Emotion sharing plays a key role in many accounts of empathy. However, some equate emotion sharing with emotional “contagion” and thereby discount it as a form of empathy. In what follows, I clarify the nature of empathic emotion sharing and differentiate it from contagion. I first reflect on the notions of sharing an object and of sharing a life, arguing that each has four core features. I then argue that emotion sharing also has those features. These characteristics allow me to (...) distinguish emotion sharing from contagion and other phenomena like it. I end by offering positive reasons for seeing emotion sharing as a form of empathy – i.e., it provides knowledge of others’ mental states and aids in moral development. (shrink)

The purpose of this study is to assess the intention of typically developing peers towards learning in the classroom with students with Autism Spectrum Disorder. In developing countries, such as Ghana, the body of literature on the relationship between students with disabilities and typically developing peers has been sparsely studied. Using Ajzen's theory of planned behaviour as a theoretical framework for this study, 516 typically developing students completed four scales representing belief constructs, attitudes, subjective norms, and perceived behavioural controls, hypothesised (...) to predict behavioural intention. The data were subjected to a t-test, analysis of variance, and structural equation modelling. The modelling confirmed the combining ability of attitude, subjective norms, and perceived behavioural controls to predict intention. We conclude by revealing the need for policymakers to consider designing programmes aimed towards promoting social relationships between students with ASD and typically developing peers. (shrink)

