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Uniting Faith and Culture: Hans Urs von Balthasar
By J. Peter Pham|2019-07-15T23:00:01-05:00July 14th, 2013|Categories: Christianity, Communio, Culture, Faith, Featured, Hans Urs von Balthasar|Tags: 1MA|
There are a number of reasons why it could be said, albeit from different perspectives, that the Swiss theologian Hans Urs von Balthasar (1905–1988) occupies a unique place among the great thinkers of the twentieth century.[1] From the point of view of sheer literary volume, he is almost alone in the sheer monumental proportions of his written corpus. Any one part of his published works could fairly be said to constitute a respectable life's work for a lesser man. His doctoral studies in German philosophy and letters culminated in the publication of a vast three-volume thesis, The Apocalypse of the German Soul,[2] on the idea of the end of the world in modern German literature from Lessing to Ernst Bloch. His Jesuit scholastic studies at Fourvière-Lyons under the influence of Henri de Lubac led to a lifelong love affair with the Church Fathers and the publication of pioneering studies of Origen, Irenaeus, Gregory of Nyssa, and Maximus the Confessor, among others. His time in France also opened up for him a new world of literature and he subsequently published a major study of Bernanos as well as translations into German of the writings of Claudel, Péguy, and Calderón. And then there are his theological writings.
As his cousin, the former dean of the Philosophy Faculty of Rome's Gregorian University, Bishop Peter Henrici, observed after his death, "He wrote more books than an ordinary man succeeds in reading in a lifetime."[3] And the remark is hardly an exaggeration: the bibliography assembled by Balthasar's literary executor, Cornelia Capol, catalogues 119 books, 532 articles, 114 contributions to other books, 110 translations (mostly into German) of authors both ancient and modern, 29 contributions to literary anthologies, 103 forewords and afterwords to books by other authors, 93 major book reviews, and 13 critical editions of various writers.[4] In this last category is included his sixty-two-volume collection of the works of Adrienne von Speyr, the physician he converted to Catholicism and whose mysticism would, in turn, influence his theology.[5]
Balthasar is also unique in the annals of thought for his theological approach, developed in the course of his great trilogy: The Glory of the Lord, a six-volume (seven in the English translation)theological aesthetics; Theo-Drama, five volumes on theological dramatics;[6] and Theo-Logic, three volumes on theological logic. Using the classical philosophical tradition's four attributes of being—unity, goodness, truth, and beauty—Balthasar sketched out a new approach to orthodox Christian theology. The trilogy has been likened to an icon triptych. The first panel explores how divine glory was reflected in the beauty, the pulchrum, of the face of Christ and how Christian tradition has sought to give creative expression to this aesthetic dimension of faith. The second panel accents the good, the bonum, in the encounter in history of divine and human freedom. The third panel focuses on the true, the verum. What is the philosophical nature of truth? How does one move from philosophical truth to Jesus Christ's claim of being truth personified? How is one led to the truth of Christ?
Finally, it could be said that Balthasar was unique in that his thought possessed, in the words of Henri de Lubac, a "highly synthetic character that tears down barriers to which the standard"[7] thought of modern times is accustomed. Not only did his religious thought effect a unity between mysticism and positive theology, spirituality and dogmatics, but also—as will be argued in this essay—his life was a remarkable witness to the unity of faith and culture in the midst of today's world. Two avenues of inquiry suggest themselves for this reconsideration: first, a study of Balthasar as an interpreter of Western culture and its present-day crisis; second, an analysis of his theological synthesis as an attempt at a new integration of theology in culture.
"This man is perhaps the most cultivated man of his time. And if one were to find a Christian culture anywhere, one would find it in him."[8] So Henri de Lubac, writing in 1967, described Hans Urs von Balthasar. Albert Chapelle called him the "prodigious dramaturge of Western culture."[9] Prefacing the French translation of Balthasar's A Theology of History, Albert Béguin noted that:
The work of this man, who has not yet reached the milestone of his fiftieth birthday, has an astonishing breadth. The list of books and articles he has published between 1938 and 1954 would cover page after page. And the sheer variety of subjects that he treats reveals such multi-faceted intellectual endeavor that one can hardly conceive one and the same man as having done it. Theology, patristics, philosophy, aesthetics, literary history, musicology, one finds all these subjects side by side with a whole series of literary translations. One might, at first glance, suppose that this is an alarming dispersion of energy. However, if one examines things more closely, one realizes that the one same design, or rather, the one same thrust, governs the whole body of work and orients it around certain key ideas.[10]
The index to any one of Balthasar's books gives evidence of his erudition, even in areas which most theologians would consider beyond their purview. He studied the message of the great poets from Homer to Claudel, examining in between the works of Dante, Calderón, Goethe, and Hölderlin, among others. In his search for the tools to express the dramatic character of Christian revelation, Balthasar delved into world theater from Sophocles to Ionesco. He analyzed the works of painters like Dürer and Rouault and commented on the music of Mozart, Mahler, and Schumann.
What characterized Balthasar's approach—and it could be said, as well, what characterizes the Western civilization he embodied—was an openness to other voices even as he concentrated on his own mission. Like a great composer, he succeeded in harmonizing different voices and orchestrated his symphonic masterpiece. (Not surprisingly, Balthasar once penned a little volume on Christian pluralism entitled Truth is Symphonic.[11]) However, as Béguin noted, "this dramaturgical talent—which calls, as always, for a setting of the stage on the personal level as well as a setting on the level of the common history of mankind—is unceasingly supernaturalized by the key to his composition, which gives him his Christo-centrism."[12] In fact, whatever the object of his immediate theological attention, Balthasar always returned to another object, the "Heart of the World," which is the second Person of the Trinity.[13] Like the great Church Fathers, Balthasar's openness to the riches of culture was rooted in his openness to the Logos, the eternal Word of God, which is the ordering center of all the wisdom of the world.
Already in The Apocalypse, his earliest work, Balthasar had developed a method of interpretation, inspired by phenomenology, through which he could elucidate how the German idealism of Kant, Hegel, Nietszche, and others evolved towards an idea of the Absolute that was reduced, little by little, to the measure of man. This phenomenological method was directed toward eliciting the form that constitutes the center of inspiration in a developing idea. In this regard, Balthasar repudiated the various schools of modern thought which sought to separate form and content (the "great divorce" as C.S. Lewis put it), such as the structuralism that arose from the linguistics of Ferdinand de Saussure, which explicitly separated word from meaning, a dichotomy that has been the driving force behind much of revisionist history.
At the center of Balthasar's own mission as interpreter of Western culture is the conviction that:
The Christian [does not] need to leave his center in Christ in order to mediate him to the world, to understand his relation to the world, to build a bridge between revelation and nature, philosophy and theology….This is what the saints are fully aware of. They never at any moment leave their center in Christ….When they philosophize, they do so as Christians, which means as believers, as theologians.[14]
Thus, unlike many contemporary theologians, Balthasar was a champion of the saints, expending not inconsiderable energies on analyses of the mysticism of Thérèse of Lisieux, Elizabeth of the Trinity, and, of course, Adrienne von Speyr. According to Balthasar, the tireless advocate of a praying theology (kniende-theologie), the saints alone, being caught up entirely by the divine word, are capable of speaking correctly about God.
Balthasar's work as an interpreter of Western culture reaches its climax in the third volume of The Glory of the Lord, subtitled The Realm of Metaphysics,[15] which deals with the historical evolution of the theme of the glory of God in metaphysics. This work is particularly significant because of the theological diagnosis which Balthasar elaborates concerning Western culture and the challenges which it faces in the contemporary world from feminism, environmentalism, relativism, and a host of other radical iconoclastic movements.
Reviewing the historical evolution of metaphysics, Balthasar traces the cultural crisis of the West back to its spiritual roots in the gradual narrowing of Western thought. According to Balthasar's panoramic analysis, from the late antiquity of Virgil and Plotinus down to the modern period, the metaphysical attitude of wonder (thaumazein) has, slowly but surely, been replaced by a rational spirit, characterized by scientism and positivism. However, without an objective foundation, rationalism quickly degenerates into anthropocentrism, the commitment to the standpoint that the correct interpretative principle for the whole of reality is the subjectivity of the person. This stand descends easily into pantheism where "man creates the eternal and yet is the most ephemeral," as Feuerbach and Marx contended. To paraphrase Eric Voegelin, the trans-temporal goal of union through the crucified Christ with the unseen Father is commuted to a temporal goal as the supreme fulfillment of existence. The eschaton has been immanentized completely.[16]
Thus the modern malaise, according to Balthasar, is the result of the darkening of the metaphysical outlook that took shape in the course of the development of Christian theology. To the extent that theology replaced love with gnosis, closing itself to the mystery of being in its totality, it lost contact with the center of the Christian mystery itself. Balthasar observed: "Where love no longer seconds the encompassing metaphysical act, it collapses sceptically and agnostically within itself to what is found within the world: the dimension of glory is lost within that of beauty."[17]
Balthasar argues that instead of posing the profound question, "Why is there something rather than nothing?," Western philosophy has strayed from the wonder (Ver-wunderung) to mere admiration (Be-wunderung) before the order of existence as a whole, the cosmos. Like Heidegger, Balthasar sees in the metaphysical attitude of wonder a counter to modern technocracy with its attendant nihilism and materialism. And, as a Christian, Balthasar is convinced that that wonder carries the name of love, which alone respects both the mystery of God and the mystery of man. According to Balthasar, the alternatives are either the love of the God of Jesus Christ in perfect discipleship or his substitution (and elimination) by man-made secondary realities. Thus he concludes his long analysis of glory in the realm of metaphysics by appealing to Christians to make themselves the guardians of metaphysics as servants of culture:
The Christians of today, living in a night which is deeper than that of the so-called Dark Ages, are given the task of performing the act of affirming Being, unperturbed by the darkness and the distortion, in a way that is vicarious and representative for all humanity: an act which is at first theological, but which contains within itself the whole dimension of the metaphysical act of the affirmation of Being. Those who are directed in this way to pray continually, to find God in all things and to glorify Him are able to do so on particular grounds (that is particular graces) which allow them to perform their "creaturely duty" (as Ruybroeck, Bérulle, and Condren understood it). But in so far as they are to shine "like the stars in the sky," they are also entrusted with the task of bringing light to those areas of Being which are in darkness so that its primal light may shine anew not only upon them but also upon the whole world; for it is only in this light that man can walk in accordance with what he is truly called to be.[18]
This mission requires more than speculative argumentation. It requires an investment of the entire force of one's personal existence in the drama of love. For Balthasar, there is no such creature as neutral or purely speculative metaphysics. The profound experience of being requires a commitment of freedom in love: in fact, the title of the final section of the volume is "Love as Custodian of Glory." Consequently, the Christian's duty is "to experience the presence of absolute love, and himself to actualize it, and to make it visible."[19]
In fact, elsewhere Balthasar argues that the original experience of being and love are coextensive, placing all human existence under the sign of love:
The infant is brought to consciousness of himself only by love, by the smile of his mother. In that encounter, the horizon of unlimited being opens itself to him, revealing four things to him: (1) that he is one in love with the mother, even in being other than his mother, therefore all Being is one; (2) that love is good, therefore all Being is good; (3) that that love is true, therefore all Being is true; and (4) that that love evokes joy, therefore all Being is beautifu1.[20]
This perspective offers a foundation to receive the Christian revelation, that is to say, the revelation of love in the person of Christ. And, for Balthasar, the renewal of Western culture requires a renewed appreciation of this vision.
Balthasar's theological project can be understood as an attempt to integrate theology and culture. To appreciate the significance of this undertaking, however, the radical character of his vocation and spiritual pilgrimage should be recalled. Scion of a patrician family, the youthful Balthasar had not had the slightest thought of becoming a priest or entering a religious order until a dramatic encounter while on retreat in 1927, when he was finishing his dissertation. Years later, he recalled his vocation came almost as lightning from a cloudless sky:
Even now, thirty years later, I could still go to that remote path in the Black Forest, not far from Basel, and find again the tree beneath which I was struck as by lightning. . . . And yet it was neither theology nor the priesthood which then came to my mind in a flash. It was simply this: you have nothing to choose, you have been called. You will not serve, you will be taken into service. You have no plans to make, you are just a little stone in a mosaic which has long been ready.[21]
Balthasar's vocation led him from Idealism and Romanticism in German literature to scholasticism in the formation of the Society of Jesus to the Fathers of the Church (thanks to the influence of Henri de Lubac) and on to the mysticism of the saints. Eventually the dialogue he pursued with the Protestant theologian Karl Barth and the common mission he undertook with Adrienne von Speyr contributed to his unique theological vision, one which was radically Christo-centric and Trinitarian.
The centerpiece of his life's work is, of course, the trilogy, a sort of the literary triptych, which, as noted earlier, opens with a theological aesthetics and closes with a Theo-Logic, a divine logic. The central panel of that triptych is the Theo-Drama where God wrestles with his rebellious creature and exercises his own freedom to the point of dying on the cross and descending into hell.
Most revealingly, Balthasar does not cast the question of eternal damnation in the traditional terms of "What does man lose in losing God?" Rather, he poses it from the divine viewpoint: "What does God lose in losing man?"[22] That he should close the Theo-Drama on such a note is not surprising when one considers the questions with which he opened it, questions long smoldering in the back of the Western mind:
What is the relationship between divine and human freedom? Should we suppose that God accepted some limit on his freedom when he created man, by whom his world could be brought either to perfection or to destruction? Is he powerless in the face of autonomous man's "No"? And how is this divine powerlessness related to the Godforsakenness of his Son on the Cross?[23]
Balthasar responds to these questions in dialogue with some of the most representative minds of modernity. In this way he achieved a new integration of theology and culture since the aesthetic and dramatic categories of his thought relate to the aspirations and context of modern man. Artistic creativity, environmental concerns, suffering, and daily life—all these challenges are met by his referring of all human situations back to the mystery of Christ.
The importance of his theological dramatics is highlighted in the anthropological implications of his work. The distinctive feature of Balthasar's depiction of the economy of salvation is the analogy which he draws with the world of theater. He adapts the classical idea of the "world theater" which conceives the history of the world as a "play" in which a transcendent power entrusts human beings with specific roles. The human "actors" assume these roles during their earthly life and are judged by the transcendent spectator according to their performance.[24]
Balthasar transforms this notion of "world theater" so that it might fit the Biblical account of salvation history. In the resulting Theo-drama, God no longer simply performs the role of the transcendent judge of the play, the "Most High Master of the Play,"[25] but rather assumes the three roles of the dramaturgical triad: author, director, and actor. Thus Balthasar notes:
Theo-drama (as distinct from merely human drama) is only possible where "God" or a "God," or some accredited representative of God, steps onto the stage of life's play as "a person" in the action, separate from the other characters.[26]
Interpreting the world as a stage, Balthasar envisions man as well as the incarnate Son as actors who have been given roles by an author, the Father, who himself remains transcendent to the drama. The actor's task is to assume freely, yet obediently, their individual role which establishes their identity on stage. Although the actors are limited by their pre-assigned role, Balthasar's theo-drama acknowledges finite human freedom by allowing individuals the possibility of interpreting their own roles.[27]
For Balthasar, the Holy Spirit is the drama's director, who transposes the author's text into the actual performance. As director, the Spirit can interpret the play independently of neither the play's author nor its actors. The Spirit is bound to the author's text insofar as he mediates the Father's will to each individual person. Furthermore, he is bound to the actors insofar as he does not appear on stage himself, but draws together the various elements of the play, constantly reminding the cast of its common endeavor.
In order to bridge the gap between the world of theater and that of salvation history, Balthasar relies on the key biblical concept of mission (Sendung), especially as it appears in the Johannine corpus of the New Testament. In the Fourth Gospel, Jesus appears as the divine Son who understands his life and death as a mission to "exegete" the eternal Father through his own person and being.[28] Balthasar claims that Jesus' mission constitutes his personal identity in the world in much the same way as an actor's role constitutes his personal identity on stage. Balthasar develops his theological anthropology on the basis of this Sendungchristologie.
Furthermore, according to Balthasar, what is true for Christ in this case also holds true for other beings: God bestows an individual mission upon every human being which constitutes one's own personality.[29] In this light, the mission of the Christian is understood to be more than a merely human search for happiness, albeit with the help of divine providence, but as a living and suffering, in and with God, for the world. While medieval man saw Christian revelation as the achievement of a cosmological order and moderns, bent on anthropocentrism, try to find in it a response to the needs of man, Balthasar locates the distinctiveness of Christianity in the absolute love of Christ and the saints. For Balthasar, the renewed integration of faith and culture occurs not at all if it does not occur under the sign of absolute love:
The plausibility of this divine love is not illumined by reducing it to and comparing it with what man has always recognized as love. Its plausibility comes only from the form of revelation itself. This form is so majestic that, without expressly demanding it, its perception exacts from the beholder the attitude of adoration.[30]
This approach is founded on two pillars of human experience: the encounter of persons, which facilitates a prior comprehension of the gratuitousness of love, and aesthetics, which offers the experience of beauty as the privileged locus for an objective vision of revelation. According to Balthasar, when "one experiences startling beauty (in nature or in art) . . . [what] confronts us is overpowering, like a miracle, and only as a miracle can it be understood; it can never be tied down by the person having the experience. The appearance of its inner unfathomable necessity is both binding and freeing, for it is seen clearly to be the appearance of freedom itself."[31] In the context of Christian revelation, this principle of "love alone is worthy of faith" is personified in Jesus Christ, the objective figure that ravishes the assent of the believer.
Balthasar's trilogy is devoted to an exposition of this approach in its aesthetic, dramatic, and logical facets which, while taking their inspiration from contemporary culture, are purified by revelation. By appealing to the great theologians and mystics of the past who were caught up by the glory of God, Balthasar calls modern man to turn away from his own petty preoccupations and to turn to the One who alone is able to fulfill him. Thus Balthasar points a way out of the spiritual crisis of modernity, characterized by a narcissistic subjectivism which closes the individual in on himself, by taking up man's aspirations to freedom and solidarity and offering a liberating communion with a greater reality, one revealed in the mystery of love.
In an age when many in the West, tired of relativism and yearning for the absolute, turn to the exotic mysticism of the Orient, Balthasar's theological aesthetics offers to the West the foundations for a new integration of faith and culture. He reminds his readers that, while they can multiply religious "experiences" and social "commitments," all striving is in vain unless it is rooted in an encounter with the objective face of the Absolute. However, he also hastens to recall that this encounter must necessarily lead to a sense of mission. Thus theological aesthetics bear fruit in the theological dramatics: if The Glory of the Lord is an attempt to turn modernity away from subjectivism, then the Theo-Drama embraces with enthusiasm the authentic freedom which is so sorely needed in Western culture today.
In the end, it is not surprising that the recognition that he was so long denied within his own Roman Catholic Church—for example, he was forced to leave the Jesuit order in 1948; the then-Roman Congregation for Seminaries and Universities inhibited him from accepting the chair of theology at Tübingen; and he was not even invited to the Second Vatican Council—came at long last to Balthasar in the pontificate of Pope John Paul II when, in 1984, he was awarded the Pope Paul VI Prize and four years later, he was raised to the dignity of a cardinal of the Roman Church, although he died three days before his formal investiture. John Paul, after all, is the philosopher par excellence of freedom in relation to truth. Influenced in many respects by the dramatic theology of the Swiss theologian, John Paul has made freedom the theme of his entire pontificate; from the religious to the political arena, from sexual to economic concerns, the Pope has repeatedly called for respect for the person of the individual in his self-determination in confrontation with and his mission from the Absolute.[32]
Of course, this message is rejected by a human reason which sees itself as absolute. In fact, Balthasar's last years were often bitter ones as he ruminated on the darkness which had overshadowed not only Western culture but which had also penetrated the Church. Particularly pointed were his attacks on the "liberal broadening of dogma, of some transformation of its content into a 'non-objective' reality"[33] which has resulted in a "whole gamut of possible new Christianities, which threaten to lose every continuity with what until now was understood as Christianity, and which would perhaps do better to change their trademark."[34] In an address in Madrid only a month before his death, Balthasar went so far as to admit that: "Humanity will prefer to renounce all philosophical questions—in Marxism, or positivism of all stripes, rather than accept a philosophy that finds its response only in the revelation in Christ."[35] Nonetheless, even as he mourned the loss of the metaphysical vision and the concomitant passing of the cultural and spiritual patrimony of the West, Balthasar looked forward to its renewal, a renewal for which he long labored to lay the foundations:
We are living in a time of collapsing images of gods and idols. The spiritual and cultural traditions of the West have become questionable. Indeed it goes beyond that, for they have been liquidated—quickly and relatively painlessly. Just as a tree in autumn drops its leaves, without pain or regret, in order to gather once more new strength from within, to renew its powers in hibernian peace, so too the tree of culture is being stripped of its leaves. Of course as autumn moves into winter, the leaves lie thickly under our feet—and the great books thickly in the bookstores. . . . But we are not deceived for a moment about the outcome. . . . A small regret may be permitted us, just as autumn is the time of elegiac lyric, but who would want to huddle up in eschatological pathos! We trust the powers of nature, her wise economy and the laws of her renewa1.[36]
As one awaits that renewal, the figure of Hans Urs von Balthasar and the integration of faith and culture which he achieved in his life and in his work guide wayfarers in that dark night, a beacon of light pointing the way to spiritual renewal of the culture—a light which Balthasar himself would, no doubt, be quick to emphasize—is but the reflected light of glory.
Repubplished with the gracious permission of Modern Age (Spring 2000).
For biographical information, cf. Elio Guerriero, Hans Urs von Balthasar ("I teologi del ventesimo secolo" Series 2; Cinisello Balsamo, Milan, 1991).
Hans Urs von Balthasar, Apokalypse der deutschen Seele. Studien zu einer Lehre von letzen Haltungen (Salzburg, 1937–1939).
Peter Henrici, "Hans Urs von Balthasar: A Sketch of His Life," in Hans Urs von Balthasar: His Life and Work, ed. David L. Schindler (San Francisco, 1991), 7.
Cornelia Capol, ed., Hans Urs von Balthasar. Bibliographie 1925–1990 (Einsiedeln, 1990).
For details on Balthasar's symbiotic intellectual and spiritual relationship with Speyr, cf. Hans Urs von Balthasar, Our Task: A Report and a Plan, trans. John Saward (San Francisco, 1994).
Noteworthy is the recent publication of the first full-length commentaries on the first two parts of the trilogy: Aidan Nichols, The Word Has Been Abroad: A Guide Through Balthasar's Aesthetics (Washington, D.C., 1998); idem., No Bloodless Myth: A Guide Through Balthasar's Dramatics (Washington, DC, 1999).
Henri de Lubac, "A Witness to Christ in the Church: Hans Urs von Balthasar," in The Church: Paradox and Mystery, trans. James R. Dunne (Staten Island, 1969), 108.
Ibid., 105.
Albert Chapelle, "La merveille de l'être," in Revue catholique internationale Communio (March-April 1989), 45.
Albert Béguin, preface to Théologie de l'histoire, by Hans Urs von Balthasar (Paris, 1955), 1.
Hans Urs von Balthasar, Truth is Symphonic: Aspects of Christian Pluralism, trans. Graham Harrison (San Francisco, 1987).
Béguin, 13.
Cf. Hans Urs von Balthasar, Heart of the World, trans. Erasmo S. Leiva (San Francisco, 1979).
Hans Urs von Balthasar, "Theology and Sanctity," in Explorations in Theology, Vol. 1, Word Made Flesh, trans A.V. Littledale and Alexandre Dru (San Francisco, 1989), 195.
The English edition of the works of Balthasar separates Volume 3 into two volumes and publishes it, respectively, as Volumes 4 and 5. Cf. Hans Urs von Balthasar, The Glory of the Lord, Vol. 4, The Realm of Metaphysics in Antiquity, trans. Brian McNeil and others, ed. John Riches (San Francisco, 1989); and idem., The Glory of the Lord, Vol. 5, The Realm of Metaphysics in the Modern Age, trans. Oliver Davies and others, ed. Brian McNeil and John Riches (San Francisco, 1991).
Cf. Eric Voegelin, The New Science of Politics (Chicago, 1952).
Ibid., 642–643.
Hans Urs von Balthasar, My Work: In Retrospect, trans. Brian McNeil and others (San Francisco, 1993), 114.
Hans Urs von Balthasar, "Pourquoi je me suis fait prêtre," in Pourquoi je me suis fait prêtre. Témoignages recueillis, ed. Jorge Sans Vila and Ramón Sans Vila (Tournai, 1961), 21.
Hans Urs von Balthasar, Theo-Drama, Vol. 5, The Last Act, trans. Graham Harrison (San Francisco, 1998), 506.
Hans Urs von Balthasar, Theo-Drama, Vol. 1, Prolegomena, trans. Graham Harrison (San Francisco, 1988), 50.
Ibid., 135–258. Decisive for Balthasar's concept of drama is the poet of the Spanish Baroque period, Calderon de la Barca (1600–1681), in whose works the notion of the "theater of the world" experienced a renaissance. Cf. ibid., 163–177, 361–369.
Hans Urs von Balthasar, Theo-Drama, Vol. 2, Dramatis Personae: Man in God, trans. Graham Harrison (San Francisco, 1990), 189.
Cf. Theo-Drama, Vol. 1, 254–257. Also cf. Theo-Drama, Vol. 2, 62–70.
Cf. ibid., 152–159.
Balthasar's notion of personhood, as it is developed in dialogue with other philosophical and sociological notions, cf. Theo-Drama, Vol. 1, 481–648.
Hans Urs von Balthasar, Love Alone: The Way of Revelation (London, 1968), 47.
Ibid., 44.
Cf. Rocco Buttiglione, Karol Wojtyla: The Thought of the Man Who Became Pope John Paul II, trans. Paolo Guietti and Francesca Murphy (Grand Rapids, Mich., 1997).
Hans Urs von Balthasar, The Moment of Christian Witness, trans. Richard Beckley (San Francisco, 1994), 148.
Balthasar, My Work, 118.
Hans Urs von Balthasar, "Patristik, Scholastik, und wir," in Theologie der Zeit, 3 (1939), 65.
About the Author: J. Peter Pham
J. Peter Pham is Director of the Africa Center at the Atlantic Council in Washington, DC. He previously held an academic appointment as tenured Associate Professor of Justice Studies, Political Science, and Africana Studies at James Madison University in Harrisonburg, Virginia.
January 9th, 2021 | 1 Comment
Pierre Cardin: The Death of a Man Who Destroyed Fashion


