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We review the current status of dimensions, as the result of a long and controversial history that includes input from philosophy and physics. Our conclusion is that they are subjective but essential concepts which provide a kind of bookkeeping device, their number increasing as required by advances in physics. The world almost certainly has more than the four dimensions of space and time, but the introduction of the fifth and higher dimensions requires a careful approach wherein known results are embedded and new ones are couched in the most productive manner. [ABSTRACT FROM AUTHOR]

This paper presents an informal discussion of the space and time of classical Newtonian theory and of the space-time of Einstein's special relativity theory, together with a comparison of them. The essential reason for the (1+3)- dimensionality of classical theory and the 4D of special relativity is described. [ABSTRACT FROM AUTHOR]

We use a new, distinctly "geometrical" interpretation of nonrelativistic quantum mechanics (NRQM) to argue for the fundamentality of the 4D block world ontology. We argue for a geometrical interpretation whose fundamental ontology is one of space-time relations as opposed to constructive entities whose time-dependent behavior is governed by dynamical laws. Our view rests on two formal results: Kaiser (1981, 1990), Bohr and Ulfbeck (1995), and Anandan (2003) showed independently that the Heisenberg commutation relations of NRQM follow from the relativity of simultaneity (RoS) per the Poincaré Lie algebra. And, Bohr, Ulfbeck, and Mottelson (2004a, 2004b) showed that the density matrix for a particular NRQM experimental outcome may be obtained from the space-time symmetry group of the experimental configuration. This shows how the block world view is not only consistent with NRQM, not only an implication of our geometrical interpretation of NRQM, but it is necessary in a nontrivial way for explaining quantum interference and "non-locality" from the space-time perspective. Together the formal results imply that contrary to accepted wisdom, NRQM, the measurement problem and socalled quantum nonlocality do not provide reasons to abandon the 4D block world implication of RoS. Rather, the deep noncommutative structure of the quantum and the deep structure of space-time as given by the Minkowski interpretation of special relativity (STR) are deeply unified in a 4D space-time regime that lies between Galilean space-time (G4) and Minkowski space-time (M4). [ABSTRACT FROM AUTHOR]

The Hamiltonian structure of general relativity (GR), for both metric and tetrad gravity in a definite continuous family of space-times, is fully exploited in order to show that: (i) the Hole Argument can be bypassed by means of a specific physical individuation of point-events of the space-time manifold M4in terms of the autonomous degrees of freedom of the vacuum gravitational field (Dirac observables), while the Leibniz equivalence is reduced to differences in the non-inertial appearances (connected to gauge variables) of the same phenomena. (ii) The chrono-geometric structure of a solution of Einstein equations for given, gauge- fixed, initial data (a 3-geometry satisfying the relevant constraints on the Cauchy surface), can be interpreted as an unfolding in mathematical global time of a sequence of achronal 3-spaces characterized by dynamically determined conventions about distant simultaneity. This result stands out as an important conceptual difference with respect to the standard chrono-geometrical view of special relativity (SR). [ABSTRACT FROM AUTHOR]

The concept of fundamental particle has been quite elusive along the history of physics. The term fundamental is commonly used as a synonymous of structureless particles. However, this assumption is clearly contradictory. For example, it is impossible to explain spin without assuming a structure for the particle. In fact, a point particle is by definition spherically symmetric, a symmetry violated by the presence of spin. This problem is usually circumvented by saying that spin is a purely quantum property, which cannot be explained by classical physics. This means to keep it as a mysterious property of nature. [ABSTRACT FROM AUTHOR]

Contrary to our immediate and vivid sensation of past, present, and future as continually shifting non-relational modalities, time remains as tenseless and relational as space in all of the established theories of fundamental physics. Here an empirically adequate generalized theory of the inertial structure is discussed in which proper time is causally compelled to be tensed within both space-time and dynamics. This is accomplished by introducing the inverse of the Planck time at the conjunction of special relativity and Hamiltonian mechanics, which necessitates energies and momenta to be invariantly bounded from above, and lengths and durations similarly bounded from below, by their respective Planck-scale values. The resulting theory abhors any form of preferred structure, and yet captures the transience of now along timelike world lines by causally necessitating a genuinely becoming universe. This is quite unlike the scenario in Minkowski space-time, which is prone to a block universe interpretation. The minute deviations from the special relativistic effects such as dispersion relations and Doppler shifts predicted by the generalized theory remain quadratically suppressed by the Planck energy, but may nevertheless be testable in the near future, for example via observations of oscillating flavor ratios of ultrahigh energy cosmic neutrinos, or of altering pulse rates of extreme energy binary pulsars. [ABSTRACT FROM AUTHOR]

