Guillaume Jacques

David Bradley

Katia Chirkova

Henriette Daudey

Randy Lapolla

Cédric Patin

Alexis Michaud

Thomas Pellard

Melanie Viljoen

Elizabeth Zeitoun

Tonal alternations in the Pumi verbal system 1

Keywords: Pumi, Tonal alternation, Directional prefixes

This paper presents two hitherto unnoticed sets of alternations in the tonal systems of Northern Pumi. First, in some verbs, rising tone in the citation form alternates with falling tone in the perfective. Second, the verb "to go", when combined with other verbs, presents tonal alternations which are not found elsewhere in the verbal system. A sample text is added in the appendix to illustrate the tonal phenomena discussed in the article.

Introduction

Like most Qiangic languages (apart from Northern Qiang and Japhug Rgyalrong), Pumi dialects 2 have tonal contrasts. On monosyllables, either two tone categories [START_REF] Shaozun | Pumiyu fangyan yanjiu[END_REF]:21, Matisoff 1997) or three [START_REF] Ding | The Pitch-accent system of Niuwozi Prinmi[END_REF][START_REF] Ding | Prinmi[END_REF], Lu 2001: 97) are found depending on the dialect.

Earlier work on Pumi [START_REF] Shaozun | Pumiyu jianzhi[END_REF]:15-20, Fu 1998:18-20, Lu 2001) analyzes the tonal system of this language on the model of Chinese, transcribing all syllables as if they had an inherent tone. They do, however, mention some cases of sandhi and interaction between stems and affixes (Fu 1998:28-9, 88-9). [START_REF] Matisoff | Dayang Pumi phonology and adumbrations of comparative Qiangic[END_REF] is the first article to use Africanist transcription instead of the Chao Yuen-ren system for transcribing Pumi tones, a system which is better suited for a language typologically closer to the African type than to the Chinese one [START_REF] Hyman | Kuki-Thaadow: an African tone system in Southeast Asia[END_REF][START_REF] Evans | African Tone in the Sinosphere[END_REF]. Although the focus of this article was mainly on comparative linguistics, he nevertheless offers a valuable description of tonal sandhi and tonal alternation phenomena at the word level. [START_REF] Ding | The Pitch-accent system of Niuwozi Prinmi[END_REF][START_REF] Ding | A typological study of tonal systems of Japanese and Prinmi: Towards a definition of pitch-accent languages[END_REF][START_REF] Ding | The use of perception test in studying the tonal system of Prinmi dialects: a speaker-centered approach to descriptive linguistics[END_REF] analyzes the Pumi tonal system as a Pitch-accent system 3 and compares it to various Japanese dialects. Since his analysis is considerably more developed than those of his predecessors, it is worthwhile to describe it in more detail.

In the variety described by Ding, as in other dialects described by [START_REF] Shaozun | Pumiyu fangyan yanjiu[END_REF], monosyllables have three surface realizations: high (H), falling (F) and rising (R). He analyzes these categories as underlying H, HL and LH respectively. This analysis is justified by the patterns of tone spreading when an enclitic is added after a monosyllabic noun, as shown in Table 1:

Table 1 : Analysis of Pumi tones in monosyllables according to Ding (2006:13) The fact that the same tonal categories are realized as contour tones in citation form but appear as series of High and Low when a clitic is attached shows that an analysis in terms of syllable tone is not tenable for Pumi dialects, and that all contour tones of this language must be analyzed as combinations of high and low tones.

In polysyllabic words, the range of attested possibilities for a word of n syllables is much lower than 3ⁿ, the number of categories which would be expected if each syllable had an underlying tone. Ding (2003:590 and2007:14) describes the following categories (where F, H, L and R represent the surface realizations of the tones, respectively falling, high, low and rising): only four patterns on disyllabic words (instead of an expected 3 2 = 9), six patterns on trisyllabic words (instead of 27) and seven on quadrisyllabic words. 2 Pumi (a.k.a. Prinmi) is a cluster of at least two distinct languages spoken across Yunnan and Sichuan, mainly in Lanping county, Ninglang county (Yunnan), Muli county and Jiulong county (Sichuan). The total number of speakers is above 30,000. In Muli county, it is the main language among ethnic Tibetans, and speakers of Shixing, Kami Tibetan and other languages also learn it as a second language (Chirkova 2009:3). 3 Since the category of 'Pitch Accent' is difficult to define rigorously cross-linguistically [START_REF] Hyman | How (not) to do Phonological Typology: The Case of Pitch-Accent[END_REF], we will avoid this term in the present paper.

The categories C and D on bisyllabic words on the one hand and E and F on trisyllabic words are distinct only when a clitic or a compound word is added: with C and E categories, the high tone does not spread onto the next syllable, while it does with D and F categories. For instance, tõ L pu H 'donkey' and tʃɨ L mɐ̃H 'dog hair' pattern differently when a subsequent syllable is added: with tõ L pu H , which belongs to category D, the subsequent syllable has a high surface tone (for instance tõ L pu H k'ʉ H "donkey head"), whereas with tʃɨ L mɐ̃H, which belongs to category C, it has a low surface tone (no example provided in Ding's papers).

According to Ding's analysis, two underlying properties are enough to yield the observed categories: the place of the high tone in the prosodic domain, and whether this high tone can spread on the next syllable. The number in the first column indicates which syllable in the prosodic domain is the locus of the high tone, and in the second column, + indicates that the high tone can spread,indicates that it cannot, and X that either is possible in free variation. The underlying high tone spreads rightwards, so that syllables situated on its left are realized with a low tone. If the number of syllables in the prosodic domain is inferior to the place of the high tone (for instance, category C with a monosyllable, the high tone should be placed on the second syllable, but only one syllable is present in the prosodic domain), a rising tone results.

This analysis elegantly accounts for the data of the Xinyingpan variety; however, it will be shown below that some tonal alternations found in other Pumi dialects are difficult to explain using this model.

Since the regular tonal alternation patterns in nominal compounding observed in these Pumi dialects do not seem to exhibit exactly the same behavior as in the variety described by Ding, I will lay out basic facts about the tonal system of Shuiluo Pumi and describe tonal alternations in nouns (which are simpler) before discussing the problematic phenomena observed in the verbal system in the following sections.

Segmental inventory.

The dialects investigated in this study are the Mudiqing (木底箐, Ninglang, Lijiang, Yunnan) and Shuiluo (水洛, Muli, Liangshan Yi autonomous prefecture, Sichuan) varieties, which belong to the Northern Pumi language (see Figure 1). 4 The tonal alternation patterns in both dialects are almost identical. These dialects are related to the Xinyingpan dialect, with which they are mutually intelligible according to my informants' opinion. The data was collected during two field trips in July-August 2008 and February-March 2009 respectively. The present paper will mostly use data from the Shuiluo dialect. Since the segmental phonology of Shuiluo Pumi differs from all previously described varieties (including the eight dialects in Lu 2001, among which it is closest to Taoba), I will first present a basic account of the vocalic and consonantal inventory of this language. The basic syllabic structure is C(r)(G)V(G), where C, V and G stand for consonant, vowel and glide respectively. This syllabic structure is simpler than the one found in Lanping of Xinyingpan, as most consonant clusters have disappeared. The only attested clusters are br-and pr-, and br-freely alternates with dʐw-in some words.

category spread 1 2 3 4 A 1 - F HL HLL HLLL B 1 + H HH HHL HHLL C 2 - R LH LHL LHLL D 2 + LH LHH LHHL E 3 - LR LLH LLHL F 3 + LLH LLHH G 4 X LLR LLLH
The list of attested rhymes is the following: The phonetic realization of these rhymes is not entirely straightforward: /wə/ is realized [ʉ],5 the glide -w-is realized [-ɥ-] after a coronal consonant and before a front vowel. /ə/ has three allophones:

[ɿ] after dental fricatives and affricates (as in tsə́"to milk" [tsɿ 55 ]), [ʅ] after retroflex fricatives and affricates (as in tʂə́"gall bladder", [tʂʅ 55 ]) and as the labial trill [ʙ] after bilabial stops (as in pə́"to dig" [pʙ 55 ]). As in other attested forms of Pumi, no final consonants are found.

There are 40 consonant phonemes in Shuiluo Pumi: 3. The tone system of Shuiluo Pumi Like the Niuwozi Pumi dialect studied by Ding, Shuiluo Pumi has three surface tonal categories on monosyllables, that can be described as high, falling and rising in isolation. This basic system on monosyllables is also remarkably similar to that of Shixing [START_REF] Chirkova | Approaching the Prosodic system of Shixing[END_REF]. For convenience, I use the Africanist tone marks (like Matisoff 1997), as summarized in the following table. Low pitch (which is either the effect of an underlying low tone or the realization of an absence of tone) will be transcribed with a grave accent.

Table 5 : Tonal categories on monosyllables in Shuiluo Pumi.

For the time being, I follow Ding's analysis. However, we will see that the analysis of the rising tone is not straightforward. On disyllabic nouns, four distinct patterns are attested: In compound words of two syllables, the following rule can be used to predict the tonal pattern of the resulting disyllable (rows indicate the first element and columns the second element of the compound): Only two patterns are found: high-high and low-high. The tonal pattern of the disyllable only depends on the first syllable. A similar, though not identical rule is found in the neighboring Shixing language (Chirkova and Michaud 2009:549). However, this productive rule has a few exceptions, such as bípɜ̀'sand' from bí 'sand' and pɜ̌'flour, powder' (the expected form would be *bípɜ). 7 These exceptions are best considered as older compounds, to which the regular rule is not applicable. The composition rules presented above do not only apply to compound nouns, but sometimes also to object+verb compounds, though the exact environment where these occur still deserves further research.

Regular alternations in the verbal system

The Pumi verb is very rich in affixes, most of which are monosyllabic. The longest suffix -mədərə is three syllables long, made from the nominalizing suffix -mə (itself a grammaticalized form of the noun mə̂'man'), the copula də̂and the progressive suffix -rə. It is an evidential marker used in stories to report facts that the speaker has not witnessed himself. Although the etymology of this trisyllabic suffix is still synchronically transparent, it is fully grammaticalized, and the lexical tone of the verb spreads over the three syllables.8 dó 'back' rə̂'skin' sěi 'blood' gwi ̃'bear' gwi ̃dó gwi ̃rə́gwi ̃séi kwɜ̂'cow' kwɜ̀dó kwɜ̀rə́kwɜ̀séi ɣwi ̃'horse' ɣwi ̃dó ɣwi ̃rə́ɣwi ̃séi

H F R H H H H H H H F L H L H L H R L H L H L H
Suffixes are toneless, and the intrinsic tone of the verbal stems spreads onto it. 9 The following table shows tonal spreading patterns with a monosyllabic suffix (the progressive -rə) and with the trisyllabic -mədərə: 10 The tonal spreading patterns of the first two tonal categories are straightforward. They can be represented in the following way:

The high tone is a single H tone underlyingly. It associates to the stem and spreads rightwards to all following suffixes, which are toneless.

