Denis Moreau

CLARIFYING THE CONCEPT OF SALVATION: A PHILOSOPHICAL APPROACH TO THE POWER OF FAITH IN CHRIST'S RESURRECTION

In this paper, I develop a philosophical clarification of the statement "faith in the resurrection of Christ saves men from sin", using some of the main arguments and hypotheses of my recent book, The Ways of Salvation (Les Voies du salut, Paris, 2010). I begin with some remarks on the theme of salvation in contemporary language and philosophy. I then sketch a conceptual analysis of the concept of salvation, first in its general sense, then in its specifically Christian one. Finally, I offer a hypothesis on the modus operandi of salvation, or at least of one aspect of salvation as understood by Christianity. I.

The concept of salvation still occurs regularly in ordinary language. It also appears, typically without being defined clearly, in a number of contemporary philosophical works far removed from Christianity.

It is striking how commonly the notion of salvation and related words (the verb 'to save' , the nouns 'saviour' , 'salvage') are used in most European languages. In French, people greet one another with the word "salut," in Italian they say "salve, " or "ti saluto, " in German they say "salü, " (or "heil, " "heil dich, " in the past). Though people using the word in such situations may not know it, this recalls an ancient practice of wishing an interlocutor 'salvation' upon meeting. For instance, Pythagorean philosophers appear to have greeted each other with the word 'health!' ugiainein, (a greeting also found in the New Testament, at the beginning of The Third Letter of John), and Seneca's letters to Lucilius often begin EUROPEAN JOURNAL FOR PHILOSOPHY OF RELIGION 3/2 (AUTUMN 2011), PP. 387-407 with the formula: "Seneca Lucilio suo salutem dat." The themes of saviour, salvage, salvation, which are etymologically as well as conceptually related to that of salvation are also increasingly common in political discourse (such and such a person is considered the country's saviour), economic discourse (the salvage of a corporation), as well as computer discourse (we save or salvage data). Finally, on a funnier, but no less meaningful note, French supermarkets sell a shower gel called "Axe. Difficult Morning, anti-hangover. " The product's packaging states quite clearly that it is intended for people who have a hard time waking up after partying, while the label describes its properties in terms that could come straight from a theology class: "miracle shower gel […] it will save your morning and bring you back to life after a short and restless night. "

Of course, the very frequency with which the concept of salvation is used means that in a certain way it is spent, close to losing its meaning from being used in too many contexts. But it might also be fair to ask whether this frequency of use doesn't echo, albeit weakly, ancient questions, long-standing concerns. In fact, if someone wanted to develop a Christian apologetic on the basis of the contemporary world's language use and dominant concerns, this theme of salvation would probably be an interesting starting point, a 'good hold' as people use the word 'hold' in rock-climbing.

All the more so because, while this notion of salvation retains, in its technical use at least, strongly religious and more specifically Christian connotations, it crops up in a surprising way in the writings of philosophers who are not particularly known for their support of Christianity, or are even quite critical of it.

Nietzsche is a striking if ambiguous example. As everyone knows, he sees himself as a fierce opponent of Christianity. But in several texts, he advocates a system of thought that, like Christianity, will lead to salvation -as long as we interpret salvation in accordance with its etymology, as a healing, the conclusion of a struggle against disease and weakness that yields 'the great health' . 1 The word also occurs in Jean Paul Sartre, in the famous last page of his autobiography The Words: "My sole concern has been to save myself -nothing in my hands, nothing up my sleeve -by work and faith. As a result, my pure choice did not raise me above anyone.

Without equipment, without tools, I set all of me to work in order to save all of me. If I relegate impossible Salvation to the prop-room, what remains?" 2 Similarly, in a rather mysterious footnote at the end of the section in Being and Nothingness called "Second attitude toward others: indifference, desire, hate, sadism", Sartre adds: "These considerations do not exclude the possibility of an ethics of deliverance and salvation. But this can be achieved only after a radical conversion which we can not discuss here. " 3 Ludwig Wittgenstein, in a text from Culture and Value (1937), for his part, wrote: "If I am to be really saved [erlöst], what I need is certainty, not wisdom, dreams, or speculation […] For it is my soul with its passions, as it were with its flesh and blood, that has to be saved [erlöst], not my abstract mind. " 4 And finally, Michel Foucault declares, in a way that is both enigmatic and fascinating, "I know that knowledge has the power to transform us, that truth is not just a way of deciphering the world […], but that, if I know the truth, then I will be transformed, maybe even saved. Or else I will die. But I believe, in any case, that for me these two are the same. " 5 These texts have three things in common: the theme of salvation is, for different reasons, unexpected; we understand, as we read them, that it is an important notion, one that reflects a concern essential to the author who uses it; but neither the context of these texts, nor, often, the entire corpus of their authors, give us a clear idea of how we should interpret 'salvation' or 'being saved' . Such conceptual blurriness, if not legitimate, is at least acceptable in the realm of ordinary language. But it is more problematic in a philosophical discourse that aims at conceptual clarity and rigor. To remedy this situation, I propose here a short clarification of the concept of salvation.

2 Jean-Paul Sartre, The Words, translated from the French by Bernard Frechtman, Vintage Books, 1981, p. 255 3 Jean-Paul Sartre, Being and Nothingness, translated and with an introduction by Hazel E. Barnes, Washington Square Press, 1956, p. 534, n. 13 4 Culture andValue, translated by Peter Winch, (Oxford: Basil Blackwell, 1974), pp. 32-33 5 "Interview, " by Stephen Riggins (1982;Dits et Ecrits, Paris, Gallimard, 2001), II, 1354

II. CLARIFICATION OF THE CONCEPT OF SALVATION

Historically, among the Greeks and Romans, the word salvation first meant the state of being or remaining whole and in good health, "safe and sound. " To be saved, then, was to be healed, and salvation, in the practical sense, meant health -not just physical, but also moral and spiritual health. In a more abstract sense, salvation meant both having reached a desirable way of life, as well as the process of attaining it, by being either removed from a situation or freed from a danger that somehow separated us from it. In a general sense, then, salvation can be understood as the return to a desirable former state that had been lost (as when one is saved from a sickness or a shipwreck), the safeguarding of this state against a threat (as one saves one's freedom from a potential oppressor, or one's life from a danger), or, finally, the improvement attaining this state represents. The meaning of the word salvation can, in short, be analyzed into two parts. Understood in its negative aspect, to be saved means to be delivered and freed, rescued and ripped away from a dangerous situation where looms a serious menace. Understood in its positive aspect, to be saved means being granted some good, reaching a state seen as beneficial or desirable, progressing from trials and wretchedness to a state of happiness and fulfilment. Therefore, I think we should find two elements in any soteriology. (A) A pessimistic or lucid diagnosis of our present situation as one that is painful and dangerous, a state we are inevitably and structurally thrown into, and out of which we must claw our way. An optimistic theory that held that everything is naturally for the best and will continue that way could not be called a soteriology.

(B) A more optimistic assessment of whether it is possible to leave this grievous state behind. If a theory accepts the pessimistic diagnosis of the human condition described in (A), but judges that we are bound to remain in this state of wretchedness, decay, and misery, then it is not describing human existence from a soteriological point of view.

Within the framework of these two elements, we can highlight a number of criteria to distinguish different kinds of soteriologies. For instance, we can distinguish different types of soteriology based on:

(a) Whether salvation is achieved through oneself (auto-salvation) or through someone else, something external to the self (hetero-salvation).

I will return to this distinction, which plays an essential role in differentiating Christian soteriology from most other forms of soteriology developed in philosophical contexts.

(b) The manner of reaching salvation. Individualistic theories hold that it is individuals who reach salvation, and holistic views assume that salvation is achieved collectively, by a group (a community, a nation, a Church, humanity as a whole).

(c) How broadly the class of the saved is extended. Some theories include only a few or a small group among the saved, some include the greater part of humanity, and some universalist or even cosmic doctrines include all of humanity, or even the entire universe, among the saved.

(d) Where salvation will take place: immanent theories hold that salvation is attained in this world, while some reserve salvation for another world.

(e) The nature of the alleged saviour: it can be a god (theo-soteriology), a man or a group of men (anthropo-soteriology), or even something else (extra-terrestrial beings, etc.).

(f) What degree of salvation is attainable: some theories hold that salvation is partial, others that it is total, others integrate the two into a process of salvation in stages or degrees.

(g) The nature of salvation, its content: most often, it is happiness, but even if we leave aside the well-known difficulties in agreeing on a common definition of happiness6 , there is no logical obstacle to imagining a different content for salvation.

Let us consider, for instance, how the Marxism of the 19 th and 20 th centuries, interpreted as a theory of salvation (or rather a secularized transposition of a theology of salvation) 7 , fits many of these categories. Marxism combines (a) auto-salvation (it is human beings who save themselves) and (e) anthropo-soteriology: it relies on a group of men (the proletariat, or its educated avant-garde) who hold the function of saviour in a period of transition (until the foundation of a classless society) to achieve a salvation which (c) all men or humanity as a whole share. This is (b) the conclusion of a collective process, which consists in (g) a happiness that is (f) complete and (d) obtained in this world.

III. THE SPECIFICALLY CHRISTIAN CONCEPT OF SALVATION

The New Testament attests that Jesus, whose Hebrew name Yeshoua means 'God saves, ' was quickly recognized by his disciples as the 'saviour' (salvator, sôter), the one who saves (salvare, sôzô) or brings salvation. 8 These early Christian texts use different themes, different images, to describe the status of the one who is saved, and the nature of salvation. The saved man is an invalid cured by Christ, a slave he frees, a debtor whose debt he forgives, a man possessed whose demonic bonds he looses, a man condemned whom he pardons, a dead man he brings back to life, etc. 9 One feature distinguishes Christian soteriology very clearly from those that can be found in ancient wisdom, the philosophical classics (e.g., Spinoza) 10 , Nietzsche, even, when he advises "independents" to "get up on their own" 11 , and contemporary thought that emphasizes human or personal autonomy. Indeed, for all other soteriologies, salvation is something that man, or in some cases humanity understood collectively, can achieve on his own, by his own actions, by making the best use of his own strengths and natural powers -where the rational powers are often singled out -in a process that is clearly a form of auto-salvation. The end of chapter 9, book IV of Epictetus's Discourses neatly synthesizes this conception of salvation: "Look, you have been dislodged though by no one else but yourself. Christian salvation, by contrast, is "salvation from elsewhere" ("hetero-salvation"). Man cannot reach it on his own, and it requires an external, divine and supernatural intervention: from a saviour, or else from a revelation -if by revelation we understand a body of knowledge that can neither be found in oneself nor gathered by human intellectual capacities alone, but must be received. The Christian approach to the concept of salvation, then, introduces a notion of, if not passivity, at least receptivity or dependence on an other (or an Other), that is, in the broad sense, a notion of heteronomy.