Writing, the exigency of writing: no longer the writing that has always (through a necessity in no way avoidable) been in the service of the speech or thought that is called idealist (that is to say, moralizing), but rather the writing that through its own slowly liberated force (the aleatory force of absence) seems to devote itself solely to itself as something that remains without identity, and little by little brings forth possibilities that are entirely other: an anonymous, distracted, deferred, (...) and dispersed way of being in relation, by which everything is brought into question – and first of all the idea of God, of the Self, of the Subject, then of Truth and the One, then finally the idea of the Book and the Work so that this writing (understood in its enigmatic rigor), far from having the Book as its goal rather signals its end: a writing that could be said to be outside discourse, outside language. – Maurice Blanchot ( The Infinite Conversation , xi) The Compendium * Beginning with the idea and practice of writing, and moving to the subject or self, then to truth and the One, and arriving at the Book and the Work, Blanchot’s words reflect the trajectory of the following work of writing. True to the embeddedness of the subject in the work of writing, the term “Compendium” has been a metonym for me over the past few years as I have thought about the concept of totality alongside its expression in figures such as the One, the Whole, or the All. In the following I aim to share some of the content of this metonym, and to enrich it by making some distinctions. 1 Here the term “Compendium” will refer to a concept of totality that is ontological (pertaining to being, and the copula) and also textual (pertaining to the symbolic, and the signifier-signified relationship). The Compendium is a figure for thinking the world as a Book, in the broadest sense—an approach to reality that is by no means new, but one that is due for renewal. 2 In partial answer to the question of the Compendium we will say that the word ‘Compendium’ stands in for expressions which seek to totalize, or concepts which seek to approach the concept of ‘everything,’ particularly when these expressions are bound up in the question of the Book (both the book as a physical object and the Book as a metaphysical figure: a way of thinking about the work in and of writing). The question of the Compendium is strongly associated with the physical and metaphysical form of the Book, like the mysterious and symbolic books which are opened in the Biblical book of Revelation, the book of life and the book of death, which together constitute an important couplet. 3 Parataxis & Hypotaxis There are two helpful distinctions that will bring us closer to an answer to the question “What is a Compendium?” The first distinction is between two figures in and for writing: parataxis and hypotaxis . Typically paired in contrast to one another as literary techniques, with parataxis indicating a side-by-side placement of textual elements and hypotaxis referring to subordinate arrangements of textual elements, the two figures can be understood as having a philosophical significance in addition to their practical function as devices for writers. For us these two terms will remain between philosophical theory and writing practice, and serve as a perspective for our writing and creation of texts. Here I take texts to refer to anything from the concrete written words in a book, to the ontological text of the world that we experience. Parataxis , as a figural way of thinking about the ontological structure of texts, refers to texts which are tightly woven and interdependent – texts within which each sentence bears the weight of the entire work. Parataxis often involves repetition, great density, and fragility. One of the best examples of this sort of text is Theodor Adorno’s posthumous magnum opus , Aesthetic Theory . In addition to the text itself, the translation history of the book may also help us to better understand parataxis and its relation to hypotaxis . The final published version of Aesthetic Theory , in German, was a text that Adorno intended to revise and rewrite, but this intention was never realized because of his untimely death. Where the German text of Aesthetic Theory certainly exemplifies the concept of parataxis , the first English edition took this densely woven text and carved Adorno’s lengthy paragraphs and lengthy sentences into manageable ‘bite-sized’ pieces of English text. The first translator took further liberty and inserted headings and new paragraph breaks where there were none in the original. Aesthetic Theory was eventually retranslated by Robert Hullot-Kentor and is now available in a form that is much more faithful to the original work. 4 The pertinent idea that this translation history points to is the distinction between parataxis and hypotaxis . Where parataxis describes texts which are repetitive, densely woven, and often fragile, hypotaxis describes texts which are hierarchical and in which the primary relation is linear – the latter of which is very similar to the first English translation of Aesthetic Theory . On the other hand, the original German text that Adorno wrote was very dense, often repetitive, and contained long sections of text unbroken by paragraphs or headings. This example of parataxis was then turned towards hypotaxis through the initial translation which took a text that was, in many ways, nonhierarchical and nonlinear, and artificially subjected it to a hierarchy that was not its own (and here I take the word of Fredric Jameson who comments on the two translations in a note at the beginning of his book Late Marxism ). 5 The point here is not about translation, but rather the distinction between parataxis and hypotaxis precisely as they are figures for discourse, written or otherwise. The concept of parataxis looks far more postmodern and rhizomic than the concept of hypotaxis which remains very modern and arborescent. This is a more figural way of talking about the distinction. However, if we wanted to take a more precise and analytical approach we could say that on the level of form parataxis involves a relationship between sentences and paragraphs in which each part bears an equally crushing responsibility to present the whole content of the text. As well, on the level of content, parataxis requires that each concept in a work take on the full conceptual weight of the total work. Continuing the analysis, we could say that parataxis describes texts in which there is no (or very little) linear or causal move from antecedent to consequent, whether in content or form. Instead, parataxis describes texts which weave together concepts via conjugations or associations. The concept of hypotaxis , on the other hand, requires that texts submit themselves to hierarchy, one example of which is logical argumentation. The contingency of the conclusion upon premises in hypotaxis is certainly distinct from the repetition, re-presentation, and conjugation of concepts in parataxis , and I think that this is evident in the difference between the writing styles prevalent in contemporary Analytic philosophy and Continental philosophy. It is not the case that the distinction between parataxis and hypotaxis absolutely corresponds with the writing styles of Continental and Analytic philosophy (respectively). There are thinkers who take exception to this pattern, such as Badiou’s more formal approach in the discourse of Continental philosophy to give one example. However, when Analytic philosophy takes its writing lessons from the sciences, and seeks to do philosophy via the clean linear move from antecedent to consequent, then I think that Analytic philosophy writes with hypotaxis in mind. On the other hand, when Continental (particularly German and French) philosophy takes its writing lessons from narrative or poetry then Continental philosophy writes with parataxis in mind. It is not fair to say that all those writing Continental philosophy are looking to narrative and poetry for stylistic direction, just as it is not the case that all those writing Analytic philosophy have mathematics and science as their writing format, however there are some striking ways in which the differing epistemologies of Continental and Analytic philosophy encourage writing with parataxis and hypotaxis in mind (respectively). For some, writing with parataxis or hypotaxis in mind is a conscious decision and a product of stylistic self-consciousness, and for others it is an unconscious discursive and epistemic requirement that must be met in order to be involved in a particular discourse. Before moving on to our second distinction, and then an examination of the question of the Book, it is important to point out that both totality and figurations like the One, the Whole, and the All, come out of very human desires to totalize or to actualize our being-towards-totality. The relationship between identity and totality then can be construed, with Heidegger in mind, as striving toward wholeness, completion, or fulfillment, each of which is a quality of compendia. 6 Compilation & Selection The first distinction between parataxis and hypotaxis pertains to both form and content, and to both the mechanics of writing and the ideas to which writing refers. The second distinction pertains to both as well. In the process of writing and in the process of perception as well, there is a tension between compilation and selection . To begin with, in the work of theory-writing there is compilation, and this is because writing theory requires a broad perspective and certain measure of totalization. On the other hand, writing theory requires that the writer select an idea or a combination of ideas to individuate out of a radical and infinite multiplicity of identities and combinations. The concern here is not primarily for the writing of a theory about a specific idea, like a secondary work or a reference text. But rather the concern is with the writing of grand theories: Marx and Marxism, Derrida and Deconstruction, Husserl and Phenomenology, Sartre and Existentialism, Saussure and Structuralism —each of which strives along a trajectory towards being all-encompassing, whether it intends to or not. Today, we could even say that Speculative Realism has embarked on this journey, and perhaps now we could say that the discourse of Speculative Realism has reached the inevitable point after which a grand theory leaves the hands of its writer or writers and becomes a possession of the collective consciousness of the academy, or (perhaps now) the mass consciousness of the blogosphere—and this is a true pharmakon , a poison and a cure, a blessing and a curse. 7 When we write theory, whether the scope is that of a grand theory or a particular theory, we write in the tension between compilation (making good on the desire to be all encompassing) and selection (being required to decide and discern and to judge what is included in the work and what is deleted or appended or abridged or given over to ellipses). Generally speaking, the work of theory-writing often comes out of a desire to totalize, that is, to develop a theory of everything that is able to apprehend new experiences and ideas while still remaining whole. I grant that this is not a universal desire, and that not everyone who sits down to write a work (or book) of theory does so because of their will-to-totality, but it remains that this drive to totalize does condition a great many writers of theory (especially those who seek to develop grand theories like those mentioned previously). At the beginning of his book of interviews, Between Existentialism and Marxism , Sartre is quoted as saying, after completing the first volume of his Critique of Dialectical Reason : “I no longer feel the need to make long digressions in my books, as if I were forever chasing after my own philosophy. It will now be deposited in little coffins, and I will feel completely emptied and at peace—as I felt after Being and Nothingness . A feeling of emptiness: a writer is fortunate if he can attain such a state. For when one has nothing to say, one can say everything .” 8 This is where our two initial distinctions come to bear on the being-towards-totality as it is expressed in the concrete practice of writing: the first being between parataxis and hypotaxis , and the second being between compilation and selection. Where this second distinction is concerned there is a certain paradox at play given that selection is inescapable, and compilation is unachievable. Selection is inescapable (with the figure of the Compendium in mind) because, even when the imperative to compile is followed to excruciating lengths, selection still has the last word. This is why the Compendium is a figure rather than some material thing that can actually be accomplished or actualized. No matter how far one goes along with the will-to-archive and the will-to-compile, it is only ever a question of minimizing selection and never eliminating it. This leads to the second point, which is that total compilation is unachievable. The closest thing to total compilation that we have before us is the ontological and textual fabric of the world. Even in the case of the world, compilation cannot be enacted in a total fashion because of the need to include both the sphere of the actual and the sphere of the possible or potential. This as another way in which compilation is always-already selection, and further evidence that total compilation is practically impossible. Discursive Figuration and Total Writing Rather than figuration referring to a figure of speech or an image, here the term points to a way in which to think about the work of writing, both the work put into writing, and the work that is the result of writing: the finished yet incomplete final piece. As figures or figurations, the Book and the Compendium are ways of thinking about the work of writing and the human desire for the wholeness, completion, and fulfillment that are made manifest in the completed form of the book (whether a published or printed or saved document put to rest by the author). These two figures bear more strongly upon works that seek to be all-encompassing, and works that strive towards expressions of totality such as the One, the Whole, or the All. Moving on to the topic of the subtitle, and on a more prescriptive note, I would say that there is more hope for theory-writing to be found in parataxis than hypotaxis . Part of the reason for this claim is the poststructuralist critique of hierarchies, and yet another part is the compelling line of thinking called “weak thought” (in theology by John D. Caputo, and in philosophy by Gianni Vattimo, among others). 