The falling tone is underlying composed of two tonal elements H and L. When suffixes are added to the stem, the high tone remains in situ, while the low tone is reassociated to the following suffix. The same analysis cannot be proposed for the rising tone if it is analyzed as LH, as the following result would be expected:

We would obtain *tà-mədəŕə, an impossible form. In order to explain the attested form tà-mədərə, (1)

H H → σ + σ σ σ σ + σ σ σ (2) HL HL H L → = → σ + σ σ σ σ + σ σ σ σ + σ σ σ (3) LH LH L H → = → σ + σ σ σ σ + σ σ σ σ + σ σ σ
9 Apart from the suffixes, directional prefixes and many discourse particles are toneless; in the appended story, toneless affixes and particles have no tone marks in the underlying form. The idea that underlying low-tone syllables must be distinguished from toneless syllables in Pumi is found in Greif (2010).

10 Many examples of verbs with this suffix can be found in the story in the appendix.

A rule against low-tone words (or rather the obligatory presence of a high tone in phonological words as in other Qiangic languages, see Evans 2008:484) would then change the last syllable of the word into a surface rising tone by insertion of a postlexical high tone.

A similar rule has been described for Yongning Na (Michaud 2008:186). 11 In this analysis, lowrising words such as rɜdʑə̌'alcohol' cited in section 1 must be analyzed with a low tone pattern L as the underlying form. The only problem with this theory is the form of verbs with monosyllabic suffixes: the suffix is realized with a high tone, not a rising one. If the analysis of rising tone monosyllabic verbs as having an underlying low tone were correct, we would expect the form *tà-rəǐ nstead of attested tà-rə:

I do not have any elegant explanation to propose to solve this problem. Two analyses are possible. First, one could analyze rising tone monosyllables as underlying LH, and add a rule according to which a high tone following a low tone is deleted if it has to spread on more than one syllable. However, this analysis is problematic as it increases the complexity of the basic form and requires the addition of an arbitrary rule.

Second, if one maintains the analysis of the monosyllables as low tones, one must account for the high tone by a special rule. A possible solution is to suppose that low tone fails to spread on onesyllable suffixes, and that the postlexical high tone (automatically added in words without underlying high tone) associates with a toneless syllable. Therefore, no surface rising tone occurs.12 I will adopt the second option in this paper.

Disyllabic verbs have a much simpler pattern: the suffixes are always realized with low pitch, except with low-rising disyllabic verbs (that is, underlying low tone verbs), as the last syllable of the verb undergoes the same change as for monosyllabic verbs.

(4) L L → σ + σ σ σ σ + σ σ σ (5) L (H) σ + σ σ σ (6) L L (H) → σ + σ σ + σ (7) L L (H) → σ + σ σ + σ
Table 10 : Tonal spreading with disyllabic verbs.

Most prefixes are toneless, and are realized with a surface low tone, 14 the default realization for toneless syllables. Among the directional prefixes, only the 'up' prefix has a high tone: Table 11 : Pumi directional prefixes.

The tone of this prefix regularly spreads rightwards onto the verb stem and the verb's own tone is disassociated (the same phenomenon has been noted in Fu 1998:28-9 for a different dialect). However, the high tone of the prefix can only cross one morpheme boundary: the suffix will remain toneless, and ends up with a surface low pitch.

This phenomenon can be observed from the following examples (the perfective forms have either the egophoric 15 -sã suffix or the non-egophoric one -ɕi).

Irregular alternations

In the previous section, I have presented a basic account of the regular tonal alternations observed in the verbal system of Shuiluo Pumi. In the present section, I will turn to two sets of alternations that cannot be accounted for synchronically by a set of simple rules and are lexically determined.

Rising tone monosyllabic verbs

We observe that several rising tone monosyllabic verbs undergo tonal alternation in the perfective form, when a directional prefix is added. The directional prefix has low surface tone, the verb stem high surface tone and the suffix low surface tone (the first example suǎ in the table is a nonalternating verb):

Table 13 : Examples of alternating rising tone verbs.

The resulting perfective forms with LHL tone pattern look like perfective forms of falling tone verbs. Unlike the alternations observed with the prefix tə-, which can be explained as cases of assimilation, there is no evidence that the tone change observed here is caused by the directional prefixes. Some disyllabic verbs with low-rising tone pattern, such as kətsěi 'small', also present this alternation: the perfective form of this stative verb is realized as nɜ-kətséi-ɕì 'it became small'.

These alternations are observed both in running speech (recorded stories, as the one in the appendix) and isolated sentences. Therefore, it cannot be an effect of intonation.

Alternating verbs should be synchronically analyzed as a fourth lexical tonal class, distinct from both falling tone and rising tone, with a distinct underlying representation. However, an analysis in terms of H and L tones cannot easily account for the data presented above. Since falling and rising tone verbs have been respectively analyzed as H, HL and L in the previous section, we are left with only two possibilities: either to consider alternating verbs as underlying LH, or to suppose the existence of a contrast between L and underlyingly toneless verbs. 16 Proposing a category LH has one obvious advantage: we could unify the four tonal patterns found on monosyllables with those of disyllables shown in Table 4. However, the discussion in ʑi ̃də-ʑi ̃-ɕì *də-ʑi ̃-ɕí It became lighter 16 One anonymous reviewer suggested that the alternation could be explained by assuming that the alternating verbs were LH underlyingly, with a leftward spread of the L tone on the prefix. He pointed out that such cases of leftward spread are attested in Pumi dialects (see for instance Ding 2001:76 concerning the ɐ-prefix with the kinship terms). However, one would still have to explain why LH spreads leftwards with some verbs, and rightwards with others, and why in HL verbs the H tone does not spread leftwards. Besides, allowing leftward spread in the verbal system would unduly increase the complexity of the analysis. We adopt here the idea that only rightward spread is allowed.

section 3 has already shown that LH would account poorly for the spreading rules observed when various suffixes are added to the verb stem.

The other alternative, which I will adopt in this article, is to hypothesize that of the alternating and non-alternating rising tone verbs, one category is toneless and receives a default low tone like affixes or sentence particles, and the second one has an underlying L tone. According to the analysis in section 3, we would expect a prefixed monosyllabic verb with underlying Low tone to have the surface form LLH:

Non-alternating verbs are therefore verbs with underlying L tone. With toneless verbs, on the other hand, the insertion of a high tone is not accounted for by the rules presented in section 3. The surface tonal pattern LHL looks as if the HL had been inserted on the toneless verbal stem (a simple H tone would not do, as it would result in an LHH pattern):

This process could be interpreted either as a kind of polarity.17 However, a simpler solution might be possible. As mentioned in section 3, a postlexical H tone must be inserted in all-low phonological words. Since toneless syllables are normally realized with a surface low tone, the high tone insertion would have to appear in a word whose syllables are all underlyingly toneless. 18 To explain the LHL tone pattern, we only need to add two specifications to the postlexical High tone insertion rule:

(i) In verbal forms where all syllables are underlyingly toneless, the postlexical H tone is inserted on last syllable of the morpheme that contains the second syllable, regardless whether it is an affix or a verbal stem. 19(ii) Postlexical H tones cannot spread.

We need an additional rule to explain why toneless and Low tone verb stems share the same tonal patterns in non-prefixed verb forms: (iii) In non-prefixed verb forms (when the verb stem is the first syllable of the phonological word), an L tone is inserted on toneless verbal stems. Without rule (iii), we would expect the bare stem form to be realized with a high tone (13), and the form suffixed with -mədərə to be realized as LLLH ( 14).

(9) L L (H) → σ + σ + σ σ + σ + σ (10) HL → σ + σ + σ σ + σ + σ (11) (H) → σ + σ + σ σ + σ + σ (13) (H) → σ σ (14) (H)
With rule (iii), the non-prefixed verbal forms of toneless stems are virtually identical to L tone stems as in ( 15) and ( 16):

These alternations have never been described for Pumi dialects with three tones, but similar phenomena have been mentioned by [START_REF] Matisoff | Dayang Pumi phonology and adumbrations of comparative Qiangic[END_REF] and Fu (1998), who both worked on two-tone dialects. Matisoff (1997:209) presents the following data: Several etyma are found in both Matisoff's and Fu's lists: "rot" and "collapse", but since their fieldwork research was based on entirely different wordlists, relatively few comparable etyma are found. By contrast, I collected vocabulary on Shuiluo and Mudiqing Pumi using the same basic list, and as a result find many common etyma between these two varieties, allowing a better evaluation of how stable the tonal alternation is across Pumi dialects. In my data, I find the following list of examples belonging to the fourth category: The tonal correspondences are even more complex if Fu's (1998) data are taken into consideration. Although a few items in her list correspond to alternating verbs in Shuiluo or Mudiqing ('give', 'hear' = 'understand', 'smoke' = 'drink'), in other cases they correspond to falling tone verbs, for instance tsʰue 24 'give back', whose cognates are tsʰwɛ̂in Shuiluo and tsʰwî in Mudiqing.

These puzzling exceptions can be accounted for by supposing a process of analogical leveling at the expense of the alternating verbs: alternating verbs, which are much less numerous than either rising tone or falling tone verbs, tend to become falling tone verbs (or in one case, a rising tone verb) by generalizing the surface tone found on the prefixed verbal forms. In this hypothesis, whenever a verb has tonal alternation in any Pumi dialect, regardless of the fact that other dialects may have falling tone in the corresponding etymon, the verb in question ought to be reconstructed in proto-Pumi as belonging to the fourth tonal category, that is alternating verbs.

We find no cases of a correspondence between a high tone verb and an alternating verb across dialects, 20 and this is expected since alternation verbs never behave like high-tone verbs either in prefixed and non-prefixed form; an analogical change from high-tone to alternating or viceversa would not be motivated phonetically.