It is often said that the relativistic fusion of time with space rules out genuine change or "becoming". I offer the classical sequential growth models of causal set theory as counterexamples. [ABSTRACT FROM AUTHOR]

Different aspects of relativity, mainly in a canonical formulation, relevant for the question "Is spacetime nothing more than a mathematical space (which describes the evolution in time of the ordinary three-dimensional world) or is it a mathematical model of a real 4D world with time entirely given as the fourth dimension?" are presented. The availability as well as clarity of the arguments depends on which framework is being used, for which currently special relativity, general relativity and some schemes of quantum gravity are available. Canonical gravity provides means to analyze the field equations as well as observable quantities, the latter even in coordinate independent form. This allows a unique perspective on the question of dimensionality since the space-time manifold does not play a prominent role. After reintroducing a Minkowski background into the formalism, one can see how distinguished coordinates of special relativity arise, where also the nature of time is different from that in the general perspective. Just as it is of advantage to extend special to general relativity, general relativity itself has to be extended to some theory of quantum gravity. This suggests that a final answer has to await a thorough formulation and understanding of a fundamental theory of space-time. Nevertheless, we argue that current insights into quantum gravity do not change the picture of the role of time obtained from general relativity. [ABSTRACT FROM AUTHOR]

A 100 years have passed since the advent of special relativity and 2008 will mark another important to all relativists anniversary - 100 years since Minkowski gave his talk "Space and Time" on September 21, 1908 in which he proposed the unifi- cation of space and time into an inseparable entity - space-time. Although special relativity has been an enormously successful physical theory no progress has been made in clarifying the question of existence of the objects represented by two of its basic concepts - space-time and world lines (or worldtubes in the case of extended bodies). The major reason for this failure appears to be the physicists' tradition to call such questions of existence philosophical. This tradition, however, is not quite consistent. In Newtonian mechanics physicists believe that they describe real objects whenever they talk about particles - one of the basic concepts of Newtonian physics. The situation is the same in quantum physics - no one questions the existence of electrons, protons, etc. Then why should the question of existence of worldtubes (representing particles in relativity) be regarded as a philosophical question? [ABSTRACT FROM AUTHOR]

1. We argue that the constancy of c implies a realistic now at a distance in all inertial systems, which in turn leads to determinism. 2. We elaborate that denying this violates physical consistency, that requires the concept "velocity". 3. Physics implies models making the world observationally and logically coherent, also as to the experiences of different observers. Velocity and now at a distance are cases in point. 4. Retroaction - which implies a realistic future - can be demonstrated to appear in some experiments. 5. We give a demonstration of determinism via reality at a distance that has an effect here and now too. 6. If the phenomenon of Einstein, Podolsky and Rosen (EPR) does not imply retroaction, it makes necessary nonlocal signals. This is highly relevant to reality and "now" at a distance. 7. The clock paradox shows concrete visible consequences of "merely metrical" relativistic coordination and time flow at a distance. Keywords: determinism, special relativity, indeterminism, clock paradox, block universe [ABSTRACT FROM AUTHOR]

The block universe idea, representing space-time as a fixed whole, suggests the flow of time is an illusion: the entire Universe just is, with no special meaning attached to the present time. This paper points out that this view, in essence represented by usual space-time diagrams, is based on time-reversible microphysical laws, which fail to capture essential features of the time-irreversible macro-physical behaviour and the development of emergent complex systems, including life, which exist in the real Universe. When these are taken into account, the unchanging block Universe view of space-time is best replaced by an evolving block Universe which extends as time evolves, with the potential of the future continually becoming the certainty of the past; space-time itself evolves, as do the entities within it. However this time evolution is not related to any preferred surfaces in space-time; rather it is associated with the evolution of proper time along families of world lines. [ABSTRACT FROM AUTHOR]