9 The consequences of these two convictions are such that if one is to strongly assert the truth of a grand theory without leaving the realm of hierarchy-critique and weak thought, then one must write paradoxically with parataxis in mind, and in so doing write weakly and non-hierarchically. This does not mean avoiding a sort of topography or topology when writing theory, rather it means avoiding both subjugation and oppression in thought by pursuing a nonviolent sort of ontology. In order to do this, theory writing does not present itself as an exercise in logical analysis where one mechanically moves from a set of premises (via contingency) to the inevitable conclusion (via necessity). Instead, theory approaches the idea of a Compendium. Given that the figural Compendium places parataxis and compilation above hypotaxis and selection, then the question becomes: is placing one part of a binary term before another not just another way of selecting, or another expression of hypotaxis ? Counter to the urge to resign oneself to the reign of selection, with some qualification one can nonetheless write without deciding whether selection and hypotaxis should be subservient to compilation and parataxis (which is to say that writing-without-decision is somewhere between and beyond possibility and impossibility). This dilemma can be disarmed by stating that theory writing is not about primacy, and not about having-decided-beforehand, both stylistically and also where content is concerned. This is the paradox of writing: always striving along a trajectory ( telos ) towards totality via parataxis and compilation, but always being drawn back to finitude and making selections, and placing one idea before another with hypotaxis in mind. In light of this paradox, the question of the Compendium as a figure for discourse and a figural Book has resonance with Jacques Derrida’s essay on Edmund Jabès’ Book of Questions , found in his Writing and Difference . The Question of the Book “Little by little the book will finish me.” 10 Derrida quotes Jabès, and proceeds to outline the reflexive relationship between the author of the book and the book itself, each of which are subjected to the other through a sort of chiasmus, both ontological and textual (not that the two are entirely separable). The Book, for Derrida like Blanchot in the introductory quotation, “infinitely reflects itself” and “develops as a painful questioning of its own possibility”, and in light of this we can draw an association with the painful tension between the practical reality of selection and the ideal trajectory of compilation. 11 This tension in writing, between the will to total compilation and the necessity of abridgment, is an ontological tension between part and whole just as it is a textual tension between signifier and signified. The ontological tension in writing that Derrida expresses in his essay on Jabès is between everything and nothing – a contradiction which Derrida finds in Jabès’ Book of Questions and also in the divine, in God. When one writes between compilation and selection one writes towards totality, and here we can follow Derrida who states that the lapse in signification, presumably in the signifier-signified discrepancy, is a “ rupture with totality itself” and furthermore that this lapse cannot be rectified through deductive reason or even philosophical discourse. 12 This rupture with totality, found in the troubled relation between signifier and signified, can also tell us a lot about the Book and about writing. Given an understanding of writing as an ontological act that strives towards (but never accomplishes) totality, we can see the written book as the manifested and given body of that striving. Perhaps in other disciplines or even other schools of philosophy there is a cultural climate within which the article is prized above the book, but I am fairly confident, especially when referring to the grand theories of Continental thought, that there is a respect for the book as a uniquely meaningful object capable of apprehending totality. Derrida writes, Between the too warm flesh of the literal event and the cold skin of the concept runs meaning. This is how it enters into the book. Everything enters into, transpires in the book. This is why the book is never finite. It always remains suffering and vigilant. 13 Here the vitality of the literal event is compromised by the deadening weight of the concept, and the figure of the Book assists in this lapse of meaning. The Book is a totalizing object, much like the figure of the Compendium, and yet this totalization lacks its final object of completed totality both because the figural Book is necessarily incomplete, and because the physical book is always-already a product of selection. Instead of achieving its end of being a place where everything takes place, it suffers from the lapse of writing, the discrepancy between event and concept. Derrida continues his exposition on Jabès by bringing to light yet another reflexive relation, a reversal of the relationship between the Book and the world. Derrida writes that, for Jabès, “the book is not in the world, but the world is in the book.” 14 In addition to what was stated before, I take figuration to mean that the concepts employed, such as the Book or the Compendium, are not subject to the supposed rigor of analysis that is so prized by Enlightenment or Capitalist realisms. The figure of the Book and the figure of the Compendium cannot be held accountable to standards imported from scientific method, given an understanding that these standards require a concept to be replicable, consistent, falsifiable, measurable, and so on… This is an important point to make, especially in the present atmosphere within which theory must justify itself under conditions that are not its own. Instead of being held to these standards, figuration serves as a way in which to think about writing that leaves thought open to idealistic speculation, imperfect analogies, and most importantly the ever-present gain and loss that occur in the relationship between thought and being. By extension the figural way of addressing writing indulges in enough generalization to accommodate the excess/lack relationship between the ideal form of the Book or the Compendium, and the manifested and given body of a text (as it is practically completed and closed). Here we speak against the hostile atmosphere which would have theory submit itself to hierarchical rigor, rather than Blanchot’s “enigmatic rigor”, by being explicit about the discursive conditions and epistemic conditions under which theory operates. To write with parataxis in mind is, to a certain degree, to write with weakness as one’s methodology (with weak thought being in opposition to hypotaxis and hierarchy as much as weak thought can be in opposition to anything). Rather than asserting the strength of hierarchical theory, and rather than employing antagonistic argumentation with the goal of refutation different discursive and epistemic conditions for theory writing must be cultivated (and these are by no means new). First the critical and theoretical spirit reveals itself as being concerned with the task of complication—especially the complication and critique of binaries, dichotomies, dualities, polarities, paradoxes, parallaxes, hybridities, and especially antinomies. Second, theory positions itself as a sort of showing or revealing, rather than being ultimately focused on coming to full agreement or disagreement. This is where figuration can help theory-writing, both by placing emphasis on teleologies and trajectories (like parataxis and compilation), and by shifting focus away from pure primacy, power, or absolute origin. Figuration then, assists theory writing by placing the concern of theory outside of the concerns of the hard or soft sciences, and into the realm of thought or the idea. To speak of ‘the’ Compendium or ‘the’ Book, here, is to generalize not regarding the perfect form of the Compendium or Book, but to engage speculatively with an abstract idea which remains singular and yet complicated by multiplicity. The question of the Book that Derrida asks through Jabès is a question of everything and nothing, totality and nonbeing, and this question is a concern for writing and a concern for the figure of the Compendium so defined by the tension between compilation and selection, and parataxis and hypotaxis . On the note of nonbeing, we can look to the final page of Derrida’s first essay on Jabès in Writing and Difference and notice the introduction of his neologism, différance (with an ‘a’). Derrida writes: Life negates itself in literature only so that it may survive better. So that it may be better. It does not negate itself any more than it affirms itself: it differs from itself, defers itself, and writes itself as différance . Books are always books of life (the archetype would be the Book of Life kept by the God of the Jews) or of afterlife (the archetype would be the Books of the Dead kept by the Egyptians). 15 This introduction of différance into the equation of the Book, alongside ontological affirmation and negation, hearkens back to the discussion of the symbolic lapse earlier in his essay. Differing and deferring, the Book is always a Book of Life and a figure for the intersection of the vital (life) and the total in writing. In a way, the writer of the Book may be someone who lives life, and in another way the writer of the book-as-object may be someone who engages in the act of writing, both practically (by inscribing words on paper, or typing script on a computer) and ontologically or symbolically (by investing their existence and inexistence into a work worthy of the figural Book). In addition to the writer as writer, the writer also serves as an editor, and the editorial role alongside the idea of the Compendium shows the editor to be one who collects and compiles, while being restrained by eventual selection and decision. Beyond this the editor of works and texts, in both the Book and the world, is engaged in a process of inscribing notation—of annotating (on) the Compendium. Eternal commentary is a feature of the textual Compendium, just as open-ended totality describes the ontological Compendium. The writer as writer does not simply write texts, but rather engages in a profoundly ontological act of inscribing their existence into the world, and the writer as editor does not merely annotate or alter texts as they are, but rather comments upon the text of the world. In/Conclusion So as we create texts, and as we write and create theory, let us be and remain attentive to the figure of the Compendium and the figure of the Book. These two concurrent metonymies are essential for total writing (grand theories, etc.), and may be forgettable for those concerned with fragmentary or hierarchical writing (which are valid in their own right). From the ontological and symbolic text of the world and phenomenological experience, to concrete texts such as books or mixed media, the figure of the Compendium and the figure of the Book are important for the actualization of the human will to be all-encompassing. The figure of the Compendium teaches writers of theory that the repetition, density, and fragility of parataxis offers a strong sort of weakness which does not become yet another variety of oppressive and hegemonic thought. Rather than write with hypotaxis in mind, subjugating one thought to another via cause and effect or antecedent and consequent, I would hope that theory-writing could guiltlessly indulge in the dialectical and contradictory conjugation of ideas, and take this as a legitimate methodology. Rather than being ashamed of showing resonances or giving way to abridgment or ellipses, it is my own authorial (but not authoritative) conviction that one need not give up on totalizing and constructing grand theories because of the fact that complete totalization is impossible, and one need not give up on totalizing and constructing grand theories because of the worry of violence, for the Book and the Compendium lack their completed object and are ever incomplete trajectories. NOTES * UPDATED 7/19/13: we've updated the HTML version to reflect the (correct) PDF version. Our apologies to the author—eds. 1. I would like extend thanks to Dr. Peter Schwenger of the University of Western Ontario for his comments on this paper and his hospitality as it was presented as a Theory Session at the Centre for Theory and Criticism on October 26th 2012. I would also like to thank Andrew Weiss for conversation and critique. 2. I should clarify my use of the term ‘figure’ at the outset. Given the use of the term by Jean-Francois Lyotard in Discourse, Figure (and also Gilles Deleuze in Francis Bacon ), I should state clearly that my use of the term will not correspond to the term ‘figure’ understood as the representation of an object. Instead, my use of the term ‘figure’ and its variations (‘figural’, ‘figurative’, ‘figuration’) will refer to the illustrative or metaphorical use of the Compendium or the Book as models or ways of thinking about writing and discourse. 3. Cf. Revelations 20:12-15. 4. Theodor Adorno, Aesthetic Theory , Trans. Robert Hullot-Kentor (London & New York: Continuum Press, 1997). 5. Fredric Jameson, Late Marxism: Adorno, or the Persistence of the Dialectic (London & New York: Verso, 1990), ix-x. 6. I have written about this concept of being-towards-totality elsewhere in two pieces of writing: the first is called Notes on the Compendium (an unfinished draft) which addresses some very wide theoretical concerns, being structured as a sort of itinerary or archive, and the second is Dialectics Unbound (Punctum Books, 2013) which outlines a concept of totality without the violence of totalization. While these works are more concerned with ontological totality, here I would like to focus on the idea of the Compendium as a textual totality. 7. Cf. Louis Morelle, “Speculative Realism: After Finitude and Beyond, A vade mecum” in Speculations: Journal of Speculative Realism . Issue 3 (Brooklyn, New York: Punctum Books, 2012), 241-272. 8. Jean-Paul Sartre, Between Existentialism and Marxism . Trans. John Matthews (London & New York: Verso, 1974), 9. 9. Cf. John D. Caputo, The Weakness of God (Indiana: Indiana University Press, 2006). 10. Jacques Derrida, “Edmond Jabès and the Question of the Book” in Writing and Difference . Trans. Alan Bass (Chicago: University of Chicago Press, 1978), 65. 11. Ibid, 65. 12. Ibid, 71. 13. Ibid, 75. 14. Ibid, 76. 15. Ibid, 78. (shrink)