Salvation as understood by Christianity should therefore be defined as "movement from a negative to a positive state brought about by an external agent [which presupposes] three elements: a terminus, a starting point, and a transforming agent;"13 or else, "a process whose beneficiaries are moved from a negative situation to a new fulfilled existence by the action of an external agent. "14 In this case, the external agent is Jesus Christ, who, according to different acceptable translations of a series of Greek verbs used in the New Testament "frees, " (eleutheroô), "saves" (sôzô), "delivers" (rhuomai, luô), "tears away" (exaireô) humanity so that it can reach a new way of life.

That Christ is the saviour and that he saves men from sin is an idea that is so obvious to those who accept the Christian faith, that, unlike other concepts also central to Christianity (the trinity, Christ's two natures, etc.), it has never been seriously contested or rejected by any important currents in Christian thought. As a consequence, the great councils that enabled Christians to clarify important but controversial aspects of their faith, and thus to discriminate between orthodox and heretical theses, never made it the object of a dogmatic clarification. 15 The creed of the councils of Nicaea and Constantinople tells us only that Christ came « for us men and for our salvation » (Τὸν δι' ἡμᾶς τοὺς ἀνθρώπους καὶ διὰ τὴν ἡμετέραν σωτηρίαν κατελθόντα ἐκ τῶν οὐρανῶν / Qui propter nos homines, et propter nostram salutem descendit de caelis).

How the process of salvation works, its modus operandi, was thus left so open that it has given rise to a number of speculative accounts, about which it is not always easy to decide whether they are rivals, or whether they bring to light, without fully coordinating or synthesizing their incomplete accounts, different aspects of the concept of salvation. In Latin Christianity, for the last thousand years, the dominant answer to the question of how salvation works was the satisfaction theory proposed by Anselm of Canterbury in the Cur Deus Homo, and developed later, most notably by Thomas Aquinas and later Thomists. 16 I find this theory quite powerful, and I think that conceptually it is still relevant. But I don't want to ignore the fact that this explanation has become almost foreign, so to speak, or "inaudible" for many of our contemporaries. It has in fact become commonplace, for Christians and non-Christians alike, to reject the satisfaction theory of salvation as "judicial", "sacrificial", "vengeful", and "vindictive". Our contemporaries criticize its characterization of salvation as a "compensatory transaction", and the way it seems to worship pain by concentrating all of Christ's saving work in his passion and the sufferings that accompany it. 17 Moreover, in focusing so exclusively on 15 Among Catholics, the "schemata from the preparatory sessions" of the first Vatican council (1869-70) had considered defining redemption, but the texts were neither discussed nor voted on by the council (which was cut short by the Italian army's arrival in Rome). See Jean Rivière, Le Dogme de la rédemption (Paris, Gabalda, 1931), 116-120. There is also a mention (without definition) of the theme of "satisfaction" in a text from the 6 th session of the council of Trent. 16 See Summa Theologiae, III, questions 46-49; see also, for example, Eleonore Stump, "Atonement According to Aquinas" in Oxford Readings in Philosophical Theology, ed. Michael Rea, (Oxford, Oxford UP, 2009), vol.1 267-293. 17 For this type of criticism among writers who attack Christianity, see for example Nietzsche, The Antichrist, § 41 "'how could God allow it!' [Jesus' death] To which the the passion and death of Christ, this theory doesn't really assign to other aspects of his existence -his incarnation and even more importantly, his resurrection considered in themselves -any significant role in the work of salvation.

Therefore, without rejecting the fundamental importance of satisfaction and atonement in the concept of salvation, it is tempting to look for another explanation, or, more modestly, a complementary explanation of the modus operandi of Christian salvation understood as Christ's freeing us from sin.

IV. AN ATTEMPT TO EXPLAIN THE STATEMENT "FAITH IN THE RESURRECTION OF CHRIST SAVES MEN FROM SIN"

It's just this type of explanation I' d like to offer. I have tried to take into account an important trend in contemporary theology, which is really a corollary of the current turn away from satisfaction theories: the rejection of an exclusively sacrificial or expiatory theory of salvation, and the desire to see that the thinking that focuses exclusively on Jesus's passion and marginalizes the resurrection stop dominating the discussion. This agenda, which has been, in a way, the height of fashion in European salvation theology since the 1950's 18 , has been clear in its critical project, clear about what it is rejecting. It has also been clear in its general theoretical aim: "to restore Christ's victory, and return Christ's deranged reason of the little community [Jesus' disciples] formulated an answer that was terrifying in its absurdity: God gave his son as a sacrifice for the forgiveness of sins. At once there was an end of the gospels! Sacrifice for sin, and in its most obnoxious and barbarous form: sacrifice of the innocent for the sins of the guilty! What appalling paganism!". And in a Christian author, see for example, René Girard, Des choses cachées depuis la fondation du monde (Paris, Grasset, 1978) 269: "God [would] not just be demanding a new victim, he [would] be demanding the most precious, most cherished victim: his own son. This claim has done more than anything else, no doubt, to discredit Christianity in the eyes of men of good will in the modern world. It might have been tolerable to the medieval mind, but it has become intolerable to ours, and has become the stumbling block par excellence for an entire world repelled by the concept of sacrifice. " 18 Examples of this tendency born in the inter-war period among protestant theologians (like Karl Barth) include: François-Xavier Durrwell, Walter Kasper, Joseph Moingt, Jürgen Moltmann (at least in Theology of Hope), Wolhart Pannenberg, Karl Rahner, Bernard Sesboüe, Michel Deneken. resurrection to the central place in treatises on redemption it should never have lost"; "[to show that] the resurrection plays a fundamental role in salvation. " 19 But when it comes to the constructive side of the agenda, to specific explanations in support of the general aim, the results have been more problematic, less successful. I am here attempting one such specific explanation of the statement "faith in the resurrection of Christ saves us from sin. "

Here, now, is a summary of the argument I have developed to provide a philosophical clarification of this proposition. I will present it in four parts, each one corresponding to a section in my book.

(1) The first section offers a defence of a pragmatic approach to belief: beliefs should be considered not only with respect to their truth value, but also with respect to their effects, how they transform the believer and the world in which he acts. I say "not only…but also" because I don't think we should lose interest in the truth of beliefs, or reject Clifford's principle with its stipulation that the fundamental maxim of the ethics of belief is "It is wrong always, everywhere, and for anyone, to believe anything upon insufficient evidence. " But nothing stands in the way of our connecting our interest in the truth of beliefs to a pragmatic investigation of beliefs. We will then supplement our interest in a belief 's orthodoxy (how is its content theoretically righteous, speculatively correct) with a further question about its eudoxia (how is it beneficial for the believer who accepts it?). I offer a set of criteria for classifying beliefs from this pragmatic perspective, among which the two most fundamental are the magnitude and the value of a belief 's effects. We can thus distinguish between weak beliefs, which have a minimal impact on the believer's life ("beliefs with weak existential implications"), and highly effective ones, which have a significant impact on the life of those who come to believe them ("beliefs with strong existential implications"). We can also distinguish beliefs that produce correct or beneficial behaviour in those who accept them (eupraxic beliefs) from beliefs that produce incorrect or harmful behaviour (dyspraxic beliefs).

(2) Section II focuses on the notion of death. It takes as its starting point the classical view that any proposition about the nature of death (and more particularly of my death) can be an object of belief only, that positive knowledge or science about my death is impossible. If we accept this, then we are led to ask, even in this life: which among the available beliefs about death are beneficial and which are harmful? In the remainder of section II, I explain, without much original thought, that we spontaneously and naturally believe that death is the end of life, and that in most cases we are afraid of death so understood. As the expression 'fear of death' is a little bland and not always clear, I try to narrow it down by distinguishing different types or different degrees of fear of death. It ranges from the instinctual reaction we share with other animals, fleeing death as a fundamental threat, an annihilation of what we are insofar as we are alive, to highly intellectualized responses like the great artistic evocations of death (Mozart's Requiem, Molière's Don Juan) or philosophical investigations like Heidegger's study of anxiety. The general idea underlying this section is that we are naturally fearful of a death we interpret as the end of life.

