Bruno Locatelli
email: bruno.locatelli@cirad.fr

Raffaele Vignola

Managing watershed services of tropical forests and plantations: Can meta-analyses help?

Keywords: ecosystem services, tropics, natural forest, planted forests, hydrology, policy, meta-analysis

The watershed services provided by tropical natural and planted forests are critical to human well-being. An increasing number of valuation studies and experiences with payment for ecosystem services have dealt with the role of ecosystems in regulating the flow of water. However, several studies have been based on misconceptions about the role of forests and plantations in the hydrological cycle, despite the publication of many reviews by hydrologists. The objective of this paper is to evaluate whether meta-analyses applied to studies comparing water flows in tropical watersheds under natural or planted forests and non-forest lands can provide useful results for valuing watershed ecosystem services and making decisions. The metaanalyses show significantly lower total flows or base flows under planted forests than non-forest land uses. Meta-analyses conducted with subsamples of the data also show lower total flow and higher base flow under natural forests than non-forest land uses. However, the available studies were restricted to humid climates and particular forest types (Pinus and Eucalyptus planted forests and lowland natural forests). The small number of available studies with sufficient original data is a major constraint in the application of metaanalyses. This represents a major technical challenge for valuation studies or payment for ecosystem services, especially in countries where financial resources for implementing field research are scarce.

Introduction

Ecosystems provide services critical to human well-being, in particular watershed services that regulate the quantity of water available for human activities. The conservation of dry season stream flows is essential for navigation, recreation, wildlife, and for rural communities, as well as for irrigation systems that lack the technology for pumping groundwater [START_REF] Aylward | Land use, hydrological function and economic valuation[END_REF]. The reduction of storm flow may benefit housing, infrastructure, or agriculture in flood-prone areas. The conservation of total annual water flow is also relevant to reservoirs for drinking water or hydroelectricity production [START_REF] Guo | An assessment of ecosystem services: water flow regulation and hydroelectric power production[END_REF].

Over the past 50 years, however, the conversion of natural ecosystems to other land uses has dramatically altered hydrological cycles (Millennium Ecosystem Assessment, 2005). The combined effects of climate and land cover changes require societies to adopt appropriate adaptation measures for reducing their vulnerability to water scarcity and excess [START_REF] Oki | Global hydrological cycles and world water resources[END_REF][START_REF] Hulme | Adapting to climate change: is there scope for ecological management in the face of a global threat?[END_REF]. These measures should include the protection or restoration of ecosystems providing watershed services, especially in developing countries with low technical and financial capacity to regulate water flows with engineered solutions [START_REF] Bergkamp | Adaptation of Water Management to Climate Change[END_REF][START_REF] Pattanayak | Valuing watershed services: concepts and empirics from Southeast Asia[END_REF]. In this respect, financial mechanisms that encourage the provision of ecosystem services, such as payments for ecosystem services (PES), are increasingly being used to manage upstream forest ecosystems for the regulation of water flows [START_REF] Wunder | Payments for environmental services: some nuts and bolts[END_REF]FAO, 2004;[START_REF] Dudley | Running pure: the importance of forest protected areas for drinking water[END_REF].

Valuation studies of watershed ecosystem services and management or policy decisions about PES are not always scientifically sound. Various misconceptions about the role of ecosystems in regulating the flow of water persist among managers and decision-makers, despite the publication of many scientific papers on this issue (e.g., [START_REF] Bosch | A review of catchment experiments to determine the effect of vegetation changes on water yield and evapotranspiration[END_REF][START_REF] Bruijnzeel | Hydrology of moist tropical forests and effects of conversion: a state of knowledge review[END_REF][START_REF] Critchley | Hydrological impacts of conversion of moist tropical forests to agriculture and plantations[END_REF][START_REF] Sahin | The effects of afforestation and deforestation on water yields[END_REF][START_REF] Bonell | Possible effects of climate variability and change on tropical forest hydrology[END_REF][START_REF] Calder | Forests and hydrological services: reconciling public and science perceptions[END_REF][START_REF] Best | A critical review of paired catchment studies with reference to seasonal flows and climate variability[END_REF][START_REF] Andreassian | Water and forests: from an historical controversy to scientific debate[END_REF][START_REF] Bruijnzeel | Hydrological functions of tropical forests: not seeing the soil for the trees?[END_REF][START_REF] Scott | Forest hydrology: impacts of forest plantations on streamflow[END_REF][START_REF] Bonell | Forests, water and people in the humid tropics: past, present and future hydrological research for integrated land and water management[END_REF][START_REF] Farley | Effects of afforestation on water yield: a global synthesis with implications for policy[END_REF][START_REF] Guillemette | Rainfall generated stormflow response to clearcutting a boreal forest : peakflow comparison with 50 world-wide basin studies[END_REF][START_REF] Scott | The hydrological and soil impacts of forestation in the tropics[END_REF]. A challenge facing managers and decision-makers is the complexity of the effect of ecosystems on water flows [START_REF] Bruijnzeel | Hydrology of moist tropical forests and effects of conversion: a state of knowledge review[END_REF][START_REF] Fujieda | Hydrological processes at two subtropical forest catchments: the Serra do Mar, Sao Paulo, Brazil[END_REF][START_REF] Bonell | Possible effects of climate variability and change on tropical forest hydrology[END_REF][START_REF] Van Noordwijk | Quantifying off-site effects of land-use change: filters, flows and fallacies[END_REF][START_REF] Waage | Payments for Ecosystem Services: Getting started[END_REF].).