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James helps Cavs rally past Nets
LeBron James played with his aching back and the Cleveland Cavaliers rallied for a much-needed win.
James, who didn't practice Monday and Tuesday because of back spasms, scored 33 points as the Cavaliers overcame a 14-point deficit to defeat the New Jersey Nets, 104-83, Wednesday night.
The Nets led 67-53 midway through the third period before the Cavaliers rallied to take a 76-74 lead going into the fourth quarter. Bostjan Nachbar's 3-pointer to begin the fourth gave New Jersey a 77-76 lead, but Cleveland scored the next nine points.
Sasha Pavlovic had a jumper, Joe Smith scored two baskets, and Wally Sczerbiak added a 3-pointer to go ahead, 85-77.
As the Cavaliers took control, New Jersey went scoreless on 10 straight possessions and didn't score again until Vince Carter's two free throws with 4:36 left. The Cavaliers took a 90-79 lead on James' three-point play with 4:23 to play. The Nets scored nine points in the fourth quarter.
James was 11-of-21 from the field with seven rebounds and eight assists.
Sczerbiak scored 14 while Zydrunas Ilgauskas 13 for the Cavaliers, who lead Washington by two games for the fourth seed in the Eastern Conference.
Carter scored 19 points and Richard Jefferson added 15 to lead New Jersey, which won the first three meetings this season against Cleveland.
The Nets lead 53-49 at halftime and scored the first seven points of the third quarter.

