a technical process of converting our paper-borne heritage into electronic form -is dramatically insufficient. To paraphrase Clemenceau's famous words about war and militaries, digitization may be too serious a thing to be left to the digitizers alone. Scholars must face the issue and understand it as one of the most important problems they have to deal with and, as I will argue, as a real opportunity to renew their practices and disciplines.

To start with some affective considerations, which, after all, matter too, digitization is a major source of anxiety for the Humanities. Not least because the world of computers was radically strange to scholars, many of whom spent a long time using computing machines and word processors often in quiet ignorance of the underlying processes. Moreover, there is some resistance from scholars, long familiar with browsing, annotating and cherishing their books, to iPads and other devices, which allow similar practices and may even evoke similar feelings in the process. Deeper than that, though, they are disturbed by the idea that, along with the possible evanescence of the final printed output, the material signs of distinction and auctoritas may also disappear. For digitization is also a transition from a universe of tangible books or academic journals, whose recognizable appearance evidences interest in content and the dignity of authors, to an undifferentiated situation of Big Commensurability, deserted by the immediate material marks of reliability and reading-worthiness. The dramatic issue is: what is the future of philological probity and other humanistic ideals in a world where the ultimate measure is that of the byte, according to which, millions of contentless electronic messages have greater weight than the Quixote? To cope with this central question, we should perceive digitization not as a technical issue, but as a whole process of appropriately integrating the tiny hand of our cultural heritage in the ocean of digitally-borne information.

The Empire of Digital

It is a commonplace that the ways of doing and communicating science are rapidly changing under the ubiquitous influence of computers. The widespread use of these machines results in an epistemic enhancement. This enhancement has several aspects. First, and most manifestly, it consists of an extension of our native capacities as paper readers: we can now be acquainted with many more items than those we previously had access to. Overcoming the necessity of travelling to libraries and of waiting for book delivery, our new databases and repositories lead us to a position of consulting and browsing more material than before. In this way, we are doing approximately the same job but at higher speed, at a wider range and in greater comfort. Second, and more importantly, this enhancement may be a qualitative augmentation of the scope of scholarship, giving us access to features of cultural heritage that we were not equipped to detect in previous times. The frequency of occurrences of given words in wide corpora, the dynamics of their evolution or the regularities of their coexistence with other words are typical features of this kind. Third, the treatment and analysis of such massive data often involves conversion of the modalities of our work; for example, when we are led to visualize these data by means of curves or other iconic ways of representation. In such cases, the very task of reading cannot be done by the human eye and it is left to machines: we simply deal with the result of their work, using a radical approach to our cultural heritage that was previously unavailable.

To grasp the meaning of the transformation just described, scholars should firstly realize that their field is not at all as specific as they use to believe, and that the DIgital Turn is today affecting the whole of scientific enterprise in a similar way. The Galilean ideal of reading Nature in mathematical language, namely of understanding and foreseeing it in the analytic way, by means of appropriate differential equations, is today in question. To restrict to some examples, dynamics of fluids has become a largely computerized discipline, not only because we use computers to solve numerically equations that are analytically unsolvable in most cases (that use of computers would correspond to the extensive meaning of "epistemic enhancement" above), but because the detail of a turbulent flow cannot even be suitably represented and analyzed except via a super-computer. 1 The same consideration of the complexity of data issue applies, even more eloquently in the present context, to the "reading" of the human genome, which can obviously not be done by man alone: the database currently used encompasses more than five times the amount of information contained in the Library of Congress. To sum up, the most intriguing features of Digital Humanities are nowadays shared by classical domains of hard sciences. One has just to ask, in the general landscape of this New Kind of Science, what is the specific object of the humanistic brand of this general evolution.

Philology aufgehoben

The best way of characterizing the Digital Humanities is probably to contrast them with the tradition of the past half-century in 'continental' philosophy. People in this trend used to describe their activity as 'interpretive' or 'hermeneutic'. Considering that the Masters of Suspicion (Marx, Nietzsche and Freud, to quote some of them) had definitely deconstructed the traditional idea of philosophy as a variety of the pursuit of truth, they have emphasized the interpretive nature of any enterprise of understanding. The salient issue, in this perspective, is the adjudication of the standards of correctness for the interpretations. Either one faces the problem seriously, and one must, therefore, be prepared to admit an exception to the move beyond True and False. Or, one assumes integrally the relativist motto of hermeneutics, and one has to make interpretation itself subject to interpretation, not to a justifiable verdict of correctness or incorrectness.

The second horn of the dilemma has been clearly preferred, positing the idea that any interpretation is legitimate, insofar as it is expressive enough of the idiosyncratic situation of the interpreter: briefly expressed, interpretation becomes a genuine act of creation by the reader. No surprise, then, that in these conditions philology or 'objectivist' scholarship were dismissed in recent times: their relegation is on a par with the fashionable idea that the meaning of a text is just what the reader attributes to it:

The relegation of writing to the indeterminate and endlessly transforming processes of textual dissemination is a by-product of Saussurian linguistics and some of the structuralist theories built upon it. In privileging the structures of speech over those of script, it displaced the older, text-based, philological, diachronic study of language, in favour of purely synchronic analysis -how people talk now. This shift in attention away from the study of historical process makes it easy to conclude that we cannot really presume to recover an authorial voice at all, or an intended meaning, from the written or printed records of it. We are left only with synchronic structures, and the conventions which regulate their meaning as we read. It follows, of course, that if the meaning we read is entirely a function of the structural relations within the verbal sign system which constitutes a text, then it is not something inherent which can be expressed at all. Meaning is not what is meant, but what we now agree to infer [11, pp. 33-34]

Rematerialization

Digitization is often regarded as immaterialization. The presiding spirit of the assimilation is simply this: books are solid and available, whereas their digitized counterparts only enjoy an intermittent and mediated existence, at the moments when, and on the condition that, the relevant electronic devices are powered up. Although it corresponds significantly with daily superficial experience and feeling, the immaterialization view is wrong in many senses. First, digital information is not floating above the material world; rather, it is always physically implemented, albeit in another manner than paper information (if this was not the case, the nagging question of the so-called obsolescence of the electronic storage devices would disappear). Second, and deeper, digitization, when strictly and adequately conceived and deployed, locates the texts in the landscape of a fine-grained ontology, closer to the usual material ontology than the paper publication does. This point, which is of fundamental significance for the Digital Humanities, needs some explanation.

Currently, a copy of Twardowski's famous essay 'Actions and Products' ( [START_REF] Twardowski | Actions and Products. Comments on the Border Area of Psychology, Grammar and Logic[END_REF]) is on my desk, open at the initial page, and I am reflecting on the ideas defended in this text and on their relevance to the topic of the Digital Humanities. It appears as if the book -I mean, my copy of the book, the book as a physical object -connects me with the thoughts Twardowski had a century earlier. Now, the question is that of the nature of the relata. No miracle, of course, is involved, because the relationship between myself, as a material creature, and the products of the mental activity of a philosopher I have never met is mediated by the book. I am physically related to my copy of that book, the inked inscriptions on the book impress my retina, I recognize in those inscriptions familiar words and sentences and, as a competent English-reader, I access the linguistic meaning of those sentences, which delineate the thoughts Twardowski had in his head at the time he wrote them. This is a well-known story: our capacity to access the thoughts of past thinkers is to be decomposed in (i) our perceptual capacity to access the token of the words they drew on paper; and (ii) our linguistic capacity to access the meaning of those words (now considered, not as concrete inscriptions or 'tokens,' but as linguistic 'types'). Nevertheless, crucial details of this familiar story differ, according to whether we are dealing with printed books or digital devices. To say that I have the text of Twardowski's essay at hand is somewhat misleading. Of course, this assertion is always misleading, except when uttered by the happy few who have the privelige to leaf through Twardowski's manuscript, contained in the Library of the Institute of Philosophy in Warsaw (Fig. 1). As for the other readers, the author has not marked the signs they read himself. The physical product of Twardowski's writing activity in Łvov lies in a blue-grey dossier T.16 in Warsaw library and nowhere else and, strictly speaking, we cannot talk of it being displayed on the pages of a printed book or on the screen of a computer. Nevertheless, the assertion about my possession of his text is particularly unacceptable when I am reading a copy of the book. The point is not that Twardowski never had any physical contact with this object -this is trivial -but rather that the material chain that connects the both of us and which allows me to access his thoughts has been seriously broken.

Twardowski had a wide command of many languages beyond Polish, but he had never written anything in English and, therefore, he has not marked anywhere any sequence of signs equiform to those in front of me. The text I have on my desk is just a translation of the Polish original and the relevant causal chain has, therefore, been broken or dashed by the translation process. And what if we suppose that the book on my desk is the Polish version [START_REF] Twardowski | Wybrane pisma filozoficzne[END_REF], republished in 1965 in Warsaw along with other writings by the author. The problem remains the same: Twardowski never collected his works in this way and the logical structure of this book as well as its physical shape cannot be considered as genuinely derivative from his intentions and activity. This, then, is the point where paper and digital publication crucially differ. This difference may be explained by Fig. 2.

S1 S2

TE1 TE2 SO1 SO2 B1 B2 The traditional circuit (in red on the figure) of paper publication is rather simple, albeit sometimes complicated to execute. One or another of the sources (S 1, S 2) are selected and a subsequent book B1 (or B2) is published, providing its reader with words and sentences that are type-equivalent to their counterparts in the sources. The organization and the structure of this material are left to the appreciation of the publisher and they are not supposed to be similar to the organization and structure of the sources. A fortiori the same applies to the physical shape of the original, which is neither conserved, nor recoverable from the printed book. In the case of a scholar edition, an apparatus criticus separate from the text refers to the source and discusses the other printed versions on this basis. With some rare exceptions (the best example of which may be Die Kritik der reinen Vernunft, whose 1781 and 1787 editions differ significantly, specifically in relation to some well-known passages), only one source is displayed in the main text and the others are simply referred to in footnotes. In other words, the object one has in hand, the printed volume, is just one book, in the sense of it being the output of the publication process from a given source. The relationship between B1 and B2, which are considered as material objects, is not the physical cohabitation in one mundane item, but rather their proximity on the shelves of our libraries.

The situation is thoroughly different with Digital Humanities editions, from the perspectives of their relationship to their sources and also of their mutual relations. The circuit (in blue on the Fig. 2) differs from the previous one in terms of the interposition of a new level between the source and the final output; namely, that of the canonical encoding TEI of the properties of the source, which extend beyond its literal content properly said. Far from being a continuous recording indifferent to the fine structure (page-breaks, etc.) of the source or even to its physical appearance, this operation keeps track of everything potentially relevant in the source, making, in principle, any consultation of the original by the reader pointless. The process can be extended to the manuscript sources, as shown in the example of Twardowski's essay.