In summary, for the Northern Pumi dialects under investigation, I propose that we must distinguish four, not three tonal classes on monosyllabic verbs: For disyllabic verb stems, we find five tonal categories: No direct evidence of a toneless category has been found in either monosyllabic or disyllabic nouns, which seem to lack this kind of tonal alternation at least in Shuiluo Pumi, but traces of it could potentially be found in seemingly irregular nominal compounds (such as those mentioned in section 2). No definite historical explanation for this phenomenon can be given until the issue of Pumi tonogenesis has been properly addressed. This topic lies beyond the scope of this paper, as it would require a systematic comparison with other Qiangic languages.

The verb 'to go'

When a monosyllabic verb is suffixed with the non-egophoric form ʂə̃of the verb 'to go', this verb undergoes unusual tone changes:

Table 19 : Tonal alternations with the verb "to go".

When the basic verb is high tone, the tone of ʂə̃is assimilated, and the high tone propagates again onto the suffix, quite unlike the high tone of the directional tə-, which, as we have seen, cannot propagate across more than one morpheme boundary.

When the verb is falling tone, ʂə̃becomes low-toned, and so does the following prefix. This form is expected from the rules proposed in section 2:

Finally, when the preceding verb is in the rising tone, we observe an entirely unexpected alternation: ʂə̃becomes high tone, and this high tone spreads onto the following suffix. No matter which underlying representation one chooses for ʂə, one cannot explain this phenomenon. Even assuming that ʂə̃were a toneless stem for instance, one would expect L-L-LLR *ʑə-ʂə-mədərə, not L-H-HHH.

As with the preceding alternation, I have no theoretical analysis to propose that would solve this problem. In present-day Shuiluo Pumi, this verb is highly irregular: it has four unpredictable forms ɕə-(basic form) ɕə-(egophoric with directional prefixes), ʂə̃(non-egophoric) and ɕot (imperative). 21 Whether these unpredictable forms are the result of preservation of ancient alternations or secondary idiosyncrasies must be left for future investigations.

Conclusion

This article has documented and analyzed several tonal alternations in the verbal system of Shuiluo Pumi. It demonstrated that four tonal classes must be distinguished for monosyllabic verbs, and five for disyllabic verbs. The issue of tonal alternations in noun phrases (including tri-and quadricsyllabic nominal compounds), and of the effect of intonation on tone realization is left for further research. An important issue would be to determine whether evidence for the contrast between toneless vs. L tonal classes found in verbal morphology can be brought to light in the domain of nominal morphology.

Properly understanding the alternation patterns of Pumi is not only of interest for historical linguistics and theoretical phonology, it also has a considerable practical importance for transcribing Pumi stories correctly, as is illustrated in the excerpt from a Pumi traditional story presented in the Appendix.

Appendix: A sample story in Shuiluo Pumi 22

The glosses generally follow the Leipzig Glossing Rules, except for the following: Clitics, indicated by = rather than -, differ from genuine suffixes in that only the rising tone can spread on them, the high tone cannot spread rightwards on clitics in this variety of Pumi, unlike in the variety described by [START_REF] Ding | Prinmi[END_REF].

The story of the one who made divination with a pig head.

Once upon a time, there was a poor beggar.

He was begging around. One day, he arrived at a place where there was a household of yak herders.

There was a shepherdess milking yaks.

He had plenty of time, and watched her milking yaks.

As she was milking yaks, one half of her turquoise pendant fell down.

It fell on the grass.

She did not notice it when it fell on the grass. At that moment, the yak defecated on her turquoise.

The beggar saw it.

After a while, as she finished milking her yak, the shepherdess took the milk to her tent.

The beggar went inside her house to beg. She searched for everywhere in the house, in the tent, and on the pasture, but could not find it.

The turquoise was lost.

The shepherdess was about to cry.

"My turquoise was so beautiful, and it is lost. He pretended to thrust it towards many yak dungs.

After a while, he thrust it towards the yak dung where the turquoise was.

He said: "Isn't it your turquoise here on the ground?"

The people of this house were very glad.

Figure 1 :

 1 Figure 1: The location of Lanping, Shuiluo, Mudiqing and Xinyingpan Pumi dialects.

a

  solution would be to analyze the surface rising tone as an underlying low tone. The form tà-mədərə̌would be as such: rə́ʑwi ̃-mədərə ʑwi ̃-mədəŕə́o bserve ʑɛ̂ʑɛ-rə ʑɛ-rə̀ʑɛ-mədərə ʑɛ-mədərə̀s tay tǎ tǎ-rə tà-rə́tǎ-mədərə tà-mədərə̌a rrive

  Basic form Perfective form (1sg or 3sg) expected meaning swǎ kʰə-swà-sãI counted tɕǐ nɜ-tɕí-sã*nɜ-tɕì-sãI poured lěi nɜ-léi-sã*nɜ-lèi-sãI ploughed ɕʰãtʰɜ-ɕʰa-sã*tʰɜ-ɕʰa-sãI scooped gǐ nɜ-gí-sã*nɜ-gì-sãI wore (clothes) těi nɜ-téi-sã*nɜ-tèi-sãI wore (a hat)

  with the verb 'to go' expected meaning dzãdza-ʂə-mədəŕə́*dza-ʂə-mədərə̀w ent to sit tô tó-ʂə-mədərə̀id. went to see ʑə̌ʑə-ʂə-mədəŕə́*ʑə-ʂə-mədərə̀w ent to sleep

  it outside the tent, on the pasture.

Table 2 : Realization of the tonal categories in Xinyingpan Pumi (Ding 2007:14).

 2

	Citation form Realization with an enclitic	meaning
	ʃi F	ʃi H ge L H.L	louse
	ʃi H	ʃi H ge H H.H	hundred
	ʃi R	ʃi L ge H L.H	new

Table 3 : The rhymes of Shuiluo Pumi.

 3

Table 4 : The initial consonants of Shuiluo Pumi.

 4

		i	e	ɛ	ə	ɜ	a	u	o	ĩ	ɛ̃ə̃ã	õ
	V	i		ɛ	ə	ɜ	a	u	o	ĩ	ɛ̃ə̃ã	õ
	Vu				-əu				-ou
	Vi		-ei
	-jV		-je	-jɛ	-jə -jɜ -ja -ju -jo		-jã -jõ
	-wV -ui		-wɛ -wə -wɜ -wa			-wĩ -wɛ̃-wã
			-wei		-jəu
	p	t			k
	pʰ	tʰ			kʰ
	b	d			g
		ts	tɕ	tʂ

Table 6 : Tonal categories on disyllables in Shuiluo Pumi.

 6 We find here the same four surface categories presented by Ding (C and D can only be distinguished if a subsequent syllable is added). For trisyllables and quadrisyllables, there are not enough words in my lexicon to provide a comprehensive account of the possible surface forms. Contour tonal patterns are restricted to the last syllable of a phonological word, following a general cross-linguistic

		tsʰ tɕʰ	tʂʰ
		dz dʑ	dʐ
		s	ɕ	ʂ	x	h
		sʰ	ɕʰ	ʂʰ
		z	ʑ	ʐ	ɣ	ɦ
	m	n			ŋ
	m̥	n̥			ŋ ̥
		l		r
		l ̥
		example meaning Underlying analysis
	v̂njɛ̂eye	A: 1, -spread
	v́njɛ́red	B: 1, +spread
	v̌njɛ̌black	C: 2, -spread
			example meaning	Ding's analysis
	vv̀HL rwəbõw illow		A: 1, -spread
	vv́HH rwɜḿi ̃mare		B: 1, +spread
	vv́LH rwɜmi ̃female yak C/D: 2
	vv̌LR rɜdʑə̌a lcohol		E: 3, -spread

Table 7 : Examples of regular tonal change in composition.

 7 The tonal patterns observed in Table7can be described as follows:

Table 8 : Regular tonal change in composition.

 8

Table 9 : Tonal spreading with monosyllabic verbs.

 9

Table 12 : Spread of the high tone from the directional prefix.

 12

	Basic form tone meaning	realization
	sɛní	LH	to hear	sɛǹí mədərəm
	əɕɛ́H	tosearch	məɕwɛ́mədərə̀1 3
	súdju	HL	to think	súdjù mədərək
	ətsěi	L	tobesmall kətsèi mədərəď
	irection	Shuiluo Pumi
	up	tə-
	down	nɜ-
	out	kʰə-
	in	hɜ-
	cislocative	də-
	translocative	tʰɜ-
	(8)	H (H,HL,L)	H (H,HL,L)
			=	→

σ + σ + σ σ + σ + σ basic form meaning perfective expected tɕú sour tə-tɕú-ɕì *tə-tɕú-ɕí ɕʰî wake up tə-ɕʰí-sã*tə-ɕʰí-sag jǒ lift tə-gjó-sã*tə-gjò-sa1

Table 14 : Alternating verbs in Matisoff (1997).

 14

	Matisoff (1997:209-10) only briefly comments on this phenomenon: "In what appears to be the
	majority of cases, a verb under the low tone acquires the high tone after a directional prefix in

Dayang [Pumi]

". In the light of the Shuiluo Pumi data, it seems that Matisoff's examples actually conflate two distinct phenomena: the systematic high-tone assimilation of the 'up' prefix tə-on the one hand (these cases have been shaded in grey in the table) and the genuine tonal alternation occurring with all other directional prefixes (only seven examples in all).

Fu (1998:88-9) notices 21 verbs in her corpus which undergo tone change when a directional

Table 15 : Alternating verbs in Fu (1998).

 15

Table 16 : Alternating verbs in Shuiluo and Mudiqing Pumi.