In this paper I put forward a new micro realistic, fundamentally probabilistic, propensiton version of quantum theory. According to this theory, the entities of the quantum domain - electrons, photons, atoms - are neither particles nor fields, but a new kind of fundamentally probabilistic entity, the propensiton - entities which interact with one another probabilistically. This version of quantum theory leaves the Schroedinger equation unchanged, but reinterprets it to specify how propensitons evolve when no probabilistic transitions occur. Probabilisitic transitions occur (...) when new "particles" are created as a result of inelastic interactions. All measurements are just special cases of this. This propensiton version of quantum theory, I argue, solves the wave/particle dilemma, is free of conceptual problems that plague orthodox quantum theory, recovers all the empirical success of orthodox quantum theory, and at the same time yields as yet untested predictions that differ from those of orthodox quantum theory. (shrink)

A fully micro realistic, propensity version of quantum theory is proposed, according to which fundamental physical entities - neither particles nor fields - have physical characteristics which determine probabilistically how they interact with one another . The version of quantum "smearon" theory proposed here does not modify the equations of orthodox quantum theory: rather, it gives a radically new interpretation to these equations. It is argued that there are strong general reasons for preferring quantum "smearon" theory to orthodox (...) quantum theory; the proposed change in physical interpretation leads quantum "smearon" theory to make experimental predictions subtly different from those of orthodox quantum theory. Some possible crucial experiments are considered. (shrink)

According to orthodox quantum mechanics, state vectors change in two incompatible ways: "deterministically" in accordance with Schroedinger's time-dependent equation, and probabilistically if and only if a measurement is made. It is argued here that the problem of measurement arises because the precise mutually exclusive conditions for these two types of transitions to occur are not specified within orthodox quantum mechanics. Fundamentally, this is due to an inevitable ambiguity in the notion of "meawurement" itself. Hence, if the problem of measurement is (...) to be resolved, a new, fully objective version of quantjm mechanics needs to be developed which does not incorporate the notion of measurement in its basic postuolates at all. (shrink)

The Maxwell equations are shown to be the one-photon spin-one quantum equations. All Maxwell equations (without sources) are derived simultaneously from first principles, similar to those which have been used to derive the Dirac relativistic electron equation. The wavefunction is a linear combination of the electric and magnetic fields. The procedure is not unique, there are ambiguities of adding a scalar field. A quaternionic representation of the Maxwell equations (with sources) is constructed, a (...) covariant reformulation of which is presented. Whittaker potentials are analysed. Conservation laws are derived using a method of “pseudo-Lagrangians.”. (shrink)

In this paper the Lorentz transformations (LT) and the standard transformations (ST) of the usual Maxwell equations (ME) with the three-dimensional (3D) vectors of the electric and magnetic fields, E and B, respectively, are examined using both the geometric algebra and tensor formalisms. Different 4D algebraic objects are used to represent the usual observer dependent and the new observer independent electric and magnetic fields. It is found that the ST of the ME differ from their LT and consequently (...) that the ME with the 3D E and B are not covariant upon the LT but upon the ST. The obtained results do not depend on the character of the 4D algebraic objects used to represent the electric and magnetic fields. The Lorentz invariant field equations are presented with 1-vectors E and B, bivectors EHv and BHv and the abstract tensors, the 4-vectors Ea and Ba. All these quantities are defined without reference frames, i.e., as absolute quantities. When some basis has been introduced, they are represented as coordinate-based geometric quantities comprising both components and a basis. It is explicitly shown that this geometric approach agrees with experiments, e.g., the Faraday disk, in all relatively moving inertial frames of reference, which is not the case with the usual approach with the 3D bf E and B and their ST. (shrink)