The first lines of the manuscript in Fig. 3 This code integrates not only the author's words, but the concrete way he wrote them on his page (e.g. the centring of the title is reflected on line 5, the deletion of the articles before the words 'grammaire' and 'logique' is recorded on line 16, etc.). In short, the TE level keeps track in coded format of every feature of the source, including the material aspects that are neglected in the traditional edition. This long-established method is more or less guided by the famous slogan WYSIWYG: What You See -you typist, on your screen -Is exactly What You Get in printed form. From the Digital Humanities standpoint, this result is unacceptably poor, because the editorial output is only what one sees in the main course of the text, obliging us to add separately a lot of paraphernalia to deal with the remaining text. One could say that, in this case, the leading slogan is the opposite of WYSIWYG; namely, WYKIWAW: What You Keep -you, editor -Is exactly What the Author Wrote. Moreover, the connectedness of the screen outputs SO (Fig. 4) for different sources or variants is physically enforced by the versatility of electronic devices: suitable hyperlinks are enough to ensure the possibility, for the reader, to navigate between the corresponding windows without changing his environment in the slightest way. One can summarize the difference between the two publishing schemes as follows: the new scheme firmly re-establishes the chain between the author and the product of his activity, which was threatened by the traditional way of proceeding. This restoration has wide consequences, often unperceived or misapprehended. [START_REF] Dubucs | Logic, Act and Product[END_REF] The end of the reader-shift Prima facie, the enterprise of digital edition involves a new avatar of the technical means of reproduction of the cultural works Walter Benjamin analyzed in his most influential essay [START_REF] Benjamin | The Work of Art in the Age of Mechanical Reproduction[END_REF] on the philosophy of culture. Benjamin's train of thoughts is well-known: the quality of the original presence -of the 'aura', in his terms -of a cultural item is always deprecated, even destroyed, by its reproduction. The impetuous development of reproduction technics in modern times induces a "sense of the universal equality of all things" ([2, p. 223]), which is a mark of the contemporary mass culture.

Benjamin even seems to have put his finger, in a premonitory manner, on the Big Commensurability characteristic of the Digital Age, with his description of a certain erosion of the very distinction between readers and writers:

For centuries a small number of writers were confronted by many thousands of readers. This changed toward the end of the last century. With the increasing extension of the press, which kept placing new political, religious, scientific, professional, and local organs before the readers. an increasing number of readers became writers -at first, occasional ones. It began with the daily press opening to its readers space for "letters to the editor." And today there is hardly a gainfully employed European who could not, in principle, find an opportunity to publish somewhere or other comments on his work, grievances, documentary reports, or that sort of thing. Thus, the distinction between author and public is about to lose its basic character. The difference becomes merely functional; it may vary from case to case. At any moment the reader is ready to turn into a writer [2, pp. 231-232].

From this perspective, the cultural impact of the contemporary Web might be viewed as a magnification, at a huge scale, of the effects that the daily press had a century ago; namely, the loss of distinction, authority and sacrality of the written work.

As far as written works are concerned, Benjamin's analysis of the effects of reproducibility can be challenged. The first objection is that if reproduction itself is to be incriminated, then the fatal turn should have been taken, not in contemporary times, but centuries ago by Gutenberg. This conclusion, which Benjamin seems ready to accept -"printing, the mechanical reproduction of writing [...] is merely a special, though particularly important, case [of the phenomenon examined here]" ([2, pp. 218-219]) -does not fit well with the background of the whole essay; namely, that of a mutual harmony between capitalism and mechanical reproducibility: the anachronism was unacceptable, even for one ready to accept that "the perspective of world history" (Loc. cit.) affords a certain liberty with historical accuracy. Besides, Benjamin claims insistently that the possibility of traceable ownership should be considered as one of the characteristic marks of an original, not a reproduced, piece of culture:

Even the most perfect reproduction of a work of art is lacking in one element: its presence in time and space, its unique existence at the place where it happens to be. This unique existence of the work of art determined the history to which it was subject throughout the time of its existence. This includes the changes which it may have suffered in physical condition over the years as well as the various changes in its ownership. The traces of the first can be revealed only by chemical or physical analyses which it is impossible to perform on a reproduction; changes of ownership are subject to a tradition which must be traced from the situation of the original. [2, p. 220].

Books can, of course, be such items, as lyrically described by Benjamin himself in a short essay dedicated to book collecting [START_REF] Benjamin | Unpacking My Library: A Talk about Book Collecting[END_REF]. To conclude, printed books, or at least some of them, should not be considered as inauthentic replicas of 'auratic' sources, but as original pieces of culture by themselves. That changes everything, because:

1. the significance of such books can no longer be viewed as only derivative of a first, inaugural (handwritten) act of creation.

2. there is no compulsory reason to consider different editions, or even different copies of the same edition,2 as just redundant tokens or realizations of the same abstract type.

Once we have rejected, in the domain of written culture, Benjamin's motto of the absolute privilege of the punctual, inaugural hic and nunc, the binary opposition between the 'auratic' start-point and the replications of this sacrosanct origin vanishes in the same way. The site of textual authority can no longer be equated with the only Ur-Text. The sources of reading-normativity are distributed in the subsequent proliferation of versions or variants and their readers are, therefore, not left to appreciate and to judge the erratic circumstances of their activity. In sum, one can escape the dramatic consequence of Benjamin's conception in the domain of textual culture; namely, that people remote from the origin are obliged, and then free, to adopt whatever interpretation of their cultural legacy. To avoid this interpretive anarchism, according to which 'everything goes' for the contemporary reader, is precisely the aim of the Digital Humanities.

5 Interpreting texts and cultural artefacts: the DH stance

As has been remarked ( [START_REF] Fitzpatrick | The Humanities, Done Digitally[END_REF]), current practitioners of the Digital Humanities deal with such a number of technological and institutional issues that one might consider this to be making (archives, tools, methods or consortia), rather than interpreting. Some (including [START_REF] Rafael | The Digital Humanities Situation. The Transducer[END_REF]) evoke the temptation of simply describing the distinctive nature of the Digital Humanities by pushing forward the practical idiosyncrasies of the people working in the field -who have a minimal acquaintance with the traditional Humanities, they use computers, they collaborate in a interdisciplinary way, and so on. This was a plainly desperate move, analogous to that of defining chemistry as merely the scientific field of those who manipulate retorts and test tubes.

The foundations of the difficulty lies, beyond the fact that Digital Humanists are supposed, to reuse Sokal's phrase, to "transgress the boundaries," in the inconsistency of the following data:

1. The hope, or wish, or certainty, that nothing should prevent Digital Humanities from dealing with digital production as well as with paper-borne heritage 32. The fact -the Big Commensurability analyzed above -that, in the digital realm, no watertight distinction between authors and readers can survive: in digital environments readers become prolix and give free and public play to their verbosity, becoming authors as well and, moreover, indiscernibly so.

3. The normative thesis that humanistic scholarship rests on a fundamental asymmetry between authors and commentators, the latter trying to grasp the meaning of texts and being apt to be right or wrong in doing so.

In sum, and paradoxically, the Digital Humanities seems to make sense only as far as paper-legacy in concerned, that means in the domain of computer-aided Humanities or, as one used to say, the 'computing Humanities.' The solution to this difficulty might be found in analyzing the canonical author/reader asymmetry more deeply than usual. Given the lack of clarity in the very notion of the 'meaning of a text' -while we have at hand a reasonable analysis of the meaning of a word, or of a sentence, it is highly questionable whether a rigorous counterpart to this is available for a text today -the simplest way is to start at the sentential level. According to the traditional picture, the author uses a certain sentence σ to express, or to refer to, some abstract entity A, which is σ's meaning. The duty of the reader is to recognize that σ has that meaning. He is reading σ as meaning B, and he is right if

B = A.
As simple as it may appear, this picture raises a series of difficulties. The first lies in the assumption of the pre-existence of the entity targeted by the author; as it were, an inert, sleeping entity waiting to be woken up by someone's statement. This view is not entirely indefensible: it has been notoriously advocated by Bolzano, who conceives his Sätze an sich in this way, i.e. propositions that are what they are independently of any speaker who is able to give them a linguistic form in his or her idiom. Nevertheless, one would prefer, ceteris paribus, to dispense with this metaphysical view of language, where authors are rarely considered either active or creative. As soon as we dismiss this kind of conception, we should also abandon the deciphering model of interpretation: readers are not trying to decode an author's message, for there is nothing for an author to encode.

Turning to more mundane entities and prosaic analysis, one could simply distinguish, (following [START_REF] Twardowski | Actions and Products. Comments on the Border Area of Psychology, Grammar and Logic[END_REF]), between two kinds of products of a mental activity such as thinking. The first is a psychic product, the thought. These thoughts, and psychic products in general, never last. Far from being eternal Gedanke an sich, they ontologically depend on the thinker and they do not survive him or even his particular relevant thinking action. On the other hand, mental activity may be the (partial) cause of physical products, such as spoken or written sentences, which survive its non-lasting psychic products. In other terms, the only enduring products of mental activity are its material traces. Twardowski calls these traces psycho-physical products, because the people who accede to them may have a similar mental activity and produce similar psychic products for themselves. That is, roughly presented, the Twardowskian explanation of the way that 'thoughts' are subtracted from the transitoriness of one's mental life and potentially transmitted to others: by the mere survival of the material traces that often accompany and express psychic activities.

Returning to the question of author/reader asymmetry, it appears in this frame in another light. This asymmetry is essentially that of anteriority, in both a temporal and a causal sense: authors leave traces for their readers that can be interpreted by them to occasion similar mental episodes as those that governed the production of these marks. This 'naturalistic' account of the transmission of culture is, however, compatible with the normative component that is currently, and rightly so, felt to be an essential part of humanistic scholarship. All readers are not on the same line: one in a position of being acquainted with the trace and its material details is the best equipped to understand the author. That does not forcefully mean that the author's contemporaries are the only ones able to do this. Contrary to the spoken discourse, which, in order to be understood, often requires hearing in praesentia, the written discourse, when suitably established and appropriately transcripted (metadata are crucial) can survive for a long time from the moment of its material production.

Far from being a renouncement of the traditional Humanities, DH may be viewed as scholarship put in the right perspective: neither platonic thoughts jointly targeted, nor abstract texts independent of their material realizations, but rather normed mental activity directed towards inscriptions and governed by them.

Figure 1 :

 1 Figure 1: Twardowski's archive, Library of Warsaw Philosophy Institute

Figure 2 :

 2 Figure 2: Two different schemes

Figure 3 :

 3 Figure 3: Manuscript of the French Version of Actions and Products

Figure 4 :

 4 Figure 4: Screenshot of the French version of Actions and Products after TEI Encoding

  are encoded as follows in XML/TEI standard:

	<teiHeader>
	<text>
	<body>
	<pb n="1"/>
	<fw place="top-center" type="pageNum">71</fw>
	<head>
	<title>Fonctions et produits</title>
	</head>
	<p>
	<lb n="1"/>Quelques remarques
	<subst>
	<add place="above">des</add>
	<del>touchant les</del>
	</subst>
	confins
	<lb n="2"/>de la psychologie, <del>la</del> grammaire
	et <del>la</del> logique
	</p>
	<p>
	<lb n="3"/><hi>& 167;1.</hi> En comparant deux expressions
	</p>

On the general scope of this extension of human capacities by computing machines, see[START_REF] Humphreys | Extending Ourselves. Computational Science, Empiricism, and Scientific Method[END_REF]

As an example, compare the two following editions of the latin translation of Diophante's Arithmétiques by Claude-Gaspard Bachet, Sieur de Méziriac: the 'original' published in Paris in 1621, and the re-edition of 1671 in Toulouse, augmented by Fermat's notes. Or, to come to the nub of the issue, the specific significance of Fermat's copy of the original, with the marvellous annotation stating Fermat's 'last theorem' as well as the impracticability of writing its demonstration within the margin of the copy.