Since the nineteenth century, it is known, through the work of Lobatchevski, Riemann, and Gauss, that spaces do not need to have a vanishing curvature. This was for sure a revolution on its own, however, from the point of view of these mathematicians, the space of our day to day experience, the physical space, was still an essentially a priori concept that preceded all experience and was independent of any physical phenomena. Actually, that was also the view of Newton and Kant with respect to time, even though, for these two space-time explorers, the world was Euclidean. [ABSTRACT FROM AUTHOR]

Copyright of Il Cielo Sopra Roma is the property of Springer eBooks and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Il Cielo Sopra Roma is the property of Springer eBooks and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Il Cielo Sopra Roma is the property of Springer eBooks and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Il Cielo Sopra Roma is the property of Springer eBooks and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Il Cielo Sopra Roma is the property of Springer eBooks and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Il Cielo Sopra Roma is the property of Springer eBooks and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

I. Among the various phenomena of corporeal nature, there are two, which, as they are very greatly distinguished almost before all others, having been examined in themselves, have occupied to a very great extent the philosophers of all time. One of these is that general tendency towards its centre of all bodies moving about the surface of the earth, which is commonly called gravity; the other is the regular gyration of planets in their orbits, which recurs with definite periods. Various hypotheses have been devised by various people for the explanation in mechanical terms of those phenomena. An impartial examination of these will prepare the way for explaining and developing that general law of universal gravitation, to which, it will be established, those two most noble effects are to be referred as a common foundation, even if at first sight they seem to have nothing in common; from this we will also seize the opportunity to consider along the way certain other forces of nature, which it is necessary to put in place for the solution of certain other phenomena, which philosophers have undertaken to explain likewise by mechanical theories. [ABSTRACT FROM AUTHOR]

The prize topic proposed by the Royal Academy of Sciences for 1740 was the tides: le Flux et Reflux de la Mer. The winners, in the order in which their essays were published by the Academy, were Father Antoine Cavalleri [23], Daniel Bernoulli [12], Colin MacLaurin [68] and Leonhard Euler [38]. Cavelleri's work was based on the Cartesian idea of vortices;47 the other three were founded on Newtonian principles and were subsequently reproduced in the "Jesuit Edition" of the Principia (1739-1742) [86] as illustrations of the Newtonian philosophy. Newton had already discussed in its earlier editions the tidal forces on the Earth which resulted from the gravitational attraction of the Sun and of the Moon; his work was of course the starting point for MacLaurin's investigations. [ABSTRACT FROM AUTHOR]

The main purpose of the present volume is to discuss, and to make generally accessible, three dissertations1 by the Scottish mathematician Colin MacLaurin (1698-1746), who was regarded both in Britain and in continental Europe as one of the leading mathematicians of his time. [ABSTRACT FROM AUTHOR]

Following the 25 years of the ‘old quantum theory' in which results of great importance were discovered, but by methods that lacked any consistent foundation,1 the modern rigorous form of quantum theory was produced in the mid-1920s.2,3 Werner Heisenberg's matrix mechanics of mid-19254,5 was followed by Erwin Schrödinger's sequence of papers developing the mathematically analogous wave mechanics, which was published through the first half of 1926.6,7 In September 1927, in a lecture at Como,8 Niels Bohr responded to concerns about the conceptual structure of the new theory by announcing his ideas on complementarity, which were to constitute what became known as the Copenhagen interpretation of quantum theory. Einstein heard Bohr expound these ideas the following month at the fifth Solvay congress in Brussels, and, at least by that stage, had clearly concluded that Bohr's views were unacceptable9; Einstein would argue against them for the remainder of his life. [ABSTRACT FROM AUTHOR]