redpajama_set_name: RedPajamaCommonCrawl


What's the best affordable, but trendy area for a young professional in Atlanta?
I've lived in the Suburbs around Marietta, but I'm looking to move further into Atlanta by 2015. I'm an esthetician, and I currently work in Buckhead, but there is no way I can afford that cost of living. Any where that's just as trendy, but cheaper? Oh and closer than the Kennesaw/Marietta area?


text:
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text:
Kettering crime rate soars over last decade
Published: 11:21 Tuesday 09 July 2019
The crime rate in Kettering has shot up by more than 50 per cent over the past decade.
Kettering councillor Mick Scrimshaw revealed the figures at the council's town forum meeting last night (Jul 8) after tracking back through crime reports given by police to Kettering Council.
Recorded crime in Kettering borough has increased by more than 50 percent in the past decade.
During 2009 there were 5,353 crimes recorded in Kettering compared to 8,231 recorded between June last year and this June, making it a 54 per cent hike in total crime during the 10-year period.
The rising crime levels in the town have been a concern for some time, with councillors regularly expressing their anxiety about how crime and anti-social behaviour has rocketed in the market town.
There are regular reports from police about criminal incidents and last week a woman was robbed by a man on a motorbike.
The town has also become under siege from drug factories with a £3m cannabis farm set up at the former Gala Bingo raided last month, and other drugs farms found in the borough.
The town has 16 county lines – organised drug gangs – operating from it with a number of suspected Kettering drug dealers caught in a police operation last month.
Cllr Scrimshaw, who is leader of the Labour opposition group, said the figures were 'startling'.
He said: "Since 2010 we have had a rising crime rate in Kettering by 54 per cent and I think that is a startling figure. Whatever the reason for that, it is clear that the people of Kettering are being let down. There is a problem there and we need to solve it. Whether this is because of social problems, lack of policing, austerity, I don't know."
Insp Scott Little, who is in charge of the Kettering and Corby police neighbourhood teams, said at the meeting that there was currently an extra robbery a week being carried out in the town compared to the same period last year. There were 139 robberies in the period from June 2018 to this June compared to 87 in the same 12 months before.
Figures in a report say that crime has risen in the town by 13 per cent in the past year.
The inspector said that resolution rates were not where he wanted them to be. Overall just 15 per cent of total crime was resolved with 1,288 incidents solved. This compares to 30 per cent of crimes resolved in 2009.
Last month (June) the Met Police Commissioner Cressida Dick said the national resolution rate was 'woefully low'.
Kettering murder trial: Body undiscovered for 12 weeks
Corby boy took weapon to retail park


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text:
The Promised War: A Thriller
Written By: Thomas Greanias
Narrated By: Piter Marek
Publisher: Simon & Schuster
Deep beneath the ancient city of Jerusalem lies a secret that knows no bounds, devastating enough to reach across time. History's greatest spy story begins here.
For a millennium, Jerusalem's Temple Mount has been at the center of war and death. There's never been a time when blood wasn't spilled upon this ancient, sacred site. Flash forward to present-day Jerusalem, where 35-year-old Israeli counterterrorism agent Sam Deker has just thwarted the most recent act of violence—an attempt by radical Palestinians to blow up the Dome of the Rock mosque and pin the blame on right-wing Orthodox Jews. The threat, however, is a diversion. Deker himself is the real target. He is captured and taken to neighboring Jordan, where he is tortured because of his deep knowledge of Israel's most closely guarded state secret.
Deker escapes with his comrade Uri Elezar, making it all the way to the border, only to be taken down at the banks of the Jordan River. This time, however, Deker wakes up in the middle of the ancient Israelite army on the eve of its historic siege of Jericho. Deker doesn't know if he is dead, in some torture-induced psychosis, or really back in time. But General Bin-Nun has declared a colossal holy war, and he's sending Deker and Elezar on a dangerous mission to spy on the Promised Land in advance of the invasion.
For Deker, it's his only hope to escape this genocidal hell. Then he finds himself in the arms of a beautiful enemy named Rahab, caught in a web of deadly betrayal, as he struggles to unlock the truth, secure Israel's future and his own, and save the twenty-first century from The Promised War.
Fiction & Literature >
Promised War: A Thriller
by Thomas Greanias
This title is due for release on June 15, 2010.
This title is due for release on June 15, 2010


text:
\section{Introduction}

The Lipkin model was originally proposed by Lipkin, Meshkov and Glick\cite{1}
and has been widely used to test several kinds of many-body theories
of strongly interacting fermion systems, for instance, the time-dependent
Hartree-Fock method,\cite{2} finite temperature dynamics,\cite{3,4}
collective dynamics of many-fermion system,\cite{5,6}
phase transitions, spin tunnelling,\cite{7} etc.
This statement can be found in the paper by the present authors with
Kuriyama.\cite{8}
Further, they have investigated the Lipkin model for thermal effect
in Ref. \citen{9}.
With the aim of participating in these researches under a certain new
viewpoint for the Lipkin model, Ref. \citen{8} was published.
On the basis of a new boson realization
proposed by the present authors with Kuriyama,\cite{10} various
numerical results were presented in Ref. \citen{8}

It is well known that, as was already mentioned, the Lipkin model is a
schematic model for understanding the dynamics of many-fermion systems
and it obeys the $su(2)$-algebra.
The interaction term in the Hamiltonian is expressed in terms of the quadratic
form with respect to the raising and the lowering operator in the
$su(2)$-generators.
On the other hand, we know that the $su(2)$-generators can be expressed
in terms of the bilinear forms of two kinds of boson creation
and annihilation operator, i.e., the Schwinger boson representation.
Therefore, we can apply the Schwinger boson representation to the
Lipkin model and the interaction term can be re-formed in terms of the
product of the boson-pair creation and annihilation.
The boson pairs and the boson number operator form the $su(1,1)$-algebra,
and then, the Lipkin model is formulated in the frame of the
$su(1,1)\otimes su(1,1)$-algebra.
With the aid of the MYT mapping method,\cite{11} we can transcribe the
system in the Holstein-Primakoff representation for the
$su(1,1)$-algebra, and in Refs. \citen{8} and \citen{10},
a certain special case
was treated.

In response to the above situation, very recently, the present authors
proposed a viewpoint that the two-level pairing model in many-fermion systems
is also described in terms of the
$su(1,1)\otimes su(1,1)$-algebra.\cite{12,13}
Hereafter, Ref. \citen{13} is referred to as (I).
The two-level pairing
model is a kind of the $su(2)\otimes su(2)$-algebraic model, and
with the aid of four kinds of bosons, we can treat this model
in the Schwinger boson representation.
In this sense, also, by applying an idea similar to that adopted in
Refs. \citen{8} and \citen{10}, we can show that the present system is
reduced to that obeying the $su(1,1)\otimes su(1,1)$-algebra.\cite{12}

A main aim of the present paper is to show again that the Lipkin model is a
kind of the $su(1,1)\otimes su(1,1)$-algebraic model.
Of course, the basic idea is similar to that adopted in Ref.\citen{12}.
In contrast to the case of the two-level pairing model, the
$su(1,1)$-generators in the Lipkin model consist of boson-pairs and
boson number operators.
Therefore, the structures of the $su(1,1)$-algebra are different from
each other.
The present case is characterized by the quantum numbers
$t=1/4$ and $3/4$ and in Refs. \citen{8} and \citen{10},
only the case $t=1/4$ was treated.
As was shown in Ref. \citen{12}, the two-level pairing model is characterized
by the quantum numbers $t=1/2,\ 1,\ 3/2,\cdots$.
This difference is not so important for applying the MYT boson mapping method
to obtain the Holstein-Primakoff representation.
However, it influences us into constructing boson coherent states which lead
to the classical counterpart of the models.
The two-level pairing and the Lipkin model are expressed in terms of four
and two kinds of bosons, respectively.
Therefore, compared with the two-level pairing model, a possibility to
construct boson coherent state in the Lipkin model is limited and a
special device is necessary.
Under the above-mentioned scheme, we arrive at the following conclusion:
We can describe the Lipkin model and the two-level pairing
model in common ring.
Of course, the comparison with the result based on the
$su(2)\otimes su(2)$-coherent state is necessary.