Meta-analysis, a statistical technique for combining the quantitative findings of several studies, has seldom been used in forest hydrology [START_REF] Ilstedt | The effect of afforestation on water infiltration in the tropics: A systematic review and meta-analysis[END_REF]. Meta-analysis has the ability to consider several studies that are in themselves inconclusive, and provide a statistically conclusive synthesis. Although it requires simplification of the observed phenomena, meta-analysis has the advantage of producing results that are more easily understandable by decision-makers than narrative reviews. The objective of this paper is to evaluate whether meta-analyses applied to studies comparing water flows in tropical watersheds under different forest or non-forest covers can provide useful results for valuing watershed ecosystem services and making decisions about PES and watershed management.

Method

We conducted several meta-analyses to combine the findings of studies comparing water flows between watersheds under natural forests vs. non-forest land uses, and planted forests vs. non-forest land uses. These studies were synchronic comparisons of two or more paired watersheds or diachronic comparisons of one watershed under changing forest cover over time. We considered natural and planted forests because both are being considered in PES for watersheds (Wunder et al., 2008). For instance, the national PES scheme in Costa Rica rewards forest conservation and plantation and recognizes explicitly hydrological services, including the provision of water for human consumption, irrigation, and energy production [START_REF] Pagiola | Payments for environmental services in Costa Rica[END_REF].

We considered three hydrological variables of interest: annual total flow, storm flow, and base flow. According to the Glossary of Hydrology of UNESCO-WMO (1992), annual total flow is the "total volume of water that flows during a year, usually referring to the outflow of a drainage area or river basin". Storm flow is "part of surface runoff which reaches the catchment outlet shortly after the rain starts; its volume is equal to rainfall excess". Base flow is the "part of the discharge (volume of water flowing through a river or channel cross-section in unit time) which enters a stream channel from groundwater from lakes and glaciers during long periods when no precipitation or snowmelt occurs" (UNESCO- WMO, 1992). Other authors have modified this definition by adding that base flow is "the more or less permanent flow supplied to drainage channels by rather invariable sources" [START_REF] Susswein | Forest watershed functions and tropical land use change. Towards integrated natural resource management in forest margins of the humid tropics: local action and global concerns[END_REF]. In areas with seasonality in rainfall, the discharge during most of the dry season results from base flow [START_REF] Smakhtin | Low flow hydrology: a review[END_REF].

We followed the generally accepted procedures for meta-analysis (e.g., Cooper and Hedges, 1994;[START_REF] Gurevitch | Meta-analysis in Ecology[END_REF]. First, we searched peer-reviewed articles comparing water flows under natural or planted forests and non-forest land uses in tropical watersheds. We searched references in CAB, ISI Web of Science, and the AGRICOLA literature database in March 2006 with the following query: "(forest OR deforestation OR reforestation OR afforestation) AND (water OR hydrology OR hydrological) AND (watershed OR catchment OR land use OR land-use) AND (tropical OR subtropical)". About 1100 references were retrieved.