Later in this last chapter we assess Einstein' s contributions to quantum theory. However we start by recognising how much the study of quantum theory has advanced in the half-century since Einstein 's death.In Part C of this book,we dis- cussed the work of John Bell,the range of novel interpretations of quantum theory, the coming of quantum information theory,and the new ideas on the classical limit of quantum theory,while in the previous chapter we reviewed a few of the current developments.As we have stressed all this serves to vindicate Einstein 's practi- cally lone opposition to Copenhagen over the last 30 years of his life,but we now ask the question—how would Einstein have reacted to these new lines of study? [ABSTRACT FROM AUTHOR]

For followers of Copenhagen, study of the foundational aspects of quantum theory was effectively completed in 1927 by Bohr's paper at Como. Bohr was to continue to write on quantum theory and complementarity for the remainder of his life, but his writings were mainly explanatory, or attempts to strengthen the philosophical base of complementarity, or applications of complementarity to areas of knowledge other than quantum theory.1 It was taken for granted that no genuine re-examination of the fundamental issues was required-or, indeed, permitted. John Clauser2 has made it particularly clear to what extent any questioning of the Copenhagen position was regarded practically as heresy: ‘Religious dogmatism then quickly promoted a nearly universal acceptance of quantum theory and its Copenhagen interpretation as gospel, along with a total unwillingness to even mildly question the theory's foundations.' [ABSTRACT FROM AUTHOR]

In Chapter 7, we studied the discussions between Einstein and supporters of Copenhagen, in particular Max Born and Wolfgang Pauli, concerning the classical limit of quantum theory and macroscopic quantum theory. We discussed the argument of Born and Pauli that the limiting procedure was straightforward, and that there were no special features in macroscopic quantum theory, such as macrorealism, that could make it different in principle from microscopic quantum theory. [ABSTRACT FROM AUTHOR]

In the first decade of the twenty-first century, quantum information theory is indisputably a ‘hot topic'. A great deal of theoretical work is being performed in the main branches of the subject—quantum computation, quantum cryptography and quantum teleportation. Many different experimental techniques are being explored with the eventual aim of producing the first useful quantum computer, though it is recognised that this will almost certainly be decades away. In the other two main branches, though, considerable progress has been made; quantum teleportation has been demonstrated in the laboratory, while quantum cryptography has reached the stage where it is capable of being applied to ensure the security, say, of the financial district of a large city; this will probably happen quite soon. [ABSTRACT FROM AUTHOR]

From the time of Bohr's Como paper in 1928, the Copenhagen interpretation reigned supreme for several decades. Indeed supporters of the Copenhagen interpretation, such as Leon Rosenfeld and Rudolf Peierls,1 disliked even the use of the term ‘Copenhagen interpretation', for it suggested that this was one interpretation, possibly among many conceivable ones. For them, the conceptual structures of Bohr and Heisenberg were not an adjunct to quantum theory, but a clear and indispensable component of the theory. Einstein, as we have seen at length, disliked the Copenhagen interpretation. Perhaps as much as or even more then the interpretation itself, he disliked the fact that its supporters considered it immune from criticism, so that no questioning would be taken seriously, and certainly any alternative interpretation would be ruled out of court without being given even fair attention. [ABSTRACT FROM AUTHOR]

The question of whether it is possible to supplement the wave-function of a system with extra parameters, known as hidden variables, has been discussed at several points earlier in this book. In Chapters 3 and 4, we saw that, though hidden variables might seem to have the potential to solve many of the apparent problems of quantum theory, orthodox approaches made a major point of excluding them from consideration. In Chapter 5 we discussed why, fairly soon after discovery of (modern) quantum theory, Einstein rejected the idea of producing a simple hidden variable theory. Nevertheless, his project of producing a (itcomplete) quantum theory was likely to lead others in the direction of adding hidden variables, even though his own ideas were considerably more grandiose. [ABSTRACT FROM AUTHOR]

Jeremy Bernstein1 has written that ‘In 1926, even before Heisenberg's paper [on the uncertainty or indeterminacy principle], Einstein abandoned the quantum theory'. Bernstein also writes of the depth of Einstein's feelings about the theory, that, at least at one period: ‘He wished to destroy it.' And indeed it is extremely commonly implied that Einstein ‘did not accept' quantum theory; faced with a shelf of popular physics books, or a series of relevant websites, it would take a very short time to gain the overwhelming impression that Einstein ‘rej ected quantum theory', usually stated without any qualification. [ABSTRACT FROM AUTHOR]