In \S 2, the Lipkin model in many-fermion systems is recapitulated with
some new aspects and in \S 3, the Schwinger boson representation is
described.
Section 4 is devoted to re-forming the Lipkin model in the Schwinger
boson representation.
Of course, the MYT mapping method and a certain boson coherent state, which is
called the $su(1,1)\otimes su(1,1)$-coherent state,
play a central role and the model
can be expressed in terms of two kinds of new bosons.
In \S 5, the model is formulated in the frame of one kind of boson.
Finally, in \S 6, the comparison with the result
based on the $su(2)\otimes su(2)$-coherent state is sketched.
In Appendix A, a possible idea for constructing the orthogonal
set for the Lipkin model in the fermion space is demonstrated.
In Appendix B, the boson-pair coherent state and a certain new boson
coherent state are formulated.

\section{The original form of the Lipkin model in the fermion space}

The model discussed in this paper is the Lipkin model, which enables
us to obtain a schematic understanding of the problems mentioned in the
opening paragraph of \S 1.
The framework of this model consists of two single-particle levels
with the same degeneracies $\Omega=2j+1$
($\Omega$; even integer, $j$; half-integer), in which $N$ fermions
are moving a certain type of the interaction.
We specify the upper and the lower level by $\sigma=+$ and
$\sigma=-$, respectively.
The Hamiltonian ${\hat {\cal H}}$ is expressed in the form
\begin{equation}\label{2-1}
{\hat {\cal H}}=\epsilon{\hat {\cal S}}_0-(G/4)
\left[\left({\hat {\cal S}}_+\right)^2+
\left({\hat {\cal S}}_-\right)^2\right] \ .
\end{equation}
Here, ${\hat {\cal S}}_{\pm,0}$ are defined as
\begin{eqnarray}\label{2-2}
& &{\hat {\cal S}}_+=\hbar\sum_{m=-j}^{j}{\hat c}_m^*(+){\hat c}_m(-) \ ,
\qquad
{\hat {\cal S}}_-=\hbar\sum_{m=-j}^{j}{\hat c}_m^*(-){\hat c}_m(+) \ ,
\nonumber\\
& &
{\hat {\cal S}}_0=(\hbar/2)\sum_{m=-j}^{j}({\hat c}_m^*(+){\hat c}_m(+)
-{\hat c}_m^*(-){\hat c}_m(-))\ .
\end{eqnarray}
The parameters $\epsilon$ and $G$ denote the difference of the single-particle
energies between the two levels and the strength of the interaction,
respectively.
The set $({\hat c}_m(\sigma),{\hat c}_m^*(\sigma))$ represents
fermion operator in the single-particle state $(\sigma, j, m;
m=-j,-j+1,\cdots , j-1, j)$.
Total fermion number operator ${\hat {\cal N}}$ is given as
\begin{equation}\label{2-3}
{\hat {\cal N}}=\sum_{m=-j}^{j}
({\hat c}_m^*(+){\hat c}_m(+)+{\hat c}_m^*(-){\hat c}_m(-))\ .
\end{equation}

We know that the set $({\hat {\cal S}}_{\pm,0})$ obeys the $su(2)$-algebra,
and then, the Hamiltonian ${\hat {\cal H}}$ is diagonalized in the
frame of the orthogonal set obtained in the well-known procedure.
In this sense, it may be not necessary to repeat the discussion on the
eigenvalue problem for the Lipkin model.
However, as was mentioned in \S 1, we will describe this model in the
framework of the Schwinger boson representation and the quantum
numbers characterizing this model formally do not connect with
those appearing in this model as a many-fermion system
we are discussing in this section.
One of the typical examples may be total fermion number,
which is not contained in the Schwinger boson representation.
Therefore, inevitably, focusing on the quantum numbers,
we have to repeat the discussion on the eigenvalue problem.

If the state with the minimum weight, which we denote $\rket{m}$,
is derived, we can construct the orthogonal set by operating the
raising operator ${\hat {\cal S}}_+$ successively on $\rket{m}$.
In the present case, $\rket{m}$ should satisfy the condition
\begin{subequations}\label{2-4}
\begin{eqnarray}
& &{\hat {\cal S}}_-\rket{m}=0 \ ,
\label{2-4a}\\
& &{\hat {\cal S}}_0\rket{m}=-S\rket{m} \ , \quad
(S=\hbar s; s=0,1/2,1,\cdots )
\label{2-4b}\\
& &{\hat {\cal N}}\rket{m}=N\rket{m} \ , \quad\ \
(N=0,1, 2,\cdots , 2\Omega)
\label{2-4c}
\end{eqnarray}
\end{subequations}
In order to understand the structure of $\rket{m}$ obeying the
condition (\ref{2-4}), first, we set up the following form of $\rket{m}$:
\begin{equation}\label{2-5}
\rket{m}={\hat X}_\nu^*{\hat Y}_s^*\rket{0} \ .
\end{equation}
Here, ${\hat X}_\nu^*$ and ${\hat Y}_s^*$ are operators which should satisfy
the condition
\begin{subequations}\label{2-6}
\begin{eqnarray}
& &[\ {\hat {\cal S}}_{\pm}\ , \ {\hat X}_\nu^*\ ]=0 \ , \qquad
[\ {\hat {\cal S}}_{0}\ , \ {\hat X}_\nu^*\ ]=0 \ , \nonumber\\
& &[\ {\hat {\cal N}}\ , \ {\hat X}_\nu^*\ ]=2\nu{\hat X}_\nu^* \ ,
\quad (\nu; {\rm positive\ integer})\ ,
\label{2-6a}\\
& &[\ {\hat {\cal S}}_{-}\ , \ {\hat Y}_s^*\ ]=0 \ , \nonumber\\
& &[\ {\hat {\cal S}}_{0}\ , \ {\hat Y}_s^*\ ]=-\hbar s{\hat Y}_s^* \ ,
\nonumber\\
& &[\ {\hat {\cal N}}\ , \ {\hat Y}_s^*\ ]=2s{\hat Y}_s^* \ ,
\label{2-6b}\\
& &{\hat X}_\nu{\hat Y}_s^*\rket{0}=0 \ .
\label{2-6c}
\end{eqnarray}
\end{subequations}
If ${\hat X}_\nu^*$ and ${\hat Y}_s^*$ obey the condition (\ref{2-6}),
we can prove easily that the form (\ref{2-5}) satisfies the condition
(\ref{2-4}).
In order to investigate ${\hat X}_\nu^*$ and ${\hat Y}_s^*$ in rather
concrete form, we note
\begin{eqnarray}\label{2-7}
& &[\ {\hat {\cal S}}_{\pm}\ , \ {\hat c}_m^*(\pm) \ ]=0 \ , \qquad
[\ {\hat {\cal S}}_{\mp}\ , \ {\hat c}_m^*(\pm) \ ]=\hbar{\hat c}_m^*(\mp)
\ , \nonumber\\
& &[\ {\hat {\cal S}}_0\ , \ {\hat c}_m^*(\pm) \ ]=
\mp (\hbar/2){\hat c}_m^*(\pm) \ , \qquad
[\ {\hat {\cal N}}\ , \ {\hat c}_m^*(\pm) \ ]={\hat c}_m^*(\pm) \ .
\end{eqnarray}
We can see that the operator ${\hat c}_m^*(\pm)$ is spinor with respect
to $({\hat {\cal S}}_{\pm,0})$.
With the use of the relation (\ref{2-7}), we are permitted to set up
\begin{subequations}\label{2-8}
\begin{eqnarray}
& &{\hat X}_\nu^*=\sum_{(k,l)}
c(k_1,\cdots ,k_\nu; l_1,\cdots ,l_\nu){\hat c}_{k_1}^*(+)\cdots
{\hat c}_{k_\nu}^*(+){\hat c}_{l_1}^*(-)\cdots {\hat c}_{l_\nu}^*(-) \ ,
\label{2-8a}\\
& &{\hat Y}_s^*={\hat c}_{m_1}^*(-)\cdots {\hat c}_{m_{2s}}^*(-) \ .
\label{2-8b}
\end{eqnarray}
\end{subequations}
Here, $c(k_1,\cdots ,k_\nu; l_1,\cdots ,l_\nu)$ may be determined by
appropriate method, which is sketched in Appendix A, but, in this paper,
the explicit form is not necessary.
Clearly, there exists the relation
\begin{equation}\label{2-9}
\nu=N/2-s\ .
\end{equation}
Then, we have the restriction
\begin{equation}\label{2-10}
0 \leq \nu \leq \Omega\ , \qquad 0 \leq \nu +2s \leq \Omega \ .
\end{equation}
The relation (\ref{2-10}) gives us $0\leq \nu \leq \Omega-2s$
and using the relation (\ref{2-9}), we have
\begin{equation}\label{2-11}
0 \leq s \leq N/2 \ , \qquad
0\leq s \leq \Omega-N/2\ .
\end{equation}
The relation (\ref{2-11}) is equivalent to
\begin{eqnarray}\label{2-12}
& &{\rm if}\quad 0\leq N \leq \Omega \ , \qquad 0\leq s \leq N/2 \ ,
\nonumber\\
& &{\rm if}\quad \Omega\leq N \leq 2\Omega \ ,
\qquad 0\leq s \leq \Omega-N/2 \ .
\end{eqnarray}
Noting that $\nu$ is a positive integer, the relation (\ref{2-12})
can be expressed as follows:
\begin{eqnarray}\label{2-13}
& &{\rm if}\quad N={\rm even\ and}\
0\leq N \leq \Omega \ , \qquad s=N/2,\ N/2-1,\ \cdots ,\ 1,\ 0 \ ,
\nonumber\\
& &{\rm if}\quad N={\rm even\ and}\
\Omega\leq N \leq 2\Omega \ , \qquad s=\Omega-N/2,\ \Omega-N/2-1,\
\cdots ,\ 1,\ 0 \ ,
\nonumber\\
& &{\rm if}\quad N={\rm odd\ and}\
0 < N < \Omega \ , \qquad s=N/2,\ N/2-1,\ \cdots ,\ 3/2,\ 1/2 \ ,
\nonumber\\
& &{\rm if}\quad N={\rm odd\ and}\
\Omega < N < 2\Omega \ , \qquad s=\Omega-N/2,\ \Omega-N/2-1,\
\cdots ,\ 3/2,\ 1/2 \ .\nonumber\\
& &
\end{eqnarray}
The relation (\ref{2-9}) tells us that $2\nu$ denotes the seniority
number, and usually, the Lipkin model is treated in the case
$N=\Omega$ with $s=\Omega/2$ ($\nu=0$).
The above means that the lower level is fully occupied by $\Omega$ fermions
and the upper is empty if the interaction is switched off.

As a final discussion in this section, we will contact with the problem how
to construct the orthogonal set.
Since $\rket{m}$ satisfies the conditions (\ref{2-4a}) and (\ref{2-4b}),
formally,
$({\hat {\cal S}}_+)^{\sigma}\rket{m}\ (\sigma=s+s_0)$ gives us the state with
the eigenvalue $(s,s_0; s_0=-s,-s+1,\cdots s-1,s)$.
However, further consideration is necessary.
The Hamiltonian (\ref{2-1}) consists of ${\hat {\cal S}}_0$,
${\hat {\cal S}}_+^2$ and ${\hat {\cal S}}_-^2$.
Therefore, the matrix elements of ${\hat {\cal H}}$ between two states with
$\Delta s_0=\pm 1$ always vanish.
This means that the whole space specified by a given value of $s$
is divided into two groups, i.e.,
$\{({\hat {\cal S}}_+)^{2n}\rket{m}\}$
and $\{({\hat {\cal S}}_+)^{2n+1}\rket{m}\}$.
Here, $n$ denotes $n=0,1,2,\cdots $ and the normalization constants
are omitted.
For the group $\{({\hat {\cal S}}_+)^{2n}\rket{m}\}$, $n$ is restricted to
\begin{subequations}\label{2-13sub}
\begin{equation}\label{2-13a}
-s \leq -s+2n \leq s\ , \qquad {\rm i.e.,}\qquad 0\leq n \leq s \ .
\end{equation}
On the other hand, for the group $\{({\hat {\cal S}}_+)^{2n+1}\rket{m}\}$,
we have
\begin{equation}\label{2-13b}
-s+1 \leq -s+2n+1 \leq s\ , \qquad {\rm i.e.,}\qquad 0\leq n \leq s-1/2 \ .
\end{equation}
\end{subequations}
Noting that $s$ is integer or half-integer, the orthogonal set for
diagonalizing ${\hat {\cal H}}$ is classified into four groups:
\begin{subequations}\label{2-14}
\begin{eqnarray}
& &{\rm (i)}\quad
\{({\hat {\cal S}}_+)^{2n}\rket{m} ; s={\rm integer},\ n=0,1,2,\cdots ,s\} \ ,
\label{2-14a}\\
& &{\rm (ii)}\quad
\{({\hat {\cal S}}_+)^{2n+1}\rket{m} ; s={\rm integer},\ n=0,1,2,\cdots ,s-1\}
\ ,
\label{2-14b}\\
& &{\rm (iii)}\quad
\{({\hat {\cal S}}_+)^{2n}\rket{m} ; s=\hbox{\rm half-integer},\
n=0,1,2,\cdots ,s-1/2\} \ ,
\label{2-14c}\\
& &{\rm (iv)}\quad
\{({\hat {\cal S}}_+)^{2n+1}\rket{m} ; s=\hbox{\rm half-integer},\
n=0,1,2,\cdots ,s-1/2\} \ .
\label{2-14d}
\end{eqnarray}
\end{subequations}
Through the above-mentioned procedure, we can prepare the orthogonal set for
the original form of the Lipkin model.