Second, we selected studies conducted in whole watersheds (i.e., not in plots) that compared several watersheds under different land-uses during several years or one catchment during several years before and after a land-use change. We selected only studies reporting field measurements (i.e., not modeling results) and providing "sufficient data". By "sufficient data", we mean that the studies reported one of the following data combinations: (1) annual values of a hydrological variable for each watershed and each year, or (2) means and standard deviations of a hydrological variable and the number of observations for watersheds under forests and non-forest land uses. This selection resulted in only 10 studies. To ensure a larger sample, we searched for documents cited by review articles that complied with our selection criteria, and retrieved 10 more studies. We characterized the studies according to forest type (planted vs. natural forests), watershed area, humidity index, and forest cover difference. Data about the humidity index, defined as the ratio of annual precipitation and potential evapotranspiration, came from [START_REF] Deichmann | Global digital data sets for land degradation studies: a GIS approach[END_REF]. Forest cover difference was defined as the difference in forest cover between the compared watersheds, and was considered to be high if it was more than 50% of the watershed area. Third, each comparison (i.e. comparison of one hydrological variable between different watersheds) found in the selected studies was converted to a dimensionless scale called the effect size. We chose the Hedge's unbiased estimator g of effect size, and used equations described by Cooper and Hedges (1994) to estimate the effect size and its variance for each comparison. Fourth, we combined the estimates of effect sizes using a random effect model that considers a random variation among the studies in the "true" effect [START_REF] Gurevitch | Meta-analysis in Ecology[END_REF]Shadish and Hoaddock, 1994). We conducted six meta-analyses for each of the three hydrological variables of interest and for natural and planted forests. We also applied meta-analyses to subsets of the data, e.g., only small watersheds. Finally, for each meta-analysis, we conducted a sensitivity analysis to determine whether some individual comparisons influenced the results, by performing n partial meta-analyses (n being the number of comparisons) on data sub-samples containing all the comparisons but the n th one. Only significant differences (at p < 0.05) observed in a meta-analysis and all associated partial meta-analyses were reported. We considered that the tendency to report only significant results -or the "file drawer problem" (Fernandez-Duque and Valeggia, 1994) -was not a relevant bias in our analysis, because the results of costly long-term hydrological studies are generally reported even if no significant differences are found.

Results and discussion

Selected studies Among the 20 selected studies, nine were conducted in Asia, eight in Africa, and three in Latin America (see Table 1 for details of the studies, and Figure 1 for their location). The small number of studies is due to the lack of comparison between watersheds under forests and non-forest land uses and the lack of sufficient data in many studies. The focus of the meta-analyses on tropical areas, the theme of this special issue of Forest Ecology and Management, also strongly reduced the number of available studies, as tropical areas are under-represented in the scientific literature about forest hydrology. For example, among 135 watershed experiments reviewed in [START_REF] Andreassian | Water and forests: from an historical controversy to scientific debate[END_REF] 1. Location of the 20 studies selected for the meta-analyses.>> Among the 20 studies, 13 studies were conducted in watersheds smaller than 1 km 2 and seven in watersheds larger than 1 km 2 . Seventeen studies compared watersheds with high forest cover difference (more than 50% of the watershed area). Seventeen studies were conducted in humid areas, and three in dry areas or transition areas between humid and dry. This result shows a bias in the meta-analyses, as no studies in semi-arid or arid areas and few studies in dry areas provided sufficient data. As the results cannot be generalized to the tropics as a whole, the scientific knowledge for decision making in dry, semi-arid, and arid areas is lacking.

In the 20 studies, we retrieved 63 comparisons between watersheds under natural or planted forests and nonforest land uses. On average, a comparison was based on 16 observations (e.g., one watershed observed during 16 years or four watersheds during four years). The numbers of studies and comparisons are within the acceptable range of sample sizes used in other meta-analyses. Indeed, in the few meta-analyses that have been applied to hydrology, the sample sizes were generally lower; for instance, 14 observations and four studies in [START_REF] Ilstedt | The effect of afforestation on water infiltration in the tropics: A systematic review and meta-analysis[END_REF].

Natural forests

In 12 of the 20 selected studies, 39 comparisons were analyzed between watersheds under natural forests and non-forest land uses. These comparisons were related mostly to watersheds larger than 1 km 2 (26 comparisons), with high forest cover differences (24), and in humid areas (36). No studies about montane cloud forests complied with our selection criteria. As these forests can have a different effect on water flows compared to other natural forests because of their capacity to collect water from the clouds, valuation of ecosystem services or decision making about cloud forests should not be based on our meta-analyses. Including all comparisons, the meta-analysis did not show significant differences in total flow between watersheds under natural forests and non-forest land uses (see Table 2). However, when selecting comparisons made in small watersheds or large differences in forest cover, the meta-analyses showed that the total flow was significantly lower in watersheds under natural forests than non-forest land uses. This result shows the interest in comparing small watersheds or watersheds with large differences in forest cover. Some authors have stated that the effects of forests on water flows are discernable only in watersheds smaller than 1000 km 2 (FAO and CIFOR, 2005) or 500 km 2 [START_REF] Pattanayak | Valuing watershed services: concepts and empirics from Southeast Asia[END_REF], while others have stated that the most significant observations have been made in small watersheds, usually less than one km 2 [START_REF] Bruijnzeel | Hydrological functions of tropical forests: not seeing the soil for the trees?[END_REF]. Separating the hydrological effects of forests is more difficult in larger catchments, which generally present diverse land uses and land-use changes.