On "Electronic Literature", namely digital born writing, not print literature digitized afterwords, [cf. 6].


Shlomo Barak
email: shlomo@tagapro.com

Where is Anti-Matter?

Keywords: Anti-matter, Quark, Meson, Baryon, Strong force

published or not. The documents may come

Where is Anti-Matter?

Introduction

Where is Anti-Matter is a long standing issue.

The New Scientist journal [START_REF] Gefter | Antimatter mysteries[END_REF] summarizes the situation:

There are two plausible solutions to this mystery.

First, there might be some subtle difference in the physics of matter and antimatter that left the early universe with a surplus of matter. While theory predicts that the antimatter world is perfect reflection of our own, experiments have already found suspicious scratches in the mirror. In 1998, CERN experiments showed that one particular exotic particle, the kaon, turned into its antiparticle slightly more often than the reverse happened, creating a tiny imbalance between the two.

Second plausible answer to the matter mystery is that annihilation was not total in those first few seconds: somehow, matter and antimatter managed to escape each other's fatal grasp.

Somewhere out there, in some mirror region of the cosmos, antimatter is lurking and has coalesced into anti-stars, anti-galaxies and maybe even anti-life.

We, in contrast, suggest that quarks are topologically twisted electrons or twisted positrons, see [START_REF] Barak | On the Essence of Electric Charge, Part 2: How Charge Curves Space[END_REF] and Section 2 here, and hence, as we show, each and every neutral atom is constructed of nothing but an equal number of topologically twisted electrons and positrons.

The spin of the proton and that of the neutron is s = ½ and, so far, it is not clear where it comes from. Our model of the nucleons suggests a possible answer to this "spin crisis".

We also raise the possibility that the strong force, between quarks and between nucleons, is an electromagnetic force. In the case of nucleons, the force, as we show, is a result of a Yukawa pion exchange. Our suggestion is based on Feynman's explanation as to how an exchange of a particle can generate an attractive force [START_REF] Feynman | Lectures on Physics III Addison-Wesley[END_REF].

Quarks

Fig. [START_REF] Barak | On the Essence of Electric Charge, Part 2: How Charge Curves Space[END_REF] shows a trio of d quarks which are sub-tracks of a twisted electron at "rest" [START_REF] Barak | On the Essence of Gravitation and Inertia A, Part 1: Inertia and Free Fall of an Elementary Particle[END_REF]. In a translational motion the trio becomes a trio of spirals [START_REF] Barak | On the Essence of Gravitation and Inertia A, Part 1: Inertia and Free Fall of an Elementary Particle[END_REF].

In our model of quarks, we assign charge to each sub-track according to the time the electron (positron) spends on this sub-track. We, of course, assume that the tangential velocity is c.

Fig. (1) The Quarks

The electron spends one third of its full revolution time in each of the sub-tracks. We therefore assign it a charge of 1/3 Q 0 . The resultant spin of this trio of quarks is S = ½ , due to the quarks individual spins, as Fig. [START_REF] Barak | On the Essence of Electric Charge, Part 2: How Charge Curves Space[END_REF] shows. Each sub-track has the same spin as that of the electron. We refer to the sub-track of an electron as the quark, d, and to the sub-track of a positron as the anti-quark, d ~.

It is still not fully clear why the track is twisted.

It is thus clear that the quarks are not independent fundamental particles, and therefore individual quarks do not exist. An electron spends two thirds of its time in the sub-track with the double radius, therefore, we assign it a charge 2/3 Q 0 and refer to it as the quark u ~. For a positron, we refer to it as the quark u.

This simple model of quarks enables us to derive, for the first time, and accurately calculate their masses [START_REF] Barak | On the Essence of Electric Charge, Part 2: How Charge Curves Space[END_REF].

A Derivation and Calculation of the Quarks Masses

We know the electron mass M e , see [START_REF] Barak | On the Essence of Electric Charge, Part 2: How Charge Curves Space[END_REF]. From this mass, based on our quark model, we derive the masses M u ̃ and M d of the first generation quarks, which are also the masses M u and M d ̃ of their anti-particles. The electron angular momentum L = M e R e 2 ω must be conserved [START_REF] Barak | On the Essence of Gravitation and Inertia A, Part 1: Inertia and Free Fall of an Elementary Particle[END_REF].

Hence it is the same L for each of the sub-tracks of the d quark, see Fig. [START_REF] Barak | On the Essence of Electric Charge, Part 2: How Charge Curves Space[END_REF]. For the u ̃ quark it is 2L, since the L of its companion d quark points in the opposite direction, see Fig. [START_REF] Feynman | Lectures on Physics III Addison-Wesley[END_REF]. For the quarks, ω is the same, but R and M are different and conversely related. In a twisted track, which is a set of three quarks, ddd, the radius of each sub-track is

e R 3 1 R  .
The length of the electron wavepacket is conserved [START_REF] Barak | On the Essence of Gravitation and Inertia A, Part 1: Inertia and Free Fall of an Elementary Particle[END_REF], hence:

M u ̃e e M 5 . 4 M 4 9 2   (2)
From ( 1) and [START_REF] Gefter | Antimatter mysteries[END_REF] and M e = 0.51 MeV we obtain the following results (3) and (4):

M d = 4.5 MeV (3)
A recent experimental value [START_REF] Olive | [END_REF] is: M d = 4.8 +/-0.5 MeV.

M u ̃ = 2. 25 MeV (4)
A recent experimental value [START_REF] Olive | [END_REF] is: M u ̃ = 2.3 +/-0.8 MeV. Overlapped dashed and solid circles represent the electromagnetic bond, of opposite charges, between the relevant quarks. These quarks that create a bond are not numbered as originating quarks of a particle, be it a meson, baryon or any other particle.

The Mesons

Note that in particle diagrams a solid line indicates an electron, and a dashed line a positron.

It seems that this electromagnetic bond, of opposite charges, between the relevant quarks is the known "strong force". But, in this paper, we do not provide any formal proof that this is indeed the case.

This subject is also discussed in Sections 8 and 9.

The П Family with Spin S=0 4.1

A single twisted track has a very short life time, as can be seen for the Π + and Π -mesons.

For the Π 0 meson, the life time is much shorter due to annihilation.

       e e 0
and even:

     e 2 e 2 0
See the table in [START_REF] Ne'eman | Particle Hunters[END_REF]. and similar reactions, it seems as if a neutrino is incorporated in the   construction (as in similar particles). We speculate that this neutrino causes the electron (positron) to change from one sub-track to another, as if it circulates around the twist.

Baryons

The baryons are composed of three quarks. As an example, consider the family, with the members:  ++  +  0  -, that have the spin S = 3/2.

On the Neutron and the Proton

We model the nucleons as kind of a motor with a neutral stator cage, which contains a large number of quarks, and a rotor, which contains only a few valance quarks, which determine the charge of the nucleon.

Our rotor models of the proton and neutron enable us to derive and calculate the "proton charge radius" [START_REF] Barak | On the Essence of Electric Charge, Part 2: How Charge Curves Space[END_REF]. Our result is well within the error range of the experimental result, see [START_REF] Barak | On the Essence of Electric Charge, Part 2: How Charge Curves Space[END_REF].

The Rotor Model 6.1

The neutron rotor, see the simplistic model in We, thus, arrive at a heavier neutron than the proton, which complies with experimental data. The disintegration reaction of a free neutron, with a life time of ~10 minutes, is: Note that the total number of quarks that construct the stator can also be 3x4 + 3x4 = 24 or 3x5 + 3x5 = 30.

The spin contribution, of the stator type cage, to the total spin of the nucleon, might be zero, since in this case the magnetic interaction lowers the cage energy. The spin of the nucleon is thus the spin of its "rotor" which is S = ½ , as where the spin of the nucleon comes from, is known as the "proton spin crisis".

Based on our understanding of what are quarks, and how the building blocks of matter are constructed, we arrive at the conclusion in Section 7.

In the Universe the Number of Electrons Equals the Number of Positrons

A Neutron is made of an equal number of electrons and positions (twisted topologically to become quarks). A Proton is made of electrons and positrons, as is the Neutron, but has one electron (three quarks) less. An atom is made of an equal number of protons and electrons Hence, an Atom is made of an equal number of positrons and electrons (as should be evident from the pair production process).

Remarks on the Strong Force

The Force between Quarks 8.1 14) shows how an electric field between three pairs of charges, of the neutron rotor and proton rotor, create an attracting force that binds them. This is an attractive force only, and spin independent (ignoring weak magnetic interactions) as is the strong force.

Figures in this paper

Note that when the neutron and proton are far apart, the inner distribution of charge is blurred and the effective field decays rapidly, which explains the short range. Note also that according to our model of the electron (positron) [START_REF] Barak | On the Essence of Gravitation and Inertia A, Part 1: Inertia and Free Fall of an Elementary Particle[END_REF] the radii r e and R e of the electron (positron) is reduced, at high velocities, by the relativistic factor γ, and hence close to their surface the field is stronger.

Fig. ( 2 )Fig

 2 Fig. (2) shows again a trio of d quarks , but in this case with S = 3/2:

  a twisted track of a pair of quarks, like u d , we get for u ~ a sub-track with e

Fig ( 3 )

 3 Fig (3) and Fig. (4) show mesons, which are a twisted electron or positron with only one

Fig. ( 5 )

 5 Fig. (5) The Π Mesons

  Fig. (6) The ρ Mesons

2 dud

 2 Fig. (9) shows the spatial charge distribution, known from scattering experiments. This is

Fig. ( 9 )

 9 Fig. (9) Neutron Charge Distribution

  Fig. (10) The Proton Rotor

  And now we know where the electron comes from.If the stator is composed of three (3) twisted positrons and three (3) twisted electrons, the total number of quarks in the stator is 3x3 + 3x3 = 18, see Fig. (12).

Fig

  Fig. (12) The Stator Structure

  Fig. (8) and Fig. (10) show. This issue, of

2 Fig. (

 2 Figures in this paper, which represent mesons and baryons, show couples of sub-tracks of an

Fig

  Fig. (14) The (NP) Strong Force

Acknowledgements

We would like to thank Mr. Roger M. Kaye for his linguistic contribution and technical assistance.

Here, as well, there is an attractive force at short range, but as the distance increases this force becomes repulsive since the particles have the same charge sign.

We consider the binding force, for both the neutron and proton, and proton and proton cases, to be the result of a Π + meson exchange. This meson is merely a twisted positron, and its movement back and forth, is the "exchange". This is the Yukawa theory of meson exchange, as the source of the Strong Force.

Feynman [START_REF] Feynman | Lectures on Physics III Addison-Wesley[END_REF] explains how an exchange of a particle can generate an attractive force. In Chapter 10-1, there is a description of the force between two protons of the ion  2 H . This force is the result of the electron exchange between the protons. The reaction is:

The attractive force comes from the reduced energy of the system due to the possibility of the electron jumping from one proton to the other. In such a jump the system changes from the configuration (hydrogen atom, proton) to the configuration (proton, hydrogen atom), or switches back.