\section{The Lipkin model in the Schwinger boson representation}

It is well known that the $su(2)$-generators in the Schwinger boson
representation are expressed in terms of two kinds of bosons
$({\hat b}_+,{\hat b}_+^*)$ and $({\hat b}_-,{\hat b}_-^*)$:
\begin{equation}\label{3-1}
{\wtilde S}_+=\hbar{\hat b}_+^*{\hat b}_-\ , \qquad
{\wtilde S}_-=\hbar{\hat b}_-^*{\hat b}_+\ , \qquad
{\wtilde S}_0=(\hbar/2)({\hat b}_+^*{\hat b}_+
-{\hat b}_-^*{\hat b}_-)\ .
\end{equation}
The operator expressing the magnitude of the $su(2)$-spin is given
in the form
\begin{equation}\label{3-2}
{\wtilde S}=(\hbar/2)({\hat b}_+^*{\hat b}_+
+{\hat b}_-^*{\hat b}_-)\ .
\end{equation}
In the fermion space, it may be impossible to describe such a simple
expression.
The state with the minimum weight, which we denote as $\kket{m}$,
is given as
\begin{equation}\label{3-3}
\kket{m}=\left(\sqrt{(2s)!}\right)^{-1}({\hat b}_-^*)^{2s}\kket{0}\ .
\qquad (s=0, 1/2, 1,\cdots )
\end{equation}
The state $\kket{m}$ satisfies
\begin{subequations}\label{3-4}
\begin{eqnarray}
& &{\wtilde S}_-\kket{m}=0 \ ,
\label{3-4a}\\
& &{\wtilde S}_0\kket{m}=-S\kket{m}\ , \qquad (S=\hbar s) \ .
\label{3-4b}
\end{eqnarray}
\end{subequations}
In the Schwinger boson representation, there does not exist any operator
which corresponds to ${\hat {\cal N}}$, and then, we cannot set up the
relation such as (\ref{2-4c}).
The successive operation of ${\wtilde S}_+$ on the state $\kket{m}$ gives us
the orthogonal set in the Schwinger boson representation.
Later, we will discuss this problem.

The Hamiltonian ${\wtilde H}$ which corresponds to ${\hat {\cal H}}$
shown in the form (\ref{2-1}) is, of course, expressed as follows:
\begin{equation}\label{3-5}
{\wtilde H}=\epsilon{\wtilde S}_0 -(G/4)
\left[\left({\wtilde S}_+\right)^2+\left({\wtilde S}_-\right)^2\right] \ .
\end{equation}
Any matrix element of ${\wtilde H}$ for the orthogonal set derived
from $\kket{m}$ is of the same form as that of ${\hat {\cal H}}$ for the
orthogonal set derived from $\rket{m}$.
However, the original form of the Lipkin model is characterized by the
quantities $\Omega$ and $N$, which give us the restriction
(\ref{2-13}).
On the other hand, the Lipkin model in the Schwinger boson representation
does not contain such quantities.
Then, introducing $\Omega$ and $N$ from the outside,
we require the relation (\ref{2-13}) as the condition that the
magnitude of the $su(2)$-spin $s$, the eigenvalue of ${\wtilde S}$
shown in the relation (\ref{3-2}), should satisfy in the Schwinger
boson representation.
Through this requirement, both frameworks connect with each other.

Now, let us construct the orthogonal set for diagonalizing ${\wtilde H}$
shown in the relation (\ref{3-5}).
The Hamiltonian ${\wtilde H}$ also consists of ${\wtilde S}_0$,
${\wtilde S}_+^2$ and ${\wtilde S}_-^2$.
Therefore, the same argument as that given in \S 2 is possible.
We can divide the whole space into two groups,
$\{({\wtilde S}_+)^{2n}\kket{m}\}$ and $\{({\wtilde S}_+)^{2n+1}\kket{m}\}$.
Except the normalization constants, both states can be rewritten as
\begin{subequations}\label{3-6}
\begin{eqnarray}
& &\left({\wtilde S}_+\right)^{2n}\kket{m}=
({\hat b}_+^*)^{2n}({\hat b}_-^*)^{2(s-n)}\kket{0} \ ,
\label{3-6a}\\
& &\left({\wtilde S}_+\right)^{2n+1}\kket{m}=
({\hat b}_+^*)^{2n+1}({\hat b}_-^*)^{2(s-n)-1}\kket{0} \ .
\label{3-6b}
\end{eqnarray}
\end{subequations}
The form (\ref{3-6a}) gives us
\begin{subequations}\label{3-7}
\begin{equation}\label{3-7a}
2n \geq 0 \ , \qquad 2(s-n) \geq 0 \ , \qquad{\rm i.e.,}\qquad
0\leq n\leq s \ .
\end{equation}
Also, the form (\ref{3-6b}) gives us
\begin{equation}\label{3-7b}
2n+1 \geq 0 \ , \qquad 2(s-n)-1 \geq 0 \ , \qquad{\rm i.e.,}\qquad
0\leq n\leq s-1/2 \ .
\end{equation}
\end{subequations}
The relation (\ref{3-7}) is identical to the relation (\ref{2-13sub}).
Under the same idea as that used in \S 2, the orthogonal set is also
classified into four groups:
\begin{subequations}\label{3-8}
\begin{eqnarray}
& &{\rm (i)}\quad
\{({\wtilde S}_+)^{2n}\kket{m}=({\hat b}_+^{*2})^n({\hat b}_-^{*2})^{s-n}
\kket{0} ; \nonumber\\
& &\qquad\qquad
s={\rm integer},\ n=0,1,2,\cdots ,s\} \ ,
\label{3-8a}\\
& &{\rm (ii)}\quad
\{({\wtilde S}_+)^{2n+1}\kket{m}=({\hat b}_+^{*2})^n({\hat b}_-^{*2})^{s-n-1}
\cdot{\hat b}_+^*{\hat b}_-^*\kket{0} ; \nonumber\\
& &\qquad\qquad
s={\rm integer},\ n=0,1,2,\cdots ,s-1\} \ ,
\label{3-8b}\\
& &{\rm (iii)}\quad
\{({\wtilde S}_+)^{2n}\kket{m}=({\hat b}_+^{*2})^n({\hat b}_-^{*2})^{s-n-1/2}
\cdot{\hat b}_-^*\kket{0} ; \nonumber\\
& &\qquad\qquad
s=\hbox{\rm half-integer},\ n=0,1,2,\cdots ,s-1/2\} \ ,
\label{3-8c}\\
& &{\rm (iv)}\quad
\{({\wtilde S}_+)^{2n+1}\kket{m}=({\hat b}_+^{*2})^n({\hat b}_-^{*2})^{s-n-1/2}
\cdot{\hat b}_+^*\kket{0} ; \nonumber\\
& &\qquad\qquad
s=\hbox{\rm half-integer},\ n=0,1,2,\cdots ,s-1/2\} \ .
\label{3-8d}
\end{eqnarray}
\end{subequations}

From the relation (\ref{3-8}), we can learn that for a given value of $s$,
the maximum number of the operation of ${\wtilde S}_+^2$ is fixed.
We also mentioned that the quantities $\Omega$ and $N$ are introduced from
the outside and they should obey the condition (\ref{2-13}).
However, it may be convenient for the practical aim to rewrite the
condition (\ref{2-13}) in the form so as to be able to know the
values of $N$ permitted for a given value of $s$.
This can be done in the form
\begin{subequations}\label{3-9}
\begin{eqnarray}
& &{\rm (a)}\
{\rm if}\ s={\rm integer},\quad N=2s,2s+2,\cdots , \Omega, \nonumber\\
& &\qquad\qquad\qquad\qquad\qquad\qquad\qquad
2\Omega-2s,
2\Omega-2s-2,\cdots ,2\Omega\ , \quad
\label{3-9a}\\
& &{\rm (b)}\
{\rm if}\ s=\hbox{\rm half-integer},
\quad N=2s,2s+2,\cdots ,\Omega-1, \nonumber\\
& &\qquad\qquad\qquad\qquad\qquad\qquad\qquad
2\Omega-2s,
2\Omega-2s-2,\cdots ,2\Omega-1\ .
\label{3-9b}
\end{eqnarray}
\end{subequations}
The above is the outline of the Lipkin model in the Schwinger boson
representation.

\section{Reformulation in the form of the $su(1,1)\otimes su(1,1)$-algebra}

In Refs. \citen{12} and \citen{13}, we showed that the two-level pairing
model
was re-formed in terms of the $su(1,1)\otimes su(1,1)$-algebraic model
in the Schwinger boson representation.
The present authors, with Kuriyama, already showed that the Lipkin model
can be also re-formed in terms of the $su(1,1)\otimes su(1,1)$-algebraic
model in the Schwinger boson representation.\cite{8,10}
In this section, we recapitulate this re-formation in the notations used
in Refs. \citen{12} and \citen{13}.
Of course, the formalism given in Refs. \citen{8} and \citen{10} is
supplemented with newly added features.

First, we note the following re-form:
\begin{subequations}\label{4-1}
\begin{eqnarray}
& &\left({\wtilde S}_+\right)^2=4(\hbar/2){\hat b}_+^{*2}\cdot
(\hbar/2){\hat b}_-^2 \ ,
\label{4-1a}\\
& &\left({\wtilde S}_-\right)^2=4(\hbar/2){\hat b}_-^{*2}\cdot
(\hbar/2){\hat b}_+^2 \ ,
\label{4-1b}\\
& &{\wtilde S}_0=(\hbar/4+(\hbar/2){\hat b}_+^*{\hat b}_+)-
(\hbar/4+(\hbar/2){\hat b}_-^*{\hat b}_-) \ .
\label{4-1c}
\end{eqnarray}
\end{subequations}
Then, we define the  boson-pairs in the set $({\wtilde T}_{\pm,0}(\sigma))$:
\begin{eqnarray}\label{4-2}
& &{\wtilde T}_+(\sigma)=(\hbar/2){\hat b}_{\sigma}^{*2} \ , \quad
{\wtilde T}_-(\sigma)=(\hbar/2){\hat b}_{\sigma}^{2} \ , \quad
{\wtilde T}_0(\sigma)=\hbar/4+(\hbar/2){\hat b}_{\sigma}^{*}{\hat b}_{\sigma}
\ . \quad (\sigma=\pm)\nonumber\\
& &
\end{eqnarray}
It is easily verified that the set $({\wtilde T}_{\pm,0}(\sigma))$
obeys the $su(1,1)$-algebra.
With the use of the set $({\wtilde T}_{\pm,0}(\sigma);\sigma=\pm)$,
the Hamiltonian (\ref{3-5}) can be rewritten as
\begin{equation}\label{4-3}
{\wtilde H}=\epsilon({\wtilde T}_0(+)-{\wtilde T}_0(-))
-G\left({\wtilde T}_+(+){\wtilde T}_-(-)
+{\wtilde T}_+(-){\wtilde T}_-(+)\right) \ .
\end{equation}
The form (\ref{4-3}) tells that the Lipkin model is a possible model
obeying the $su(1,1)\otimes su(1,1)$-algebra.