<< INSERT Table 2. Results of the meta-analyses. >>

The effect of natural forests on total flow has been demonstrated by other authors in diverse situations, although they may not be valid for some very old forests or cloud forests [START_REF] Calder | Forests and hydrological services: reconciling public and science perceptions[END_REF][START_REF] Bruijnzeel | Forest hydrology: impacts of forest conversion on streamflow[END_REF]. These effects can be explained by the higher evapotranspiration rates in forests compared to pasture or annual cropping land uses [START_REF] Bruijnzeel | Hydrology of moist tropical forests and effects of conversion: a state of knowledge review[END_REF]. Trees with deep roots and high transpiration rates may act as pumps that remove water from the soil and transpire it. Although conversion of natural forests to nonforest land uses is immediately followed by a period of increased total flow, the subsequent period may not be characterized by a high total flow, for instance with forest recovery or high evapotranspiration rates in the subsequent vegetation.

The meta-analysis showed no significant differences in base flow between watersheds under natural forests and non-forest land uses (see Table 2). The effect of natural forests on base flow results from two competing processes [START_REF] Calder | Forests and hydrological services: reconciling public and science perceptions[END_REF][START_REF] Bruijnzeel | Forest hydrology: impacts of forest conversion on streamflow[END_REF]: high transpiration by the forest, contributing to a low base flow, and high infiltration under the forest, generally contributing to soil water recharge and high base flow. If the infiltration capacity of the soil is increased when converting natural forests to non-forest land uses, base flow can be increased. Natural forests conserve base flow compared to other land uses in situations where the alternative land use decreases the infiltration capacity [START_REF] Bruijnzeel | Forest hydrology: impacts of forest conversion on streamflow[END_REF]). In the metaanalysis, the base flow was higher under natural forests for the subset of large watersheds; but this result is difficult to explain. This may due to the regional recycling of rainfall by forests [START_REF] Vanclay | Managing water use from forest plantations[END_REF] but this interpretation is controversial [START_REF] Bruijnzeel | Hydrological functions of tropical forests: not seeing the soil for the trees?[END_REF].

Regarding storm flow, no significant difference was found between watersheds under natural forests and non-forest land uses (see Table 2). Forest hydrologists agree that the relationship between forests and storm flow is not clear [START_REF] Bosch | A review of catchment experiments to determine the effect of vegetation changes on water yield and evapotranspiration[END_REF]. Even if higher infiltration and evapotranspiration under forests may reduce storm flow, our results showed that this effect was not always significant. The role of forest cover for storm flow reduction is a debated issue, especially for the part of the storm flow that is released during extreme rainfall events (peak flow or maximum flow) and is responsible for floods. According to FAO and CIFOR (2005), there is no hard evidence that tropical forests reduce floods, while [START_REF] Bradshaw | Global evidence that deforestation amplifies flood risk and severity in the developing world[END_REF] showed that flood frequency is negatively correlated with forest cover. The effect of forests on large scale floods seems particularly weak; [START_REF] Bruijnzeel | Hydrological functions of tropical forests: not seeing the soil for the trees?[END_REF] concluded that floods occurring in extreme conditions (e.g., extreme rainfall at the end of the rainy season when soils are wet) are not regulated by vegetation cover. Our analyses did not distinguish storm flows from rainfall events with different intensities, and thus it was not possible to determine the role of forests in regulating the peak flow resulting from extreme rainfall events. The regulation of annual storm flow is different from the regulation of peak flow resulting in large-scale floods [START_REF] Bruijnzeel | Hydrological functions of tropical forests: not seeing the soil for the trees?[END_REF]. However, even if forests do not prevent large scale floods, their role in preventing average and most frequent floods should not be undervalued.

