despite this fact, nobody accurately states what space is all about. As a result, the (...) concept of space remains ambiguous, challenging to transmit, and a point of fundamental misunderstandings, with consequences on how we move, inhabit, and transform the world around us (1).In the attempt to provide order in today’s theory of space in architecture, De Matteis’ Affective Spaces invites the reader to explore the question of space from the subject’s point of view by asking what the encounter with architecture is like, and how do I feel, here, now?, borrowed from Eugene Minkowski (7). De Matteis offers an interconnected yet clear and instructive way to explore further what first-hand, direct experience with architecture can tell us about the effects it produces on us. Through this exciting intellectual journey, De Matteis thus shows how architecture, generally understood as an art of building space, does not regard the built environment alone. As people are also emotionally affected by the (built) world they live in, this book can ultimately be read as a quest to understand the relation between the feelings and moods arising from the bodily, corporeal experience of the built environment and the way they invite us to respond and act within environments we inhabit and modify (130).To accept De Matteis’ invitation to join him on his quest entails two things. The first is to take the position that the root of the emergence of sense and feelings is the subject’s corporeity. We must thus acknowledge the primacy of feeling (130). The second, by extension, is to put aside preconceived notions and conceptual tools for representing space inherited from modernity, as they are grounded in dualistic contraposition between physical reality, which is seen as objective (9). In contrast, experience and feeling are seen as subjective (9). Consequently, [End Page 142] they cannot adequately grasp the non- measurable dimensions of the lived space and thus the experience that the subject makes of it.Instead, De Matteis argues that one cannot describe space as a thing separate from us and that we are always a part of the space we inhabit. For this purpose, the book derives its theoretical model of space from phenomenology, which considers the felt body as a sounding board for the built environment and human experience of the world.The book is arranged into fourteen chapters that show the thematic progression of the argument, and is structured in three parts that explore space, the living body and corporeal dynamics, and the experiential character of some specific families of space. Titles of the chapters provide clear signposts to the reader about where the story is going and are much appreciated in this unique exploration. They also show that the author carefully weaves the conceptual intersections between phenomenology and its model of space with investigations in several fields of knowledge. Drawing across a multitude of disciplines, ranging from geography, anthropology, psychology, philosophy of mind, and aesthetics to neurosciences, and offering many examples from art, literature, film, and everyday life, the book helps the reader visualize the content discussed. This panoramic character and accessible language and format make it an engaging read for experts in the field and to a broader audience interested in topics addressed in the book.Perhaps one of the most compelling arguments the book brings forward is that the corporeal dynamics of space enable understanding of space as a thing that is not separate from us. It is from the coming together of persons and things that atmospheres arise. They are not objective, yet they exist as the natural qualities of things; neither are they subjective, yet they belong to sensing beings. This understanding implies a possible workable definition with no boundary between the subject and object. However, it also clearly points out that the book emerged in response to our modern understanding and critique... (shrink)

In this paper, we use the differential forms of three-dimensional Euclidean space to realize a Clifford algebra which is isomorphic to the algebra of the Pauli matrices or the complex quaternions. This is an associative vector-antisymmetric tensor algebra with division: We provide the algebraic inverse of an eight-component spinor field which is the sum of a scalar + vector + pseudovector + pseudoscalar. A surface of singularities is defined naturally by the inverse of an eight-component spinor and corresponds to a (...) generalized Minkowski “double” light cone in the parameter space. A general description of finite spatial rotations, which utilizes the Baker-Campbell-Hausdorff formula, generalizes the usual infinitesimal treatments of the rotation group. We derive an explicit expression for the angle corresponding to two successive finite rotations in any direction. We also discuss Lorentz transformations and duality rotations of the electromagnetic field and exhibit a relationship between the algebraic inverse and a duality rotated field. Using a combined transformation, one can always transform an arbitrary electromagnetic field (E≠0) into a pure electric field, but never into a pure magnetic field. (shrink)

The purpose of this paper is twofold: a) to explore the compatibility of Minkowski’s space-time representation of the Special theory of relativity with a dynamic conception of space-time; b) to locate its roots in invariant features - like entropic relations - of the propagation of signals in space-time. From its very beginning Minkowski’s four-dimensional space-time was associated with a static view of reality, e.g. a block universe. Einstein added his influential voice to this conception when he wrote: ‘From (...) a “happening” in three-dimensional space, physics becomes (…) an “existence” in the four-dimensional “world”.’ (Einstein, Relativity 1920, 122) Yet it is by no means clear that Minkowski himself was a believer in the block universe. In his 1908 Cologne lecture on ‘Space and Time’ he speaks of a four-dimensional physics but concedes that a ‘necessary’ time order can be established at every world point. Although the conception of the block universe has gained much currency, an alternative view has been in circulation since the 1910s according to which the trajectories of particles constitute histories in space-time. (Robb 1914, Cunningham 1915, Carathéodorys 1924, Schlick 1917, Reichenbach 1924). (shrink)