 16 From this list, we observe that not all verbs are alternating in both dialects, even when we find cognates. Five exceptions are found (out of 13 verbs with cognates in both dialects). In Mudiqing there are two falling tone verbs dʐẽ'break' and tsʰə̂'fat' corresponding to alternating tone verbs in Shuiluo, and one rising tone verb dʑẽ'light' without alternation (past kʰə-dʑe-ɕí) whereas its

	Meaning			Basic form	Meaning	Basic form
	烂 rotten			bʑi	换 change	ʒdʒu
	天阴 dark			ɖəuŋ	还 give back	tsʰue
	(棍子)断 break			ɖən	垮 collapse	bie
	脱落 to get off			gɑ	撕 tear off	ʈʰe
	脱臼 dislocate			ɬi	卖 sell	ʃtʃi
	剩余 remain			xɑ	弄垮 cause to collapse	pʰie
	听见 hear			noŋ	拿 take	ɹuɑ
	灭 destroy			go	送 give	ʃtʃyn
	采 pick up			qʰɑ	跑 run	bʑin
	抽烟 smoke (= drink)		tʰin	扔 throw	vbɑ
	连接 link			ʃtʃʰuɑ
		Shuiluo Mudiqing
	scoop	ɕʰãtʂʰal
	eak	ʐə̌dʑəř
	ide	dzěi	dzɛǐ
	plant	lěi			lɛǐ
	wear (hat)	těi			tɛǐ
	pour	tɕǐ
	wear	gǐ			gǔ
	remember	tʂi	drink
		tʰi ̃tʰief
	lee	tsʰi	break
		dʐi ̃dʐel
	ight	ʑi ̃dʑẽ(no alternation)
	fat	tsʰwə̌tsʰəf
	lee				pʰef
	ly				bier
	un				dʐɛ̃

cognate in Shuiluo has alternation. Similarly, in Shuiluo we find two falling tone verbs ɕî 'sell' and ʂû 'hide' corresponding to alternating verbs in Mudiqing.

Table 17 : Monosyllabic verbs.

 17

Table 18 : Disyllabic verbs.

 18

	bear on the back		kǔ
	do		dzǔ
	sell	ɕî	kǐ
	hide	ʂû	tʂǔ
	understand		noũ
	se	ʐəʐá	ʐɑǒ
	btain		rǐ
	give		tɕyet

20

As was pointed out by an anonymous reviewers.

  He thrust it in all directions, towards the inside of the tent, towards the outside, on the altar, He thrust it on the corner of the door.After a while, as he knew that the turquoise was under the yak dung,

	38	think-NAR waste tsɜkʰuə̌tjɜ-jã́ǎ-kʰi ̃̌bo PFV-become-N.EGO say =then cry-about_to-NAR dʐã-kei tɜbó gùtṍ=wõ̀pəŕó-mədərəǹ
	32 51	What a waste !" she said, and was about to cry. pá rû hálṍti dei ŋû" tsɜkʰuə̌tjɜ-jã́à-kʰi ̃́bò dʐã-kéi ow hearth-LOC burn-NAR hjǔ=wõ təra, gṍ təra,	tɕ<w>ə-mədərə tə-bjie tɜ-lwəȟ
		pá pig_head one-CL rû Then, he cooked it on the hearth. hálṍtì dèi Q-give TOP fine-OPT ŋû" jù=wṍ tərà, gṍ tərà,	tɕ<w>ə-mədərəď tə-bjé tɜ-lwəť
	21 27	məɕ<w>ɛ-mədərə "tʰɜʑɛ" hmi ̃̌=mə-bje ivination divinate a_little PART know say<N.EGO:VOL>-NAR tɕə̃=wõ hiǔ=wõ ljãlja "njə̂kʰətí tʰô ɕǐ mǎ-a-wõ" if you give me the head of a pig, it will be just fine. ent=LOC these:GE inside these:GE here one-thrust
	45	məɕ<w>ɛ-mədərə̀tɕə̃=wõ̀hiù=wṍ ljãljãš tʰɜʑɛ̂hmi ̃̀=mə-bjè njə̂kʰətí tʰô ɕǐ má-à-wṍ" He said "I know divination a little." tɜtɕǐ kʰətí swǎ-rõ "...." tɕ<w>ə-mədərə N N
	earch< N.EGO:VOL >-NAR məɕ<w>ɛ-mədərə məɕ<w>ɛ-mədərəs njə̂pá rû T earch< N.EGO:VOL >-NAR She told the beggar: "Don't you have some trick, house-LOC tent-LOC all nã̀ú-ní má-à-ŋú please beg=NMLZ-DAT you some method have NEG-Q-modal njə̂pá rû nã ú-ní mǎ-a-ŋû" 33 nãdjɛ́tʰɜʑɛ̂njə̂=njɛ 28 pá ti rû-ku nãdjɛ́tʰɜʑɛ̂njə̂=njɛ̀pá tì rú-kù then please you =AGEN divination a_little divinate-IMP Bring me a pig's head, 39 tsɜkʰwə̌tjɜ-jã́jɜ" tɕ<w>ə-mədərə tsɜkʰwə̌tjɜ-jã́jɜ" tɕ<w>ə-mədərəp ig_head one-CL bring say<N.EGO:VOL>-NAR tɜtɕǐ kʰətí swà-rṍ "...." tɕ<w>ə-mədərəǎ lso some recite-DUR "...." únṍ tɜ-lwə̌tʰɜḱʰwə tɜ-lwəú say<N.EGO:VOL>-NAR nṍ tɜ-lwə̌tʰɜḱʰwə̀tɜ-lwəǒ He also recited something like "....", utside one-thrust altar one-thrust
	40	you divination divinate etc "In this case, please make a divination, this_way tsɜkʰwə̌tjɜ-jã́kʰǔ" tɕ<w>ə-mədərə NEG-Q-know
	22 46 34 41 52	kí-wõ kí-wõ̀məɕ<w>ɛ-mədərə̀mi ̃́-tʂʰwɜ-mədəŕəṕ məɕ<w>ɛ-mədərə mi ̃́-tʂʰwɜ-mədərə asture-LOC search<N.EGO:VOL>-NAR NEG.PFV-find-NAR wouldn't you know how to practice divination, by any chance ?" tsɜkʰwə̌tjɜ-jã́kʰǔ" tɕ<w>ə-mədərəp swǎ-ji bo tsã-bje tjǒ la mǎ-bõ-mədərə tɕ<w>ə-mədərə tɕ<w>ə-mədərəš ay<N.EGO:VOL>-NAR we will provide you with anything you need, njə̂mjɛ̂kʰǔ la ǎ-bə kʰi ̃̌-ʂa bo njə̂mjɛ̂kʰǔ là à-bə́kʰi ̃̀-ʂá bò you what need all my-house give-FUT TOP I need a pig's head." ig_head one-CL need say<N.EGO:VOL>-NAR "áláɕi" tɕ<w>ə-mədərə tʂʰã́=kʰjɛ tsɜkʰwəá swà-jí bò tsã-bjé tjǒ là mà-bṍ-mədərər ecite-NMLZ:MOD ǔ=po kjã̌zə-bje tɜ-lwə, úní =kʰjɛ TOP he-DAT any all NEG-be_there-NAR ù=pó kjã̌zə-bjè tɜ-lwə, úní =kʰjɛì it was not at all what had to be recited. t=under door corner-DAT one-thrust this_way =time
	23 29 ǔ=kʰjɛ tɜbó ɜbó ow ù=kʰjɛ́=bò rǐ rǐ turquoise PFV-disappear niɜ-mî niɜ-mî =bo utí hmi ̃̌=mə dɜ̂dɜt dɜ̂dɜn ASSER T ASSER T =njɛ "pá rû ùtí hmi ̃̀=mə́=njɛ̀"pá rú this_time =TOP this beg=NMLZ =AGEN T divination divinate 35 njə̂pá rû =kʰjɛ mjɛ̂ti kʰǔ wõ, njə̂pá rû =kʰjɛ̀mjɛ̂tì kʰǔ wõ, you divination divinate =when what a_little need modal láɕì tɕ<w>ə-mədərə̌tʂʰã́=kʰjɛ̀tsɜkʰwəť hank_you say<N.EGO:VOL>-NAR quick =when pig_head tjɜ-jã́tsã-bje tʰɜ-kʰ<w>i ̃̌-mədərə tjɜ-jã́tsã-bjé tʰɜ-kʰ<w>i ̃̀-mədərəǒ He was reciting nonsense, deceiving them. 47 xaswaməswá pə̂tʂɜ̂pâ-mədərə xàswàməswá lwə-rə̃ʂə̃-mədərə pə̂tʂɜ̂pá-mədərər ecite_nonsense do lwə-rə̃́ʂə̃-mədərət trick do:N.EGO:VOL-NAR hrust-PROG go:N.EGO:VOL-NAR
	24 25 30 tsã́pədʑí tɜbó utí tɜbó ùtí ow this yak rwɜ-rwɜ-â rǐ â rǐ turquoise this_way ɬǐ-ɬí-herd girl hõbǎ hṍbà ú-ní tsʰwí=mə xwéi-tɕɛtɕɛ-mədərə xwéi-tɕɛtɕɛ-mədərəǹ cry-about_to-NAR =də =kʰjɛ ú-ní tsʰwí=mə̀=də̀=kʰjɛÌ good=NMLZ =CO P =when tɜbó njɜ-mî-ʂət ɜbó njɜ-mí-ʂə̃ǹ Then the beggar said "I know divination a little." Just before, he had seen the turquoise going under the yak dung. hálṍti =bo ŋû" tɕ<w>ə-mədərə hálṍtì =bò ŋû" tɕ<w>ə-mədərəǎ _little =TOP know say<N.EGO:VOL>-NAR rwɜ-xêi =pɜ̌=po rǐ ǔ =po tsã́pədʑí rwɜ-xéi =pɜ́=pò rǐ ù=pó he just_before yak-dung =CL =under turquoise it=under whatever you need when you divinate, whatever you use to divinate." pá r û-sə̃mjɛ̂kʰǔ wõ'' pá rú-sə̃̀mjɛ̂kʰǔ wõ'' divination divinate-NMLZ:OBL what need modal tɕ<w>ə-mədərə They said "Thank you", and immediately gave him a pig's head. ne-CL he-DAT PFV-give<N.EGO:VOL>-NAR 42 tɜbó tsã́=njɛ tsɜkʰuə-bje sɜkɜradʐwi ̃́tɜ-tsá tɜbó tsã́=njɛ̀tsɜkʰuə-bjé 48 tɜbó tsɜkʰwə̌tʰjɛ̌hmi ̃̌-rə tsʰɜ̂bo tɜbó 53 tɜ-tʂʰɛ̃̌bo tsã́ti rwɜ-xêi =pɜ̌=po rǐ ǔ=po tsɜkʰwə̌tʰjɛ̌hmi ̃̀-rə́tsʰɜ̂bò now pig_head almost cooked-PROG tɜ-tʂʰɛ̃̌bò tsã́tì rwɜ-xéi =pɜ́=pò rǐ ù=pó time TOP one-moment TO he this yak-dung =C =under turquoise under_it As the pig head was about to be cooked, P L sɜkɜràdʐwi ̃́tɜ-tsá tɕ<w>ə-mədərəš ay<N.EGO:VOL>-NAR 36 utí hmi ̃̌-mə ti tsá tʂéi-mədərə now he =AGEN T pig_head-DAT wood_stick one-CL hɜ-tsî təbo hɜ-tsî 49 tsɜkʰwə̌kɜradʐwi ̃́=kʰu djã-mədərə, utí kʰə-tʂǒ təbo tsɜkʰwə̌kɜràdʐwi ̃́=kʰù djã-mədərə, ùtí kʰə-tʂǒ kuî tsã́tã-mədərə tsã́xá kǔ-mədərə təbò pig_head stick =on be_there-NAR this PFV-take_out kuî tsã́tã-mədəŕə́tsã́xá kù-mədərəb then e_there he see-NAR he know know-NAR he took out the pig head from the stick where it was. təbò nɜ-ʂə̃̂tsã́tã-mədərə nɜ-ʂə̃̂tsã́tã-mədəŕəṔ FV:DOWN-go.N.EGO he see-NAR The beggar wanted to eat meat. ùtí hmi ̃̀-mə́tì tsá tʂéi-mədəŕət his beg-NMLZ this meat want_to_eat-NAR Then, he inserted a wood stick into the pig head, PFV-insert then 43 tsɜkʰuə̌hɜ-ʂʰwěi təbo ǔ-kʰu hɜ-tsî təbo 50 swǎ-rə̃pə̂nã bo kõ̂lwəš wà-rə̃́pə̂nã̀bò 54 hjǔ kjã=wõ kʰə-lwə̌kʰə-ʑî, kõ̂lwəř ecite-SIMULT do etc hjǔ kjã=wṍ kʰə-lwə̌kʰə-ʑî, TOP valley thrust tent door=LOC PFV:OUT-thrust PFV:OUT-come
	26 31 44 37	ow tɕʰápá tɕʰápá He was not worried at all. PFV-disappear-N.EGO nɜ-dʐã-ʂə" nɜ-dʐã-ʂə̃̀tɕə̌=təbò xwéi-tɕɛtɕɛ-mədərə̀ t ɕə̌=təbo xwéi-tɕɛtɕɛ-mədərə tsã́gjã=wõ xwɜ̂=po dʑwá tʂǎ-mədərə tsã́gjã=wṍ xwɜ̂=pò dʑwá tʂà-mədərəȟ e heart=LOC heart =under smooth very-NAR tɜbó gutṍ=wõ pəŕó-mədərə While he was reciting, he thrust (the stick) around in a disorderly way. "To do divination, I don't need anything else, pá rû-sə̃bo tɜnjɛ̌bo kʰətí kʰǔ mǎ-wõp á rú-sə̃̀bò tɜǹjɛ̌bò kʰətí kʰǔ mà-wṍ divination divinate-NMLZ:OBL TOP other TOP some need NEG-modal tsɜkʰuə̌hɜ-ʂʰwěi təbò ù-kʰú hɜ-tsî təbò pig_head PFV-pierce then this-above PFV-insert jã́lwə̌pâ-mədərə then jã́lwə̌pá-mədərəm he pierced the pig head, and inserted (the wood stick) into it. ountain_crest thrust do:N.EGO:VOL-NAR