In Chapter 10-2, Feynman shows how the force between a proton and a neutron can be explained by the exchange of a  + meson. Feynman concludes:

Now we might ask the following question: could it be that forces between other kinds of particles have an analogous origin? What about, for example, the nuclear force between a neutron and a proton, or between two protons? In an attempt to explain the nature of nuclear forces, Yukawa proposed that the force between two nucleons is due to a similar exchange effect -only, in this case, due to the virtual exchange, not of an electron, but of a new particle, which he called a meson.

The exchange in this case is:

The Π + meson is emitted from the proton to convert it into a neutron, and by combining with the other neutron to convert it into a proton, and vice versa.

The Standard Model, however, considers meson exchange as a complex effect of color interactions, and not a primary cause.

Based on the above we suggest to reassess the idea of a Strong Force and to explore again the possibility that it is an electromagnetic force.

Summary

So far, there is no evidence for the existence of stars and galaxies made of anti-matter (antihydrogen was first created in the laboratory in 1997).

The question: Why, at large, is the universe composed of matter only?

Now becomes: Why we do not see in nature atoms made of anti-protons and positrons?

In the GDM there is no charge conjugation. A negative charged nucleus curves space around it negatively. This might be the reason for instability in the orbital motion of a positron around it (hence anti-hydrogen created in the laboratory should have a limited lifetime).


Shlomo Barak
email: shlomo@tagapro.com

The Photon and the Quantum Enigma

Keywords: Photons, Electromagnetism, Geometrodynamics, Spacetime

come

Introduction

The reading of papers [START_REF] Barak | On the Essence of Electric Charge, Part 1: Charge as Deformed Space[END_REF], [START_REF] Barak | On the Essence of Electric Charge, Part 2: How Charge Curves Space[END_REF] and [START_REF] Barak | On the Essence of Gravitation and Inertia, Part 1: Inertia and Free Fall of an Elementary Particle[END_REF] is a must, otherwise the following statements will be considered an absurdity.

We contend that the photon is a confined oscillating dipole with its field. This dipole is an oscillation of the bivalent elementary charges, which in the pair production process are converted into the stable bivalent elementary charges of the electron and positron (as it is for other particles). The photon in our theory holds all the basic features that appear in all the other elementary particles.

We further contend that EM waves are ensembles of closely packed photons that oscillate in phase and hence are classical waves. Similarly ensembles of closely packed ground state photons -photomsas we call them, oscillating in phase are also classical waves. As we show, photons and photoms condense when they are in phase and disperse when in antiphase. These features dispel, as we explain, the need to attribute a dualistic nature to photons.

In modern literature [START_REF] Loudon | The Quantum Theory of Light[END_REF] [5] [START_REF] Milonni | The Quantum Vacuum[END_REF] the photon is considered a wavepacket, but its shape, structure and size are not discussed.

We remind the reader that GDM stands for the "Geometrodynamic Model of Reality"our new theory of the foundations of physics, see [START_REF] Barak | On the Essence of Electric Charge, Part 2: How Charge Curves Space[END_REF].

The GDM Model of the Photon

Paper [START_REF] Barak | On the Essence of Electric Charge, Part 1: Charge as Deformed Space[END_REF] shows that contracted space is positive charge and dilated space is negative charge.

It also shows that the electric field E is proportional to u, the Elastic Displacement Vector of the elastic space lattice. E is defined as E = -Hu, and points in the opposite direction to u. the cells below are contracted and those above are dilated, and the electric field points upwards. But contracted space and dilated space are the bivalent electric charges [START_REF] Barak | On the Essence of Electric Charge, Part 1: Charge as Deformed Space[END_REF]. Thus the GDM space vibrational photon, as described, is an oscillating electric dipole with its electric field. This dipole and its field, as we show, are confined in a finite volume of space and the oscillating charges are the elementary electric charges. The fine structure constant α, where Q is the elementary charge, is:

α = Q 2 / ℏc (1)
From (1) we get:

ℏc = Q 2 / α → 2πℏc/λ = 2π Q 2 / αλ → ℏω = 2π Q 2 / λ α → ℏω = 2Q 2 / 2 r e (2)
where the photon energy is ℏω and r e is defined as:

r e = α/2π •λ (3)
As if the energy of the photon is "concentrated" in two "spheres" of the bivalent elementary charge Q with a radius r e .

The factor 2 in the numerator of equation ( 2) is accounted for the energy of contraction (one "sphere") and that of dilation (other "sphere"). We relate to r e as the "radius" of the "cylinder", which contains the two "spheres".

The Oscillation of the Center of Energy of the Photon

For a spring with a constant K and mass M: y = y 0 Sinωt v = ωy 0 Cosωt a = -ω 2 y 0 Sinωt , where the axis y is perpendicular to the line of propagation.

U k = 1/2 Mv 2 = 1/2M ω 2 y 0 2 Cos 2 ωt U p = 1/2Ky 2 = 1/2 Ky 0 2 Sin 2 ωt U= U k +U p = constant U=U k (max)= U p (max) = M ω 2 y 0 2 = Ky 0 2 ω =√K/M (4)
Since U is a constant we are allowed to replace M, see [START_REF] Barak | On the Essence of Gravitation and Inertia, Part 1: Inertia and Free Fall of an Elementary Particle[END_REF], in the above equations ( 4), by M = U/c 2 , which gives:

U = M ω 2 y 0 2 = U/c 2 ω 2 y 0
Hence y 0 ω = c, but c = λ/(2πω) and the maximum amplitude of the center of energy is:

y 0 = λ/2π (5)
We denote y 0 as R e and relate to it as half the "cylinder height". Note that (3) and ( 5) give:

r e /R e = α (6)
This is the same ratio, as it is for the radii of the electron [START_REF] Barak | On the Essence of Gravitation and Inertia, Part 1: Inertia and Free Fall of an Elementary Particle[END_REF] (their sizes are different). From the geometry of both the electron and the photon it seems that this ratio determines the probability for the photon to interact with the electron. Hence 𝑗 = -√α = -0.08542455 is the probability amplitude of the QED Junction, Coupling, which is "… one of the greatest damn mysteries of physics: a magic number"; Feynman [START_REF] Feynman | QED[END_REF] in his own words.

Note that the contraction or dilation move at the same velocity c, as the propagation does, and hence the maximum possible half height of the disturbance is

H = c•1/4T = 1/4•λ ,
whereas as [START_REF] Grynberg | Quantum Optics[END_REF] shows: R e = 1/2π•λ.

Photon Energy and the Planck Constant ℏ

Using (4) and ( 5) we get U = K λ 2 /2π 2 where K is an attribute of space similar to the K of a spring. If the "Space Spring Constant K" is a function of ω only we guess that it can be:

K = Mω 2 = (U/c 2 )ω 2 = ℏω 3 / c 2
where ℏ, not yet identified, is a constant of proportionality.

Necessarily U = ℏω where ℏ becomes the known quantum constant.

Thus a classical functionality of K(ω) ∝ ω 3 leads to the basic quantum mechanical result.

The Photon's Volume

Taking the "cylinder height" as 2R e , gives for the maximum of the oscillatory volume:

V Photon = πr e 2 2R e = α 2 /8π 2 • λ 3 = 6.75•10 -7 • λ 3 cm 3 [START_REF] Feynman | QED[END_REF] The shorter is λ, the more compact is the photon.

For λ = 500nm we get: V Photon = 6.75•10 -7 • λ 3 ~ 8•10 -20 cm 3 .

The Photon Spin

The photon spin L, for a circular polarized photon, is its linear momentum P = U/c times its radius of rotation R e = λ/2π, see ( 5), L = PR e = U/c•λ/2π but U = ℏω. Hence the known result:

L= ℏ (8)
8 Photoms -the Ground State Photons QED, [START_REF] Milonni | The Quantum Vacuum[END_REF], attributes to a photon in its ground state one half of the energy, half of the momentum, and half of the spin. To distinguish a photon from a ground state photon, we name it a Photom (a photon at the bottom is a photom). In the GDM, however, there are two types of photoms. These photoms are the split of the photon in Fig. [START_REF] Barak | On the Essence of Electric Charge, Part 1: Charge as Deformed Space[END_REF]. The GDM considers them as the pair of an electron and positron of the Dirac Sea, and vacuum polarization is due to them. This way we dispel the need for a separate ground state (vacuum state) for the electromagnetic field and that for the electron and positron field (Dirac sea). Note that we do not know the exact shape of the photon or the photom. All we know, at this stage, is their geometric and dynamic features. This is the reason why we use terms like cylinder-like.

The

Spatial Density of Photoms

The spatial density of an ensemble of photoms of a given λ and a bandwidth dλ, see [START_REF] Milonni | The Quantum Vacuum[END_REF], is: n(ν) = 8πν 2 /c 3 •dν but ν=c/λ and dν=c/λ 2 •dλ hence:

n(λ)=8π/λ 4 •dλ (9)
For photoms of λ = 500nm and a bandwidth dλ = 0.5nm the spatial density is: General Relativity shows that light velocity is slower close to a star than at a distance from it [3] [8]. As a result, a falling photon in a gravitational field gains linear momentum (P = U/c).

In order to retain its angular momentum (L = P•R e ), it has to reduce its R e = λ/2π , see [START_REF] Barak | On the Essence of Electric Charge, Part 2: How Charge Curves Space[END_REF],

hence the blue shift. And in climbing the gravitational field, it will be red shifted.

Condensation and Dispersion

Fig. ( 4), shows a symbolic space lattice, which is like a 3D fisherman's net. Imagine that each vertical plane, in Fig. ( 4), going into the page, is a plane in which a photon or a photom are moving. If two adjacent photons or photoms of the same frequency oscillate in anti-phase side by side they stretch the lattice horizontally and move apart to ease the tension. The result is dispersion -a reduced spatial density of the photons or photoms. If, however, they oscillate in phase, they move together to ease the tension and the result is condensation -higher density.

A similar situation occurs if they move one below or above the other. In this section we dispel the need for attributing a dualistic nature to photons.

Interference with Classical Waves -Ensembles of Photons

Photons or photoms, of the same wavelength, when they are in phase and closely packed, they move as a classical wave. Let a wave front, of such a classical wave, hit the double slit; if the photons or photoms, which are coming from the two slits and hit the screen, are in phase (on the screen) they create maxima zones on the screen, and minima if out of phase. In other words; intensity is reduced in zones of destructive interference whereas in zones of constructive interference intensity is enhanced. Photons and photoms entering these zones, however, are neither annihilated nor created and the total energy is conserved. No annihilation or creation takes place, only a spatial displacement of the particles that result in a reduction in their density (intensity) in one zone and an increase in their density (intensity) in the other.

Interference with Individual Photons

There is a body of both experimental and theoretical work that shows how the vacuum state of the electromagnetic field (photoms) induces "spontaneous" emission [START_REF] Milonni | The Quantum Vacuum[END_REF]. Thus, the basic state is that of a single photon plus a single photom. When one photom, out of the many that construct a wave front of photoms with a frequency that matches a transition in an exited atom (in the source), hits the exited atom it has a probability to induce the emission in phase of a photon. The probability of the photon to hit a certain spot on the screen is thus the classical probability of the inducing photom to arrive at this same spot. Hence, the many, one at a time, photon accumulation effect is the interference pattern.

This dispels the need for attributing a dualistic nature to photons and solves the collapse issue of the "wavefunction" (although there is no such thing for a photon). Note that the width and the height of the slits, as well as the distance between them, comply with our understanding of the photon and photom, and the interference experiment.