(3) Section III shows that this standard belief about death is fundamentally dyspraxic, that is, it leads the believer to behave in ways that are bad and harmful, and which depending on one's lexical preferences, one can call faults, "bad deeds" or "sins". I will call this thesis, that there is a causal connection between the fear of death and sin, "the Lucretius hypothesis" because the idea is expressed, albeit without any detail about the precise nature of the connection, at the beginning of book III of De rerum natura.

and the old fear of Acheron driven headlong away, which utterly confounds the life of men from the very root, clouding all things with the blackness of death, and suffering no pleasure to be pure and unalloyed (…) Avarice and the blind craving for honours, which constrain wretched men to overleap the boundaries of right, and sometimes as comrades or accomplices in crime to struggle night and day with surpassing toil to rise up to the height of power-these sores in life are fostered in no small degree by the fear of death. For most often scorned disgrace and biting poverty are seen to be far removed from pleasant settled life, and are, as it were, a present dallying before the gates of death; and while men, spurred by a false fear, desire to flee far from them, and to drive them far away, they amass substance by civil bloodshed and greedily multiply their riches, heaping slaughter on slaughter. Hardening their heart they revel in a brother's bitter death, and hate and fear their kinsmen's board. In like manner, often through the same fear, they waste with envy that he is powerful, he is regarded, who walks clothed with bright renown; while they complain that they themselves are wrapped in darkness and the mire. Some of them come to ruin to win statues and a name; and often through fear of death so deeply does the hatred of life and the sight of the light possess men, that with sorrowing heart they compass their own death, forgetting that it is this fear which is the source of their woes, which assails their honour, which bursts the bonds of friendship, and overturns affection from its lofty throne. For often ere now men have betrayed country and beloved parents, seeking to shun the realms of Acheron. For even as children tremble and fear everything in blinding darkness, so we sometimes dread in the light things that are no whit more to be feared than what children shudder at in the dark, and imagine will come to pass. 20 In a number of analyses that I can't repeat here in any detail, I go on to show that the fear of death understood as the end of life leads to a series of evil actions: avarice or greed, gluttony, lust, homicide, disrespect to father and mother, and pride. By way of example, here is how we can establish a connection between fear of death and avarice, using a text from Karl Marx as support:

That which is for me through the medium of money -that for which I can pay (i.e., which money can buy) -that am I myself, the possessor of the money. The extent of the power of money is the extent of my power. Money's properties are my -the possessor's -properties and essential powers. Thus, what I am and am capable of is by no means determined by my individuality. I am ugly, but I can buy for myself the most beautiful of women. Therefore I am not ugly, for the effect of ugliness -its deterrent power -is nullified by money […] I am bad, dishonest, unscrupulous, stupid; but money is honoured, and hence its possessor. Money is the supreme good, therefore its possessor is good. Money, besides, saves me the trouble of being dishonest: I am therefore presumed honest. I am brainless, but money is the real brain of all things and how then should its possessor be brainless? Besides, he can buy clever people for himself, and is he who has power over the clever not more clever than the clever? Do not I, who thanks to money am capable of all that the human heart longs for, possess all human capacities?[...]That which I am unable to do as a man, and of which therefore all my individual essential powers are incapable, I am able to do by means of money. 21 The text reminds us of the specific function traditionally attributed to money: it is the universal mediator, the converter that makes all things commensurate by translating disparate realities and use-values into the same yard stick (the exchange value). To this classical analysis, Marx adds the idea that the act of buying, understood as an act of appropriation, causes the attributes of the property to be transferred to its owner. By merging the two driving ideas of this Marxist analysis, we can answer the question: in this mode of production and exchange, what is the object whose appropriation money fundamentally allows, and whose properties an owner claims for himself, at least at the level of fantasy?

It is time.

Wage-labour, after all, is the employer's use of his capital to "buy himself " his employees' time as well as the product of their work during that time. A commodity, likewise, is just the fruit of the work-time needed to produce it. It follows that buying and hoarding money (avarice in the strict sense), or objects (in particular, manufactured objects), in other words, being miserly in the broad sense, is really amassing human time, in so far as it is instantiated, and has been, in a certain sense, deposited in those objects. The more money we have, therefore, the more able we are to appropriate other people's time, buying it with wages, or buying it through the mediation of the commodities it has produced, and the more justified we feel in thinking of all this time as potentially our own.

Whether he consumes or saves, and whether his saving is an end in itself or the means to future consumption, the miser doesn't believe that "time is money"; he is rather moved by the belief that "money is time", that in the world in which he acts, having money means being able to acquire other people's time, literally, "saving time" or "buying time".

We can thus interpret the amassing of money as a more or less conscious fantasy promise of an indefinite heap of time, the illusory assurance that our existence will continue indefinitely, and so, as a fantasized attempt to escape our fear of death.

This, then, is how a causal and explanatory connection can be established between the fear of death on the one hand, and avarice or greed on the other. Using a number of different theoretical tools, I try to show in section III of my book that the same connection can be established between fear of death understood as the end of life on the one hand, and gluttony, lust, homicide, disrespect to mother and father, and pride on the other. Section III's general conclusion, then, is that the common belief about death has the characteristic effect of leading human beings to this type of evil act. The fear of death tends to land us in a kind of existential mediocrity, or, in the worst cases an existential incompetence. In other words, the standard belief about death is fundamentally and globally dyspraxic: it causes us to settle in a negative state that we can also call a state of sin, from which we need to be "saved", right now, in this life.

(4) In the fourth section, I finally turn to the question of salvation. To be saved from the negative state described in the previous section, one might adopt an orthopraxic belief about death -a belief that frees us from the evil acts we commit out of fear of death, and sets in motion a series of intellectual and emotional reactions that improve human existence.

Here my remarks turn avowedly Christian: for I believe that the Christian belief that death has been defeated by Christ is such a belief, orthopraxic in the highest degree.

I do not discuss the question of the truth or epistemic reliability of this belief in my book: I have nothing new to say on this subject.22 I accept this belief in a hypothetical way, following a method sometimes called philosophical theology. I ask: "If someone believes that Christ is resurrected, thereby signalling to us that death is not in fact the end of life, then what happens?"

All the elements I have discussed so far are falling into place to form a philosophical explication of how Christian salvation works. The belief in Christ's resurrection abolishes the ordinary representation of death as an absolute end, as well as the fearful relationship that follows. 23 This belief, then, is able to free us from the morally bad consequences (the sins) of the ordinary representation of death and our reaction to it. Thus, going back to the definition already spelled out, salvation is:

A process whose beneficiaries are moved from a negative situation… That is, humanity's situation as depicted in section III: "led astray" by the ordinary representation of death, and with a propensity to morally undesirable acts.

…to a new fulfilled existence…A man who no longer acts as described in section III has settled into a better realm of existence, one that is qualitatively superior to his former existence, not only because he is now rid of certain negative features, but also because he finds himself in a new set of circumstances conducive to leading a new life, one where dynamisms and capacities that could not be developed in his former existence can flourish. Ancient authors summarized it this way: "Christ killed the death that was killing man"; "he cast death's tyranny out of our nature completely by rising from the dead. " 24 …by the action of an external agent. Salvation is brought about by the power of the belief in Christ's resurrection. From an objective or historical standpoint, the external agent, the saviour, can be identified as Jesus Christ. If, on the other hand, we focus on the information contained in the proposition "Christ is resurrected, " then the external agency is a revealed body of faith. For this proposition cannot be deduced from 23 Cf. Athanasius of Alexandria, The Incarnation of the Word, 27 (Paris, Cerf, 1973, "Sources chrétiennes" n. 199) 362-365 : "Death has been destroyed, and the cross represents the victory won over it. It has no strength left, it is really dead. […] Ever since the Saviour resurrected his own body, death is no longer frightening. All those who believe in Christ […] really know that if they die, they do not perish but live. "

24 Melito of Sardis, Peri Pascha (Paris, Cerf, 1966 "Sources chrétiennes", n. 123), 96-97; Nicholas Cabasilas, The Life in Christ, III, 7 (Paris, Cerf, 1989-1990 "Sources chrétiennes" n. 355 et 361) 243. The text of this 14 th century Byzantine author deserves to be quoted more completely: "Thus, while men were cut off from God in three ways -through nature, through sin and through death -the saviour allowed them to meet him perfectly […], by removing one by one all the obstacles that kept them apart: [he removed the obstacle of] nature by sharing in humanity, the obstacle of sin by dying on the cross, and the last obstacle, the tyranny of death, he completely expelled from our nature by rising from the dead. " natural principles of human knowledge, nor can it be demonstrated a priori, and it is no doubt quite different from an ordinary piece of historical or experiential knowledge. It must therefore be a "revelation", which means that it presents for belief a body of knowledge that does not and cannot come from "me", but is given from elsewhere. Of course it belongs to the world, it appears in it, but from a source that is divine, or claimed to be so.

To use a different language: all the analyses I've developed so far now make it possible to explain the proposition "faith in the resurrection of Christ frees men from sin". If these earlier hypotheses are granted, then faith in the resurrection of Christ will, or should, set in motion in the individual who accepts it a series of intellectual and emotional transformations that improve his existence. If, as I have argued, the central problem of human life, what leads us astray and ruins our lives, is precisely a certain fearful relationship to death understood as the end of life, then the belief that death has been vanquished -a belief central and unique to Christianity25 -must be an excellent way to reach salvation -where salvation is understood as an improvement of existence that begins in this life, not in its eschatological sense (though, of course I don't reject that sense of the word).

CONCLUSION

In conclusion, let me make four qualifications to the thesis I am defending.

(a) I am not, of course, claiming that we are saved through knowledge, and that salvation is available only to the "experts" who think seriously about how salvation works. That would be Gnosticism, and I am not a Gnostic. It is faith that saves in my view, and it is "enough", so to speak, to believe that Christ is resurrected to enter into the process of salvation I describe. Someone who thinks seriously about the problem just adds an explanation of how the dynamics works, or tries to make the process intelligible.