  same one COP-EVD These people believed it was true and said: "Thank you so much, how extraordinary, you are an uncommon man, thank you !" They said that he knew divination, that the poor beggar knew divination.

		áláɕi"	tɕ<w>ə-mədərə
	55	rwɜ-xêi=rə áláɕì tɕ<w>ə-mədərəť =to rwɜ-xéi-rə́=tò hank_you say<N.EGO:VOL>-NAR la lwə-rõtɕə là lwə-rõtɕəý
			ak-dung-PL	=on all thrust-pretend
	56 60		tɜ-tʂʰɛ̃̌=kʰjɛ bo tɜbó tsã́pá rû tɜ-tʂʰɛ̃̌=kʰjɛ̀bò tɜbó tsã́pá rû one-moment =when TOP now he divination divinate know-PROG say utí rwɜ-xêi ŋû-rə tɕə, ùtí rwɜ-xéi ŋú-rə̀tɕə, this yak-dung	rǐ rǐ turquoise
		kwî-sə̃ti mədʐɜ́pá r û kwí-sə̃̀tì ədʐɜ́pá r û be_there-NMLZ:OBL this =on one-thrust =to tɜ-lwə̌=njɛ ŋû-rə tɕəm bo ŋú-rə̀tɕəp =tò tɜ-lwə̌=njɛ̀bò =AGENT TOP oor_man divination divinate know-PROG say
	57		njə-bǎ		rǐ	təpǒ	â-mə-kwî-rə	ǎ "
	61		njə-bǎ you-house:GEN ljowá-bje thɜ-lóz<w>ə-mədərə rǐ təpǒ turquoise on_the_ground Q-NEG-be_there-PROG á-mə-kwí-rə̀ǎ" INT ljòwá-bjé thɜ-lóz<w>ə-mədərəà
		tɕ<w>ə-mədərə ll-DAT
		tɕ<w>ə-mədərəš	They told everyone.
		ay<N.EGO:VOL>-NAR
	58		mɜkɜ-bə̌njɜ-gjǔ	=njɛ	bo
			mɜkɜ-bə́njɜ-gjǔ	=njɛ̀bò
			house	PFV-glad =AGEN	TOP
						T
	59		mɜkɜ-bə̌dʑi ̃́njɜ-ʑû,	njɜ-gjǔ	=njɛ	"áláɕi,
			mɜkɜ-bə́dʑi ̃́njɜ-ʑû,	njɜ-gjǔ	=njɛ̀áláɕì
			house	true PFV-believe	PFV-glad	=AGEN	thank_you
							T
			tɕósõtɕʰí	niə̂=bo mə̂mǎ-dʐâ	tǐ	də-ɕi,
			tɕ ósotɕʰí	niə̂=bò mə̂mà-dʐâ	tǐ	də-ɕì
			my_respects you =TO

P man NEG-

PFV-tell<N.EGO:VOL>-NAR

My Mudiqing Pumi consultant Mr. Cao was born in 1962 and lives in Yongning township in a community of speakers of the Na language (see[START_REF] Michaud | Phonemic and tonal analysis of Yongning Na[END_REF] for more information about this language). He speaks Chinese, Na and Pumi. My Shuiluo Pumi consultant Ngag-dbang was a retired cadre. He lived in Qiaowa (the seat of the government in Muli county), was married to a speaker of Taoba Pumi (a very close variety) and spoke the language on a daily basis, though Chinese was naturally his main language.

This analysis is preferred because the non-egophoric infix <w> added to a stem in -ə yields the surface vowel [ʉ]: ʂə' to jump' > [ʂʉ] ʂwə.

These aspirated fricatives come from {s+aspirated affricate} clusters.

The word M tsɑ.bɤ 'flour, powder' in Yongning Na, whose second syllable is possibly related to pɜ̌(either as a cognate or as a loanword) also presents irregular tonal patterns(Michaud and Latami 2009:14).

If it were not fully grammaticalized, one would not expect the tone spreading patterns described in this section. The copula, when it appears with nouns, does not show such peculiar behaviour: the high tone of monosyllabic nouns does not spread on the copula. In isolation, the copula followed by the progressive is realized də-rə̀(or rather də-rə), but in fluent speech it is often realized as low tone on both syllables as an effect of the intonation.

'[…] the L+L sequence cannot surface as such, due to a general prohibition against all-L phonological words (and a fortiori all-L utterances) in Na. The all-L sequence is avoided by post-lexical addition of a final M tone, which modifies L+L to L+LM[…]'

This idea was suggested by Alexis Michaud, but I remain responsible for any error in the analysis set out in this section.

This form has the non-egophoric volitive infix <w>.14 Unlike Shixing where the tone of the verb spreads leftwards onto the prefix(Chirkova and Michaud 2009:550-551).15 I adopt here Tournadre's (2008) terminology rather than the terms 'conjunct/disjunct'.

Presence of polarity in Pumi would be one further argument in favor ofEvans' (2008:484-5) idea that Qiangic languages, including Pumi, present tonal features thought to be characteristic of African tone languages.

Note that in this hypothesis, toneless verb stems do not follow the same tone rules as toneless affixes or sentence particles.

In disyllabic verbal stems such as nɜ ̀-kə ̀tséi-ɕì "it became small' mentioned above, the high tone is placed on the second syllable.

Most of these forms are attested in the story in the appendix.

A partially similar story is found in the Tibetan ro-sgrung cycle(Robin and Klu-rgyal 2005:177-193). This story was recorded from Ngag-dbang in Muli in February 2009. We present here only one fourth of the entire story, for want of space. The recording will be made available on the LACITO Archive (http://lacito.vjf.cnrs.fr/archivage/) eventually.

This sentence shows an example of the alternation treated in section 5.2. Note also the irregular form hɜ-ɕə̌of the verb "to go".

my Shuiluo Pumi consultant Ngag-dbang [ŋawṍ] 昂 翁 who unfortunately passed away in late 2009, a few months after my field trip. This paper was completed was funded by the ANR (Agence Nationale de la Recherche) project PASQi (What defines Qiang-ness: Towards a phylogenetic assessment of the Southern Qiangic languages of Muli 07-JCJC-0063).


Guillaume Jacques Crlao

REVIEW OF ZHANG JICHUAN 张济川 2009 藏语词 族研究 ZANGYU CIZU YANJIU (A STUDY ON WORD FAMILIES IN TIBETAN), BEIJING: SHEHUI KEXUE WENXIAN CHUBANSHE

Unlike Classical Indo-European languages like Greek and Sanskrit, whose morphology is well understood in spite of their intricate complexity, the morphology of Sino-Tibetan languages is still poorly studied, even for a major literary language such as Tibetan. This sad state of affair is due in part to the fact that the languages whose historical phonology is best studied, Chinese and Lolo-Burmese, have poorly preserved proto-Sino-Tibetan morphology. 2 The book under review is the first systematic attempt at an exhaustive analysis of Tibetan derivational and flexional morphology. Building on previous work, in particular Li's (1933) seminal article on Tibetan historical phonology and Coblin's (1976) work on verbs, Zhang Jichuan provides a detailed survey of Tibetan morphology.