It is shown that the usual procedures of obtaining the macroscopic Maxwell equations from the microscopic Maxwell-Lorentz equations by performing averages contain an arbitrary choice of gauge. By a suitable different choice of the gauge the so-obtained Maxwell equations can be cast back to the form of the starting Maxwell-Lorentz equations. Therefore one cannot consider the Maxwell equations to be obtainable from the Maxwell-Lorentz equations by simply performing averages. (...) The implication of this result is that besides the electromagnetic fields produced by the moving electric charges, as given by the Maxwell-Lorentz equations, there may be some other agents that cannot be identified as some kind of motion of the electric charges and that participate in the production of the electromagnetic fields. (shrink)

On the basis of the exact solutions to the non-stationary spherically symmetric Einstein and Maxwell equations for dust matter and radial electromagnetic field, a model of a wormhole with the pulsating in time inner world and two static throats has been developed. It has been shown that such a wormhole with an arbitrary radius of the Gaussian curvature can connect both two different asymptotically flat space-times and two regions of the selfsame space-time. The problem of the fulfilment of (...) the energy conditions in this wormhole has been investigated, as well as the problem of its traversability investigation has been set. (shrink)

In a rather old paper, Mario Liu described a hydrodynamic Maxwell equations. While he also discussed potential implications of these new approaches to superconductors, such a discussion of electrodynamics of superconductors is made only after Tajmar’s paper. Therefore, in this paper we present for the first time a derivation of fluidic Maxwell-Proca equations. The name of fluidic Maxwell-Proca is proposed because the equations were based on modifying Maxwell-Proca and Hirsch’s theory of electrodynamics of (...) superconductor. It is hoped that this paper may stimulate further investigations and experiments in superconductor. It may be expected to have some impact to cosmology modeling too, for instance we consider a hypothetical argument that photon mass can be origin of gravitation. Then, after combining with the so-called chiral modification of Maxwell equations (after Spröessig), then we consider chiral Maxwell-Proca equations as possible alternative of gravitation theory. Such a hypothesis has never considered in literature to the best of our knowledge. (shrink)

A detailed outline is presented of several convergent points of view connecting the self-dual and anti-self-dual fields with their free data. This is done for the Maxwell and for linearized gravity as exemplifying the approaches. The Sparling equation provides one tool of great power and characterizes one approach. The twistor theory of Penrose yields another equally powerful point of view. The links between these two basic approaches given in this paper provide a unification that allows workers and others with (...) interest in this area to proceed more readily toward the goal of understanding the full nonlinear Einstein equations. (shrink)

In recent years, there are a number of proposals to consider collision-based soliton computer based on certain chemical reactions, namely Belousov-Zhabotinsky reaction, which leads to soliton solutions of coupled Nonlinear Schroedinger equations. They are called Manakov System. But it seems to us that such a soliton computer model can also be based on solitary wave solution of Maxwell-Dirac equation, which reduces to Choquard equation. And soliton solution of Choquard equation has been investigated by many researchers, therefore it seems (...) more profound from physics perspective. However, we consider both schemes of soliton computer are equally possible. More researches are needed to verify our proposition. (shrink)

We present a mechanical model of a quasi-elastic body which reproduces Maxwell’s equations with charges and currents. Major criticism against mechanical models of electrodynamics is that any presence of charges in the known models appears to violate the continuity equation of the aether and it remains a mystery as to where the aether goes and whence it comes. We propose a solution to the mystery—in the present model the aether is always conserved. Interestingly it turns out that the (...) charge velocity coincides with the aether velocity. In other words, the charges appear to be part of the aether itself. We interpret the electric field as the flux of the aether and the magnetic field as the torque per unit volume. In addition we show that the model is consistent with the theory of relativity, provided that we use Lorentz–Poincare interpretation of relativity theory. We make a statistical-mechanical interpretation of the Lorentz transformations. It turns out that the length of a body is contracted by the electromagnetic field which the molecules of this same body produce. This self-interaction causes also delay of all the processes and clock-dilation results. We prove this by investigating the probability distribution for a gas of self-interacting particles. We can easily extend this analysis even to elementary particles. (shrink)

The aim of this paper is to illustrate four properties of the non-relativistic limits of relativistic theories: that a massless relativistic field may have a meaningful non-relativistic limt, that a relativistic field may have more than one non-relativistic limit, that coupled relativistic systems may be "more relativistic" than their uncoupled counterparts, and that the properties of the non-relativistic limit of a dynamical equation may differ from those obtained when the limiting equation is based directly on exact Galilean kinematics. These properties (...) are demonstrated through an examination of the non-relativistic limit of the familiar equations of first-quantized QED, i.e., the Dirac and Maxwell equations. The conditions under which each set of equations admits non-relativistic limits are given, particular attention being given to a gauge-invariant formulation of the limiting process especially as it applied to the electromagnetic potentials. The difference between the properties of a limiting theory and an exactly Galilean covariant theory based on the same dynamical equation is demonstrated by examination of the Pauli equation. (shrink)