(b) The thesis I am defending does assert that salvation or "justification" is brought about by faith, or, more precisely, by faith in Christ's resurrection with its message that death has been defeated. 26 I am here using the world faith in a strong sense, the sense of the credere in Deum, of strong conviction, a deep and sincere adherence that causes noticeable changes in the believer's interactions with the world. It is faith used in this sense that has certain consequences for the believer's salvation in my remarks above. It is again, faith used in this strong sense that contemporary authors call performative: "the Christian message was not only 'informative' but 'performative'"; "is the Christian faith […] 'performative' for us -is it a message which shapes our life in a new way, or is it just 'information' which, in the meantime, we have set aside and which now seems to us to have been superseded by more recent information?"27 If, by 'performative" (in the broad sense) 28 we understand the property whereby certain beliefs not only represent a state of affairs ("information"), but also produce a change by forming or transforming other beliefs and behaviours, then we can indeed speak of the saving character of performative faith in Christ's resurrection: the belief that death has been defeated has the characteristic effect of producing salvation. Being freed from sin -that is, being less tempted, better able to resist temptation, committing fewer or even none of the morally reprehensible acts described above -flows directly from the radical modification in the meaning of death that comes from accepting the belief in Christ's resurrection. c) On the other hand, in this defence of salvation through faith, I do not mean to commit myself to a specific position in the age-old (though nowadays mostly becalmed) theological debate about the respective roles of faith and works (i.e., individual actions) in salvation. Even if my thesis can nominally evoke the sola fide of the Reformation, since it gives faith the essential role in the production of salvation, it is in fact closer to the position generally thought of as the Catholic one: where the emphasis is on how man is brought into a situation where he can act well, and, using his freedom correctly, (subjectively) appropriate the salvation Christ brings (objectively). In this light, justification becomes the fact of finding oneself in a new practical context, one where obstacles and obstructions to right action no longer bind us when we act, so that it becomes possible for us to be just. On this interpretation, then, justification is more a journey than a consequence of faith in the resurrection that is acquired once and for all; it is not so much a state in which we find ourselves, as the lifelong tension required for becoming just. Collectively, it is less an event we can describe as having happened, than the process in which, according to Christianity, human beings have been involved since the discovery of the empty tomb. The idea that what leads to salvation is revealed implies a certain receptivity, or even passivity, as the intervention of a God or saviour implies heteronomy. But, it turns out, none of these are incompatible with the individual's having, or needing to have some personal agency in the process of his salvation, that is, with the possibility that the individual's actions will have the character of an auto-salvation. Receptivity and passivity are the starting point, necessary conditions for a new mode of action that an individual can't adopt on his own. d) Finally, I want to say more about what it means for this discussion of Christ's saving action to shift the focus from the passion to the resurrection -something scholastic theology often saw as nothing more than a happy ending (it's always better when the good guy wins) or else a "miracle" meant to elicit or strengthen faith. 29 First of all, the shift of focus does not imply that the darkness of Good Friday was useless or superfluous, that Jesus's dying on the cross after a degrading agony was irrelevant, or that nothing essential would be missing if he had died peacefully in his bed before being resurrected a few days later, or even more to the point, if he had proclaimed his victory over death without dying himself. First, as theologians in the first few centuries explained in their arguments against the docetist heresy, the reality of Christ's resurrection and humanity require the reality of his death. Second, the passion insofar as it is sorrowful and negative, and the resurrection, are like two facets of the same event -an event that brings salvation 30 and revelation -which can only be fully understood by focusing alternatively on one or the other of its facets. In its initial phase, this event reveals that salvation is not obtained easily; it is not obtained through the means usually promoted in the world (power, riches, honour, will to dominate), through violence, the desire to punish or to seek vengeance. Christ's passion shows that the road to salvation is hard, and that the fight against evil and everything that leads to it is sometimes painful and can require great sacrifice. The passion reminds us that "there is here an ordered sequence that Christ himself followed: first the passion, then the glorification. […] As long as our life here on earth lasts, suffering and death come before joy and resurrection. " 31 As the theologians say, it was "appropriate, " from this point of view, that Christ should die in the agony of the cross. However, the passion (and this includes its role in salvation) can be fully understood only in the light of the later event that gives it meaning. A story in which Jesus was only crucified would have a completely different meaning. Or it might not have any meaning at all, like a symbol of the absurdity and cruelty of the world. So, following Karl Barth, I want to warn those who think seriously about Christian justification and salvation against the Nordic Melancholy of a (good) Friday theology, abstractly focused on the cross alone and forgetful of Resurrection Sunday.

The seriousness with which we insist on the starting point of justification is a good and necessary thing, that is, on the fact […] that we can only go in one direction: from the death [of Jesus] on the cross to his resurrection. And so we must consider first what is past, that is, our death which he suffered, then what is future, that is, the life he received. […] But we must 30 Ultimately, Thomas Aquinas doesn't disagree: "as to efficiency, which comes of the Divine power, the Passion as well as the Resurrection of Christ is the cause of justification. " Summa Theologiae III, question 56, article 2 ad 4. This same question in the Summa parcels out each one's role by distinguishing two aspects in the "complete" concept of redemption: the passion and death of Christ cause the forgiveness of sins by providing satisfaction, while the resurrection institutes a new life. In every case, finding "the correct dose" of passion and resurrection respectively seems to be one of the central concerns of Christian soteriology.

31 Hans Urs von Balthasar, Theodramatik, 5 vol., Fribourg-Einsiedeln, Johannes Verlag, 1973-1983, vol. 4 see to it that this seriousness -there are examples of this both in Roman Catholic and also in Protestant circles -does not, at a certain point which is hard to define, become a pagan instead of a purely Christian seriousness, changing suddenly into a "Nordic morbidity", losing the direction in which alone it can have any Christian meaning, suddenly beginning to look backwards instead of forwards, transforming itself into the tragedy of an abstract Theologia crucis which can have little and finally nothing whatever to do with the Christian knowledge of Jesus Christ.

[…] The knowledge of our justification as it has taken place in Him can not possibly be genuinely serious except in this joy, the Easter joy. [START_REF] Barth | Die Kirchliche Dogmatik[END_REF] Finally, I do not for a moment claim that this explanation of how salvation works is the only valid one or that it excludes all others. It is, for instance, perfectly compatible with the satisfaction or atonement theory. Ultimately, I think that the best description of the Christian conception of salvation is this: "Christ's incarnation, his life and acts, his passion, death and resurrection, are what make salvation from sin possible. " Of course, one could say that all this constitutes just one event of salvation, the "Jesus Christ event". But as soon as we try to explain how this salvation really works, we end up distinguishing different explanatory frameworks; some that focus more on the incarnation's saving power as is the case with so-called deification-theories, others like satisfaction theories, that focus more on the passion. As a philosopher, and following in the footsteps of some of Saint Paul's texts 33 , I have wanted to draw attention to the soteriological value of faith in the resurrection. 34 I don't think that anyone can be certain with categorical certainty, that the proposition "Christ is risen from the dead" is true. Accepting it will always imply an irreducible element of faith, something like a bet that answers the existential question "what may I hope?" as much as the historical one "what can I know?". But if this proposition is true, if death really has been conquered by a fully human person, this implies an unprecedented existential transformation, whose full array of consequences, in my view, contemporary philosophers, including those who have paid a lot of attention to the relation between life and death (phenomenology, for instance, in Heidegger) have not analyzed. I have tried here to sketch out a few of those consequences, while at the same time suggesting that there is a real existential benefit to the belief that Christ is risen from the dead, in betting, as Pascal understood the word, that he is really risen.

speaking, the fact of dying and, in an analogical sense, the "spiritual death" brought about by the breakdown in the relationship with God. If we take "death" in the strict and biological sense, achievements of modern science seem difficult to reconcile with this view (as, in all cases, the theme of a unique and temporally determined peccatum originale originans). My interpretation avoids this problem, considering that sin, as a situation, and from there the sins, as actions, follow from mortality as it is spontaneously understood as a fundamental characteristic, both biological and existential , of humanity. The point is not to identify "sin" and "finiteness", but to show how sin stems from a form of spiritual negativity inherent to a certain understanding of finiteness, probably de facto inevitable, but not insurmountable.

  […] Turn yourself away to return […] to freedom […]. And now, are you not willing to come to your own rescue? […] If you seek something better, go ahead and continue what you are doing. Not even a god can save you. " 12

See, for example, Ecce Homo, "Why I am so clever, " I; Thus Spoke Zarathustra, I and II.

Cf. Aristotle, Nichomachean Ethics, I, 2

See Karl Löwith, Meaning in History: The Theological Implications of the Philosophy of History (Chicago, University of Chicago Press, 1949), ch. 2

See, for example, Acts of the Apostles, 4:12, and 13:23; I John, 4:14; Gospel of Luke, 2:11.

For more fully developed typologies, as well as detailed studies on the theme of salvation in different New Testament texts, see, for example, Le Salut Chrétien. Unité et diversité des conceptions à travers l'histoire, ed. Jean-Louis Leuba(Paris, Desclée, 1995); Salvation in the New Testament, Perspectives on Soteriology, ed. Jan G. van den Watt(Leiden-Boston, Brill, 2005); Alloys Grillmeier, "Die Wirkung des Heilshandelns Gotte in Christus" in Mysterium Salutis, ed. Johannes Feiner and Dan Magnus Löhrer(Einsiedeln, Benziger, 1969), vol. III-2, 327-390.

See, for example, Jean Lacroix, Spinoza et le problème du salut,(Paris, PUF, 1970). The explicit goal of Spinoza's Ethics is to "lead, as if by the hand, to knowledge of the human mind and its supreme blessedness" (beginning of part II) which is identified as "salvation" (V, 36, scolie).

Posthumous text cited in Didier Franck, Nietzsche et l' ombre de Dieu (Paris, PUF, 1998) 427. Cf. Ecce Homo, "Why I am so wise, " § 2: "I took myself in hand and I healed myself. " We can also remember the taunt "Save yourself " shouted at Christ on the cross according to the Gospels (Matthew 27:40).

I am following, with some modifications, W. A. Oldfather's translation in Epictetus, The discourses as Reported byArrian, The Manual, and Fragments, LOEB Classical Library, volume II,

Paul-Évode Beaucamp, Supplément au Dictionnaire de la Bible(Paris, Letouzey et Ané), vol.11, col. 516.

The different components of this definition are taken from Raymond Winling, La Bonne nouvelle du salut en Jésus-Christ. Sotériologie du Nouveau Testament(Paris, Cerf, 2007).

In this order: Henri de Lubac, Le Mystère du surnaturel,([1965], Paris, Cerf,

2000) 20 ; Bernard Sesboüe, Jésus-Christ dans la tradition de l'Église([1982], Paris, Desclée de Brouwer, 2000) 238.

Lucretius, De natura rerum, III, translation Cyril Bayley

Marx, Economic and Philosophic Manuscripts of 1844. Third Manuscript, Translation Martin Milligan.

On this theme, see for example Richard Swinburne, The Resurrection of God Incarnate,(Oxford, Clarendon Press, 2003).

See the famous claim by Paul of Tarsus in I Corinthians, 15:14: "if Christ had not been raised, then our proclamation has been in vain and your faith has been in vain. "(New Revised Standard Version)

Cf. Saint Augustin, Contra Faustum, 16, 29, Migne PL, vol. 42, col. 336 : "The very resurrection in which we believe justifies us. "

Joseph Ratzinger/Benedict XVI, Encyclical Spe Salvi, §2 et §10.