The author is particularly well placed to write such a book. He is not only an able linguist, well known for his descriptive work on Tshangla [START_REF] Zhang | Cangluo Menbayu Jianzhi (A Brief Description of the Cangluo Menba Language)[END_REF]) and for his study of Baima Tibetan historical phonology [START_REF] Jichuan | Particularités Phonétiques du Baima[END_REF], but he also speaks Lhasa Tibetan beautifully. Unlike many other mainland Chinese scholars of his generation, is also fluent in English and familiar with western Tibetology.

The book is divided into three main chapters. The first chapter (word families, 词族, pp. 3-187) includes a list of all the word families that Zhang Jichuan could find in the known lexicon of Classical Tibetan. The second chapter (morphology 形态, pp. 189-158) presents a general account of derivational and flexional morphology, including affixes, consonant and vowel alternation. The third chapter (phonology 声韵, pp. 159-360) discusses both the reconstruction of proto-Tibetan phonology and various phonetic changes from Old Tibetan to modern Tibetan languages.

WORD FAMILIES

This chapter, which constitutes the bulk of the book (more than half of the page numbers), includes a clearly structured classification of the entire Tibetan lexicon. Related verbs roots and their derived nouns are classified according to their root initial consonant. With a little practice, it is easy for someone familiar with Classical Tibetan to locate a particular word in this index of roots.

Here is the example of a typical entry (p.132). I keep here Zhang's transcription, where homorganic prenasalization is translitterated as v-(see Hill 2005a on the Tibetan letter in question, sometimes incorrectly called va-chung); the English glosses are slightly adapted, not direct translations of Zhang's Chinese glosses:

(1) vbye-bye 'to be opened' 分开 compounds, however, do not appear by themselves in the main subentries. No detailed comment is provided in this section to explain the alternations and affixes observed here; explanations only appear in the chapters on morphology and phonology.

The entry starts with a pair of intransitive / transitive verbs. In this particular case, the intransitive verb is the anticausative of the transitive verb, derived by voicing the initial consonant of the root. The transitive verb vbye, past phye itself presents voicing alternation.

Most of the nouns are derived from the voiced stems. bye-s 'place far from home', d-bye-s 'breath, size' and d-bye-n 'dissention' appear with different prefixes and suffixes. -s and -n are well-known nominalization suffixes, discussed by Zhang pp. 251-4. The prefix dhowever is not accounted for in the book. It is most likely to be related the nominalizations prefixes kɯ-or tɯ-of Rgyalrong languages (see Jacques 2010a): Tibetan preinitials d-and g-, as Li (1933) has shown, are in complementary distribution and therefore d-bye-n points either to proto-Tibetan *t-byen or *k-byen (preinitials are conventionally represented as voiced stops in Tibetan orthography, but there is no voicing contrast in this position, so that we reconstruct them as unvoiced stops). The spelling g.yen represents a dialectal variant of dbyen, as g.y-and dby-merge as j-with high tone in various Tibetan languages from Lhasa to Cone county in Gansu.

The noun phyed 'half' is derived from the unvoiced stem with the nominalizing -d suffix, also studied by Zhang (p. 252).

It is unclear whether the words vbyer 'to disperse' and g.ya 'bolt' which Zhang included in this word-family are really related. There is no known verbal -r suffix, so that any etymological relationship between vbyer and vbyed would be difficult to account for. Note however the intriguing Cone Tibetan form ndʑer², past ɕʰer¹ for the verb 'to open' (see Jacques ms.), a paradigm that points to proto-Cone present *vbyer, past *phyer, which presents similarities with both that of vbyed and that of vbyer.

This example shows how much information can be gained by analyzing each of the word families proposed by Zhang. As seen above, some of Zhang's comparison are not entirely convincing. Other doubtful examples include for instance p.8 mkhal 'kidney' with khal 'charge (on an animal)', p.24 deb 'book', sdeb 'to put together' (deb is an abbreviation of deb.ther, a noun ultimately of Greek origin, while sdeb is a genuine Tibetan verb), p.100. smyan.ga 'marriage', nye 'near', gnyen 'kin' (gnyen is indeed related to nye, but smyan.ga is unrelated, it is rather potentially cognate with Chinese 婚 *hmˁən; Chinese *ə regularly corresponds to Tibetan /a/). These minor issues have however no impact on the overall value of Zhang's book: on the contrary, it is fortunate that Zhang tried to gather as many words as possible in his word families, leaving it to the reader to accept or not his etymologies. Criticizing an etymology is always easier than proposing a new one.

In some cases, Zhang's word families miss obvious cognates. For instance, p.16 in the word family of bdag 'to belong to' (bdag.po 'possessor') he does not cite bdog 'to be there, to belong to' (bdog.po 'possessions'); in the entry of vdzin, bzung 'to seize' p.76 he does not mention the derived noun gzungs *k-dzuŋ-s 'dhāraṇī-', 3 an important example because it proves that the nominalization by a combination of *k-and *-s was still productive when Buddhist terminology was introduced in Tibet; note that the verb vdzin regularly translates Sanskrit √dhṛ-, the root from which dhāraṇī-is derived, as in grags.vdzin.ma = Yaśodharā (Mahāvyutpatti #1070, [START_REF] Ishihama | A New Critical Edition of the Mahāvyutpatti. Sanskrit-Tibetan-Mongolian Dictionary of Buddhist Terminology[END_REF].

MORPHOLOGY

The chapter on morphology (the shortest of the three) includes a list of attested alternations and affixes. For each affix, an extensive (though not entirely exhaustive) list of examples is given. These lists are of great value for the study of ancient non-productive Tibetan morphology. Although most of the affixes presented in Zhang's book have been discussed in previous work, existing grammars of classical Tibetan typically only cite a handful of examples for each formative.

Zhang also presents a survey of Tibetan verb conjugation, which follows Li (1933) and [START_REF] Coblin | Notes on Tibetan Verbal Morphology[END_REF] but does not provide reconstructions of paradigms in proto-Tibetan by internal reconstruction as Coblin did. In this section, it is unfortunate that the paradigms discussed by Zhang are taken from dictionaries, not from Old Tibetan texts. The recent articles by [START_REF] Hahn | Blags und Verwandtes (Miscellanea etymologica tibetica, VI)[END_REF] and Hill (2005b) have brought to light two types of conjugation that differ considerably between Oldand Classical Tibetan: the lh-stems (such as klog 'to read', CT past bklags, OT past blags or phlags, imperative lhogs) and the r-stems (vbri 'write', OT present vdri, past bris, imperative ris, see also Jacques 2010b on this topic).

  vbyed, phye, dbye, phyes 'to open, to separate' 使分开、打开 Title 3 vbyer, byer 'to disperse' 逃散、蔓延 vbyes 'separated' 分清 byes 'place far from home' 异乡、远方 dbyes 'breath, size' 间隔、宽度 dbyen 'dissention' 离间 phyed 'half' 一分为二、分开 g.yen 'dissention' 离间 g.ya.shing 'bolt (door)' 门闩 ʔa.shing 'bolt (door)' 门闩、插销 Some of these examples are illustrated by sentences and compounds;

This review has been written during my stay at the Research Centre for Linguistic Typology, LaTrobe University, Melbourne. I wish to thank Randy LaPolla for his invitation.

Using Chinese or Lolo-Burmese to reconstruct proto-Sino-Tibetan morphology is an exercise comparable to reconstructing Indo-European morphology based on Haitian Creole.

The sound change *dz > z in Tibetan word initially and after g-and b-is regular, see[START_REF] Coblin | Notes on Tibetan Verbal Morphology[END_REF].

Author

Author 6

PHONOLOGY

This chapter mainly discusses the historical phonology of Tibetan from Old Tibetan (as represented by the classical orthography) to various modern Tibetan languages. It also discusses proto-Tibetan internal reconstruction based on the work of Li and Coblin as well as various articles by Simon (for instance [START_REF] Simon | Loss of l or r in Tibetan Initial Consonantal Clusters[END_REF].

This section contains a wealth of valuable data and ideas, even if some recent references are inevitably missing. For instance, in the discussion on the initials v-and w-, no mention is made of the recent work by [START_REF] Coblin | On certain functions of 'a-chung in early Tibetan transcriptional texts[END_REF][START_REF] Coblin | Two Notes on Táng-time Tibetan Transcriptions of Chinese[END_REF] and [START_REF] Hill | Tibetan vwa 'fox' and the sound change Tibeto-Burman *wa -> Old Tibetan o[END_REF] see also [START_REF] Hill | Tibetan <ḥ-> as a plain initial and its place in Old Tibetan Phonology[END_REF], an article which Zhang could not have taken into account in his work anyway since his book was published the same year).

CONCLUSION

Zhang Jichuan's book is an invaluable reference that will become indispensable for any further research on Tibetan morphology. Along with [START_REF] Hill | A Lexicon of Tibetan Verb Stems as Reported by the Grammatical Tradition[END_REF] recent dictionary of verbs, it marks the end of the traditional approach to the study of Tibetan morphology based on dictionaries. This book constitutes the blueprints on the basis of which a full etymological dictionary of Tibetan and Bodic languages can be built on the model of [START_REF] Rix | Lexikon der Indogermanischen Verben[END_REF]. Now that traditional dictionaries have been thoroughly analyzed, further work on Tibetan historical morphology will have to concentrate on three topics. First, detailed descriptions of individual dialects, on the model of [START_REF] Bielmeier | Das Märchen von Prinzen Čobzaṅ[END_REF] and [START_REF] Sun | Phonological profile of Zhongu: A new Tibetan dialect of Northern Sichuan[END_REF]. Second, careful studies of Tibetan grammar based on Old Tibetan texts such as Hill (2005b). Third, comparison with other Sino-Tibetan languages.

Author


Michel Gondran
email: michel.gondran@polytechnique.org

Alexandre Gondran

The two limits of the Schrödinger equation in the semi-classical approximation: discerned and non-discerned particles in classical mechanics

We study, in the semi-classical approximation, the convergence of the quantum density and the quantum action, solutions to the Madelung equations, when the Planck constant h tends to 0. We find two different solutions which depend to the initial density . In the first case where the initial quantum density is a classical density ρ 0 (x), the quantum density and the quantum action converge to a classical action and a classical density which satisfy the statistical Hamilton-Jacobi equations.

These are the equations of a set of classical particles whose initial positions are known only by the density ρ 0 (x). In the second case where initial density converges to a Dirac density, the density converges to the Dirac function which corresponds to a unique classical trajectory. Therefore we introduce into classical mechanics non-discerned particles (case 1), which explain the Gibbs paradox, and discerned particles (case 2). Finally we deduce a quantum mechanics interpretation which depends on the initial conditions (preparation), the Broglie-Bohm interpretation in the first case and the Schrödinger interpretation in the second case.