Our concern is to compare the Lipkin model with the two-level pairing model
in the framework of the $su(1,1)\otimes su(1,1)$-algebra.
For this purpose, further, we re-form the present frame.
The state with the minimum weight, $\kket{m(\sigma)}$, for the algebra
$({\wtilde T}_{\pm,0}(\sigma))$ is determined under the condition
\begin{subequations}\label{4-4}
\begin{eqnarray}
& &{\wtilde T}_-(\sigma)\kket{m(\sigma)}=0 \ ,
\label{4-4a}\\
& &{\wtilde T}_0(\sigma)\kket{m(\sigma)}=T(\sigma)\kket{m(\sigma)} \ ,
\qquad (T(\sigma)=\hbar t_{\sigma}) \ .
\label{4-4b}
\end{eqnarray}
\end{subequations}
The condition (\ref{4-4}), with the definition (\ref{4-2}), gives
\begin{equation}\label{4-5}
\kket{m(\sigma)}=({\hat b}_{\sigma}^*)^{2t_{\sigma}-1/2}\kket{0}=
\begin{cases} \kket{0} \ , \qquad {\rm for}\quad t_\sigma=1/4 \ , \\
{\hat b}_{\sigma}^*\kket{0} \ , \quad{\rm for}\quad t_{\sigma}=3/4 \ .
\end{cases}
\end{equation}
It should be noted that there does not exist any other type.
Then, the state with the minimum weight in the present
$su(1,1)\otimes su(1,1)$-algebra is specified by $(t_+,t_-)$ and
given in the form
\begin{equation}\label{4-6}
\kket{t_+,t_-}=({\hat b}_+^*)^{2t_+-1/2}({\hat b}_-^*)^{2t_--1/2}\kket{0} \ .
\end{equation}
Then, we can construct the orthogonal set by operating
${\wtilde T}_+(+)$ and ${\wtilde T}_+(-)$ successively in the form
\begin{eqnarray}\label{4-9}
& &\kket{\kappa_+,\kappa_-;t_+,t_-}=
\left(\sqrt{(2\kappa_++2t_+-1/2)!(2\kappa_-+2t_--1/2)!}\right)^{-1}\nonumber\\
& &\qquad\qquad\qquad\qquad\qquad
\times
({\hat b}_+^*)^{2\kappa_+}({\hat b}_-^*)^{2\kappa_-}\kket{t_+,t_-} \ ,
\nonumber\\
& &\qquad
\kappa_+,\ \kappa_-=0,1,2,\cdots .
\end{eqnarray}
It may be interesting to compare the state (\ref{4-9}) with the form
(\ref{3-8}).
We can see that both are identical with each other under the
following correspondence:
\begin{subequations}\label{4-10}
\begin{eqnarray}
& &{\rm (i)}\ \ t_+=1/4 \ , \quad t_-=1/4\ , \quad \kappa_+=n \ , \quad
\kappa_-=s-n \ ,
\label{4-10a}\\
& &{\rm (ii)}\ \ t_+=3/4 \ , \quad t_-=3/4\ , \quad \kappa_+=n \ , \quad
\kappa_-=s-n-1 \ ,
\label{4-10b}\\
& &{\rm (iii)}\ \ t_+=1/4 \ , \quad t_-=3/4\ , \quad \kappa_+=n \ , \quad
\kappa_-=s-n-1/2 \ ,
\label{4-10c}\\
& &{\rm (iv)}\ \ t_+=3/4 \ , \quad t_-=1/4\ , \quad \kappa_+=n \ , \quad
\kappa_-=s-n-1/2 \ .
\label{4-10d}
\end{eqnarray}
\end{subequations}
We know that the quantum number $(t_+,t_-)$ plays a role of classifying the
four cases (i)$\sim$(iv).

Next, as was done in Ref. \citen{13}, we apply the MYT mapping method\cite{11}
to the present system.
For this purpose, we prepare the other boson space constructed by new
boson $({\hat c}_{\sigma}, {\hat c}_{\sigma}^*; \sigma=\pm)$.
The orthogonal set is given as
\begin{equation}\label{4-11}
\dket{\kappa_+,\kappa_-}=\left(\sqrt{\kappa_+!\kappa_-!}\right)^{-1}
({\hat c}_+^*)^{\kappa_+}({\hat c}_-^*)^{\kappa_-}\dket{0} \ .
\end{equation}
Then, the MYT mapping operator $\maru{U}$ is defined in the form
\begin{equation}\label{4-12}
\maru{U}=\sum_{\kappa_+,\kappa_-}\dket{\kappa_+,\kappa_-}
\bbra{\kappa_+,\kappa_-;t_+,t_-} \ .
\end{equation}
With the use of $\maru{U}$, we have the following relations:
\begin{eqnarray}
& &\maru{U}\left({\wtilde T}_0(+)-{\wtilde T}_0(-)\right)\maru{U}^{\dagger}
=T(+)-T(-)+\hbar{\hat c}_+^*{\hat c}_+-\hbar{\hat c}_-^*{\hat c}_- \ ,
\label{4-13}\\
& &\maru{U}{\wtilde T}_+(+){\wtilde T}_-(-)\maru{U}^{\dagger}
=\sqrt{2T(-)+\hbar{\hat c}_-^*{\hat c}_-}\cdot\hbar{\hat c}_+^*{\hat c}_-
\cdot\sqrt{2T(+)+\hbar{\hat c}_+^*{\hat c}_+} \ ,
\label{4-14}\\
& &T(+)=\hbar t_+ \ , \qquad T(-)=\hbar t_- \ .
\label{4-15}
\end{eqnarray}
With the help of the forms (\ref{4-13}) and (\ref{4-14}), the Hamiltonian
(\ref{4-3}) can be mapped to the following:
\begin{eqnarray}\label{4-16}
\maru{H}&=&
\epsilon(T(+)-T(-))+\epsilon(\hbar{\hat c}_+^*{\hat c}_+
-\hbar{\hat c}_-^*{\hat c}_-)\nonumber\\
& &-G\biggl(
\sqrt{2T(-)+\hbar{\hat c}_-^*{\hat c}_-}\cdot \hbar{\hat c}_+^*{\hat c}_-\cdot
\sqrt{2T(+)+\hbar{\hat c}_+^*{\hat c}_+} \nonumber\\
& &\qquad\quad
+\sqrt{2T(+)+\hbar{\hat c}_+^*{\hat c}_+}\cdot\hbar{\hat c}_-^*{\hat c}_+\cdot
\sqrt{2T(-)+\hbar{\hat c}_-^*{\hat c}_-}\biggl) \ .
\end{eqnarray}
The magnitude of the $su(2)$-spin, ${\wtilde S}$, shown in the relation
(\ref{3-2}) is mapped to
\begin{equation}\label{4-17}
\maru{S}=\maru{U}{\wtilde S}\maru{U}^{\dagger}
=T(+)+T(-)-\hbar/2+2\cdot(\hbar/2)({\hat c}_+^*{\hat c}_+
+{\hat c}_-^*{\hat c}_-) \ .
\end{equation}
Thus, we can construct the Lipkin model in the framework of the
$su(1,1)\otimes su(1,1)$-algebra in the Holstein-Primakoff representation.

Comparison of the form (\ref{4-16}) with the Hamiltonian (I$\cdot$13) is
interesting.
Except a certain term related to $\hbar{\hat c}_+^*{\hat c}_+$ and
$\hbar{\hat c}_-^*{\hat c}_-$, both coincide with each other.
Of course, the form (I$\cdot$13) is characterized by
\begin{equation}\label{4-18}
T(\sigma)=\hbar/2, \hbar, 3\hbar/2,\cdots \ .
\end{equation}
The present case is characterized by $T(\sigma)=\hbar/4$ and $3\hbar/4$.
From the above argument, we can conclude that the two-level pairing
and the Lipkin model can be treated in the common ring.
The Hamiltonian is expressed as
\begin{eqnarray}\label{4-19}
\maru{H}&=&
\epsilon(T(+)-T(-))+\epsilon(\hbar{\hat c}_+^*{\hat c}_+
-\hbar{\hat c}_-^*{\hat c}_-)\nonumber\\
& &-2g(T(+)\cdot\hbar{\hat c}_+^*{\hat c}_+
+ T(-)\cdot\hbar{\hat c}_-^*{\hat c}_-)
-2g\cdot\hbar{\hat c}_+^*{\hat c}_+\cdot \hbar{\hat c}_-^*{\hat c}_-
\nonumber\\
& &-G\biggl(
\sqrt{2T(-)+\hbar{\hat c}_-^*{\hat c}_-}\cdot \hbar{\hat c}_+^*{\hat c}_-\cdot
\sqrt{2T(+)+\hbar{\hat c}_+^*{\hat c}_+} \nonumber\\
& &\qquad\quad
+\sqrt{2T(+)+\hbar{\hat c}_+^*{\hat c}_+}\cdot\hbar{\hat c}_-^*{\hat c}_+\cdot
\sqrt{2T(-)+\hbar{\hat c}_-^*{\hat c}_-}\biggl) \ .
\end{eqnarray}
The cases $g=G$ and $g=0$ correspond to the two-level pairing and the
Lipkin model, respectively.
As was shown in (I), the Hamiltonian (\ref{4-19}) can be treated in the
$su(2)$-algebra.

Finally, we will show that a certain wave packet defined in the form
(\ref{4-23}), which we call the $su(1,1)\otimes su(1,1)$-coherent state,
presents
a classical counterpart of the Lipkin model re-formed in this section.
First, extending the state (\ref{b10}), we introduce the following
state:
\begin{eqnarray}
& &\kket{c_0}=\left(\sqrt{U_+}\right)^{-1}
\exp\left(\frac{V_+}{\hbar U_+}{\wtilde T}_+(+)\right)\cdot
\left(\sqrt{U_-}\right)^{-1}
\exp\left(\frac{V_-}{\hbar U_-}{\wtilde T}_+(-)\right)\kket{0} \ ,
\label{4-20}\\
& &U_{\sigma}=\sqrt{1+|V_{\sigma}|^2} \ . \qquad (\sigma=\pm)
\label{4-21}
\end{eqnarray}
Here, $V_{\sigma}$ denotes a complex parameter.
The state $\kket{c_0}$ is a vacuum of boson operator $({\hat \beta}_{\sigma},
{\hat \beta}_{\sigma}^*)$, i.e., ${\hat \beta}_{\sigma}\kket{c_0}=0$:
\begin{subequations}\label{4-22}
\begin{eqnarray}
& &{\hat \beta}_{\sigma}=U_{\sigma}{\hat b}_{\sigma}
-V_{\sigma}{\hat b}_{\sigma}^*
\ , \nonumber\\
& &{\hat \beta}_{\sigma}^*=-V_{\sigma}^*{\hat b}_{\sigma}
+U_{\sigma}{\hat b}_{\sigma}^* \ ,
\label{4-20a}\\
{\rm i.e.,}\qquad
& &{\hat b}_{\sigma}=U_{\sigma}{\hat \beta}_{\sigma}
+V_{\sigma}{\hat \beta}_{\sigma}^*\ , \nonumber\\
& &{\hat b}_{\sigma}^*=V_{\sigma}^*{\hat \beta}_{\sigma}
+U_{\sigma}{\hat \beta}_{\sigma}^* \ .
\label{4-20b}
\end{eqnarray}
\end{subequations}
With the use of $\kket{c_0}$ and ${\hat \beta}_{\sigma}^*$, we define the
$su(1,1)\otimes su(1,1)$-coherent state $\kket{c^0}$ in the form
\begin{equation}\label{4-23}
\kket{c^0}=\left(\sqrt{3/2-2t_+}+\sqrt{2t_+-1/2}{\hat \beta}_+^*\right)
\left(\sqrt{3/2-2t_-}+\sqrt{2t_--1/2}{\hat \beta}_-^*\right)\kket{c_0} \ .
\end{equation}
The state (\ref{4-23}) is a possible extension of the state (\ref{b13})
to the case of the $su(1,1)\otimes su(1,1)$-algebra.
By calculating the expectation value of ${\wtilde H}$ given in the relation
(\ref{4-3}) for $\kket{c^0}$ and using various relations shown in Appendix,
we have the following form:
\begin{eqnarray}\label{4-24}
{H}^0&=&
\epsilon(T(+)-T(-))+\epsilon(\hbar{c}_+^*{c}_+
-\hbar{c}_-^*{c}_-)\nonumber\\
& &-G\biggl(
\sqrt{2T(-)+\hbar{c}_-^*{c}_-}\cdot \hbar{c}_+^*{c}_-\cdot
\sqrt{2T(+)+\hbar{c}_+^*{c}_+} \nonumber\\
& &\qquad\quad
+\sqrt{2T(+)+\hbar{c}_+^*{c}_+}\cdot\hbar{c}_-^*{c}_+\cdot
\sqrt{2T(-)+\hbar{c}_-^*{c}_-}\biggl) \ .
\end{eqnarray}
We can see that $H^0$ is equivalent to $\maru{H}$ shown in the
relation (\ref{4-16}).
Combining the above result with that derived in (I),
we obtain the classical Hamiltonian of $\maru{H}$ given in the relation
(\ref{4-19}).
Of course, for the present case, $T(\sigma)$ should be used as
$T(\sigma)=\hbar/4$ or $3\hbar/4$.