We use Bub's (2016) correlation arrays and Pitowksy's (1989b) correlation polytopes to analyze an experimental setup due to Mermin (1981) for measurements on the singlet state of a pair of spin-12 particles. The class of correlations allowed by quantum mechanics in this setup is represented by an elliptope inscribed in a non-signaling cube. The class of correlations allowed by local hidden-variable theories is represented by a tetrahedron inscribed in this elliptope. We extend this analysis to pairs of particles of arbitrary (...) spin. The class of correlations allowed by quantum mechanics is still represented by the elliptope; the subclass of those allowed by local hidden-variable theories by polyhedra with increasing numbers of vertices and facets that get closer and closer to the elliptope. We use these results to advocate for an interpretation of quantum mechanics like Bub's. Probabilities and expectation values are primary in this interpretation. They are determined by inner products of vectors in Hilbert space. Such vectors do not themselves represent what is real in the quantum world. They encode families of probability distributions over values of different sets of observables. As in classical theory, these values ultimately represent what is real in the quantum world. Hilbert space puts constraints on possible combinations of such values, just as Minkowski space-time puts constraints on possible spatio-temporal constellations of events. Illustrating how generic such constraints are, the equation for the elliptope derived in this paper is a general constraint on correlation coefficients that can be found in older literature on statistics and probability theory. Yule (1896) already stated the constraint. De Finetti (1937) already gave it a geometrical interpretation. (shrink)

This useful anthology comprises seventy-nine selections arranged under three headings. Part I is titled "Ancient and Classical Ideas of Space"; part II, "The Classical and Ancient Concepts of Time"; part III, "Modern Views of Space and Time and their Anticipations." According to the general editors of the Boston series, R. S. Cohen and Marx W. Wartofsky, Capek’s choice of contents was governed by the desire to show that "parts of our view of nature greatly and mutually influence other parts, and (...) that our conception of the world keeps evolving. Thus, ideas of time intertwine with ideas of space, and both with ideas of matter and force." F. M. Cornford’s essays, "The Invention of Space" and "The Elimination of Time by Parmenides," introduce parts I and II respectively. In part I, while Descartes, Pascal, Gassendi, Newton, etc., speak for themselves, the ancients and medievals are given other mouthpieces—Duhem’s monumental Le système du monde, for Plato, Aristotle, and the medievals; C. Bailey, for Leucippus, Epicurus and Lucretius; Koyré, Höffding, Jammer, for the period from Bradwardine to William Gilbert. On the subject of time, however, both ancient and classical authors are allowed to express their opinions in their own words, with the understandable exception of the Stoics. Part III begins with the prerelativistic critique of Newton as well as the well-known Clarke-Leibniz discussion on the nature of space and time. This third part, however, is dominated largely by the implications of Einstein’s general and special relativity theories. For instance, we have Minkowski’s "Union of Space and Time." Four items, at least, are devoted to the twin-paradox. Some argue for a static, subjective, or even idealistic conception of space-time whereas positivists like Frank attack Sir James Jeans’ and other idealistic interpretations of relativity as instances of meaningless metaphysics. Part III concludes with Weyl’s objectivistic interpretation of quantum-mechanical indeterminacy. While it is difficult to find an anthology that will please all readers or teachers, this work goes far towards dealing comprehensively with the subject of space and time. As Cohen and Wartofsky note, what is distinctive of Capek’s approach within the field of philosophy of nature and its history is that "he is greatly appreciative of Bergson, James, Peirce and Whitehead" and though influenced by them, he is also critical because of the "understanding of the philosophical import that contemporary physics brings into our picture of the world." In a lengthy introduction, the editor explains his rationale for each item he includes. In general this is very helpful, but in some instances the suggested historical connections are misleading at best, if not simply false or questionable. Einstein for instance is said to have "rejected any absolute frame of reference which would be a substrate of absolutely simultaneous events... within the context of Michelson’s vain search for the absolute motion of the earth." This seems to imply—as many others have claimed—that the negative results of the ether-drift experiment influenced Einstein in formulating his relativity theory, whereas historians have pointed out that if Einstein referred to the experiment, it would have to be as a confirmation of his theory, since he did not learn of the results obtained by Michelson and Morley until after the publication of his revolutionary paper in 1905. Though there are other generalizations in the introduction which might be open to dispute, Capek’s prefatory remarks are in general helpful, and the essay collection as a whole stands on its own feet. It is valuable, if for no other reason, that a number of the selections appear in English translation for the first time.—A.B.W. (shrink)