In the narrow sense (that of J.L. Austin) used in the philosophy of language, only public utterances (and not mental states) that actualize what they describe, or "do what they say", are called 'performative' .

See, for example, Cajetan's commentary on Saint Paul (On Romans 4:25) Epistolae Pauli et aliorum apostolorum [...] juxta sensum literalem enarratae [1531].


Andrey Surzhykov

José Paulo Santos

Pedro Amaro

Paul Indelicato

Negative continuum effects on the two-photon decay rates of hydrogen-like ions

Keywords: numbers: 31, 30, J-,32, 80, Wr

Two-photon decay of hydrogen-like ions is studied within the framework of second-order perturbation theory, based on relativistic Dirac's equation. Special attention is paid to the effects arising from the summation over the negative-energy (intermediate virtual) states that occurs in such a framework. In order to investigate the role of these states, detailed calculations have been carried out for the 2s 1/2 → 1s 1/2 and 2p 1/2 → 1s 1/2 transitions in neutral hydrogen H as well as for hydrogen-like xenon Xe 53+ and uranium U 91+ ions. We found that for a correct evaluation of the total and energy-differential decay rates, summation over the negative-energy part of Dirac's spectrum should be properly taken into account both for high-Z and low-Z atomic systems.

I. INTRODUCTION

Experimental and theoretical studies on the twophoton transitions in atomic systems have a long tradition. Following seminal works by Göppert-Mayer [1] and by Breit and Teller [2] a large number of investigations have been performed in the past which focused on the decay of metastable states of light neutral atoms and low-Z ions. These investigations have dealt not only with the total and energy-differential decay rates [3,4,5] but also with the angular distributions [6,7,8,9] and even polarization correlations between the two emitted photons [10,11,12]. Detailed analysis of these two-photon properties have revealed unique information about electron densities in astrophysical plasmas and thermal Xray sources, highly precise values of physical constants [13], structural properties of few-electron systems including subtle quantum electrodynamical (QED) effects [14] as well as about the basic concepts of quantum physics such as, e.g., non-locality and non-separability [15].

Beside the decay of metastable states of low-Z systems, much of today's interest is focused also on the two-photon transitions in high-Z ions and atoms which provide a sensitive tool for improving our understanding of the electron-photon interactions in the presence of extremely strong electromagnetic fields [16]. In such strong fields produced by heavy nuclei, relativistic and retardation effects become of paramount importance and may strongly affect the properties of two-photon emission. To explore these effects, therefore, theoretical investigations based on Dirac's equation have been carried for the total and energy-differential decay rates [17,18,19,20,21] as well as for the angular and polarization correlations [22,23,24]. In general, relativistic predictions for the two-photon total and differential properties have been found in a good agreement with experimental data obtained for the decay of inner-shell vacancies of heavy neutral atoms [25,26] and excited states of high-Z fewelectron ions [27].

Although intensive experimental and theoretical efforts have been undertaken recently to understand relativistic effects on the two-photon transitions in heavy ions and atoms, a number of questions still remain open. One of the questions, which currently attracts much of interest, concerns the role of negative energy solutions of Dirac's equation in relativistic two-photon calculations. Usually, these calculations are performed within the framework of the second-order perturbation theory and, hence, require summation over the (virtual) intermediate ion states. Such a summation, running over the complete spectrum, should obviously include not only positive-(discrete and continuum) but also negative-eigenenergy Dirac's states. One might expect, however, that since the energy release in two-photon bound-bound transitions is less than the energy required for the electron-positron pair production, the contribution from the negative part of Dirac's spectrum should be negligible even for the decay of heaviest elements. From practical viewpoint, this assumption justifies the restriction of the intermediate-state summation to the positive-energy solutions only. Exclusion of the negative continuum would lead, in turn, to a significant simplification of the the second-order relativistic calculations especially for many-electron systems for which the problem of (many particle) negative continuum still remains unsolved.

Despite the (relatively) small energy of two-photon transitions, the influence of Dirac's negative continuum in second-order calculations should be further questioned because of possibility for production and subsequent annihilation of the virtual anti-particles. It has been ar-gued, for example, that transitions involving positron states have to be taken into account for the proper description of Thomson scattering [START_REF] Sakurai | Advanced Quantum Mechanics[END_REF], interaction of ions with intense electromagnetic pulses [START_REF] Boca | [END_REF]30] in the "undercritical" regime as well as magnetic transitions in twoelectron ions [31,32,33]. Moreover, the first step towards the analysis of negative-energy contributions to the two-photon properties has been done by Labzowsky and co-workers [34] who focused on E1M1 and E1E2 2p 1/2 → 1s 1/2 total decay probabilities. The relativistic calculations have indicated the importance of negativeenergy contributions not only for high-Z but also for low-Z hydrogen-like ions.

In this work, we apply the second-order perturbation theory based on relativistic Dirac's equation in order to re-analyze atomic two-photon decay. We pay special attention to the influence of negative continuum solutions on the evaluation of the transition amplitudes and, hence, on the total and energy-differential decay rates. For the sake of clarity, we restrict our analysis to the decay of hydrogen-like ions for which both the positiveand negative-energy parts of Dirac's spectrum can be still studied in a systematic way by making use of a finite basis set method [19]. Implementation of this method for computing relativistic second-order transition amplitudes is briefly discussed in Sections II A and II B. Later, in Section II C, we consider an alternative, semiclassical, approach which allows analytical evaluation of the negative-energy contributions to the two-photon matrix elements and transition rates. These two-semiclassical and fully relativistic-approaches are used in Section III to calculate the energy-differential and total decay rates for several multipole terms in the 2s 1/2 → 1s 1/2 and 2p 1/2 → 1s 1/2 two-photon decay of neutral hydrogen as well as hydrogen-like xenon Xe 53+ and uranium U 91+ ions. Based on the results of our calculations, we argue that both the total transition probabilities and the photon energy distributions can be strongly affected by the negative-state contributions; this effect is most clearly observed for the non-dipole transitions not only in high-Z but also in (non-relativistic) low-Z domain. Brief summary of these findings and outlooks are given finally in Section IV.

II. THEORY A. Differential and total decay rates

Not much has to be said about the basic formalism for studying the two-photon transitions in hydrogen-like ions. In the past, this formalism has been widely applied in order to investigate not only the total decay probabilities [17,18,19,34] but also the energy as well as angular distributions [23] and even the correlation in the polarization state of the photons [15,24]. Below, therefore, we restrict ourselves to a rather brief account of the basic expressions, just enough for discussing the role of negative-energy solutions of Dirac's equation in computing of the two-photon (total and differential) rates.

The properties of the two-photon atomic transitions are evaluated, usually, within the framework of the second-order perturbation theory.

When based on Dirac's equation, this theory gives the following expression for the differential in energy decay rate:

dw dω 1 = ω 1 ω 2 (2π) 3 c 3 ν f |A * 2 | ν ν |A * 1 | i E ν -E i + ω 1 + f |A * 1 | ν ν |A * 2 | i E ν -E i + ω 2 2 dΩ 1 dΩ 2 , (1)
where the transition operators A * j with j = 1, 2 describe the (relativistic) electron-photon interaction. For the emission of photons with wave vectors k j and polarization vectors êj these operators read as:

A * j = α • (ê j + Gk j ) e -ikj r -Ge -ikj r , ( 2
)
where α is a vector of Dirac matrices and G is an arbitrary gauge parameter. In the calculations below, following Grant [35], we employ two different gauges that are known to lead to well known non-relativistic operators. First, we use the so-called Coulomb gauge, when G =0, which corresponds to the velocity form of electronphoton interaction operator in the non-relativistic limit.

As the second choice we adopt G = (L + 1)/L in order to obtain Babushkin gauge which reduces, for the particular case of L=1, to the dipole length form of the transition operator. In Eq. (1), |i ≡ |n i κ i µ i and |f ≡ |n f κ f µ f denote solutions of the Dirac's equation for the initial and final ionic states while E i ≡ E niκi and E f ≡ E n f κ f are the corresponding one-particle energies. Because of energy conservation, E i and E f are related to the energies ω 1,2 = ck 1,2 of the emitted photons by:

E i -E f = ω 1 + ω 2 .
(

) 3
¿From this relation, it is convenient to define the so-called energy sharing parameter y = ω 1 /(ω 1 + ω 2 ), i.e., the fraction of the energy which is carried away by the "first" photon.

As usual in atomic physics, the second-order transition amplitudes in Eq. ( 1) and, hence, the two-photon transitions rates can be further simplified by applying the techniques of Racah's algebra if all the operators are presented in terms of spherical tensors and if the (standard) radial-angular representation of Dirac's wavefunctions are employed. For the interaction of electron with electromagnetic field, the spherical tensor components are obtained from the multipole expansion of the operator A * j (see Refs. [18,19,[START_REF] Varshalovich | Quantum theory of angular momentum[END_REF] for further details). By using such an expansion, we are able to re-write Eq. ( 1) as a sum of partial multipole rates

dw dω 1 = Θ1L1Θ2L2 dW Θ1L1Θ2L2 dω 1 , (4)
which describe the emission of two photons of electric (Θ j = E) or/and magnetic (Θ j = M) type carrying away the angular momenta L 1 and L 2 . For the decay of unpolarized ionic state |n i κ i , in which the emission angles as well as polarization of both photons remain unobserved, these partial multipole rates are given by [18]:

dW Θ1L1Θ2L2 dω 1 = ω 1 ω 2 (2π) 3 c 3 λΘ 1 λΘ 2 κν S jν (1, 2) 2 + S jν (2, 1) 2 + 2 κ ′ ν d(j ν , j ′ ν ) S jν (2, 1)S j ′ ν (1, 2) , (5)
where the angular coefficient d(j ν , j ′ ν ) is defined by the phase factor and 6j Wigner symbol:

d(j ν , j ′ ν ) = (2j ν + 1)(2j ′ ν + 1) × (-1) 2jν +L1+L+2 j f j ′ ν L 1 j i j ν L 2 , (6)
and the radial integral part is expressed in terms of the reduced matrix elements of the multipole (electric and magnetic) field operators:

S jν (1, 2) = nν n f κ f âλΘ 1 * L1 n ν κ ν n ν κ ν âλΘ 2 * L2 n i κ i E ν -E i + ω 2 . (7)
The summation over λ Θj in Eq. ( 5) is restricted to λ Θj = ±1 for the electric (Θ j = E) and λ Θj = 0 for the magnetic (Θ j = M) photon transitions.