\section{Re-formation in terms of one kind of boson operator}

As was discussed in (I), the Hamiltonian (\ref{4-16}) is expressed
in terms of the Schwinger boson representation for the $su(2)$-algebra:
\begin{eqnarray}
& &{\maru M}_+=\hbar{\hat c}_+^*{\hat c}_- \ , \qquad
{\maru M}_-=\hbar{\hat c}_-^*{\hat c}_+ \ , \qquad
{\maru M}_0=(\hbar/2)({\hat c}_+^*{\hat c}_+-{\hat c}_-^*{\hat c}_-) \ ,
\label{5-1}\\
& &{\maru M}=(\hbar/2)({\hat c}_+^*{\hat c}_++{\hat c}_-^*{\hat c}_-) \ .
\label{5-2}
\end{eqnarray}
Then, in the same idea as that shown in (I), we can re-form our present
system in terms of one kind of boson.
For example, we obtain the following form, which is the same as that shown
in the relation (I$\cdot$30):
\begin{eqnarray}\label{5-3}
& &\sqrt{2T(-)+\hbar{\hat c}_-^*{\hat c}_-}\cdot\hbar{\hat c}_+^*{\hat c}_-
\cdot\sqrt{2T(+)+\hbar{\hat c}_+^*{\hat c}_+} \nonumber\\
&\Longrightarrow& \sqrt{\hbar}{\hat c}^*
\sqrt{2T(+)+\hbar{\hat c}^*{\hat c}}\sqrt{2M-\hbar{\hat c}^*{\hat c}}
\sqrt{2T(-)+2M-\hbar-\hbar{\hat c}^*{\hat c}}\ .
\end{eqnarray}
Using the canonical transformation
$(\sqrt{\hbar}c_+=\sqrt{\hbar}ce^{-i\chi/2},
\sqrt{\hbar}c_-=
\sqrt{2M-\hbar c^*c}e^{-i\chi/2}$) adopted in (I),
the classical counterpart obtained by the state $\kket{c^0}$ shown in
the relation (\ref{4-23}) can be re-formed in terms of
one kind of boson-type canonical variable.
For example, we have
\begin{eqnarray}\label{5-4}
& &\sqrt{2T(-)+\hbar{c}_-^*{c}_-}\cdot\hbar{c}_+^*{c}_-
\cdot\sqrt{2T(+)+\hbar{c}_+^*{c}_+} \nonumber\\
&\Longrightarrow& \sqrt{\hbar}{c}^*
\sqrt{2T(+)+\hbar{c}^*{c}}\sqrt{2M-\hbar{c}^*{c}}
\sqrt{2T(-)+2M-\hbar{c}^*{c}}\ .
\end{eqnarray}
We can see that the form (\ref{5-4}) does not agree with that obtained by
the $c$-number replacement of the relation (\ref{5-3}).
This is also in the same situation as that in the case of the two-level
pairing model.
The form (\ref{5-4}) is obtained by the $su(1,1)\otimes su(1,1)$-coherent
state and the other
by the $su(2)\otimes su(1,1)$-coherent state.
Then, we have a question:
What type of the coherent state reproduces the $c$-number replaced
form of the relation (\ref{5-3}) ?

In order to give an answer to the above question, let us remember the case
of the two-level pairing model.
In this case, four kinds of bosons are divided into two groups.
The $su(2)\otimes su(1,1)$-coherent
state is constructed under the following idea:
This state consists of the product of two parts.
First consists of the exponential form for the raising operator of the
$su(1,1)$-algebra and the second is expressed in terms of the
Glauber form
for two kinds of bosons.
In the present case, the first part can be expressed in the same form as the
above.
However, if adopting the $su(2)\otimes su(2)$-coherent state for the second,
we cannot derive the expected form.
Under the above mentioned background, we adopt the following coherent state:
\begin{equation}\label{5-5}
\kket{c_0'}=\left(\sqrt{U_+}\right)^{-1}\!\!
\exp\!\left(\frac{V_+}{\hbar U_+}{\wtilde T}_+(+)\right)
\left(\sqrt{U_-}\right)^{-1}\!\!\exp\!\left(\frac{\gamma_-}{|\gamma_-|}
\frac{V_-}{\hbar U_-}{\wtilde T}_+(-)+\frac{\sqrt{\gamma}_-}{U_-}{\hat b}_-^*
\right)\!\kket{0} \ .
\end{equation}
Here, $V_+$ and $\gamma_-$ are complex, but $V_-$ is real and $U_+$
and $U_-$ are given as
\begin{equation}\label{5-6}
U_+=\sqrt{1+|V_+|^2} \ , \qquad U_-=\sqrt{1+V_-^2}\ .
\end{equation}
The detail properties of the state $\kket{c_0'}$ are discussed separately in
Appendix B.
Further, we define the state $\kket{{c^0}'}$ in the form
\begin{equation}\label{5-7}
\kket{{c^0}'}=
\left(\sqrt{3/2-2t_+}+\sqrt{2t_+-1/2}{\hat \beta}_+^*\right)
\left(\sqrt{3/2-2t_-}+\sqrt{2t_--1/2}{\hat \beta}_-^*\right)\kket{c_0'} \ .
\end{equation}
Using the various relations shown in \S 4 and
Appendix B and noting the relation $M=(\hbar/2)(c_+^*c_++c_-^*c_-)$,
we get the following result for the Hamiltonian ${H^0}'$:
\begin{eqnarray}\label{5-8}
{H^0}'&=&\bbra{{c^0}'}{\wtilde H}\kket{{c^0}'} \nonumber\\
&=&\epsilon(T(+)-T(-)-M)+2\epsilon\cdot \hbar c^* c \nonumber\\
& &-G\biggl( \sqrt{\hbar}c^*\cdot \sqrt{2T(+)+\hbar c^* c}
\sqrt{2M-\hbar c^* c}\sqrt{2T(-)+2M-\hbar-\hbar c^* c} \nonumber\\
& &\qquad +
\sqrt{2T(-)+2M-\hbar-\hbar c^* c}\sqrt{2M-\hbar c^* c}
\sqrt{2T(+)+\hbar c^* c}
\cdot \sqrt{\hbar}c \biggl) \ .\quad
\end{eqnarray}
Of course, the relation (\ref{5-3}) presents us the quantized form of
${H^0}'$:
\begin{eqnarray}\label{5-9}
{\maru{H'}}
&=&\epsilon(T(+)-T(-)-M)+2\epsilon\cdot \hbar {\hat c}^* {\hat c} \nonumber\\
& &-G\biggl( \sqrt{\hbar}{\hat c}^*\cdot \sqrt{2T(+)+\hbar{\hat c}^*{\hat c}}
\sqrt{2M-\hbar{\hat c}^*{\hat c}}\sqrt{2T(-)+2M-\hbar-\hbar{\hat c}^*{\hat c}}
\nonumber\\
& &\qquad +
\sqrt{2T(-)+2M-\hbar-\hbar{\hat c}^*{\hat c}}\sqrt{2M-\hbar{\hat c}^*{\hat c}}
\sqrt{2T(+)+\hbar{\hat c}^*{\hat c}}
\cdot \sqrt{\hbar}{\hat c} \biggl) \ .\quad
\end{eqnarray}
The Hamiltonians ${H^0}'$ and $\maru{H'}$ contain the quantity $M$,
which is an eigenvalue of $\maru{M}$ defined in the relation (\ref{5-2}).
Further, we note the relation (\ref{4-17}) and $\maru{S}$ is related
to ${\wtilde S}$ defined in the relation (\ref{3-2}).
Therefore, we have the following relation:
\begin{equation}\label{5-10}
S=T(+)+T(-)-\hbar/2+2M \ .
\end{equation}
Through the relation (\ref{3-9}), $M$ connects with $\Omega$ and $N$
in the original fermion system.
For example, the most popular condition ($s=N, N=\Omega, T(+)=T(-)=\hbar/4$)
gives
\begin{equation}\label{5-11}
M=(\hbar/2)\Omega \ .
\end{equation}
As was mentioned in \S 2, the case (\ref{5-11}) shows that $\Omega$
fermions occupy fully in the lower level, if the interaction is
switched off.

\section{Discussion}

Finally, in this discussion, we will sketch the results
obtained in the frame of the $su(2)\otimes su(2)$-coherent
state $\kket{c_0''}$,
which is expressed as follows:
\begin{equation}\label{6-1}
\kket{c_0''}=e^{-|c_+|^2}\exp\left(\sqrt{2c_+|c_+|}{\hat b}_+^*\right)
\cdot e^{-|c_-|^2}\exp\left(\sqrt{2c_-|c_-|}{\hat b}_-^*\right)\kket{0} \ .
\end{equation}
Here, $(c_\sigma, c_{\sigma}^*)$ plays the same role as that in the two cases
we already discussed;
the boson-type canonical variables.
The state $\kket{c_0''}$ satisfies
\begin{equation}\label{6-2}
{\hat b}_{\sigma}\kket{c_0''}=\sqrt{2c_{\sigma}|c_{\sigma}|}\kket{c_0''} \ .
\qquad (\sigma=\pm)
\end{equation}
Differently from the previous two cases, we must pay a special attention
on the present one.
The expectation values of ${\wtilde T}_{\pm}(\sigma)$ are given by
\begin{eqnarray}\label{6-3}
& &\bbra{c_0''}{\wtilde T}_+(\sigma)\kket{c_0''}
=\sqrt{\hbar}c_{\sigma}^*\sqrt{\hbar|c_{\sigma}|^2} \ ,
\nonumber\\
& &\bbra{c_0''}{\wtilde T}_-(\sigma)\kket{c_0''}
=\sqrt{\hbar|c_{\sigma}|^2}\cdot\sqrt{\hbar}c_{\sigma} \ .
\end{eqnarray}
However, for $\bbra{c_0''}{\wtilde T}_0(\sigma)\kket{c_0''}$, formally,
we have
\begin{equation}\label{6-4}
\bbra{c_0''}{\wtilde T}_0(\sigma)\kket{c_0''}=\hbar/4+\hbar |c_{\sigma}|^2\ .
\end{equation}
The problem concerning the expression is that we should include the term
$\hbar/4$ into the expression (\ref{6-4}) or not.
In Ref. \citen{14}, quantal fluctuation around the expectation value
calculated by $\kket{c_0'}$ was discussed.
If we follow the conclusion of Ref. \citen{14}, the term $\hbar/4$
should be included in the quantal fluctuations.
Then, we should set up
\begin{equation}\label{6-5}
\bbra{c_0''}{\wtilde T}_0(\sigma)\kket{c_0''}=\hbar |c_{\sigma}|^2\ .
\end{equation}
However, for the calculation of the expectation value of the
Hamiltonian ${\wtilde H}$, we can escape from this problem, because
${\wtilde H}$ consists of the difference
$(\hbar{\hat b}_+^*{\hat b}_+-\hbar{\hat b}_-^*{\hat b}_-)$ and the
term $\hbar/4$ disappears.