In the year 2011 it was shown [START_REF] Steinberg | [END_REF] in a "Weak-Experiment" [10] that a single photon, in the Double Slit experiment, goes all the way from the source to the screen through only one slit.

The photon follows a trajectory known as the D. Bohm trajectory [11].

Our "mechanism" by which the photon is "guided" is also relevant to other elementary particles, atoms, molecules and even microscopic bodies.

On EM Waves

Waves we encounter in nature are, most of the time, spherical waves rather than plane waves.

However, far from a radiator, a spherical wave appears to be approximately a plane wave. Far from the radiator the electrical field of the wave E decreases like 1/r and hence its intensity I, which is proportional to E 2 decreases as 1/r 2 . Thus the classical approach complies with the quantum approach. In this case, the intensity at a far distance r depends on the number of photons that are crossing a unit area and hence the dependence on 1/r 2 .

Summary

The GDM photon has a finite volume V Photon = 6.75•10 -7 • λ 3 cm 3 . This photon is a confined oscillating dipole of the bivalent elementary charges with their fields.

Note that our discussion is based on the GDM, which is a new approach to the fundamentals of physics in general, and electromagnetism in particular.

The GDM photon owns all the attributes necessary to be converted, directly or indirectly, to each and every one of the other elementary particles.

Fig. ( 1 )

 1 Fig. (1) shows an oscillating cylinder-like photon passing in front of us to the right, with a

Fig. ( 1 )

 1 Fig. (1) The Photon's Oscillating Space Lattice

  Positive Photom: This photom, as Fig. (2) shows, is an oscillating contraction of space (positive charge), close to the line of propagation. This oscillation is in a plane or in a rotating plane, around the line of propagation. The Negative Photom: This photom, as Fig. (3) shows, is an oscillating dilation of space (negative charge) close to the line of propagation. This oscillation is in a plane or in a rotating plane, around the line of propagation.

Fig. ( 2 )

 2 Fig. (2) shows our cylinder-like photom passing in front of us to the right, with a velocity c.

Fig. ( 2 )Figure ( 3 )

 23 Fig. (2) The Positive Photom

  ) ~2•10 11 photoms per cubic centimeter. Note that a one joule laser pulse, of λ = 500nm contains ~ 2.5•10 18 photons. 10 Light Velocity and Red/Blue Shift in a Gravitational Field

Fig. ( 4 )

 4 Fig. (4) Coupled Photoms and Coupled Photons

Acknowledgements

We would like to thank Mr. Roger M. Kaye for his linguistic contribution and technical assistance.


Giuseppe Iurato
email: iurato@dmi.unict.it

Some critical remarks on Landau's (macroscopic) phase transition theory *

Keywords: 64.60.av, 97.10.kc As2], [AK], [Bi2], [Bo], [CLP], [GL]

This paper explains the existence of a particular formal model (drawn from Theoretical Astrophysics) whose thermodynamical phenomenology shows a possible second order phase transition (according to Landau's Thermodynamical Theory) that seems does not verify the (Birman-Goldrich-Jari¢) chain subduction criterion and the (Ascher's) maximality criterion of Landau's Phenomenological Theory. Therefore, it follows that Landau's Phenomenological Theory is more restrictive than the Landau's Thermodynamical Theory.

1 Introduction Lev D. Landau ([La], [START_REF] Landau | Fisica Teorica 5: Fisica Statistica[END_REF]) given two theoretical formulations of the (macroscopic) second order phase transition theory, the rst called (Landau's) macroscopic Thermodynamical Theory, and the second called (Landau's) macroscopic Phenomenological Theory. These theories are based (especially the second) on some symmetry principles and elementary methods of the representation theory of groups.

In this context, there exist criteria that give necessary and/or sucient conditions for a second order phase transition (according to Landau).

The Maclaurin-Jacobi pattern

The ) is a simplied mathematical model of the Self-Gravitating Systems Theory (cf. [BF], [Ch], [CMR], [FP], [Jar], [Ko], [Lam], [START_REF] Lebovitz | Rotating Fluid Masses[END_REF], [START_REF] Lebovitz | Bifurcation and Stability Problems in Astrophysics[END_REF], [Lyt], [Pa], [START_REF] Stiavelli | Modelli Dinamici di Galassie Ellittiche[END_REF], [START_REF] Tassoul | Theory of Rotating Stars[END_REF], [START_REF] Tassoul | Stellar Rotation[END_REF]), and of the Physics of Compact Objects (cf. [KW], [Pa], [ST]).

In Astrophysics, there exists the problem of determining the (closed) stationary equilibrium congurations of a self-gravitating rotating uid. If we suppose that the uid is subjected to a rigid rotation around one of its symmetry axis, under the further simplied hypotheses of homogeneity, incompressibility and non-viscosity of the rotating uid, we then obtain a rst linear solution to this problem, i.e. the Maclaurin-Jacobi pattern. Applied to polytropic uid congurations with index n ≤ 0.808 ( [Va], [START_REF] Stiavelli | Symmetry and Symmetry Breaking in Astrophysics[END_REF]), to the theory of elliptic galaxies (cf. [OP], [START_REF] Stiavelli | Symmetry and Symmetry Breaking in Astrophysics[END_REF]) and to the Black Holes thermodynamics (cf. [START_REF] Davies | The Thermodynamic Theory of Black Holes[END_REF], [START_REF] Davies | Thermodynamics of Black Holes[END_REF], [Wa]), this model provides theoretical predictions in good agreement with the corresponding experimental data, so that the Maclaurin-Jacobi pattern is a valid model of Theoretical Astrophysics " .

See [START_REF] Birman | Simplied Theory of Symmetry Change in Second-Order Phase Transitions: application to V 3 Si[END_REF], [CLP], [GB], [START_REF] Jari¢ | Group Theory and Phase Transitions[END_REF], [START_REF] Jari¢ | Spontaneous Symmetry Breaking and the Chain Criterion[END_REF], [START_REF] Jari¢ | Structural Phase Transitions in Crystals: Broken-Symmetry (Isotropy) Groups[END_REF], [START_REF] Jari¢ | Symmetry Breaking and the Chain Criterion[END_REF], [START_REF] Jari¢ | Symmetry Breaking and the Chain Criterion[END_REF], [JB], [SA], [START_REF] Tolédano | The Landau Theory of Phase Transitions[END_REF]. ! See [As1], [START_REF] Ascher | Permutation Representations, Epikernels and Phase Transitions[END_REF], [AK], [START_REF] Birman | Group Theory of the Landau-Thermodynamic Theory of Continuous Phase Transitions in Crystals[END_REF], [CLP], [START_REF] Jari¢ | Group Theory and Phase Transitions[END_REF], [START_REF] Jari¢ | Counterexamples to the Maximality Conjecture of Landau-Higgs Models[END_REF], [START_REF] Tolédano | A Counterexample to the Maximal Subgroup Rule for Continuous Crystalline Transitions[END_REF]. " Furthermore, there exist satisfactory applications of this model to the Theory of Collective Models of atomic nuclei (see [ALLMNRA], [BK], [START_REF] Cohen | Equilibrium shapes of a rotating charged drop and consequences for heavy ion induced nuclear reactions[END_REF], [START_REF] Cohen | Equilibrium Congurations of Rotating Charged or Gravitating Liquid Masses with Surface Tension[END_REF], [IA], [RS], [START_REF] Rosenkilde | Stability of Axisymmetric Figures of Equilibrium of a Rotating Charged Liquid Drop[END_REF], [START_REF] Rosenkilde | Surface-Energy Tensors[END_REF]).

Following [CMR], let us consider a homogeneous, incompressible and nonviscous uid in rigid rotation around one of its symmetry axes, for example the Ox 3 axis of a Cartesian coordinates system (x 1 , x 2 , x 3 ). We choose a normalized orthogonal reference frame {O, î, ĵ, k}, whose origin O is the center of mass of the rotating uid. We suppose that a reference frame is rotating with the uid (with respect to which it is at rest # ). If ⃗ ω is the angular velocity of rotation, let us suppose that ⃗ ω = ω k, in order that the angular momentum is

⃗ J = M ( (⃗ ω × ⃗ r) × ⃗ r )
, where ⃗ r is the position vector (measured in the rest reference frame co-rotating with the uid) of the generic material point of the system, and M is the total mass of the uid. ⃗ J and ⃗ ω are both parallel to the Ox 3 axis.

Historically, C. Maclaurin ([Mc]), C.G.J. Jacobi ( [Jc]) and H. Poincaré ([Po]) dealt with the dynamical problem of determining the equilibrium congurations of the uid system. The results so obtained by the rst two authors, forms the so-called Maclaurin-Jacobi model (or M-J pattern).

For such a uid system, in the above mentioned rest reference frame corotating with the uid, the (Euler's) hydrostatic equilibrium equation is

(1) ∇(p(⃗ r) -φ(⃗ r)) = 0,
where p is the hydrostatic pressure and φ the potential of the uid system.

In the rest reference frame, let S(⃗ r; J 2 ) = 0 be the surface equation of the unknown equilibrium conguration. If we suppose S(⃗ r; J 2 ) > 0 inside the uid, then the region V S ⊂ R 3 is bounded, connected and take up by the uid, and represented by S(⃗ r; J 2 ) ≥ 0. This surface is parametrical dependent by J 2 , where J is the modulus of the angular momentum. The potential φ of this well-dened self-gravitating uid, is given by the sum of the gravitational potential φ g and of the centrifugal potential φ c , so that

(2) φ[S](⃗ r; J 2 ) = φ g [S](⃗ r) + φ c [S](⃗ r; J 2 ),
where

φ g [S](⃗ r) = 1 4π ∫ S(⃗ s)≥0 d⃗ s |⃗ r -⃗ s|
is the gravitational potential, and

φ c [S](⃗ r; J 2 ) = J 2 2I 2 (x 2 1 + x 2 2 ), I[S] = 15 8π ∫ S(⃗ r)≥0 (x 2 1 + x 2 2 )d⃗ r
# In this case, such a reference is properly called co-rotating rest reference frame of the given uid system. are, respectively, the centrifugal potential and the inertial moment of the uid system, computed with respect to the axis Ox 3 , with ⃗ r = (x 1 , x 2 , x 3 ). The gravitational potential satises the Poisson's equation

∆φ g [S](⃗ r) = -ρ, ρ(⃗ r) = 1 if S(⃗ r) ≥ 0, ρ(⃗ r) = 0 if S(⃗ r) < 0
where ρ is the (constant) density of the uid. Moreover, the associated boundary conditions are the following

p(⃗ r) S(⃗ r)=0 = 0, lim r→∞ φ g (⃗ r) = 0, φ[S](⃗ r; J 2 ) S(⃗ r)=0 = const., r = |⃗ r|.
Through the bifurcation method (see [CMR], [START_REF] Perdang | Irregular Stellar Variability[END_REF], [START_REF] Perdang | On Some Group-Theoretical Aspects of the Study of Non-Radial Oscillations[END_REF], [Dy]), and neglecting, for simplicity, the various stability questions (see [CMR], [Lyt]), a rst (linear) approximated solution to the non-linear dierential system

(3)                      ∇(p -φ) = 0 ∆φ g = -ρ p(⃗ r) S(⃗ r)=0 = lim r→∞ φ g [S](⃗ r) = 0 φ[S](⃗ r; J 2 ) S(⃗ r)=0 = const. ,
gives the so-called Maclaurin-Jacobi pattern, that consists of the rst two sequences of (innite and stationary $ ) solutions S(⃗ r; J 2 ) of (3), for suitable values of the parameter % J 2 . Precisely, we have a rst sequence of (innite and stationary & ) solutions, called Maclaurin's sequence S M acl = {S(⃗ r; J 2 ); 0 < J 2 < 0, 384436}, and a second sequence of (innite and stationary ' ) solutions, called Jacobi's sequence S Jac = {S(⃗ r; J 2 ); 0, 384436 ≤ J 2 < 0, 632243}.