I. INTRODUCTION

Quantum mechanics came to exist in 1900 with the introduction of the Planck constant h = 6, 62 × 10 -34 m 2 kg /s. Its value is very small and close to 0 in the units of classical mechanics. But, = h/2π =1 in the atomic units system (ua) used to represent the atomic scale. What is the connection between quantum mechanics and classical mechanics? We will particulary study the semi-classical approximation when the Planck constant tends towards 0. This approach usually presents two major difficulties:

-a mathematical difficulty to study the convergence of these equations, -a conceptual difficulty: the particles are considered to be indistinguishable in quantum mechanics and distinguishable in classical mechanics.

We will see how these two problems can be overcome.

Let us consider the wave function solution to the Schrödinger equation Ψ(x, t):

iℏ ∂Ψ ∂t = - ℏ 2 2m △Ψ + V (x, t)Ψ (1)
Ψ(x, 0) = Ψ 0 (x).

(

With the variable change Ψ(x, t) = ρ (x, t) exp(i S (x,t) ), the quantum density ρ (x, t) and the quantum action S (x, t) depend on the parameter . The Schrödinger equation can be decomposed into Madelung equations 1 (1926) which correspond to:

∂S (x, t) ∂t + 1 2m (∇S (x, t)) 2 + V (x, t) - 2 2m △ ρ (x, t) ρ (x, t) = 0 (3) ∂ρ (x, t) ∂t + ∇ • (ρ (x, t) ∇S (x, t) m ) = 0 ∀(x, t) (4)
with initial conditions ρ (x, 0) = ρ 0 (x) and S (x, 0) = S 0 (x).

(

) 5
The Madelung equations are mathematically equivalent to the Schrödinger equation if the functions ρ 0 (x) and S 0 (x) exist and are smooth. It will be physically the case if Ψ 0 (x) is a wave packet.

In this paper we study the convergence of the density ρ (x, t) and the action S (x, t), solutions to the Madelung equations when tends to 0. Its convergence is subtle and remains a difficult problem. We find, according to the assumptions on the initial probability density ρ 0 (x), two very different cases of convergence due to a different preparation of the particles.

Definition 1 -A quantum system is prepared in the statistical semi-classical case if its wave function satisfies the two following conditions -its initial probability density ρ 0 (x) and its initial action S 0 (x) are regular functions ρ 0 (x) and S 0 (x) not depending on .

-its interaction with the potential field V (x, t) can be described classically. The simplest case corresponds to particles in a vacuum with only geometric constraints.

It is the case of a set of particles that are non-interacting and prepared in the same way:

a free particles beam in a linear potential, an electronic or C 60 beam in the Young's slits diffraction, an atomic beam in the Stern and Gerlach experiment.

Definition 2 -A quantum system is prepared in the determinist semi-classical case if its wave function satisfies the two following conditions -its initial probability density ρ 0 (x) converges, when → 0, to a Dirac distribution and its initial action S 0 (x) is a regular function S 0 (x) not depending on .

-its interaction with the potential field V (x) can be described classically.

This situation occurs when the wave packet corresponds to a quasi-classical coherent state, introduced in 1926 by Schrödinger 2 . The field quantum theory and the second quantification are built on these coherent states 3 . The existence for the hydrogen atom of a localized wave packet whose motion is on the classical trajectory (an old dream of Schrödinger's) was predicted in 1994 by Bialynicki-Birula, Kalinski, Eberly, Buchleitner et Delande [START_REF] Bialynicki-Birula | [END_REF][5][START_REF] Buchleitner | Non-dispersive wave packets in periodically driven quantum systems[END_REF] , and discovered recently by Maeda and Gallagher 7 on Rydberg atoms.

The separation of deterministic and statistical semi-classical cases causes a strong reduction of the mathematical difficulties of the convergence study.

In section 2, we show how, in the statistical semi-classical case, the density ρ (x, t) and the action S (x, t), solutions to the Madelung equations, converge, when the Planck constant goes to zero, to a classical density and a classical action which satisfy the statistical Hamilton-Jacobi equations. These are the equations of a set of classical particles whose initial positions are known only by the density ρ 0 (x). Therefore we introduce non-discerned particles into classical mechanics and the Broglie-Bohm interpretation of the statistical semiclassical case.

In section 3, we show how, in the determinist semi-classical case, the density ρ (x, t)

and the action S (x, t), solutions to the Madelung equations, converge, when the Planck constant goes to zero, to an unique classical trajectory and an action which satisfy the determinist Hamilton-Jacobi equations. Therefore we introduce discerned particles into classical mechanics and the Schrödinger interpretation of the determinist semi-classical case:

the wave function is then interpreted as the state of a single particle similar to a soliton.

II. CONVERGENCE IN THE STATISTICAL SEMI-CLASSICAL CASE

In the statistical semi-classical case, we have:

THEOREM 1 For particles in the statistical semi-classical case, the probability density ρ (x, t) and the action S (x, t), solutions to the Madelung equations (3)(4)(5), converge, when → 0, to the classical density ρ(x, t) and the classical action S(x, t), solutions to the statistical Hamilton-Jacobi equations:

∂S (x, t) ∂t + 1 2m (∇S(x, t)) 2 + V (x, t) = 0 (6)
S(x, 0) = S 0 (x).

∂ρ (x, t) ∂t

+ div ρ (x, t) ∇S (x, t) m = 0 ∀ (x, t) (8)
ρ(x, 0) = ρ 0 (x).

We will demonstrate in the case where the wave function Ψ(x, t) at time t is written as a function of the initial wave function Ψ 0 (x) by the Feynman formula:

Ψ(x, t) = F (t, ) exp( i S cl (x, t; x 0 )Ψ 0 (x 0 )dx 0
where F (t, ) is an independent function of x and of x 0 and where S cl (x, t; x 0 ) is the classical action min u(s) t 0 L(x(s), u(s), s)ds, the minimum is taken over all trajectories x(s) with velocity u(s) from x 0 to x between 0 and t. Feynman 8 (p. 58) shows that the general paths integral formula is simplified in this form, especially when the potential V (x, t) is a quadratic function in x.

4

Let us consider the statistical semi-classical case where Ψ 0 (x) = ρ 0 (x) exp(i S 0 (x) ) with ρ 0 (x) and S 0 (x) are non-dependent functions of . The wave function is written

Ψ(x, t) = F (t, ) ρ 0 (x 0 ) exp( i (S 0 (x 0 ) + S cl (x, t; x 0 ))dx 0 .
The theorem of the stationary phase shows that, if tends towards 0, we have

Ψ(x, t) ∼ exp( i min x 0 (S 0 (x 0 ) + S cl (x, t; x 0 )).
that is to say that the quantum action S h (x, t) converges to the function

S(x, t) = min x 0 (S 0 (x 0 ) + S cl (x, t; x 0 )). (10)
However, S(x, t) given by ( 10) is the solution to the Hamilton-Jacobi equation ( 6) with the initial condition (7). This is a consequence of the principle of the least action and a fundamental property of the minplus analysis we have developed 9,10 following Maslov [START_REF] Maslov | Idempotent Analysis[END_REF] .

Moreover, as the quantum density ρ h (x, t) verifies the continuity equation ( 4) of the Madelung equations, we deduce, since S h (x, t) tends towards S(x, t), that ρ h (x, t) converges to the classical density ρ(x, t), which satisfies the continuity equation ( 8) of the statistical Hamilton-Jacobi equations. We obtain both announced convergences. This theorem will have major implications in classical and quantum mechanics. The first one is to provide an interpretation of the classical particles which satisfy the statistical Hamilton-Jacobi equations.

A. Non-discerned particules in classical mechanics

The statistical Hamilton-Jacobi equations correspond to a set of independent classical particles, in a potential field V (x, t), and for which we only know at the initial time the probability density ρ 0 (x) and the velocity v(x)

= ∇S 0 (x,t) m .
Let us consider N particles that satisfy the statistical Hamilton-Jacobi equations. We propose the following definition:

Definition 3 -N indentical particles, prepared in the same way, with the same initial density ρ 0 (x), the same initial action S 0 (x), and evolving in the same potential V (x, t) are called non-discerned.

We refer to these particles as non-discerned and not as indistinguishable because, if their initial positions are known, their trajectories will be known as well. Nevertheless, when one counts them, they will have the same properties as the indistinguishable ones. Thus, if the initial density ρ 0 (x) is given, and one randomly chooses N particles, the N! permutations are strictly equivalent and do not correspond to the same configuration as for indistinguishable particles. This means that if X is the coordinate space of a non-discerned particle, the true configuration space of N non-discerned particles is not X N but rather X N /S N where S N is the symmetric group.

The introduction of these non-discerned particles into classical mechanics solves the conceptual difficulty announced in the introduction; indiscernibility also exists in classical mechanics. These non-discerned particles in classical mechanics also give a simple solution to the Gibbs paradox. This view is not new: it features particularly in Landé In the statistical semi-classical case, the Madelung equations converge to statistical

Hamilton-Jacobi equations. The uncertainity about the position of a quantum particle corresponds in this case to an uncertainity about the position of a classical particle, only whose initial density has been defined. In classical mechanics, this uncertainity is removed by giving the initial position of the particle. It would not be logical not to do the same in quantum mechanics.

We assume that for the statistical semi-classical case, a quantum particle is not well described by its wave function. One needs therefore to add its initial position and it becomes natural to introduce the so-called de Broglie-Bohm trajectories. In this interpretation, its velocity is given by [START_REF] De Broglie | [END_REF][START_REF] Bohm | A suggested interpretation of the quantum theory in terms of "hidden" variables[END_REF] :

v (x, t) = 1 m ∇S (x, t). (11)
We have the classical property: if a system of particles with initial density ρ 0 (x) follows de Broglie-Bohm trajectories defined by the velocity field v (x, t), then the probability density of those particles at time t is equal to ρ (x, t), the square of the wave function. Using this velocity, the Heisenberg inequalities correspond to a dispersion relation position and velocity between the different non-discerned particles. This shows that, in the statistical semi-classical case, the Broglie-Bohm interpretation reproduces the predictions of standard quantum mechanics.