We present the Dirac and Laplacian operators on Clifford bundles over space–time, associated to metric compatible linear connections of Cartan–Weyl, with trace-torsion, Q. In the case of nondegenerate metrics, we obtain a theory of generalized Brownian motions whose drift is the metric conjugate of Q. We give the constitutive equations for Q. We find that it contains Maxwell’s equations, characterized by two potentials, an harmonic one which has a zero field (Bohm-Aharonov potential) and a coexact term that (...) generalizes the Hertz potential of Maxwell’s equations in Minkowski space.We develop the theory of the Hertz potential for a general Riemannian manifold. We study the invariant state for the theory, and determine the decomposition of Q in this state which has an invariant Born measure. In addition to the logarithmic potential derivative term, we have the previous Maxwellian potentials normalized by the invariant density. We characterize the time-evolution irreversibility of the Brownian motions generated by the Cartan–Weyl laplacians, in terms of these normalized Maxwell’s potentials. We prove the equivalence of the sourceless Maxwell equation on Minkowski space, and the Dirac-Hestenes equation for a Dirac-Hestenes spinor field written on Minkowski space provided with a Cartan–Weyl connection. If Q is characterized by the invariant state of the diffusion process generated on Euclidean space, then the Maxwell’s potentials appearing in Q can be seen alternatively as derived from the internal rotational degrees of freedom of the Dirac-Hestenes spinor field, yet the equivalence between Maxwell’s equation and Dirac-Hestenes equations is valid if we have that these potentials have only two components corresponding to the spin-plane. We present Lorentz-invariant diffusion representations for the Cartan–Weyl connections that sustain the equivalence of these equations, and furthermore, the diffusion of differential forms along these Brownian motions. We prove that the construction of the relativistic Brownian motion theory for the flat Minkowski metric, follows from the choices of the degenerate Clifford structure and the Oron and Horwitz relativistic Gaussian, instead of the Euclidean structure and the orthogonal invariant Gaussian. We further indicate the random Poincaré–Cartan invariants of phase-space provided with the canonical symplectic structure. We introduce the energy-form of the exact terms of Q and derive the relativistic quantum potential from the groundstate representation. We derive the field equations corresponding to these exact terms from an average on the invariant state Cartan scalar curvature, and find that the quantum potential can be identified with 1 / 12R(g), where R(g) is the metric scalar curvature. We establish a link between an anisotropic noise tensor and the genesis of a gravitational field in terms of the generalized Brownian motions. Thus, when we have a nontrivial curvature, we can identify the quantum nonlocal correlations with the gravitational field. We discuss the relations of this work with the heat kernel approach in quantum gravity. We finally present for the case of Q restricted to this exact term a supersymmetric system, in the classical sense due to E.Witten, and discuss the possible extensions to include the electromagnetic potential terms of Q. (shrink)

The fact that the same equations or mathematical models reappear in the descriptions of what are otherwise disparate physical systems can be seen as yet another manifestation of Wigner's “unreasonable effectiveness of mathematics.” James Clerk Maxwell famously exploited such formal similarities in what he called the “method of physical analogy.” Both Maxwell and Hermann von Helmholtz appealed to the physical analogies between electromagnetism and hydrodynamics in their development of these theories. I argue that a closer historical examination (...) of the different ways in which Maxwell and Helmholtz each deployed this analogy gives further insight into debates about the representational and explanatory power of mathematical models. (shrink)

Close insight into mathematical and conceptual structure of classical field theories shows serious inconsistencies in their common basis. In other words, we claim in this work to have come across two severe mathematical blunders in the very foundations of theoretical hydrodynamics. One of the defects concerns the traditional treatment of time derivatives in Eulerian hydrodynamic description. The other one resides in the conventional demonstration of the so-called Convection Theorem. Both approaches are thought to be necessary for cross-verification of the standard (...) differential form of continuity equation. Any revision of these fundamental results might have important implications for all classical field theories. Rigorous reconsideration of time derivatives in Eulerian description shows that it evokes Minkowski metric for any flow field domain without any previous postulation. Mathematical approach is developed within the framework of congruences for general four-dimensional differentiable manifold and the final result is formulated in form of a theorem. A modified version of the Convection Theorem provides a necessary cross-verification for a reconsidered differential form of continuity equation. Although the approach is developed for one-component (scalar) flow field, it can be easily generalized to any tensor field. Some possible implications for classical electrodynamics are also explored. (shrink)

Maxwell has shown in his field equations that electricity and magnetism, which are thought to be two separate phenomena, are in fact two distinct components of a single phenomenon. Maxwell not only developed these fundamental equations but used them to predict the existence of electromagnetic waves and to show that light is an electromagnetic wave. In this study, I researched on the emergence and effects of Maxwell's field equations using data from the history of (...) physics. As a result of the evaluation, I emphasized the revolutionary character of Maxwell's way of explaining the functioning of the universe and identified certain insufficient points. In conclusion, I tried to show that new explanations are inevitable in order to understand the universe and to show the role of mathematics in these explanations. (shrink)

This paper is a review of the canonical proper-time approach to relativistic mechanics and classical electrodynamics. The purpose is to provide a physically complete classical background for a new approach to relativistic quantum theory. Here, we first show that there are two versions of Maxwell’s equations. The new version fixes the clock of the field source for all inertial observers. However now, the (natural definition of the effective) speed of light is no longer an invariant for all observers, (...) but depends on the motion of the source. This approach allows us to account for radiation reaction without the Lorentz-Dirac equation, self-energy (divergence), advanced potentials or any assumptions about the structure of the source. The theory provides a new invariance group which, in general, is a nonlinear and nonlocal representation of the Lorentz group. This approach also provides a natural (and unique) definition of simultaneity for all observers.The corresponding particle theory is independent of particle number, noninvariant under time reversal (arrow of time), compatible with quantum mechanics and has a corresponding positive definite canonical Hamiltonian associated with the clock of the source.We also provide a brief review of our work on the foundational aspects of the corresponding relativistic quantum theory. Here, we show that the standard square-root and Dirac equations are actually two distinct spin- $\frac{1}{2}$ particle equations. (shrink)

A bivertical classical field theory includes the Newtonian mechanics and Maxwell's electromagnetic field theory as the special cases. This unification allows one to recognize the formal analogies among Newtonian mechanics and Maxwell's electrodynamics.

Requiring covariance of Maxwell's equations without a priori imposing charge invariance allows for both spin-1 and spin-1/2 transformations of the complete Maxwell field and current. The spin-1/2 case yields new transformation rules, with new invariants, for all traditional Maxwell field and source quantities. The accompanying spin-1/2 representations of the Lorentz group employ the Minkowski metric, and consequently the primary spin-1/2 Maxwell invariants are also spin-1 invariants; for example, Φ2−A2, E2−B2+2iE⋅B−2. The associated Maxwell Lagrangian density (...) is also the same for both spin-1 and spin-1/2 fields. However, in the spin-1/2 case, standard field and source quantities are complex and both charge and gauge invariance are lost. Requiring the potentials to satisfy the Klein–Gordon equation equates the Maxwell and field-potential equations with two Dirac equations of the Klein–Gordon mass, and thus one complex Klein–Gordon Maxwell field describes either two real vector fields or two Dirac fields, all of the same mass. (shrink)

It is shown that an approach to quantum phenomena in which charged particles are treated as macroscopically extended periodic disturbances in a nonlinear c-number field, interacting with each other via massless excitations of that field, leads almost uniquely to the five basic equations of classical electrodynamics: the Lorentz force law and Maxwell's equations. The fundamental electromagnetic quantity in this approach is the 4-vector potential Aα—interpreted absolutely as a measure of the local shift of each particle off its (...) mass shell—rather than theE andB fields, and it thus provides a new viewpoint on the questions of Aharonov-Bohm phase shifts, the existence of magnetic monopoles, and the role of gauge invariance. (shrink)