The rst sequence of solutions is geometrically formed by two-axial ellipsoids (or spheroids) whose symmetry axis of rotation is Ox 3 , whereas the second sequence of solutions is geometrically formed by three-axes ellipsoids. The gures of the rst sequence are called Maclaurin's ellipsoids (or spheroids), $ But no static. % With respects to suitable natural units, relative to the physical and geometrical parameters of an ellipsoid, the parameter J 2 takes well-dened values (see [CMR]).

& But no static.

' But no static.

whereas the gures of the second sequence are called Jacobi's ellipsoids.

When ω → 0, by a corollary of an important theorem of L. Lichtenstein (cf. [Le1], [Li]), the only stationary and static equilibrium congurations, compatible with (3), are those spherics (corresponding to J 2 = ω = 0), so that the relative symmetry group is O(3). Little by little that ω increases (and hence J 2 increases as well), the uid mass takes, at rst, a two-axial ellipsoidal shape, with symmetry axis the rotation axis and symmetry group D ∞h , whereas, as soon as J 2 has values not lower than 0,384436, the uid mass takes the shape of a three-axes ellipsoid, with relative symmetry group D 2h .

Increasing J 2 beyond the value 0,632243, we obtain another sequence of (innite and stationary ) solutions (of the system (3)), represented by new geometrical gures, called pearlike congurations of Poincaré, whose symmetry group is C 2v . If the latter sequence, called Poincaré's sequence, is

S P oin = {S(⃗ r; J 2 ); 0, 632243 ≤ J 2 < 1, 346350},
then {S M acl , S Jac , S P oin } will be called Maclaurin-Jacobi-Poincaré pattern.

The main interest of this paper is restricted at discussing the Maclaurin-Jacobi pattern, that includes the rst two sequences {S M acl , S Jac } and their temporal evolution S M acl → S Jac .

3 The second order phase transitions

Let us briey recall the main denitions of Landau's phase transition theories.

• Landau's Thermodynamical Theory. A rst semi-empirical theory of phase transitions, was given by P. Ehrenfest (1933) on the basis of the previous thermodynamical results achieved by J.W. Gibbs and F. Van der Waals. It is based on the discontinuity (of various orders) of the free energy F (P, T ) = U (P, T ) -T • S(P, T ) (with U internal energy, S entropy, P pressure), so that a phase transition is of order n ≥ 1 if there exists, at least, a partial derivative of F , of order n, that exhibits a discontinuity, while all previous partial derivatives of order m < n, are continuous. However, this classication was incomplete and not physically consistent.

As regard the various notions and group notations, let us follows [CMR], [START_REF] Landau | Fisica Teorica 3: Meccanica Quantistica (Teoria Non Relativistica[END_REF], Chap. XII., and [Ga], [MZ].

But no static. Among the thermodynamical variables (P, T, V ), we choose (P, T ) as independent.

In 1937, L.D. Landau ([La]), starting from the results of the predecessors (above mentioned), gave a rst version of a continuum phase transition theory, called Landau's thermodynamic theory (see [START_REF] Landau | Fisica Teorica 5: Fisica Statistica[END_REF], Chap. XIV, 143). This theory, before all, distinguishes between phase transitions 'with order parameter' and phase transitions 'without order parameter'. The transitions of the rst type are those involving phases having the same symmetry group ! or dierent symmetry groups not connected by any relation of the type group-subgroup. The transitions of the second type, instead, regard phases whose symmetry groups are dierent but related by some relation of the type group-subgroup. These are characterized (dierently from those of the rst type) by a symmetry change of the order parameter, that must be invariant with respect to each symmetry groups of the two phases.

Besides the symmetry change, the thermodynamical functions, characterizing the physical state of the system subjected to such phase transition, may not assume any values because it must change according to the above mentioned symmetry change, and in dependence on such order parameter.

The latter critical analysis has been systematically developed by Landau in a second time after his rst Thermodynamical Theory, and represents the so-called Landau's Phenomenological Theory. Is is stronger (as physical theory) than his initial Thermodynamical Theory. The (possible) counterexample described in this paper, just conrms this non-equivalence.

In the framework of the Thermodynamical Theory, for phase transitions with order parameter, Landau rst assumes that the free energy must be function of the order parameter, as an independent and extensive variable η, typical of the thermodynamical system under examination " . In addition, he assumes (Landau's hypothesis ) that F (P, T, η) must be analytic # in a neighborhood of the critical point (P c , T c ) (where such a transition occurs), that must be η = 0 in the initial phase (T < T c ) and that must be η ̸ = 0 in the nal phase (T > T c ). Hence, under such conditions, in a neighborhood of the critical point (P c , T c ), we must have (4) F (P, T, η) = F 0 + α(P, T )η + A(P, T )η 2 + B(P, T )η 3 + C(P, T )η 4 + ....

! For instance, the two phases of a liquid-gas transition, has O(3) as a symmetry group. " Examples of order parameter are the spontaneous polarization in a ferroelectric transition, or the magnetic susceptibility in a ferromagnetic transition.

# This is a weak-point of the Landau's theory, since (P c , T c ) is a simple singularity where, formally, there is no functional analyticity.

where F 0 = F (P, T, 0) and α, A, B, C, ... are independent of η. Landau considers only a fourth order expansion of the type (4) (called Landau's polynomial ), and assumes $ the symmetry η → -η, so that, in the expansion (4), the odd degree terms are zero. Moreover, since we consider equilibrium phase transitions, thermodynamically F should be a minimum in (P c , T c ) for η = 0, so that the expansion (4) reduces to the following Landau's polynomial (of fourth degree)

(5)

F (P, T, η) = F 0 + A(P, T )η 2 + C(P, T )η 4 .
In particular, by the minimum necessary conditions for F in (P c , T c , 0), we must have

(∂F/∂η) (Pc,Tc,0) = 0, (∂ 2 F/∂η 2 ) (Pc,Tc,0) > 0,
that is (via ( 4))

(6) A(P, T ) = a(P )(T -T c ), η = η 0 (T -T c ) 1/2 , C(P c , T c ) > 0,
with η 0 constant. Landau puts (6) as necessary and sucient conditions for classifying, as continuum, a phase transition.

• Landau's Phenomenological Theory (see [START_REF] Landau | Fisica Teorica 5: Fisica Statistica[END_REF], Chap. XIV, 145). In Landau's Thermodynamical Theory (as well as in the previous theories), many important theoretical questions about continuum phase transitions with order parameter remained unsolved; in this theory, subsisted only some vague and unfounded assumptions. The main question concerns the determination of the functional dependence laws of the values of the thermodynamical functions on the typical symmetry change of a continuum phase transition with order parameter. For this purpose, if G 0 G 1 denotes such a generic phase transition, from the initial phase with symmetry group G 0 to the nal phase with symmetry group G 1 , Landau puts η = 0 (and G 0 -invariance) in the initial phase, and η ̸ = 0 (and G 1 -invariance) in the nal phase. Moreover, he assumes that the symmetry G 1 of the nal phase (of the transition), must be the isotropy group of the order parameter η ̸ = 0, with respect to a welldened irreducible representation % D G 0 of G 0 , satisfying further conditions, respectively called Landau's condition and Lif²its's condition (see next 4). Hence, as regard what has been said above (and that forms the essence of the so-called Landau's Phenomenological Theory), it is correct to denote this transition in the compact notation 17 G 0 D G 1 .

$ This symmetry is motivated by theoretical physics reasons (since we must have the same solution considering both η and -η), and by stability reasons.

% From now on, for simplicity sake's, if not otherwise specied, any irreducible representation D G0 of G 0 , will be denoted only by D, without any other indices. Furthermore, we suppose that it denotes always an irreducible representation of G 0 .

The chain subduction criterion and the maximality criterion

Let G 0 D G 1 be a generic continuum phase transition from an initial phase with symmetry group G 0 (where η = 0 in the pre-existed phase) to the nal phase with symmetry group G 1 (where η ̸ = 0 in the resulting phase), associated to the irreducible representation D of G 0 , and having order parameter η ∈ R + ∪ {0}. In this case, we say (with Landau) that G 1 is a allowed subgroup & with respect to D.

Subsequently (see [CLP]), according to Landau and other authors, if this phase transition is of second order, then the following necessary or sucient conditions must hold ' : Successively, F.E. Goldrich and J.L. Birman ([GB]) introduce, for nite (or countable) groups, a necessary stronger condition than condition 4. of above & This is coherent with Landau's denition of continuum phase transition with order parameter, and in agreement with the fact that G 0 and G 1 must be in some relation of the type group-subgroup. See, also, the condition 1. that follows. ' The original sources are: for Landau's condition, see [La], for Lif²its's condition, see [Lif], for Birman's condition, see [START_REF] Birman | Simplied Theory of Symmetry Change in Second-Order Phase Transitions: application to V 3 Si[END_REF], and, for Ascher's condition, see [As1].

1. G 1 is a subgroup of G 0 , 2. (Landau's condition) ([D] 3 |I(G 0 )) = 0,
( [START_REF] Birman | Simplied Theory of Symmetry Change in Second-Order Phase Transitions: application to V 3 Si[END_REF]). The integer number

(7) i D (G 1 ) = 1 |G 1 | ∑ g∈G 1 χ D (g),
that gives the number of times that D| G 1 (=restriction of D to G 1 ) contains the identity representation of G 1 , is called subduction index (or frequency) of D, with respect to G 1 . Therefore, the (Birman-Goldrich)

chain subduction criterion ([GB]) states that, if D is an irreducible representation of G 0 , G 1 and G ′ 1 are subgroups of G 0 such that G ′ 1 is a subgroup of G 1 (that is G ′ 1 ⊆ G 1 ⊆ G 0 ) and i D (G 1 ) = i D (G ′ 1 ) = 1, then G ′
1 is not a permitted subgroup (in accordance with Landau) with respect to D. In consequence of the criticisms moved in [LPC], M.V. Jari¢ ([Ja2]) extendes this chain subduction criterion to the more general context including the case where i D (G 1 ) = i D (G ′ 1 ) is any non-negative integer, reaching the following, more general (Birman-Goldrich-Jari¢)

chain subduction criterion. If D is an irreducible representation of G 0 , and G ′ 1 , G 1 are subgroups of G 0 such that G ′ 1 ⊂ G 1 ⊆ G 0 , with G ′ 1 subgroup of G 1 and i D (G ′ 1 ) = i D (G 1 ) ∈ N ∪ {0}, then G ′
1 is not a (Landau's) allowed subgroup with respect to D. In the following section, these results will be applied to the Maclaurin-Jacobi pattern.