Until now, we have discussed the general expressions for the two-photon transition rates which are differential in energy ω 1 of one of the photons. By performing an integration over this energy one may easily obtain the total rate that is directly related to the lifetime of a particular excited state against the two-photon decay. As it follows from Eq. ( 4), such a total rate can be represented as a sum of its multipole components:

w tot = Θ1L1Θ2L2 W Θ1L1Θ2L2 ≡ Θ1L1Θ2L2 ωt 0 dW Θ1L1Θ2L2 dω 1 dω 1 , (8)
where

ω t = E i -E f is the transition energy.
As seen from Eqs. ( 4)-( 8), any analysis of the differential as well as total two-photon decay rates can be traced back to the (reduced) matrix elements that describe the interaction of an electron with the (multipole) radiation field. Since the relativistic form of these matrix elements is applied very frequently in studying the various atomic processes, we shall not discuss here their evaluation and just refer the reader for all details to references [18,19,35]. Instead, in the next section we will focus on the summation over the intermediate states |n ν κ ν which appears in the second-order transition amplitudes (see Eq. ( 7)).

B. Summation over the intermediate states

The summation over the intermediate states in Eq. (7) runs over the complete one-particle spectrum |n ν κ ν , including a summation over the discrete part of the spectrum as well as an integration over the positive and negative-energy continuum. In practice, of course, performing such an infinite-state summation is a rather demanding task. A number of methods have been developed over the last decades in order to evaluate the second-order transition amplitudes consistently. Apart from the Green's function approach [23,36] whichin case of a purely Coulomb potential-allows for the analytical computation of Eq. ( 7), the discretebasis-set summation is widely used nowadays in twophoton studies [19]. A great advantage of the latter method is that it allows to separate the contributions from the positive-and negative-energy solutions in the intermediate-state summation. Since the effects that arise from the negative-energy spectrum are in the focus of the present study, we apply for the calculations below the finite (discrete) basis solutions constructed from the B-spline sets.

Although the B-spline basis set approach has been discussed in detail elsewhere [19,37,38], here we briefly recall its main features. In this way, we shall consider the ion (or atom) under consideration to be enclosed in a finite cavity with a radius R large enough to get a good approximation of the wavefunctions with some suitable set of boundary conditions, which allows for discretization of the continua. Wavefunctions that describe the quantum states |ν ≡ |n ν κ ν of such a "particle in box" system can be expanded in terms of basis set functions φ i ν (r) with i = 1, .., 2N which, in turn, are found as solutions of the Dirac-Fock equation,

V (r) c d dr -κν R -d dr + κν R -2c + V (r) c φ i ν (r) = ǫ i ν c φ i ν (r) , (9)
where ǫ i ν = E i ν -mc 2 and V (r) is a Coulomb potential of a uniformly charged finite-size nucleus. Due to computational reasons, each of φ i ν (r) function is expressed as a linear combination of B-splines as it was originally proposed in Ref. [37] by Johnson and co-workers.

For each quantum state |ν the set of basis functions φ i ν (r) spans both positive and negative energy solutions. Solutions labeled by i = 1, .., N describe the negative continuum with ǫ i ν < -2mc 2 while solutions labeled by i = N + 1, .., 2N correspond to the first few states of the bound-state spectrum as well as to positive continuum with ǫ i ν > 0. Thus, by selecting the proper sub-set of basis functions φ i ν (r) we may explore the role of negative continuum in computing of the properties of two-photon emission from hydrogen-like ions.

C. Semi-relativistic approximation

Based on the relativistic theory, the expressions obtained in the previous section allow to study the influence of the Dirac's negative continuum on the properties of two-photon emission from hydrogen-like ions with nuclear charge in the whole range 1 ≤ Z ≤ 92. For the low-Z ions, moreover, it is also useful to estimate the negative-energy contributions within the semirelativistic approach as proposed in the work by Labzowsky and co-workers [34]. To perform such a semirelativistic analysis let us start from Eq. ( 1) in which we retain the sum only over the negative-energy continuum states. Since the total energy of these states is E ν = -(T ν + mc 2 ), the corresponding energy denominator of the second-order transition amplitude can be written as E ν -E i + ω j ≈ -2mc 2 which leads to the following expression for the differential decay rate: 2: (Color online) Energy-differential decay rates for the (sum of the) 2M1, 2E2 and E2M1 2s 1/2 → 1s 1/2 multipole two-photon transitions in hydrogen and hydrogen-like ions. Relativistic calculations have been carried out by performing intermediate-state summation over complete Dirac's spectrum (solid line) as well as by restricting this summation to the positive-(dashed line) and negative-energy (dotted line) states only. Results of relativistic calculations are compared also with the semi-relativistic prediction (dot-dashed line) as given by Eq. (14).

dw (-) dω 1 = ω 1 ω 2 3 c 3 1 4(mc 2 ) 2 ν∈(-) f |A * 2 | ν ν |A * 1 | i 0 0.
+ f |A * 1 | ν ν |A * 2 | i 2 dΩ 1 dΩ 2 . ( 10
)
For the further simplification of this expression we shall make use of the multipole expansion of the electronphoton interaction operators (2). For the sake of simplicity, we restrict this semi-relativistic analysis to the case of Coulomb gauge (G = 0) in which operator A * j can be written as:

A * j = α • êj (1 -ik • r + 1/2 (-ik • r) 2 + ...) , (11)
if one expand the photon exponential exp(ik • r) into the Taylor series.

In contrast to the "standard" spherical tensor expansion [18,[START_REF] Varshalovich | Quantum theory of angular momentum[END_REF], the series (11) usually does not allow one to make a clear distinction between the different multipole components of the electromagnetic field. For instance, while the first term in Eq. ( 11) describes-within the non-relativistic limit-electric dipole (E1) transition, the term (-ik • r) gives rise both, to magnetic dipole (M1) and electric quadrupole (E2) channels. Such an approximation, however, is well justified for our (semirelativistic) analysis which just aims to estimate the role of negative continuum states in the different (groups of ) multipole two-photon transitions in light hydrogen-like ions. In particular, by adopting A * j = -α • êj (ik • r) for both operators in Eq. (10) we may find the contribution from the negative spectrum to the 2M1, 2E2 and E2M1 2s 1/2 → 1s 1/2 transition probabilities:

dw (-) M1,E2 dω 1 = ω 1 ω 2 (2π) 3 c 3 1 4(mc 2 ) 2 × ν∈(-) f |α • ê2 (k 2 • r)| ν ν |α • ê1 (k 1 • r)| i + f |α • ê1 (k 1 • r)| ν ν |α • ê2 (k 2 • r)| i 2 ×dΩ 1 dΩ 2 . ( 12
)
Here, summation over the intermediate states |ν is restricted by the negative-energy solutions of the Dirac equation for the electron in the field of nucleus. In the non-relativistic limits these states form a complete set of solutions of the Schrödinger equation for the particle in a repulsive Coulomb filed [START_REF] Landau | Quantum mechanics (Course of theoretical physics v[END_REF]. By employing a closure relation for such a set we re-write Eq. ( 12) in the form:

dw (-) M1,E2 dω 1 = ω 1 ω 2 (2π) 3 c 3 1 (mc 2 ) 2 × |(ê 1 ê2 ) f |(k 1 • r)(k 2 • r)| i | 2 dΩ 1 dΩ 2 , ( 13
)
where |i and |f denote now the solutions of the Schrödinger equation for the initial and final ionic states, respectively. For the particular case of 2s 1/2 → 1s 1/2 two-photon transition, i.e., when |i = |2s and |f = |1s , this expression finally reads:

dw (-) M1,E2 dω 1 = 2 22 3 13 α 10 5πZ 4 ω 3 1 ω 3 2 , (14)
if one performs an integration over the photon emission angles as well as a summation over the polarization states (see Ref. [34] for further details). Eq. ( 14) provides the differential rate for the 2M1, 2E2 and E2M1 two-photon transitions as obtained within the non-relativistic framework and by restricting the summation over the intermediate spectrum |ν to the negative energy states only. Being valid for low-Z ions, this expression may also help us to analyze the negative-energy contribution to the total decay rate, 3: (Color online) Energy-differential decay rates for the (sum of the) E1M1 and E1E2 2p 1/2 → 1s 1/2 multipole two-photon transitions in hydrogen and hydrogen-like ions. Relativistic calculations have been carried out by performing intermediate-state summation over complete Dirac's spectrum (solid line) as well as by restricting this summation to the positive-(dashed line) and negative-energy (dotted line) states only. Results of relativistic calculations are compared also with the semi-relativistic prediction (dot-dashed line) as given by Eq. ( 16).

w (-) M1,E2 = dw (-) M1,E2
where the integration over the photon energy ω 1 is performed.

Apart from the 2M1, 2E2 and E2M1 2s 1/2 → 1s 1/2 two-photon transitions, Eqs. ( 10) and (11) may also be employed to study other decay channels. For example, the negative energy contributions to the differential as well as total rates for the E1M1 and E1E2 2p 1/2 → 1s 1/2 decay read as:

dw (-) E1,M1,E2 dω 1 = 2 17 3 12 α 8 πZ 2 ω 1 ω 2 (ω 2 1 + ω 2 2 ) , (16)
and

w (-) E1,M1,E2 = (αZ) 8 2 5π 3 7 = 5.822 10 -5 (αZ) 8 , (17)
respectively [34]. Together with Eqs. ( 14) and ( 15), we shall later use these non-relativistic predictions in order to check the validity of our numerical calculations in low-Z domain.