From the above argument, we have the Hamiltonian
\begin{eqnarray}\label{6-6}
{H^0}''&=&\bbra{c_0''}{\wtilde H}\kket{c_0''} \nonumber\\
&=&\epsilon\cdot(\hbar c_+^*c_+-\hbar c_-^*c_-)\nonumber\\
& &-G\left(\sqrt{\hbar c_-^*c_-}\cdot \hbar c_+^*c_-\cdot
\sqrt{\hbar c_+^*c_+}
+\sqrt{\hbar c_+^*c_+}\cdot \hbar c_-^*c_+\cdot
\sqrt{\hbar c_-^*c_-} \right) \ .
\end{eqnarray}
If taking into account the relation $M=(\hbar/2)(c_+^*c_++c_-^*c_-)$,
we have another form
\begin{eqnarray}\label{6-7}
{H^0}''&=&-\epsilon M+2\epsilon\cdot \hbar c^*c \nonumber\\
& &-G\left(
\sqrt{\hbar}c^*\cdot\sqrt{\hbar c^*c}(2M-\hbar c^*c)
+(2M-\hbar c^*c)\sqrt{\hbar c^*c}\cdot\sqrt{\hbar}c\right) \ .
\end{eqnarray}
As was done in Ref. \citen{12}, we introduce the canonical variable
$(\psi, K)$ in the form
\begin{equation}\label{6-8}
\sqrt{\hbar}c=\sqrt{K}e^{-i\psi} \ , \qquad
\sqrt{\hbar}c^*=\sqrt{K}e^{i\psi} \ .
\end{equation}
Then, ${H^0}''$ in the form (\ref{6-7}) can be rewritten as
\begin{eqnarray}\label{6-9}
{H^0}''&=&-2M+2\epsilon K-2G\cdot K(2M-K)+4GK(2M-K)(\sin \psi/2)^2 \ .
\end{eqnarray}
On the other hand, ${H^0}'$ shown in the form (\ref{5-8}) is rewritten in
the form
\begin{eqnarray}\label{6-10}
{H^0}'&=&\epsilon(T(+)-T(-)-M)+2\epsilon K \nonumber\\
& &-2G\sqrt{K(2T(+)+K)}\cdot\sqrt{2M-K}\sqrt{2T(-)+2M-\hbar-K} \nonumber\\
& &-4G\sqrt{K(2T(+)+K)}\cdot\sqrt{2M-K}\sqrt{2T(-)+2M-\hbar-K}
(\sin \psi/2)^2 \ .\qquad
\end{eqnarray}
Both Hamiltonians are valid only in the region
\begin{equation}\label{6-11}
0 \leq K \leq 2M-\hbar/2 \quad {\rm or}\quad 2M \ .
\end{equation}
The minimum point for both Hamiltonians (energies) appears at least at
\begin{equation}\label{6-12}
\psi=0 \ .
\end{equation}
Therefore, it may be interesting to investigate at which points for
$K$ the minimum points appear.
For the Hamiltonian (the energy) (\ref{6-9}), the behavior near $K=0$
is determined by the factor $2(\epsilon-2GM)$.
If $\epsilon > 2GM$, the energy increases in any region of $K$,
and then, the minimum point appears at $K=0$.
If $\epsilon < 2GM$, the energy decreases and afterward increases.
Then, the minimum point appears at a certain value of $K$ ($>0$).
This means that, at $\epsilon=2GM$, the phase change appears.
In the case of the Hamiltonian (the energy) (\ref{6-10}), the behavior near
$K=0$ is determined by the factor $2\epsilon K-2G\sqrt{K(2T(+)+K)}$.
In this case, the energy decreases at any point of $K$ near $K=0$ and
afterward increases.
Therefore, the minimum point appears at a certain point ($>0$) except $K=0$
in the case $G=0$.
From this argument, we cannot expect the sharp phase change.
This is in the same situation as that in the case of the
two-level pairing model.\cite{12}
In the succeeding paper, we will repeat the above-mentioned point
qualitatively including other various features.

redpajama_set_name: RedPajamaArXiv


ALMA/DICYT Astronomers using the Atacama Large Millimeter/submillimeter Array (ALMA) and the Very Large Telescope (VLT) have discovered that both, starburst galaxies in the early Universe, and a star-forming region in a nearby galaxy, contain a much higher proportion of massive stars than is found in more peaceful galaxies. These findings challenge current ideas about how galaxies evolved, changing our understanding of cosmic star-formation history and the build-up of chemical elements.
Probing the distant Universe a team of scientists, led by University of Edinburgh astronomer Zhi-Yu Zhang, used ALMA to investigate the proportion of massive stars in four distant gas-rich starburst galaxies. These galaxies are seen when the Universe was much younger than it is now so the infant galaxies are unlikely to have undergone many previous episodes of star formation, which might otherwise have confused the results.
Zhang and his team developed a new technique — analogous to radiocarbon dating (also known as carbon-14 dating) — to measure the abundances of different types of carbon monoxide in four very distant, dust-shrouded starburst galaxies. They observed the ratio of two types of carbon monoxide containing different isotopes.
"Carbon and oxygen isotopes have different origins," explains Zhang. "18O is produced more in massive stars, and 13C is produced more in low- to intermediate-mass stars." Thanks to the new technique the team was able to peer through the dust in these galaxies and assess for the first time the masses of their stars.
The mass of a star is the most crucial factor determining how it will evolve. Massive stars shine brilliantly and have short lives and less massive ones, such as the Sun, shine more modestly for billions of years. Knowing the proportions of stars of different masses that are formed in galaxies, therefore, underpins astronomers' understanding of the formation and evolution of galaxies throughout the history of the Universe. Consequently, it gives us valuable insights about the chemical elements available to form new stars and planets and, ultimately, the number of seed-black-holes that may coalesce to form the supermassive black holes that we see in the centers of many galaxies.


text:
Home › endangered marine species › New Threats to Orca Population
New Threats to Orca Population
By Charmaine Coimbra on August 2, 2018 • ( 0 )
Public domain photo
For Marine Life, New Threats from a Fast-Tracked Canadian Pipeline
A new Canadian government-backed pipeline that will triple the amount of thick Alberta tar sands oil flowing to a British Columbia port poses significant risks for a threatened population of killer whales and other coastal marine life.
BY JIM ROBBINS • AUGUST 2, 2018
Nearly everyone involved in the controversy over Canada's troubled Trans Mountain Pipeline was surprised when Prime Minister Justin Trudeau announced in May that his government would take over the construction from a private company to ensure that additional tar sands crude oil can move from northern Alberta to a port in British Columbia.
The 715-mile Trans Mountain pipeline expansion would add a parallel pipeline to an existing one, increasing the route's capacity from 300,000 to 890,000 barrels per day and helping producers sell crude and refined oil to Asian markets. Trudeau's action means that a pipeline many thought might never be built is now on a fast track to completion by 2020. Construction is scheduled to begin this month.
The expansion of the Trans Mountain pipeline poses a range of environmental impacts and risks — from the possibility of leaks as the new line crosses hundreds of streams, rivers, and lakes across the breadth of the British Columbia wilderness, to the fact that it will allow Alberta's massive tar sands reserves to be further exploited and contribute large amounts of CO2 to the atmosphere. But the most immediate and serious impacts may be to the marine environment along the coast of British Columbia, where the pipeline would terminate at the Westridge Marine Terminal at Burnaby near Vancouver.
The pipeline is expected to lead to a sharp increase in oil tanker traffic in the Salish Sea — a network of inland ocean waterways shared by British Columbia and Washington State — from four tankers a month to 34, along with associated construction and other ship traffic. Marine biologists are especially concerned about the impact of increased ship noise on a highly endangered population of 75 killer whales, known as the "southern residents," shared by the two countries.
To read the complete story on 360 Yale, click this link: Canadian Pipeline Could Threaten a Highly Endangered Species
‹ Less Hypoxia along Gulf Coast–But…
Florida's Red Tide Toll Grows ›
Categories: endangered marine species, Endangered Species, Marine Mammals, Noise Pollution, Oil Spill, Pacific Ocean
Tags: Environment, Orcas, Salish Sea, Tar Sands Oils


Stained and discolored teeth can dull an otherwise beautiful smile. Teeth Whitening for Orlando and Winter Park patients is an easy, economical and safe way to regain a bright, attractive smile. In less than an hour you can have a whiter, brighter and younger looking smile. Why wait?
A single tooth or all teeth can be whitened and brightened. Current styles are whiter than that of previous years and are attainable with modern methods. Professional grade peroxides are used for the bleaching and commonly change tooth color from five to seven shades whiter, in some cases up to twelve shades brighter. In one hour you can have that bright, white smile you've wanted.
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One month ago today we posted an article about the crackdown on unreported Korean bank accounts. IRS and Korean National Tax Service (NTS) officials have been cooperating – and prosecuting – taxpayers with unreported foreign accounts. Our post evidently struck a nerve as we have now learned of several new investigations into unreported foreign accounts.
U.S. taxpayers, which include Korean nationals holding a green card and dual nationals, must annually report all foreign financial holdings. Foreign bank and brokerage accounts are reported each year on a Report of Foreign Bank and Financial Accounts or "FBAR." Owning the account is completely legal if properly reported.
Evidently, many Korean nationals in Korea are hiding money in Swiss accounts and many Korean – Americans are failing to properly report their offshore bank accounts. Since we last wrote our story, the NTS has disclosed that it has recovered $432 million USD in taxes from previously unreported foreign accounts, many of them Swiss. That sum was recovered just in the first 6 months of this year.
IRS and U.S. Justice Department prosecutors have criminally charged several Korean Americans for failing to report their foreign holdings.
With both countries now cooperating and the new Foreign Account Tax Compliance Act (FATCA) about to take effect several months from now, the window of opportunity to come into compliance is closing quickly. Beginning next year, foreign banks must disclose the identities of account holders with ties to the United States. Once that happens, holders of unreported foreign accounts may not be eligible for the current IRS amnesty programs.
Korean Americans and Koreans here on green card status often hold their money in a Korean bank with offices in the United States. If your account is at a U.S. branch of BBCN or Hanmi, for example, you probably have no worries. If your monies are in a Swiss account or in an account located in Korea, however, you must file FBARs annually and pay tax on any income or dividends. Accounts in non-U.S. branches of Hana, Kookmin or Woori banks should be declared.
The penalty for an unreported foreign account can include prison and 50% per year of the account's high balance for each year the account was not properly declared. With the IRS going back 8 years, that means your entire account could quickly and completely be wiped out by these penalties.
Criminal prosecution and huge penalties are generally reserved for those who acted willfully. If you deliberately hid money, consider the current amnesty program as the only way of avoiding prosecution.
The IRS is offering an amnesty plan called the Offshore Voluntary Disclosure Program (sometimes called "OVDI" or "OVDP"). That program allows taxpayers with unreported foreign accounts to come forward and avoid audit, criminal prosecution and most penalties. Participants must pay a one time 27.5% penalty.
There are special amnesty programs for taxpayers with accounts worth less than $75,000 USD, those who inherited an offshore account and those not residing in the United States. Of course, there is much "fine print" with all of these alternatives.
Many of our clients simply had no idea that their offshore account had to be declared. For these folks, a traditional disclosure may be the best route. Generally penalties are greatly reduced or even waived. There are no guarantees, however.
The offshore reporting rules are very complex. Our advice is to have your returns prepared by an accountant well versed in foreign reporting requirements. If you are not currently in compliance, however, find a good tax lawyer to help navigate you through the several options and alternatives. Do not delay, however. If the IRS finds you first or if your bank discloses your name under the new FATCA law, all bets are off. The IRS amnesty programs all operate on a first contact basis meaning you must come forward first.
The tax lawyers at Mahany & Ertl have helped many foreign born Americans and Americans with foreign tax reporting problems. Offshore accounts, unreported Swiss accounts, FBAR compliance, foreign gift tax returns, OVDP and foreign corporations, we can help. In most instances, our services are can be offered at an affordable flat fee basis.