Applications to Maclaurin-Jacobi pattern

We recall that G. Bertin and L.A. Radicati (see [BR]) prove the symmetry change D ∞h → D 2h , associated to the dynamical evolution S M acl → S Jac of the Maclaurin-Jacobi pattern, may be formally interpreted as a second order phase transition in the Landau's Thermodynamical Theory. Therefore, this transition should be, also, the same according to Landau's Phenomenological Theory (if we assume valid the equivalence between these With χ D (g), we will denote the character of the element g ∈ G 1 , computed with respect to the irreducible representation D of G 0 , whereas |G 1 | denotes the order of G 1 . For not nite groups, if we denote the usual Haar's measure on G 1 with µ, then we have

i D (G 1 ) = ∫ G1
χ D (g)dµ(g), whereas, for countable groups, by means of the so-called Born-Von Kàrmàn cyclicity (boundary) conditions, a particular expression of the type (7) is possible (for further details, see [START_REF] Jari¢ | Structural Phase Transitions in Crystals: Broken-Symmetry (Isotropy) Groups[END_REF]).

In the context of Landau's theory, this means that G ′ 1 cannot be, with respect to the irreducible representation D, the symmetry group of the nal phase of a continuum phase transition of the type G 0 In the Maclaurin-Jacobi pattern, we assume the Oz axis as a rotation axis h of the rst order. Moreover, let us denote with θ(t) = ωt the angle of the rotation C θ , with E the identity, with I the inversion and with C 2ϕ the rotations of ϕ/2 radians around to the rotation axis of the second order, placed in the Oxy plane. Hence, with these notations and hypotheses, the table of characters of D ∞h is the follows (see [Co], vol. I; [ITO], App. A,B; [START_REF] Jari¢ | How to Calculate Isotropy Subgroups of a Crystallographic Space Group[END_REF]):

irr. repr. E C θ , C -θ C 2ϕ I IC θ , IC -θ IC 2ϕ A 1g 1 1 1 1 1 1 A 1u 1 1 1 -1 -1 -1 A 2g 1 1 -1 1 1 -1 A 2u 1 1 -1 -1 -1 1 E ng 2 2 cos(nθ) 0 2 2 cos(nθ) 0 E nu 2 2 cos(nθ) 0 -2 -2 cos(nθ) 0 E (n+1/2)g 2 2cos(n + 1/2)θ 0 2 2cos(n + 1/2)θ 0 E (n+1/2)u 2 2cos(n + 1/2)θ 0 -2 -2cos(n + 1/2)θ 0 with A ig , A iu i = 1, 2, real one-dimensional irreducible representations and E ng , E nu , E (n+ 1 2 )g , E (n+ 1
2 )u n ∈ N, real two-dimensional irreducible representations. Therefore, considering the group operations D 2h , D 4h , D 6h (see [Co], vol. I), by (7) it is immediate to verify that

i A 1u (D 6h ) = i A 2g (D 6h ) = i A 2u (D 6h ) = i Enu (D 6h ) = i E (n+ 1
2 )u (D 6h ) = 0, ∀n ∈ N; ! This criterion is applicable to these group chains because D ∞h is countable and

D 4h , D 6h , C 2v are nite. i A 1u (D 4h ) = i A 2g (D 4h ) = i A 2u (D 4h ) = i Enu (D 4h ) = i E (n+ 1 2 )u (D 4h ) = 0, ∀n ∈ N; i A 1u (D 2h ) = i A 2g (D 2h ) = i A 2u (D 2h ) = i Enu (D 2h ) = i E (n+ 1 2 )u (D 2h ) = 0, ∀n ∈ N; 1 ≤ i Eng (D 6h ) = 1 6 ( 1 + 5 cos(nθ) ) ≤ i Eng (D 2h ) = 1 2 ( 1 + cos(nθ) ) , ∀n ∈ N; 1 ≤ i E (n+1/2)g (D 6h ) = 1 6 ( 1 + 5 cos(n + 1/2)θ ) ≤ ≤ i E (n+1/2)g (D 2h ) = 1 2 ( 1 + cos(n + 1/2)θ ) , ∀n ∈ N; 1 ≤ i Eng (D 4h ) = 1 4 ( 1 + 3 cos(nθ) ) ≤ i Eng (D 2h ) = 1 2 ( 1 + cos(nθ) ) , ∀n ∈ N; 1 ≤ i E (n+1/2)g (D 4h ) = 1 4 (1 + 3 cos(n + 1/2)θ) ≤ ≤ i E (n+1/2)g (D 2h ) = 1 2 (1 + cos(n + 1/2)θ), ∀n ∈ N, so that ( i Eng (D lh ) < i Eng (D 2h ) ) ( i E (n+1/2)g (D lh ) < i E (n+1/2)g (D 2h ) )    ⇔ ( 0 < θ < 2kπ n ) ∀n ∈ N, k ∈ Z
with l = 4, 6, and, xing arbitrarily k, we have

( lim n→∞ 2kπ n = 0 ) ⇒ ( θ(t) = ωt = 0 ) ,
hence ω = 0 for t > 0, which is impossible because the case ω = 0 corresponds to a static self-gravitating uid mass of spherical symmetry O(3) (see 2, and [START_REF] Lebovitz | Rotating Fluid Masses[END_REF], [Li]). Therefore 6 Prospects and possible applications

i Eng (D lh ) = i Eng (D 2h ) ≥ 1, i E (n+1/2)g (D lh ) = i E (n+1/2)g (D 2h ) ≥ 1, ∀n ∈ N with l = 4, 6. In conclusion, i D (D 2h ) = i D (D lh ) l = 4,
The previous critical analysis, leads, among other, to a deeper critical study of the formal setting of the general (macroscopic) phase transition theory, eventually with the support of the (microscopic) statistical theory of phase transitions, also in connection with the theory of the rst order phase transitions. Indeed, further motivations for the present analysis come from possible applications of the Maclaurin-Jacobi pattern to particular phase transitions involved in some astrophysical models of stars " : for example, phase transitions may be invoked in the explanation of some aspects of the complex physical phenomenology of the solid external crust of a radio-pulsar (or of a rotating neutron star, in general), as, for instance, the (possible) formation of a supercial thin-layer of gases if we have a rst order phase transition instead of a second order one.

On the other hand, a similar question is already connected with the problem of the supernova explosions (see [Gr], vol. II, Chap. XVIII, 9, sect. a)), where (following W.A. Fowler and F. Hoyle) a particular phase transition neutrons producing occurs, involving a photodisintegration of 56 F e-atoms, forming a (solid) crystalline lattice imbedded in a degenerate (and strongly anisotropic if there is in action an intense magnetic eld # ) Coulomb electron gas (see [Gr], [KW], [Pa], [ST], [START_REF] Tassoul | Stellar Rotation[END_REF]). Again, a similar question enters into the problem of the determination of the shape left by a nova remnant from a rotating white dwarf parent (see [FH]).

and esteem to them for all this. Of course, the possible errors and mistakes present in this paper are only due to me.

13

  that is the symmetric cube of D does not contain the identity representation I of G 0 , 3. (Lif²its's condition) ({D} 2 |V (G 0 )) = 0, that is the anti-symmetric square of D does not contain any arbitrary vectorial representation V of G 0 , 4. (Birman's subduction criterion) D subduces the identity representation of G 1 , 5. (Ascher's maximality criterion) G 1 is a maximal subgroup of G 0 .

  so that it may be classied as a transition of the type D ∞h D D 2h with respect to some irreducible representation D of D ∞h , with D ∞h = G 0 , D 2h = G 1 and D 2h allowed subgroup (of D ∞h ) with respect to D. Nevertheless, it is possible to prove that such transition does not verify chain subduction criterion of Birman-Goldrich-Jari¢, for any given irreducible representation D of D ∞h . In fact, if we consider the groupsG 1 = D 4h , G 1 = D 6h , with G ′ 1 = D 2h proper subgroup of both, we have D 2h ⊂ D 4h ⊂ D ∞h = G 0 , D 2h ⊂ D 6h ⊂ D ∞h , and since we'll see that i D (D 2h ) = i D (D 4h ), i D (D 2h ) = i D (D 6h) for any given irreducible representation D of D ∞h , by the above mentioned Birman-Goldrich-Jari¢ criterion, it follows that ! D 2h is not a (Landau's) allowed subgroup, that is D ∞h D D 2h is not a continuum phase transition (in the sense of Landau's Phenomenological Theory), for any given irreducible representation D.

  6, for any given irreducible representation D of D ∞h , and hence, by the chain subduction criterion, D ∞h → D 2h is not a continuum phase transition, in contrast with the conclusion outlined in[BR]. It follows a restriction to the validity of the chain subduction criterion. Finally, the continuum phase transition D ∞h → D 2h invalidates too Ascher's maximality criterion because D 2h is not a maximal subgroup of D ∞h (hence, also of O(3)) since D 2h ⊂ D lh ⊂ D ∞h (⊂ O(3)) l = 4, 6, whereas, in conformity with such criterium, D 2h should be a maximal subgroup of D ∞h .

" Besides the possible applications that these arguments might have in the theory of fragmentation of atomic nuclei, and related phase transitions (see[GMS],[Mig],[RW]).# The magnetic eld congurations (see[Kc],[OPF]) of a star, are in relation with the occurrence of a phase transitions of the rst order or of the second order, with dierent physical implications in both cases.

Acknowledgements

The early origins of this paper basically spring out after have had many illuminating and interesting private discussions with both Professor Piero Villaggio, emeritus professor of structural engineering and mechanics of continua at the Faculty of Engineering of the University of Pisa, IT, and Professor Luigi A. Radicati di Bròzolo, emeritus professor of theoretical physics at the Scuola Normale Superiore, Pisa, IT, to whom I owe a great part of my extra-curricular training in mechanics, theoretical physics and high energy astrophysics, during my studies in Pisa. So, my sincere and deep thanks

* This work is an enlarged version of a presentation given at the XCI-th National Con-


Fran De

Aquino Maranhao

The Gravelectric Generator Conversion of Gravitational Energy directly into Electrical Energy

Keywords: Gravitational Electromotive Force, Gravitational Energy, Electrical Energy, Generation of Electrical Energy

The electrical current arises in a conductor when an outside force acts upon the free electrons of the conductor. This force is called, in a generic way, of electromotive force (EMF). Usually, it has electrical nature. Here, we show that it can have gravitational nature (Gravitational Electromotive Force). This fact led us to propose an unprecedented system to convert Gravitational Energy directly into Electrical Energy. This system, here called of Gravelectric Generator, can have individual outputs powers of several tens of kW or more. It is easy to be built, and can easily be transported from one place to another, on the contrary of the hydroelectric plants, which convert Gravitational Energy into Hydroelectric energy.

Introduction

The research for generate Gravitational Electromotive Force (Gravelectric Effect) began with Faraday [START_REF] Cantor | Faraday's search for the Gravelectric effect[END_REF]. He had a strong conviction that there was a correlation between gravity and electricity, and carried out several experiments trying to detect this Gravelectric Effect experimentally. Although he had failed several times, he was still convinced that gravity must be related to electricity. Faraday felt he was on the brink of an extremely important discovery. His Diary on August 25, 1849 shows his enthusiastic expectations:

"It was almost with a feeling of awe that I went to work, for if the hope should prove well founded, how great and mighty and sublime in its hitherto unchangeable character is the force I am trying to deal with, and how large may be the new domain of knowledge that may be opened up to the mind of man."