In this paper I examine the notion and role of metaphors and illustrations in Maxwell's works in exact science as a pathway into a broader and richer philosophical conception of a scientist and scientific practice. While some of these notions and methods are still at work in current scientific research-from economics and biology to quantum computation and quantum field theory-, here I have chosen to attest to their entrenchment and complexity in actual science by attempting to make some conceptual (...) sense of Maxwell's own usage; this endeavour includes situating Maxwell's conceptions and applications in his own culture of Victorian science and philosophy. I trace Maxwell's notions to the formulation of the problem of understanding, or interpreting, abstract representations such as potential functions and Lagrangian equations. I articulate the solution in terms of abstract-concrete relations, where the concrete, in tune with Victorian British psychology and engineering, includes the muscular as well as the pictorial. This sets the basis for a conception of understanding in terms of unification and concrete modelling, or representation. I examine the relation of illustration to analogies and metaphors on which this account rests. Lastly, I stress and explain the importance of context-dependence, its consequences for realism-instrumentalism debates, and Maxwell's own emphasis on method. (shrink)

Building upon work by Mary Hesse (1974), this paper aims to show that a single method of investigation lies behind Maxwell’s use of physical analogies in his major scientific works before the Treatise on Electricity and Magnetism. Key to understanding the operation of this method is to recognize that Maxwell’s physical analogies are intended to possess an ‘inductive’ function in addition to an ‘illustrative’ one. That is to say, they not only serve to clarify the equations proposed (...) for an unfamiliar domain with a working interpretation drawn from a more familiar science, but can also be sources of defeasible yet relatively strong arguments from features of the more familiar domain to features of the less. Compared with the reconstructions by Achinstein (1991), Siegel (1991), Harman (1998) and others, which postulate a discontinuity in Maxwell’s approach to physical analogy, the account defended in this paper i) makes sense of the continuity in Maxwell’s remarks on scientific methodology, ii) explains his quest for a “mathematical classification of physical quantities” and iii) offers a new and more plausible interpretation of the debated episode of the introduction of the displacement current in Maxwell’s “On Physical Lines of Forces”. (shrink)

Recent advances on quantum foundations achieved the derivation of free quantum field theory from general principles, without referring to mechanical notions and relativistic invariance. From the aforementioned principles a quantum cellular automata theory follows, whose relativistic limit of small wave-vector provides the free dynamics of quantum field theory. The QCA theory can be regarded as an extended quantum field theory that describes in a unified way all scales ranging from an hypothetical discrete Planck scale up to the usual Fermi scale. (...) The present paper reviews the automaton theory for the Weyl field, and the composite automata for Dirac and Maxwell fields. We then give a simple analysis of the dynamics in the momentum space in terms of a dispersive differential equation for narrowband wave-packets. We then review the phenomenology of the free-field automaton and consider possible visible effects arising from the discreteness of the framework. We conclude introducing the consequences of the automaton dispersion relation, leading to a deformed Lorentz covariance and to possible effects on the thermodynamics of ideal gases. (shrink)

Maxwell accounted for the apparent elastic behavior of the electromagnetic field by augmenting Ampere’s law with the so-called displacement current, in much the same way that he treated the viscoelasticity of gases. Maxwell’s original constitutive relations for both electrodynamics and fluid dynamics were not material invariant. In the theory of viscoelastic fluids, the situation was later corrected by Oldroyd, who introduced the upper-convective derivative. Assuming that the electromagnetic field should follow the general requirements for a material field, we (...) show that if the upper convected derivative is used in place of the partial time derivative in the displacement current term, Maxwell’s electrodynamics becomes material invariant. Note, that the material invariance of Faraday’s law is automatically established if the Lorentz force is admitted as an integral part of the model. The new formulation ensures that the equation for conservation of charge is also material invariant in vacuo. The viscoelastic medium whose apparent manifestation are the known phenomena of electrodynamics is called here the metacontinuum. (shrink)

We discuss, within the framework provided by a recently developed variational method, transposition-invariant field equations for unified field theories. Systems that are, in addition, invariant under Weyl-type gauge transformations or lambda transformations are derived. It is found that in a weak field limit two of the systems contain the equations of general relativity and the covariant Maxwell equations for a charge-free region.

Relativistic quantum mechanics has been formulated as a theory of the evolution ofevents in spacetime; the wave functions are square-integrable functions on the four-dimensional spacetime, parametrized by a universal invariant world time τ. The representation of states with spin is induced with a little group that is the subgroup of O(3, 1) leaving invariant a timelike vector nμ; a positive definite invariant scalar product, for which matrix elements of tensor operators are covariant, emerges from this construction. In a previous study (...) a second-order equation was introduced similar to the second-order Dirac equation, based on a quadratic function of two operators which are the self-adjoint parts, in this new scalar product, of γμpμ. It is shown in this paper that one of these operators, in fact the one from which the gyromagnetic moment is obtained, can be used to construct a first-order equation. The corresponding quantum theory is somewhat analogous to Dirac's spinor form; the Hamilton equations appear to describe dynamical degrees of freedom in a spacelike hyperplane orthogonal to nμ (in Dirac's theory the motion appears to be lightlike). It is shown that the integration over nμ required by unitarity results in timelike motion (as in the expectation value of Dirac's α). Explicit forms are obtained for the wave functions and currents for free motion. The general form of the theory is written for the (five-dimensional) pre-Maxwell fields required by gauge invariance. (shrink)

Maxwell's equations are formulated as a relativistic “Schrödinger-like equation” for a single photon of a given helicity. The probability density of the photon satisfies an equation of continuity. The energy eigenvalue problem gives both positive and negative energies. The Feynman concept of antiparticles is applied here to show that the negative-energy states going backward in time (t → −t) give antiphoton states, which are photon states with the opposite helicity. For a given mode, properties of a photon, such (...) as energy, linear momentum, total angular momentum, orbital angular momentum, and spin are equivalent in both classical electromagnetic field theory and quantum theory. The single-photon Schrödinger equation is field (or “second”) quantized in a gauge-invariant way to obtain the quantum field theory of many photons. This approach treats the quantization of electromagnetic radiation in a way that parallels the quantization of material particles and, thus, provides a unified treatment. (shrink)

James Clerk Maxwell’s 1865 paper, “A Dynamical Theory of the Electromagnetic Field,” is usually remembered as replacing the mechanical model that underpins his 1862 publication with abstract mathematics. Up to this point historians have considered Maxwell’s usage of Lagrangian dynamics as the sole important feature that guides Maxwell’s analysis of electromagnetic phenomena in his 1865 publication. This paper offers an account of the often ignored mechanical analogy that Maxwell used to guide him and his readers in (...) the construction of his new electromagnetic equations. The mechanical system consists of a weighted flywheel geared into two independently driven crank wheels in what amounts to a mechanical differential. I will demonstrate how Maxwell made use of the analogy between his flywheel system and electromagnetic induction to ground his study of electromagnetism in clear mechanical conceptions and to structure the derivation of the equations that together are now recognized as Maxwell’s equations for electrodynamics. By reconceiving specific components of his model in electromagnetic terms, while at the same time retaining many of the relations between concepts in the mechanical case, Maxwell gradually assembled increasingly generalized equations for electromotive force. Maxwell thus realized a much sought after balance between physical analogy and abstract mathematics in this, the last of his three seminal papers on electromagnetism. (shrink)