We formulate Suppes predicates for various kinds of space-time: classical Euclidean, Minkowski's, and that of General Relativity. Starting with topological properties, these continua are mathematically constructed with the help of a basic algebra of events; this algebra constitutes a kind of mereology, in the sense of Lesniewski. There are several alternative, possible constructions, depending, for instance, on the use of the common field of reals or of a non-Archimedian field (with infinitesimals). Our approach was inspired by the work of (...) Whitehead (1919), though our philosophical stance is completely different from his. The structures obtained are idealized constructs underlying extant, physical space-time. (shrink)

In quantum relativistic Hamiltonian dynamics, the time evolution of interacting particles is described by the Hamiltonian with an interaction-dependent term (potential energy). Boost operators are responsible for (Lorentz) transformations of observables between different moving inertial frames of reference. Relativistic invariance requires that interaction-dependent terms (potential boosts) are present also in the boost operators and therefore Lorentz transformations depend on the interaction acting in the system. This fact is ignored in special relativity, which postulates the universality of Lorentz transformations and their (...) independence of interactions. Taking into account potential boosts in Lorentz transformations allows us to resolve the “no-interaction” paradox formulated by Currie, Jordan, and Sudarshan [Rev. Mod. Phys. 35, 350 (1963)] and to predict a number of potentially observable effects contradicting special relativity. In particular, we demonstrate that the longitudinal electric field (Coulomb potential) of a moving charge propagates instantaneously. We show that this effect as well as superluminal spreading of localized particle states is in full agreement with causality in all inertial frames of reference. Formulas relating time and position of events in interacting systems reduce to the usual Lorentz transformations only in the classical limit (ħ→0) and for weak interactions. Therefore, the concept of Minkowski space-time is just an approximation which should be avoided in rigorous theoretical constructions. (shrink)

The way, in which quantum information can unify quantum mechanics (and therefore the standard model) and general relativity, is investigated. Quantum information is defined as the generalization of the concept of information as to the choice among infinite sets of alternatives. Relevantly, the axiom of choice is necessary in general. The unit of quantum information, a qubit is interpreted as a relevant elementary choice among an infinite set of alternatives generalizing that of a bit. The invariance to the axiom of (...) choice shared by quantum mechanics is introduced: It constitutes quantum information as the relation of any state unorderable in principle (e.g. any coherent quantum state before measurement) and the same state already well-ordered (e.g. the well-ordered statistical ensemble of the measurement of the quantum system at issue). This allows of equating the classical and quantum time correspondingly as the well-ordering of any physical quantity or quantities and their coherent superposition. That equating is interpretable as the isomorphism of Minkowski space and Hilbert space. Quantum information is the structure interpretable in both ways and thus underlying their unification. Its deformation is representable correspondingly as gravitation in the deformed pseudo-Riemannian space of general relativity and the entanglement of two or more quantum systems. The standard model studies a single quantum system and thus privileges a single reference frame turning out to be inertial for the generalized symmetry [U(1)]X[SU(2)]X[SU(3)] “gauging” the standard model. As the standard model refers to a single quantum system, it is necessarily linear and thus the corresponding privileged reference frame is necessary inertial. The Higgs mechanism U(1) → [U(1)]X[SU(2)] confirmed enough already experimentally describes exactly the choice of the initial position of a privileged reference frame as the corresponding breaking of the symmetry. The standard model defines ‘mass at rest’ linearly and absolutely, but general relativity non-linearly and relatively. The “Big Bang” hypothesis is additional interpreting that position as that of the “Big Bang”. It serves also in order to reconcile the linear standard model in the singularity of the “Big Bang” with the observed nonlinearity of the further expansion of the universe described very well by general relativity. Quantum information links the standard model and general relativity in another way by mediation of entanglement. The linearity and absoluteness of the former and the nonlinearity and relativeness of the latter can be considered as the relation of a whole and the same whole divided into parts entangled in general. (shrink)

We formulate Suppes predicates for various kinds of space-time: classical Euclidean, Minkowski's, and that of General Relativity. Starting with topological properties, these continua are mathematically constructed with the help of a basic algebra of events; this algebra constitutes a kind of mereology, in the sense of Lesniewski. There are several alternative, possible constructions, depending, for instance, on the use of the common field of reals or of a non-Archimedian field. Our approach was inspired by the work of Whitehead, though (...) our philosophical stance is completely different from his. The structures obtained are idealized constructs underlying extant, physical space-time. (shrink)

Should physicists deal with the question of the reality of Minkowski space (or any relativistic spacetime)? It is argued that they should since this is a question about the dimensionality of the world at the macroscopic level and it is physics that should answer it.