Here we show how to generate Gravitational Electromotive Force, converting Gravitational Energy directly into electricity. This theoretical discovery is very important because leads to the possibility to build the Gravelectric Generator, which can have individual outputs powers of several tens of kW.

Theory

In a previous paper, we have proposed a system to convert Gravitational Energy directly into Electrical Energy [START_REF] De Aquino | A System to convert Gravitational Energy directly into Electrical Energy[END_REF]. This system uses Gravity Control Cells (GCCs). These GCCs can be replaced by the recently discovered Quantum Controllers of Gravity (QCGs) [START_REF] De Aquino | Quantum Controller of Gravity[END_REF], which can produce similar effect. In this paper, we propose a simplification for this system, without using GCCs or QCGs.

Under these conditions, the system shown in the previously mentioned paper reduces to a coil with iron core. Through the coil passes a electrical current , with frequency . Thus, i f there is a magnetic field through the iron core. If the system is subject to a gravity acceleration (See Fig. 1 The expressions of the correlation factors Be χ , Bp χ and Bn χ are deduced in the previously mentioned paper, and are given by ( )

4 1 56 . 45 1 2 1 2 2 2 2 0 4 4 2 ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - + - = f c m B r e rms e Be μ π χ ( ) 5 1 56 . 45 1 2 1 2 2 2 2 0 4 4 2 ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - + - = f c m B r p rms p Bp μ π χ ( ) 6 1 56 . 45 1 2 1 2 2 2 2 0 4 4 2 ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - + - = f c m B r n rms n Bn μ π χ
According to the free-electron model, the electric current is an electrons gas propagating through the conductor (free electrons Fermi gas). If the conductor is made of Iron, then the free-electrons density is

3 28 / 10 4 . 8 m electrons V N × ≅ * .
The theory tells us that the total energy of the electrons gas is given by

5 3 F NE
, where

e F F m k E 2 2 2 h = ; ( ) 3 1 2 3 V N k F π =
(The Fermi sphere). Consequently, the electrons gas is subjected to a pressure ( )( )

V NE dV dE N F F 5 2 5 3 = -
. This gives a pressure of about 37GPa † . This enormous pressure puts the free-electrons very close among them, in such way that if there are inside , then we can conclude that the each electron occupies a volume:

. Assuming

( ) ( ) 8 3 1 1 3 1 2 V N k v m F e e π = = h In the case of Iron, ( 3 28 / 10 4 . 8 m electrons V N × ≅
), Eq.( 8) gives 2

x r e Δ = ≳ ( )

9 10 1 10 m - ×
Comparing with the value given by Eq. ( 7), we can conclude that is a highly plausible value for the electrons radii, when the electrons are inside the mentioned free electrons Fermi gas. On the other hand, the radius of protons inside the atoms (nuclei) is It is known that, in some materials, called conductors, the free electrons are so loosely held by the atom and so close to the neighboring atoms that they tend to drift randomly from one atom to its neighboring atoms. This means that the electrons move in all directions by the same amount. However, if some outside force acts upon the free electrons their movement becomes not random, and they move from atom to atom at the same direction of the applied force. This flow of electrons (their electric charge) through the conductor produces the electrical current, which is defined as a flow of electric charge through a medium [START_REF] Valkengurg | Basic Electricity[END_REF]. This charge is typically carried by moving electrons in a conductor, but it can also be carried by ions in an electrolyte, or by both ions and electrons in a plasma [START_REF] Fischer-Cripps | The electronics companion[END_REF]. ( )

⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - × + - = f B rms Be χ ( ) 11 1 10 35 . 2 1 2 1 2 4 9 ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - × + - = ≅ - f B rms Bp Bn
14 ∫ = = El l d E V r r
Since the current i and the area are constants, then the current density

S J r is also constant. Therefore, it follows that ( ) ( ) 15 . ∫ = = = S l V ES S d J i σ σ r r
By substitution of E , given by Eq.( 14), into Eq.( 13) yields V Fig. 2 -The voltage V between the ends of a wire when it is subjected to a uniform magnetic field B with frequency f and gravity g (as shown above). ) with insulation paint. Thus, the nucleus of the Helmholtz coil is filled with iron wires, which are subjected to a uniform magnetic field with frequency , produced by the Helmholtz coil. Then, according to Eq. ( 10), we have that

H B H f ( ) ( ) 18 1 10 46 . 1 1 2 1 2 4 18 ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - × + - = H rms H Be f B χ
By substitution of Eq.(18) into Eq.( 16), we obtain

( ) ( ) 19 1 10 46 . 1 1 2 1 0 2 4 18 gl e m f B V e H rms H ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - × + - = If ( ) T B rms H 8 . 0 = ‡ and Hz f H 2 . 0 = , then Eq. (19) gives ( ) 20 43 . 0 l V ≅ In order to obtain volts V 220 =
, the total length of the iron wire used in the Toroidal coil must l have 511.63m. On the other hand, the maximum output current of the system (through the Toroidal coil ) (See Fig. 3 (a)) will be given by (Eq.17 Note that, here the Helmholtz coil has two turns only, both separated by the distance R (See Fig. 3 (a)). By using only 2 turns, both separated by a large distance, it is possible strongly to reduce the capacitive effect between the turns. This is highly ‡ The magnetic field at the midpoint between the coils Let us now consider a new design for the Gravelectric Generator. It is based on the device shown in Fig. 4 (a). It contains a central pin, which is involved by a ferromagnetic tube. When the device is subjected to gravity g r and a magnetic field with frequency and intensity (as shown in Fig. 4 (a)) the magnetic field lines are concentrated into the ferromagnetic tube, and a Gravitational Electromotive Force is generated inside the tube, propelling the free electrons through it. The pin is made with diamagnetic material (Copper, Silver, etc.,) in order to H f H B expel the magnetic field lines from the pin (preventing that be generated a Gravitational Electromotive Force in the pin, opposite to that generated in the ferromagnetic tube).

Several similar devices ( N devices) are jointed into a cylinder with radius R and height R . The pin, the ferromagnetic tube and the cylinder have length R (See Fig. 4 27) that ( )

28 9mm r pin =
Then, the area of the cross-section of the pin is ( )

29 10 5 . 2 2 4 2 m r S pin pin - × = = π
On the other hand, the maximum output current of the Gravelectric Generator shown in Fig. 4, according to Eq. 17, will be given by ( )

Conclusion

These results show that the discovery of the Gravelectric Generators is very important, because changes completely the design of the electricity generators, becoming cheaper the electricity generation. They can have individual outputs powers of several tens of kW or more. In addition, they are easy to be built, and can easily be transported. § Note that, if the ferromagnetic tube is made with 4) expresses Be χ in the particular case of However, the values obtained in Eqs. ( 7) and ( 8) led us to think that, in the Fermi gas, . This conclusion is based on the following: while (See Eq.( 7)),

⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - + - = = ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - + - = 1
the
II f B rms Be ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - × + - ≅ 1

Making

) and (See Eq.

(18)), then Eq. (II) can be rewritten as follows (rms The equations ( 5) and ( 6) are also similarly derived from the general expression, i.e., With the dimensions previously mentioned the free electrons, if the Fermi gas, have the size of atoms, and therefore they can not cross the atoms of the conductor. Thus, passing far from the atoms nuclei, they practically do not affect the structures of the protons which are in the nuclei of these atoms. In this way, we have that

H rms B B = H f f = ( ) ( ) III f B H rms H Be ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - × + - = 1
p xp r r ≅ → 1 ≅ xp k
. Substitution of this value into Eq. (V) leads to Eq. ( 5).

In the case of the neutrons, due to its electric charge be null, we obviously have 1 = xn k . Substitution of this value into Eq. (VI) leads to Eq. ( 6).

FFig. 1 -

 1 Fig. 1 -An iron core of a coil subjected to a gravity acceleration g r and magnetic field B r

  electron's volume, then we get ( )

  rms intensity of the magnetic field through the iron core and is its oscillating frequency.

==

  This is the voltage V between the ends of a wire, when the wire with conductivityσ and crosssection area, is subjected to a uniform magnetic field S B with frequency and a gravity (as shown in Fig.(2)) (The expression of f g Be χ isgiven by Eq. (4)).Substitution of Eq. (16) into Eq. (15), gives This is the maximum output current of the system for a given value of Be χ .

σ

  Now consider the system shown in Fig.3. It is an electricity generator (Gravelectric Generator). It was designed to convert a large amount of Gravitational Energy directly into electrical energy. Basically, it consists in a Helmholtz coil placed into the core of a Toroidal coil. The current through the Helmholtz coil has frequency . The wire of the toroidal coil is made of pure iron (

=

  Since the electrical conductivity of the iron is and, assuming that the diameter of the iron wire is This is therefore, the maximum output current of this Gravelectric Generator. Consequently, the maximum output power of the generator is ( )

Fig. 3 -

 3 Fig. 3 -Schematic Diagram of the Gravelectric Generator (Based on a gravity control process

.

  (a) and (b)). On the external surface of the cylinder there is a Helmholtz coil with two turns only. The current through the Helmholtz coil has frequency in order to generate the magnetic field connected as shown in Fig.4 (c).Since the pin and the ferromagnetic tube have length R , then the total length of the conductor through the Gravelectric Generator, , is given by . Equation (20) tells us that in order l If the area of the cross-section of the cylinder above mentioned is , and the area of the cross-section of the device is , Substitution of Eq. (25) into Eq. (24) gives ( )Equation (24) tells us that for devices N 640 = , the cylinder radius must be . On the other hand, Eq. (26) tells us that in this case, we must have . Thus, if the ferromagnetic tube is covered with insulation of thickness, then the outer radius of the ferromagnetic tube is The area of the crosssection of the ferromagnetic tube, , must be equal to the area of the pin cross-section, i.e.,

  the electrical conductivity of the ferromagnetic tube. If the ferromagnetic tube is made with Mumetal() This is therefore, the maximum output current of this Gravelectric Generator. Consequently, the maximum output power of this generator is

Fig. 4 -

 4 Fig. 4 -Schematic Diagram of another type of Gravelectric Generator

  value that possibly should be very close of the real value of . Then, assuming that into Eq. (I), we get an approximated value for Be χ , given by ( )

  is about times greater than the equivalent coefficient of Eq. (10).

χFig. 5 -

 5 Fig. 5 -The voltage V (and current i ) with frequency equal to H f , now Hz f H 60 = .

  Thus, the electrical current arises in a conductor when an outside force acts upon its ) free electrons. This force is called, in a generic way, of electromotive force (EMF). Usually, it is of electrical nature (

									F e =	eE	. However, if the
	nature of the electromotive force is gravitational ( ) g m F ge e = then, as the corresponding force of
	electrical nature is	, we can write that ( ) 12 eE eE m ge = F e = g
	According to Eq. (1) we can rewrite Eq. (12) as
	follows		χ	Be	m e	0	g	=	eE	( ) 13
	Now consider a wire with length l ; cross-section area and electrical conductivity S σ . When a
	voltage	V	is applied on its ends, the electrical
	current through the wire is . Electrodynamics i
	tell us that the electric field,	E , through the wire
	is uniform, and correlated with V and	l	by
	means of the following expression [ 8]