III. RESULTS AND DISCUSSION

Having discussed the theoretical background for the two-photon studies, we are prepared now to analyze the influence of the Dirac's negative continuum on the total as well as energy-differential decay rates. We shall start such an analysis from the 2s 1/2 → 1s 1/2 transition, which is well established both in theory [18,19,23] and in experiment. For all hydrogen-like ions this transition is dominated by the 2E1 decay channel while all the higher multipoles contribute by less than 0.5% to the decay probability. The energy-differential decay rate given by Eq. ( 5) for the emission of two electric dipole photons is displayed in Fig. 1 for the decay of neutral hydrogen (H) as well as hydrogen-like xenon Xe 53+ and uranium U 91+ ions. For these ions, relativistic secondorder calculations have been done within the Coulomb gauge and by performing intermediate-state summation over the complete Dirac's spectrum (solid line) as well as over the positive-(dashed line) and negative-energy (dotted line) solutions only. As seen from the figure, the negative-energy contribution to the energy-differential decay rate is negligible for low-Z ions but becomes rather pronounced as the nuclear charge Z is increased. For the 2E1 decay of hydrogen-like uranium, for example, exclusion of the negative solutions from the intermediate-state summation in Eq. ( 7) leads to about 20 % reduction of the decay rate when compared with the "exact" result.

While for the leading, 2E1 2s 1/2 → 1s 1/2 transition the negative continuum effects arise only for rather heavy ions, they might strongly affect properties of the higher multipole decay channels in low-Z domain. In Fig. 2, for example, we display the energy distributions of photons emitted in 2M1 and 2E2 transitions. As seen from the upper panel of the figure corresponding to the decay of neutral hydrogen, negative energy part of the Dirac's spectrum gives the dominant contribution to the (sum of the) differential rates for these decay channels. With the increasing nuclear charge Z, the role of positive energy solutions also becomes more pronounced. However, these solutions allow one to describe reasonably well the differential rates (5) only if one of the photons is much more energetic than the second one, i.e., when either y < 0.1 or y > 0.9. For a nearly equal energy sharing (y ≈ 0.5), in contrast, accurate relativistic calculations of the 2M1 and 2E2 rates obviously require summation over both, the negative and the positive energy states.

Apart from the results of relativistic calculations, we also display in Fig. 2 the (sum of the) negativeenergy contributions to the 2M1, 2E2, M1E2 and E2M1 2s 1/2 → 1s 1/2 transition probabilities as obtained within the semi-relativistic approach discussed in Section II C. As expected, for low-Z ions both the relativistic (dotted line) and semi-relativistic (dot-dashed line) results basically coincide and are well described by Eq. ( 14). As the nuclear charge Z is increased, however, semirelativistic treatment leads to a slight underestimation of the negative-energy contribution to the two-photon (differential) transition probabilities. For the 2s 1/2 → 1s 1/2 decay of hydrogen-like uranium ion, for example, results obtained from Eq. ( 14) is about 30 % smaller than the corresponding relativistic predictions.

Up to now, we have been considering the 2s 1/2 → 1s 1/2 two-photon decay of the hydrogen-like ions. Apart from this -experimentally well studied-transition, recent theoretical interest has been focused also on the 2p 1/2 → 1s 1/2 two-photon decay [34]. Although such a channel is rather weak comparing to the leading onephoton E1 transition, its detailed investigation is highly required for future experiments on the parity violation in simple atomic systems [START_REF] Drukarev | [END_REF]. A number of calculations [34,42] have been performed, therefore, for the transition probabilities of the dominant E1M1 and E1E2 multipole components. In order to discuss the role of Dirac's negative continuum in these calculations, we display in Fig. 3 the energy-differential rate for the sum of the E1M1 and E1E2 2p 1/2 → 1s 1/2 two-photon transitions. Again, the calculations have been carried out within the Coulomb gauge for the electron-photon coupling and for three nuclear charges Z = 1, 54 and 92. As seen from the figure, negative-energy summation in the second-order transition amplitude ( 7) is of great importance for accurate evaluation of 2p 1/2 → 1s 1/2 transition probabilities both for low-Z and high-Z ions. That is, restriction of the intermediate-state summation to positive part of Dirac's spectrum results in an overestimation of the E1M1 and E1E2 differential in energy decay rates by factors of about 2 and 2.5 for the neutral hydrogen and hydrogen-like uranium, respectively.

Similarly to the 2s 1/2 → 1s 1/2 multipole transitions, we make use of semi-relativistic formulae from Section II C to cross-check our relativistic computations for the negative-energy contribution to the E1M1 and E1E2 2p 1/2 → 1s 1/2 decay rates in low-Z domain. Again, while for neutral hydrogen both, semi-relativistic (16) and relativistic approximations produce virtually identical results, they start to differ as the nuclear charge Z is increased.

So far we have discussed the energy-differential decay rates both for 2s 1/2 → 1s 1/2 and 2p 1/2 → 1s 1/2 two-photon transitions. Integration of these rates over the energy of one of the photons (see Eq. ( 8)) will yield the total decay rates. In Table I we display the total decay rates for the various multipole channels of 2s 1/2 → 1s 1/2 two-photon decay. In contrast to the photon energy distributions from above, here relativistic calculations have been performed in Coulomb (velocity) as well as Babushkin (length) gauges. In both gauges, negative-energy contribution to the (total) probability of the leading 2E1 transition is about eight orders of magnitude smaller than positive-energy term if decay of low-Z ions is considered but is significantly in- creased for higher nuclear charges. For the hydrogenlike uranium, for example, the total 2E1 decay rate is enhanced from 2.9041×10 12 s -1 in the velocity gauge and 2.3939×10 12 s -1 in the length gauge to the-gauge independent-"exact" value of 3.8256×10 12 s -1 if, apart from the positive-energy states, the Dirac's states with negative energy are taken into account in transition amplitude Eq. ( 7). These results clearly indicate the importance of the negative-state summation for the accurate evaluation of 2E1 2s 1/2 → 1s 1/2 total rates in both, velocity and length gauges. It worth mentioning, however, that while for velocity gauge our findings are in perfect agreement with results reported in Ref. [34] some discrepancy was found for calculations performed in length gauge for which Labzowsky and co-workers have argued that the contribution from the Dirac's negative continuum is negligible even for heaviest ions. The reason for this discrepancy is not apparent for the moment and, hence, further investigations are highly required.

In Table I, besides the leading 2E1 decay channel, we present the results of relativistic calculations for the higher multipole contributions to the 2s 1/2 → 1s 1/2 twophoton transition. The influence of Dirac's negative continuum is obviously different for various multipole combinations. While, for example, the negative-energy contribution to the intermediate-state summation in low-Z domain is negligible for the E1M2 decay it becomes of paramount importance for the 2E2 and 2M1 decay channels; an effect that has been already discussed for the case of the energy-differential decay rates (see upper panel of Fig. 2). Moreover, 2s 1/2 → 1s 1/2 transition with emission of two magnetic dipole (2M1) photons in light ions seems to happen almost exclusively via the negative energy (virtual) intermediate states. The total decay rate for this transition together with the negative-energy contribution to the probability of the 2E2 channel (evaluated in Coulomb gauge) gives in atomic units:

w 2M1 + w (-) 2E2 = 1.248 10 -6 (αZ) 10 , (18)
which is in perfect agreement with the semi-relativistic formula (15). As mentioned above for the computation of the photon energy distributions in low-Z domain, negativeenergy contribution to the intermediate-state summation is rather pronounced not only for the higher multipole terms of 2s 1/2 → 1s 1/2 decay but also for the leading E1M1 and E1M2 (two-photon) channels of 2p 1/2 → 1s 1/2 transition. Our relativistic calculations displayed in Table II indicate that one should account for negativecontinuum summation also for an accurate evaluation of the total decay rates for these two decay channels. For the decay of light elements, sizable contribution from the negative-continuum intermediate states arises both in length and velocity gauges. Again, these results partially question the predictions by Labzowsky and co-workers [34] who claimed a minor role of negative energy terms for E1M1 and E1M2 calculations in length gauge. For the velocity gauge, in contrast, our relativistic calculations:

are in a good agreement both, with the semi-relativistic prediction (17) and data presented in Ref. [34].

IV. SUMMARY AND OUTLOOK

In conclusion, the two-photon decay of hydrogen-like ions has been re-investigated within the framework of second-order perturbation theory, based on Dirac's relativistic equation. Special attention has been paid to the summation over the intermediate ionic states which occurs in such a framework and runs over complete oneparticle spectrum, including a summation over discrete (bound) states as well as the integration over the positive and negative continua. In particular, we discussed the role of the negative energy continuum in an accurate evaluation of the second-order transition amplitudes and, hence, the energy-differential as well as total decay rates. Detailed calculations of these rates have been presented for the 2s 1/2 → 1s 1/2 and 2p 1/2 → 1s 1/2 two-photon transitions in neutral hydrogen as well as hydrogen-like xenon and uranium ions. As seen from the results obtained, both the total decay probabilities and the energy distributions of the simultaneously emitted photons can be strongly affected by the negative-state summation not only for heavy ions but also for low-Z domain.

We demonstrate, however, that the role of Dirac's negative continuum becomes most pronounced for the higher (non-dipole) terms in the expansion of the electronphoton interaction; similar effect has been recently reported for the theoretical description of hydrogen-like systems exposed to intense electromagnetic pulses [30].

In the present work, we have restricted our discussion of the negative energy contribution to the second-order calculations of the total and energy-differential decay rates. Even stronger effects due to the Dirac's negative continuum can be expected, however, for the angular and polarization correlations between emitted photons. Theoretical investigation of these correlations which requires also detailed analysis of interference terms between the various (two-photon) multipole combinations is currently underway and will be published soon.

FIG. 1 :

 1 FIG.1:(Color online) Energy-differential transition rates for the 2E1 2s 1/2 → 1s 1/2 two-photon decay of hydrogen and hydrogen-like ions. Relativistic calculations have been carried out by performing intermediate-state summation over complete Dirac's spectrum (solid line) as well as by restricting this summation to the positive-(dashed line) and negativeenergy (dotted line) states only.

E1E2 = 5 .

 5 822 10 -5 (αZ) 8 ,

TABLE I :

 I Total rates (in s -1 ) for the several multipole combinations of 2s 1/2 → 1s 1/2 two-photon decay. Relativistic calculations have been performed within the velocity and length gauges and by carrying out intermediate-state summation over the complete Dirac's spectrum (Wt) as well as over the positive-(W+) and negative-energy (W-) solutions only.