We review the relativistic classical and quantum mechanics of Stueckelberg, and introduce the compensation fields necessary for the gauge covariance of the Stueckelbert–Schrödinger equation. To achieve this, one must introduce a fifth, Lorentz scalar, compensation field, in addition to the four vector fields with compensate the action of the space-time derivatives. A generalized Lorentz force can be derived from the classical Hamilton equations associated with this evolution function. We show that the fifth (scalar) field can be eliminated through the (...) introduction of a conformal metric on the spacetime manifold. The geodesic equation associated with this metric coincides with the Lorentz force, and is therefore dynamically equivalent. Since the generalized Maxwell equations for the five dimensional fields provide an equation relating the fifth field with the spacetime density of events, one can derive the spacetime event density associated with the Friedmann–Robertson–Walker solution of the Einstein equations. The resulting density, in the conformal coordinate space, is isotropic and homogeneous, decreasing as the square of the Robertson–Walker scale factor. Using the Einstein equations, one see that both for the static and matter dominated models, the conformal time slice in which the events which generate the world lines are contained becomes progressively thinner as the inverse square of the scale factor, establishing a simple correspondence between the configurations predicted by the underlying Friedmann–Robertson–Walker dynamical model and the configurations in the conformal coordinates. (shrink)

In this article some intriguing aspects of electromagnetic theory and its relation to mathematics and reality are discussed, in particular those related to the suppositions needed to obtain the wave equations from Maxwell equations and from there Helmholtz equation. The following questions are discussed. How is that equations obtained with so many irreal or fictitious assumptions may provide a description that is in a high degree verifiable? Must everything that is possible to deduce from a theoretical (...) mathematical model occur in the world? Does everything that takes place in the world have a mathematical description? (shrink)

It has recently been shown by Vargas, (4) that the passive coordinate transformations that enter the Robertson test theory of special relativity have to be considered as coordinate transformations in a seven-dimensional space with degenerate metric. It has also been shown by Vargas that the corresponding active coordinate transformations are not equal in general to the passive ones and that the composite active-passive transformations act on a space whose number of dimensions is ten (one-particle case) or larger (more than one (...) particle).In this paper, two different (families of) electrodynamics are constructed in ten-dimensional space upon the coordinate free form of the Maxwell and Lorentz equations. The two possibilities arise from the two different assumptions that one can naturally make with respect to the acceleration fields of charges, when these fields are related to their relativistic counterparts. Both theories present unattractive features, which indicates that the Maxwell-Lorentz framework is unsuitable for the construction of an electrodynamics for the Robertson test theory of the Lorentz transformations. It is argued that this construction would first require the formulation of Maxwell-Lorentz electrodynamics in the form of a connection in Finsler space. If such formulation is possible, the sought generalization would consist in simply changing bases in the tangent spaces of the manifold that supports the connection. In addition, the number of dimensions of the space of the Robertson transformations would be ten, but not greater than ten. (shrink)

We explore certain difficulties in the covariant classical mechanics associated with off-shell electrodynamics, through an examination of the classical Coulomb problem. We present a straightforward solution of the classical equations of motion for a test event traversing the field induced by a “fixed” event (an event moving uniformly along the time axis at a fixed point in space). This solution reveals the essential difficulties in the formalism at the classical level. We then offer a new model of the particle, (...) as a certain distribution of events on the worldline, which eliminates these difficulties and permits comparison of classical off-shell electrodynamics with the standard Maxwell theory In this model, the “fixed” event induces a Yukawa-type potential, permitting a semiclassical identification of the pre-Maxwell time scale λ with the inverse mass of the intervening photon. Numerical solutions to the equations of motion are compared with the standard Maxwell solutions—they are seen to coincide when λ > 10–6 sec, providing an initial estimate of this parameter. (shrink)

We explore certain difficulties in the covariant classical mechanics associated with off-shell electrodynamics, through an examination of the classical Coulomb problem. We present a straightforward solution of the classical equations of motion for a test event traversing the field induced by a “fixed” event (an event moving uniformly along the time axis at a fixed point in space). This solution reveals the essential difficulties in the formalism at the classical level. We then offer a new model of the particle, (...) as a certain distribution of events on the worldline, which eliminates these difficulties and permits comparison of classical off-shell electrodynamics with the standard Maxwell theory In this model, the “fixed” event induces a Yukawa-type potential, permitting a semiclassical identification of the pre-Maxwell time scale λ with the inverse mass of the intervening photon. Numerical solutions to the equations of motion are compared with the standard Maxwell solutions—they are seen to coincide when λ > 10–6 sec, providing an initial estimate of this parameter. (shrink)

The quantum corrections related to the ideal gas model often considered are those associated to the bosonic or fermionic nature of particles. However, in this work, other kinds of corrections related to the quantum nature of phase space are highlighted. These corrections are introduced as improvements in the expression of the partition function of an ideal gas. Then corrected thermodynamics properties of the ideal gas are deduced. Both the non-relativistic quantum and relativistic quantum cases are considered. It is shown that (...) the corrections in the non-relativistic quantum case may be particularly useful to describe the deviation from the Maxwell–Boltzmann model at low temperature and/or in confined space. These corrections can be considered as including the description of quantum size and shape effects. For the relativistic quantum case, the corrections could be relevant for confined space and when the thermal energy of each particle is comparable to their rest energy. The corrections appear mainly as modifications in the thermodynamic equation of state and in the expressions of the partition function and thermodynamic functions like entropy, internal energy and free energy. Expressions corresponding to the Maxwell–Boltzmann model are shown to be asymptotic limits of the corrected expressions. (shrink)

Originally published in 1973, this book shows that methods developed for the semantics of systems of formal logic can be successfully applied to problems about the semantics of natural languages; and, moreover, that such methods can take account of features of natural language which have often been thought incapable of formal treatment, such as vagueness, context dependence and metaphorical meaning. Parts 1 and 2 set out a class of formal languages and their semantics. Parts 3 and 4 show that these (...) formal languages are rich enought to be used in the precise description of natural languages. Appendices describe some of the concepts discussed in the text. (shrink)

A Dirac-like equation for a massive field obeying the classical Proca equations of motion (PMO) is proposed in close analogy with Majorana’s construct for Maxwell electrodynamics. Its underlying algebraic structure is examined and a plausible physical interpretation is discussed. The behavior of the PMO equations in the presence of an external electromagnetic field is also investigated in the low energy limit, via unitary transformations similar to the Foldy-Wouthuysen canonical transformation for a Dirac fermion.