In a letter to David Bohm which was written on 24th November 1954 and has been quoted by Jammer,1 Einstein wrote: ‘I do not believe in micro and macro laws, but only in (structural) laws of general validity.' This remark reflects one of the central tenets of his belief, and Einstein used it to substantiate his arguments which attempted to demonstrate the incompleteness of quantum mechanics. Einstein put forward various versions of these arguments, but there were two main themes. The first, which has certainly received the most attention in books and articles, concerns the question of hidden variables, entanglement and so on. We have discussed it in the previous two chapters, and, of course, this theme led to the work of John Bell, which is discussed in Chapter 9, and much that is in later chapters of this book. [ABSTRACT FROM AUTHOR]

The Einstein-Podolsky-Rosen paper, 1 usually just called EPR, was written in 1935. Following his discussions with Bohr, which were described in the previous chapter, Einstein was now prepared to admit that it was not possible to measure momentum and position simultaneously to a better accuracy that allowed by the Heisenberg principle. The EPR argument was, at the very least, an ingenious attempt to show that nevertheless both observables do actually have simultaneous values at all times. Bohr fairly quickly produced what he claimed to be rebuttals of the EPR thesis,2,3 and it is clear that, with very few exceptions, physicists practically at once decided that Bohr's arguments had succeeded yet again in defeating those of Einstein. By this time, of course, Einstein had long since nailed his colours to the mast of opposition to Copenhagen, and was thus almost universally regarded as being completely out of touch with the requirements of modern science. His further criticisms of orthodoxy were taken only as demonstrating yet again his lack of understanding and appreciation; it is doubtful if many physicists of the day bothered to read either EPR or Bohr's response before deciding who was right and who was wrong. [ABSTRACT FROM AUTHOR]

Heisenberg's seminal paper initiating quantum theory was written in July 1925. It is sometimes suggested that Einstein was initially unclear in his views on the theory; Pais1 suggests that he ‘vacillated.' It seems more likely that he combined at much the same time great excitement about the mathematical content and potential of the theory, with growing concern that philosophical conclusions were being drawn from the theory that he found unacceptable. In March 1926 he wrote2 to Max Born's wife that: ‘The Heisenberg-Born concepts leave us all breathless, and have made a deep impression on all theoretically oriented people. Instead of dull resignation, there is now a singular tension in us sluggish people.' Indeed it must have been as exciting for Einstein as for any other physicist to perceive a clear mathematical route forward after the conceptual turmoil of the previous quarter-century. In 1912, Einstein1 had written: "The more success the quantum theory has, the sillier it looks‘, and his feelings had probably not changed much during the 12 years before 1912 and the 13 afterwards up to 1925. Now at last it seemed that one could hope to study atoms and light from a well-defined theory rather than by guesswork and subterfuge, however brilliant. [ABSTRACT FROM AUTHOR]

In Chapter 3 we explored the various conceptual problems and indeed apparent paradoxes present in the new quantum theory of Heisenberg and Schrödinger. For a comparatively short period, discussion on how to resolve these difficulties was quite open; indeed during this period some of those who were later to support the Copenhagen interpretation without question, such as Werner Heisenberg and Max Born, showed quite clear divergences from Bohr's position. However this period ended quite abruptly after Bohr's famous Como lecture1 of 1927, the content of which was presented in similar form shortly afterwards at the fifth Solvay conference. At this lecture, Bohr expounded his approach on the way in which these problems should be resolved with the use of the conceptual framework he called complementarity, which was the basis of what was to become known as the Copenhagen interpretation of quantum theory. [ABSTRACT FROM AUTHOR]

The ‘new' quantum theory of Heisenberg and Schrödinger was immediately seen to be extremely successful in practice. It could be used to predict energy levels for a considerable number of systems, and Schrödinger's approach in particular was capable of describing many features of the atom, and predicting the results of a great variety of experiments. Yet it was also clear that there were many surprising and puzzling aspects to the theory, aspects that were certain to cause controversy. [ABSTRACT FROM AUTHOR]