This paper argues that the Einstein-Minkowski space-time of special relativity provides an adequate model for classical tense logic, including rigorous definitions of tensed becoming and of the logical priority of proper time. In addition, the extension of classical tense logic with an operator for predicate-term negation provides us with a framework for interpreting and defending the significance of future contingency in special relativity. The framework for future contingents developed here involves the dual falsehood of non-logical contraries, only one of (...) which becomes true. This has several methodological, metaphysical and physical advantages over the alternative traditional frameworks for handling future contingents. (shrink)

We calculate the mixing of real and imaginary components of space and time under the influence of superluminal boosts in thex direction. A unique mixing is determined for this superluminal Lorentz transformation when we consider the symmetry properties afforded by the inclusion of three temporal directions. Superluminal transformations in complex six-dimensional space exhibit unique tachyonic connections which have both remote and local space-time event connections.

Minkowski geometry is axiomatized in terms of the asymmetric binary relation of optical connectibility, using ten first-order axioms and the second-order continuity axiom. An axiom system in terms of the symmetric binary optical connection relation is also presented. The present development is much simpler than the corresponding work of Robb, upon which it is modeled.

Under the eternalist hypothesis that objects or events exist temporally, but independently of being present two different views of persistence are on the market: Persisting objects endure if they are multiply located in time, and persisting objects perdure if they are singly located by having numerically different temporal parts. In the framework of the special theory of relativity, the metaphysics of persistence is confronted with peculiar difficulties. Things persist by being “wholly present” at more than one time; but what are (...) times within a temporally non-separated spacetime? Things persist by having different temporal parts at different times; but what are the temporal parts of a four-dimensional object in Minkowski spacetime? Recently, several authors have argued that SR favours perdurantism over its endurantist rival. In this paper, I intend to show that the purported arguments are only those against endurantism. The first simply fails, but the second, more convincing one, is such that with a similar strategy we should argue against perdurantism as well: Enduring and perduring entities are hence both in conflict with SR which undermines the eternalist hypothesis. (shrink)

Under the eternalist hypothesis that objects or events exist temporally, but independently of being present two different views of persistence are on the market: Persisting objects endure if they are multiply located in time, and persisting objects perdure if they are singly located by having numerically different temporal parts. In the framework of the special theory of relativity, the metaphysics of persistence is confronted with peculiar difficulties. Things persist by being “wholly present” at more than one time; but what are (...) times within a temporally non-separated spacetime? Things persist by having different temporal parts at different times; but what are the temporal parts of a four-dimensional object in Minkowski spacetime? Recently, several authors have argued that SR favours perdurantism over its endurantist rival. In this paper, I intend to show that the purported arguments are only those against endurantism. The first simply fails, but the second, more convincing one, is such that with a similar strategy we should argue against perdurantism as well: Enduring and perduring entities are hence both in conflict with SR which undermines the eternalist hypothesis. (shrink)

A-theoretic presentness is commonly regarded as non-solipsist and non-relative. The non-solipsism of a non-relative, A-theoretic presentness requires at least two space-like separated things to be present simpliciter together – this co-presentness further implies the global, non-relative, non-conventional simultaneity of them. Yet, this implication clashes with the general view that there is no global, non-relative, non-conventional simultaneity in Minkowski space-time. In order to resolve this conflict, this paper explores the possibility that the non-solipsism of a non-relative, A-theoretic presentness does not (...) require at least two space-like separated things to be present simpliciter together. This can be done by holding exclusive disjunctivism –that mutually space-like separated things are present simpliciter exclusively disjunctively, and each one of them gets to be present simpliciter in a non-successive way (just like mutually time-like related things are present simpliciter exclusively disjunctively, and each one of them gets to be present simpliciter, but in a successive way). (shrink)