TABLE II :

 II Total rates (in s -1 ) for the several multipole combinations of 2p 1/2 → 1s 1/2 two-photon decay.

			Z=1		Z=54	Z=92
			length	velocity	length	velocity	length	velocity
		W+	4.1934 (-05)	3.2256 (-05)	3.0381 (+09)	2.2422 (+09)	2.1280 (+11)	1.4126 (+11)
	E1M1	W-	1.9355 (-05)	9.6773 (-06)	1.5701 (+09)	7.7417 (+08)	1.3745 (+11)	6.5902 (+10)
		Wt	9.6767 (-06)	9.6767 (-06)	6.3731 (+08)	6.3731 (+08)	3.8633 (+10)	3.8633 (+10)
		W+	3.6716 (-05)	1.2339 (-06)	2.6077 (+09)	8.6699 (+07)	1.7827 (+11)	6.3367 (+09)
	E1E2	W-	4.5159 (-05)	9.6769 (-06)	3.1980 (+09)	6.7698 (+08)	2.1653 (+11)	4.4598 (+10)
		Wt	6.6117 (-06)	6.6117 (-06)	4.2942 (+08)	4.2942 (+08)	2.3584 (+10)	2.3584 (+10)

Acknowledgements A.S. acknowledges support from the Helmholtz Gemeinschaft and GSI under the project VH-NG-421. This research was supported in part by FCT Project No. POCTI/0303/2003 (Portugal), financed by the European Community Fund FEDER, by the French-Portuguese collaboration (PESSOA Program, Contract No. 441.00), and by the Acções Integradas Luso-Francesas (Contract No. F-11/09) and Luso-Alemãs (Contract No. A-19/09). Laboratoire Kastler Brossel is " Unité Mixte de Recherche du CNRS, de l' ENS et de l' UPMC No. 8552." P.A. acknowledges the support of the FCT, under Contract No. SFRH/BD/37404/2007. This work was also partially supported by Helmholtz Alliance HA216/EMMI.


Emmanuel Munch

Mobility is the link allowing the spatial and temporal coordination of daily activities. Based on the precepts set out in the Italian Turco Law of 2000, urban time policies seek to coordinate, reconcile and harmonise the operating hours of urban transport with all the other activities that make up urban life. According to this philosophy, for instance, the arrival time of trains at the station should be coordinated with bus schedules, which in turn are coordinated with the hours of hospital nurses, children's day-care centres, public services and shops so that the various activities can be performed one after the other.

Since the early 2000s, fragmented social rhythms have been a phenomenon with a profound effect on the lifestyles of Western populations. From the point of view of social temporalities, devices, individuals, and institutions seem to function increasingly autonomously and according to an increasingly "personalised" rhythm.

We are witnessing a phenomenon of widespread desynchronisation of daily and weekly temporalities: shops open Sundays and nights, atypical working hours, shorter lunch breaks, longer daily travel time. The synchronisation of social rhythms which segmented daily life in the industrial era is now dislocated in favour of increasingly dispersed travel timing.

However, when we look at these situations from a territorial perspective, the findings are much more ambiguous than the average statistics and leading narratives would suggest.

For example, in many cities in France, Switzerland, and the United States, the increased proportion of employees able to choose their working hours -i.e. autonomy in working hours-comes alongside an increase in the synchronisation of arrivals at work during the morning rush hour.

So we are far from finished with the phenomenon of rush hour.

URBAN TIME POLICIES AND MOBILITY

On the road to "Sustimability" Since their inception, urban time policies have addressed the ambivalent impacts of empowered travel rhythms. On the one hand, they have sought to adapt the mobility offer to temporally evolved and diversified lifestyles: night buses, on-demand and intermodal transport… Strasbourg's "Revolution of Mobility", which can be found in this publication, offers an example of such diversification.

On the other hand, faced with the persistence of peak times, time offices are designing policies to manage transport demand. These take the form, among other things, of local policies to relieve peak-time transport congestion by staggering working hours. In Europe, time offices¹ were the first local authorities to successfully institute staggered working hours, mainly in universities. To our knowledge, the first such initiative took place in Poitiers in 2001. But without a doubt, the most impactful and most documented is a University of Rennes-led initiative from 2014².

Beyond these issues relating to the re-synchronisation of travel rhythms, now well mastered by time policies, I would like to point out a new issue for the temporal planning of urban spaces. It is today a blind spot in urban time policies, especially when we are interested in mobility.

PREFACE TO THE AGENDA

Transport and (re-)synchronised urban rhythms From "faster, cheaper" to "slower, closer" Often, when an urban time office analyses the temporal dimension of mobilities, it is done, implicitly, with a view to 'saving time' and maximising opportunities for interactions and exchanges between individuals, goods, and capital. Time offices' work fits into the usual framework of local action geared towards acceleration and economic growth. In this sense, someti-¹ Time offices are an institutional formulation for the Departments in charge of coordinating urban times. This designation can be found in Italy, where they first appear due to the "Turco Law" of the 2000s, and in France, since the "Aubry II" Act. ² Munch, E. (2014). "Could harmonised working times spell an end to the rush hour?". Métropolitiques.eu, 5. Available at: https://metropolitics.org/IMG/pdf/met-munch-en.pdf autonomy in working hours-comes alongside an increase in the synchronisation of arrivals at work during the morning rush hour.

So we are far from finished with the phenomenon of rush hour.

times in spite of themselves, time offices follow the precepts of the functional city. When I surveyed French, Spanish and Italian cities³, heads of time offices frequently depicted the same tendency: "the community's vocation is to absorb ever more flows".

On the upside, by enabling individuals to carry out activities previously hindered by time constraints, accelerated travel flows make it possible to diversify lifestyles. In this way, acceleration could also be conducive to a broadening of life horizons.

The downside, however, is that the effects of accelerated, and particularly motorised, mobility can be questioned in terms of road deaths and urban sprawl. Observed since the first suburban motorway constructions in the 1960s, sprawl takes people away from centres of activity rather than bringing them closer, and ultimately requires them to consume ever more fossil fuels or electricity to get around. Moreover, it is now clear that transport acceleration has a direct and intense impact on the ecological crisis we face in the 21st century. Transport speed, distance travelled, GDP growth, and CO2 consumption are historically closely linked.

Is it not time to move gradually from a model of always "faster, cheaper" to "slower, closer" to live better? The interest of this line of thought in urban spaces appears validated by the aspirations of their populations.

In 2015, an international survey⁴ on city dwellers' aspirations regarding their future lifestyles revealed that, irrespective of respondents' origin, aspirations to slow down the pace of life and return to close relationships were systematically ranked highest. More recent surveys in France confirm this⁵, and awareness-raising drawing on the Covid-19 pandemic would surely reinforce it. What form might mobility take to match these desires? Living at a less frantic pace with close relationships necessarily requires us to rethink our relationship with speed, productivism, and territorial organisation.

These readjustments can already be seen in the practices of certain catego-PREFACE TO THE AGENDA ⁴ Descarrega, B., Moati, P., 2016. « Modes de vie et mobilité. Une approche par les aspirations. Phase quantitative », Rapportde recherche, Forum Vies Mobiles, ObsoCo, Paris. Available at https://forumviesmobiles.org/recherches/3240/aspirations-liees-la-mobilite-et-auxmodes-de-vie-enquete-internationale ries of people, for whom, in 2023, shopping at local stores is likely more important than going to a distant shopping centre. In the same vein, others may now value going on holiday close to home, or a staycation. While we must be cautious about the reasons of this phenomenon, the latest national survey on household travel in France (2019) attests to this finding. It shows an increase in daily travel time, in parallel with a stagnation, or slight decrease, in distances travelled locally. For the first time in the history of national surveys, we are witnessing a slowdown in the average French person's travel speed.

In the field of mobility policies, it must be noted that local politicians have been involved, in some cases for many years, in organising mobility with a view to easing urban travel, particularly in town centres (switching to walking, cycling, boating, limiting car speeds, etc.).

PREFACE TO THE AGENDA

Slow mobility for desired sustainability: the quest for conviviality

Nevertheless, we lack a global planning model that clearly places the emphasis on preserving forms of slowness in urban metropolises, where intense paces of life are sometimes too pervasive.

The invitation, then, is to collectively define the rhythmic thresholds beyond which temporally intense activities become counterproductive for quality of life and the environment, both for individuals and the community. We are reminded of the seminal work of Ivan Illich (1974)⁶, for whom any travel exceeding 25 km/h wastes time for the community as a whole and therefore for the individual on average.

Where are the rhythmic thresholds of technical progress and economic growth, beyond which "acceleration" becomes too harmful for society? How can we define and evaluate sustainable urban rhythms based on maximum travel speed and limited accessibility? Because of their historical weight, urban time policies have an important role in creating a unified framework to regulate the mobility times in the city. Embracing sustainable objectives means putting quality before quantity and shifting priority from material to temporal prosperity. It requires transfor-Emmanuel Munch is a chrono-urbanist and socio-economist of transport. His work focuses on the problem of rush hours, the acceleration of life rhythms and the socio-ecological bifurcation.

ming our breathless functional cities into convivial ones, making clear room for a holistic vision of well-being and health in the city. This would necessarily put the spotlight on children, the elderly, and people with disabilities who are often sidelined in urban life because they are too slow to fit into the ever-accelerating course.

Finally, this new framework for urban time policies means reconciling the political with the individual. It means bringing together collective 'constraints' and individual aspirations by making slower urban mobilities a project for the common good, desirable for both the individual and the collective.

⁶ Illich, I. 1974. Energy and Equity. Harper and Row, New York.


Amplified and pluri-disciplinary approaches in need

Conclusion

• Not only to re-establish certain historical facts about the relationship between these two traditions • but more generally because a better undertanding of the coexistence and exchanges between different traditions may help shed light upon the cultural plurality of Chosŏn society, a plurality that served to stimulate its social and cultural dynamism "A rose by any other name would smell as sweet" Shakespeare

•

  Personal and social evolution -Observation of person's whole life -Observation of sociocultural changes upon long term