Therefore, when → 0, we deduce that v (x, t) given from equation ( 11) converges to the classical velocity v(x, t) = 1 m ∇S(x, t) and we obtain:

THEOREM 2 For particles in the statistical semi-classical case,when → 0, the de Broglie-Bohm trajectoires converge to the classical ones.

Figure 1 shows a simulation of the Broglie-Bohm's trajectories in the Young slits experiment of Jönsson [START_REF] Jönsson | Elektroneninterferenzen an mehreren künstlich hergestellten Feinspalten[END_REF] where an electron gun emits electrons one by one through a hole with a radius of a few micrometers. We are in a statistical semi-classical case where the electrons, prepared similarly, are represented by the same initial wave function, but not by the same initial position. In the simulation, these initial positions are randomly selected in the initial wave packet. We have represented 100 possible quantum trajectories through one of two slits: we do not show the trajectories of the electron when it is stopped by the first screen. Figure 2 shows the 100 previous trajectories when the Planck constant is divided by 10, 100, 1000 and 10000 respectively. We obtain, when h tends to 0, the convergence of quantum trajectories to classic trajectories.

III. CONVERGENCE IN THE DETERMINIST SEMI-CLASSICAL CASE

The convergence study of the determinist semi-classical case is mathematically very difficult. We only study the example of a coherent state where an explicit calculation is possible. This example will enable to understand the convergence in the determinist semi-classical case and the difference with the statistical semi-classical case.

For the two dimensional harmonic oscillator, V (x) = 1 2 mω 2 x 2 , coherent states are built 18 from the initial wave function Ψ 0 (x) which corresponds to the density and initial action :

ρ 0 (x) = (2πσ 2 ) -1 e - (x-x 0 ) 2 2σ 2 and S 0 (x) = S 0 (x) = mv 0 • x ( 12
)
with σ = 2mω . Here, v 0 and x 0 are still constant vectors and independant from , but σ will tend to 0 as .

For this harmonic oscillator, the density ρ (x, t) and the action S (x, t),solutions to the Madelung equations (3)(4)( 5) with initial conditions (12), are equal to [START_REF] Cohen-Tannoudji | Quantum Mechanics[END_REF] :

ρ (x, t) = 2πσ 2 -1 e - (x-ξ(t)) 2 2σ 2 and S (x, t) = +m dξ(t) dt • x + g(t) -ωt (13)
where ξ(t) is the trajectory of a classical particle evolving in the potential V (x) = 1 2 mω 2 x 2 , with x 0 and v 0 as initial position and velocity and g(t)

= t 0 (-1 2 m( dξ(s) ds ) 2 + 1 2 mω 2 ξ(s) 2 )ds. Because we have 2V (ξ(s)) = m d 2 ξ(s)
ds 2 • ξ(s), it yields the following theorem:

THEOREM 3 -When → 0, the density ρ (x, t) and the action S (x, t) converge to

ρ(x, t) = δ(x -ξ(t)) and S(x, t) = m dξ(t) dt • x + g(t) (14)
where S(x, t) and the trajectory ξ(t) are solutions to the determinist Hamilton-Jacobi equations:

∂S (x, t) ∂t | x=ξ(t) + 1 2m (∇S(x, t)) 2 | x=ξ(t) + V (x)| x=ξ(t) = 0 ∀t ∈ R + (15) dξ(t) dt = ∇S(ξ(t), t) m ∀t ∈ R + (16)
S(x, 0) = mv 0 • x and ξ(0) = x 0 .

Therefore, the kinematic of the wave packet converges to the single harmonic oscillator one described by ξ(t). Because this classical particle is completely defined by its initial conditions

x 0 and v 0 , it can be considered as a discerned particle.

When → 0, the "quantum potential"

Q (x, t) = - 2 2m △ √ ρ √ ρ = ω -1 2 mω 2 (x -ξ(t)) 2 tends to Q(x, t) = -1 2 mω 2 (x -ξ(t)) 2 .
It is then zero on the trajectory (x = ξ(t)). It is then possible to consider, unlike the statistical semi-classical case, that the wave function can be viewed as a single quantum particle. The determinist semi-classical case is in agreement with the Copenhagen interpretation of the wave function which contains all the information on the particle.

A. Interpretation for the determinist semi-classical case

In the determinist semi-classical case, the Broglie-Bohm interpretation is not relevant mathematically unlike the statistical semi-classical case. Other assumptions are possible. A natural interpretation is the one proposed by Schrödinger 2 in 1926 for the coherent states of the harmonic oscillator. In Schrödinger interpretation, the quantum particle in the determinist semi-classical case is a spatially extended particle, represented by a wave packet whose center follows the classical trajectory. In this interpretation, the velocity in each point of the wave function is given by [START_REF] Bohm | The Undivided Universe[END_REF][START_REF] Holland | The quantum Theory of Motion[END_REF][START_REF] Holland | Uniqueness of paths in quantum mechanics[END_REF][START_REF] Gondran | Revisiting the Schrödinger Probability current[END_REF] :

v (x, t) = 1 m ∇S (x, t) + 2m ∇ ln ρ (x, t) × k, ( 18
)
where k is the unit vector parallel to the particle spin vector. This spin current 2m ∇ρ (x, t) × k corresponds to Gordon's current when one changes from Dirac's equation to Pauli's equation and after that to Schrodinger's equation [START_REF] Holland | Uniqueness of paths in quantum mechanics[END_REF] . This current is important because it allows to revisit quantum mechanics at small scales, in particular Compton's wavelength as in the Foldy and Wouthuysen transformation [START_REF] Foldy | [END_REF] . Using this velocity, the Heisenberg inequalities correspond to a dispersion relation position and velocity between the different points of the extended particle.

For the coherent states of the harmonic oscillator in two dimensions, the velocity field ( 18) is equal to:

v (x, t) = v(t) + Ω × (x -ξ(t)) (19)
with Ω = ωk. They behave as extended particles which have the same evolution as spinning particles in two dimensions. But this can not be generalized easily in three dimensions. It seems that it is not possible to consider in three dimensions the particle as a solid in motion. This is the main difficulty in the Schrödinger interpretation: does the particle exist within the wave packet? We think that this reality can only be defined at the scale where the Schrödinger equation is the effective equation. Some solutions are nevertheless possible at smaller scales [START_REF] Gondran | Processus complexe stochastique non standard en mécanique[END_REF][START_REF] Gondran | Schrödinger Equation and Minplus Complex Analysis[END_REF] , where the quantum particle is not represented by a point but is a sort of elastic string whose gravity center follows the classical trajectory ξ(t).

B. Interpretation for the non semi-classical case

The Broglie-Bohm and Schrödinger interpretations correspond to the semi-classical approximation. But there exist situations where the semi-classical approximation is not valid. It is in particular the case of state transitions for a hydrogen atom. Indeed, since Delmelt'experiment [START_REF] Nagournay | Shelved optical electron amplifier: Observation of quantum jumps[END_REF] in 1986, the physical reality of individual quantum jumps has been fully validated. The semi-classical approximation, where the interaction with the potential field can be described classically, is not possible anymore and it is necessary to electromagnetic field quantization since the exchanges are done photon by photon.

In this situation, the Schrödinger equation cannot give a deterministic interpretation and the statistical Born interpretation is the only valid one. What is new is to consider that these interpretations depend on the preparation of the particles.

Thus, it is because de Broglie and Schrödinger keep the Schrödinger equation that Einstein, who considers it as fundamentaly statistical, rejected each of their interpretations.

Einstein thought that it was not possible to obtain an individual deterministic behavior from the Schrödinger equation. It is the same for Heisenberg who developped matrix mechanics and the second quantization from this example.

This doesn't mean that one has to renounce determinism and realism, but rather that Schrödinger's statistical wave function does not permit, in that case, to obtain an individual behavior.

IV. CONCLUSION

The study of the convergence of the Madelung equations when → 0, has encouraged us to introduce the concept of non-discerned and discerned particles in classical mechanics and has given us the three following results:

-In the statistical semi-classical case the quantum particles converge to classical nondiscerned ones verifying the statistical Hamilton-Jacobi equations. The wave function is not sufficient to represent the quantum particles. One needs to add it to the initial positions, as for classical particles, in order to describe them completely. Then, the Broglie-Bohm interpretation is relevant.

-In the determinist semi-classical case the quantum particles converge to classical discerned ones verifying the determinist Hamilton-Jacobi equations. The Broglie-Bohm interpretation is not relevant because the wave function is sufficient to represent the particles as in the Copenhagen interpretation. However, one can make a realistic and deterministic assumption such as the Schrödinger interpretation.

-In the case where the semi-classical approximation is not valid anymore, as in the transition states in the hydrogen atom, the two interpretations are wrong as claimed by Heisenberg. Consequently, Born's statistical interpretation is the only possible interpretation of the Schrödinger equation. This doesn't mean that it is necessary to abandon determinism and realism, but rather that the Schrödinger wave function doesn't allow, in that case, to obtain an individual behavior of a particle. An individual interpretation needs to use creation and annihilation operators of the quantum Field Theory.

Therefore, as Einstein said, the situation is much more complex than de Broglie and Bohm thought.

Each founding father of quantum mechanics held a piece of the truth; but the overgeneralization of their different truths has led to incompatible interpretations!

  FIG.1: 100 Broglie-Bohm's trajectories through one of two slits.

FIG. 2 :

 2 FIG. 2: Zoom of the 100 previous trajectories when the Planck constant is divided by 10, 100, 1000 and 10000 respectively.

  The fact that the Schrödinger equation associated to the Born interpretation does not lead to a description of the "real states" of an individual system, naturally incites one to find a theory that is not subject to this limitation. Up to now, the two attempts have in common that they conserve the Schrödinger equation and abandon the Born interpretation. The first one, which marks de Broglie's comeback, was continued by Bohm.... The second one,

	These three interpretations are not new, as Einstein points out in one of his last articles
	(1953), "Elementary reflexion concerning the quantum mechanics foundation" in a homage
	to Max Born:
	"

which aimed to get a "real description" of an individual system and which might be based on the Schrödinger equation is very late and is from Schrödinger himself. The general idea is briefly the following : the function ψ represents in itself the reality and it is not necessary to add it to Born's statistical interpretation.[...] From previous considerations, it results that the only acceptable interpretation of the Schrödinger equation is the statistical interpretation given by Born. Nevertheless, this interpretation doesn't give the "real description" of an individual system, it just gives statistical statements of entire systems."