I examine the issue of persistence over time in thecontext of the special theory of relativity (SR). Thefour-dimensional ontology of perduring objects isclearly favored by SR. But it is a different questionif and to what extent this ontology is required, andthe rival endurantist ontology ruled out, by thistheory. In addressing this question, I take theessential idea of endurantism, that objects are whollypresent at single moments of time, and argue that itcommits one to unacceptable conclusions regardingcoexistence, in the context of SR. (...) I then propose anddiscuss a plausible account of coexistence forperduring objects, which is free of these defects. This leaves the endurantist room for some maneuvers. I consider them and show that they do not really helpthe endurantist out. She can accommodate the notionof coexistence in the relativistic framework only atthe cost of renouncing central endurantist intuitions. (shrink)

We present a general derivation of the Duffin-Kemmer-Petiau (D.K.P) equation on the relativistic phase space proposed by Bohm and Hiley. We consider geometric algebras and the idea of algebraic spinors due to Riesz and Cartan. The generators βμ (p) of the D.K.P algebras are constructed in the standard fashion used to construct Clifford algebras out of bilinear forms. Free D.K.P particles and D.K.P particles in a prescribed external electromagnetic field are analized and general Liouville type equations for these cases are (...) obtained. Choosing particular values for the label p we classify the different types of the D.K.P Liouville operators. (shrink)

Non-commuting quantities and hidden parameters – Wave-corpuscular dualism and hidden parameters – Local or nonlocal hidden parameters – Phase space in quantum mechanics – Weyl, Wigner, and Moyal – Von Neumann’s theorem about the absence of hidden parameters in quantum mechanics and Hermann – Bell’s objection – Quantum-mechanical and mathematical incommeasurability – Kochen – Specker’s idea about their equivalence – The notion of partial algebra – Embeddability of a qubit into a bit – Quantum computer is not Turing machine – (...) Is continuality universal? – Diffeomorphism and velocity – Einstein’s general principle of relativity – „Mach’s principle“ – The Skolemian relativity of the discrete and the continuous – The counterexample in § 6 of their paper – About the classical tautology which is untrue being replaced by the statements about commeasurable quantum-mechanical quantities – Logical hidden parameters – The undecidability of the hypothesis about hidden parameters – Wigner’s work and и Weyl’s previous one – Lie groups, representations, and psi-function – From a qualitative to a quantitative expression of relativity − psi-function, or the discrete by the random – Bartlett’s approach − psi-function as the characteristic function of random quantity – Discrete and/ or continual description – Quantity and its “digitalized projection“ – The idea of „velocity−probability“ – The notion of probability and the light speed postulate – Generalized probability and its physical interpretation – A quantum description of macro-world – The period of the as-sociated de Broglie wave and the length of now – Causality equivalently replaced by chance – The philosophy of quantum information and religion – Einstein’s thesis about “the consubstantiality of inertia ant weight“ – Again about the interpretation of complex velocity – The speed of time – Newton’s law of inertia and Lagrange’s formulation of mechanics – Force and effect – The theory of tachyons and general relativity – Riesz’s representation theorem – The notion of covariant world line – Encoding a world line by psi-function – Spacetime and qubit − psi-function by qubits – About the physical interpretation of both the complex axes of a qubit – The interpretation of the self-adjoint operators components – The world line of an arbitrary quantity – The invariance of the physical laws towards quantum object and apparatus – Hilbert space and that of Minkowski – The relationship between the coefficients of -function and the qubits – World line = psi-function + self-adjoint operator – Reality and description – Does a „curved“ Hilbert space exist? – The axiom of choice, or when is possible a flattening of Hilbert space? – But why not to flatten also pseudo-Riemannian space? – The commutator of conjugate quantities – Relative mass – The strokes of self-movement and its philosophical interpretation – The self-perfection of the universe – The generalization of quantity in quantum physics – An analogy of the Feynman formalism – Feynman and many-world interpretation – The psi-function of various objects – Countable and uncountable basis – Generalized continuum and arithmetization – Field and entanglement – Function as coding – The idea of „curved“ Descartes product – The environment of a function – Another view to the notion of velocity-probability – Reality and description – Hilbert space as a model both of object and description – The notion of holistic logic – Physical quantity as the information about it – Cross-temporal correlations – The forecasting of future – Description in separable and inseparable Hilbert space – „Forces“ or „miracles“ – Velocity or time – The notion of non-finite set – Dasein or Dazeit – The trajectory of the whole – Ontological and onto-theological difference – An analogy of the Feynman and many-world interpretation − psi-function as physical quantity – Things in the world and instances in time – The generation of the physi-cal by mathematical – The generalized notion of observer – Subjective or objective probability – Energy as the change of probability per the unite of time – The generalized principle of least action from a new view-point – The exception of two dimensions and Fermat’s last theorem. (shrink)