Planted forests

In 10 of the 20 selected studies, 24 comparisons between watersheds under planted forests and non-forest land uses were analyzed. These comparisons were related mostly to watersheds smaller than 1 km 2 (22 comparisons), with high forest cover differences (all 24), and in humid areas (23). The selected studies referred to planted forests with species of two genera: Eucalyptus and Pinus. No studies were available for other species, including native species. This shows that there is a lack of scientific data for making decisions about planting species other than Eucalyptus and Pinus genera, as the results of the meta-analyses cannot be generalized to any planted forest in the tropics.

The meta-analyses showed that total flow and base flow were lower in watersheds under planted forests than non-forest land uses (see Table 2). These results can be explained by the high transpiration rates of exotic species, especially Eucalyptus [START_REF] Vertessy | Factors determining relations between stand age and catchment water balance in mountain ash forests[END_REF]. Infiltration capacity may be higher under planted forests than non-forest land uses [START_REF] Ilstedt | The effect of afforestation on water infiltration in the tropics: A systematic review and meta-analysis[END_REF], but may not be enough to offset the higher loss of water by transpiration. However, some authors have reported that in degraded soils, planted forests may increase infiltration more than transpiration, thus increasing base flow, compared to non-forest land uses [START_REF] Bruijnzeel | Hydrological functions of tropical forests: not seeing the soil for the trees?[END_REF]). However, the available studies did not compare water flows between planted forests and non-forest land uses on degraded soils. Including all comparisons, the meta-analysis did not show significant differences in storm flow between watersheds with planted forests and non-forest land uses. This analysis of planted forests and storm flow was based on only five comparisons, compared to eight to 16 for other analyses.

Usefulness of the meta-analyses

The results of the meta-analyses allow the analysis of some misunderstandings about the effects of natural and planted forests on water flows in decision making on PES or watershed management. According to the conventional beliefs analyzed by various authors [START_REF] Chomitz | The domestic benefits of tropical forests[END_REF][START_REF] Susswein | Forest watershed functions and tropical land use change. Towards integrated natural resource management in forest margins of the humid tropics: local action and global concerns[END_REF][START_REF] Calder | Forests and hydrological services: reconciling public and science perceptions[END_REF][START_REF] Bruijnzeel | Forest hydrology: impacts of forest conversion on streamflow[END_REF][START_REF] Kaimowitz | Useful Myths and Intractable Truths: the Politics of the Link Between Forests and Water in Central America[END_REF], natural and planted forests are thought to increase total flows. This belief is deeply rooted in public perception, as was shown by [START_REF] Kosoy | Payments for environmental services in watersheds: Insights from a comparative study of three cases in Central America[END_REF] in Central America, where more than 90% of the population involved in a survey perceived that more forests would lead to higher total water flows. According to FAO (2004), many hydrological PES are based on assumptions that have not been verified in the particular case. For example, in the PES scheme in Pimampiro (Ecuador), the effect of land-use of hydrological services has not been studied [START_REF] Wunder | Decentralized payments for environmental services: The cases of Pimampiro and PROFAFOR in Ecuador[END_REF]. In Costa Rica, a private hydro-energy firm pays for ecosystem services provided by forestation and forest conservation with the objective of increasing its energy production, in part through increasing total flow [START_REF] Miranda | Voluntary agreements in watershed protection experiences from Costa Rica[END_REF]. Three PES schemes studied by [START_REF] Kosoy | Payments for environmental services in watersheds: Insights from a comparative study of three cases in Central America[END_REF] in Central America are built upon the belief that forests increase total flows. [START_REF] Aylward | Land use, hydrological function and economic valuation[END_REF] showed that several valuations of forest hydrological services were based on the unverified assumption that natural forests would decrease storm flow and increase base flow. Among the studies valuing forest watershed services in the FAO Forest Valuation Database (FAO, 2007), three were based on unsubstantiated relationships between forests and water, for instance that forests decrease floods or increase total flows. To supply water and reduce sediment in the Panama Canal, a plan developed in the 1990s promoted planting forests among other activities, and was based on the belief that planted forests would increase total and base flows [START_REF] Calder | Forests and hydrological services: reconciling public and science perceptions[END_REF][START_REF] Kaimowitz | Useful Myths and Intractable Truths: the Politics of the Link Between Forests and Water in Central America[END_REF]. After hurricane Mitch in Central America in October 1998, the public, decision-makers, environmentalists, and international agencies were convinced that deforestation had increased the damage, and proposed forest planting as a watershed management measure [START_REF] Kaimowitz | Useful Myths and Intractable Truths: the Politics of the Link Between Forests and Water in Central America[END_REF]. Similarly, [START_REF] Bruijnzeel | Hydrological functions of tropical forests: not seeing the soil for the trees?[END_REF] reported that large-scale forest planting in upper watersheds was proposed in Bangladesh and China to reduce damage caused by floods.

Even though the results of the meta-analyses should not be generalized to any tropical watershed, total flows appeared to be lower in watersheds under natural forests than non-forest land uses (for small watersheds or with high differences in forest cover). The results show no significant differences in storm flow between watersheds under natural forests and non-forest land uses. Regarding planted forests, the results showed that they reduced base flow and total flow and had no significant effect on storm flow compared to non-forest land uses. These results highlight that the effects of natural or planted forests on water flows are not the same as perceived by some decision makers. Meta-analyses can help correct misunderstandings and show that some of the effects of natural or planted forests on water flows are unclear, and that inappropriate generalizations should be avoided [START_REF] Tognetti | A knowledge and assessment guide to support the development of payment arrangements for watershed ecosystem services[END_REF].

Limitations

The meta-analyses faced several limitations. First, the small number of available studies limited the number of significant differences and prevented from taking into account other factors, for instance soil characteristics, geology, topography, rainfall pattern, location of land-use change in the watershed and land management in non-forest land areas, which are important factors in explaining the difference in water flows between forested and non-forested watersheds [START_REF] Bruijnzeel | Hydrological functions of tropical forests: not seeing the soil for the trees?[END_REF][START_REF] Aylward | Land use, hydrological function and economic valuation[END_REF]. Applying meta-analyses in a rigorous way is difficult with a small number of studies, especially with regards to the sensitivity analysis which is conducted on subsamples of the dataset. The following results were discarded by the sensitivity analysis: more base flow in watersheds under natural forests than under non-forest land uses (with all watersheds), less storm flow in watersheds under natural forests than under non-forest land uses (only with watersheds where forest cover differed by more than 50% of the watershed area), and less storm flow in watersheds under natural forests than under non-forest land uses (only with small watersheds).

Second, the available studies had been mostly conducted in humid areas and in some forest types (e.g., no cloud forests or planted forests with species other than of the Pinus and Eucalyptus genera). For these reasons, the results cannot be generalized to the entire tropics. Third, the available studies used different methods (e.g. for separating base flow and storm flow) and different time periods for calculating flow-rainfall ratios. A more thorough analysis of original hydrological data would be necessary to extend our simplified meta-analyses.

Conclusions

Meta-analysis appears to be a promising way to combine results from studies comparing water flows between tropical watersheds under natural or planted forests and non-forest land uses. It can help decision makers understand the effects of forests on water flows, and also highlight the effects that remain unclear. This help is necessary because even though hydrologists have published many narrative reviews about these effects, not all economists or decision makers have attempted to integrate this knowledge into the process of determining the value of hydrological ecosystem services or for making decisions. The meta-analyses show significantly lower total flows or base flows under planted forests than non-forest land uses. Meta-analyses conducted with subsamples of the data also show lower total flow and higher base flow under natural forests than non-forest land uses. the available studies were restricted to humid climates and particular forest types (Pinus and Eucalyptus planted forests and lowland natural forests).

The small number of available studies with sufficient data is a major constraint in the use of meta-analysis for analyzing the effects of natural or planted forests on water flows. It impedes analyzing the interacting effect of important factors for water flow regulation, such as soil, geology, topography, or land management practices. Furthermore, the available studies with sufficient data are restricted to some local conditions or forest types and knowledge is lacking about dry areas, cloud forests, native species plantations, and plantations on degraded lands.

The lack of measurements of water flow under different land-uses and of empirical data on hydrological services represents a major technical challenge for valuation studies or payment for ecosystem services [START_REF] Kremen | A call to ecologists: measuring, analyzing, and managing ecosystem services[END_REF], especially in countries where financial resources for implementing field research are scarce. More empirical data on underrepresented local conditions or forest types and a facilitated access to hydrological data would be valuable for watershed managers and decision-makers. Analyzing original data about water flows, climate and watershed characteristics in a consistent way could improve further application of meta-analysis. (1) Two studies are reported in the same reference (Bailly et al., 1974), one in Périnet (1974p) and another in Manankazo (1974m).

(2) HU=humid area, HU-DR= humid area near the transition between humid and dry (less than 50km), DR=dry area. Humidity index taken from [START_REF] Deichmann | Global digital data sets for land degradation studies: a GIS approach[END_REF].

(3) S=Small (< 1 km 2 ), L=Large (> 1 km 2 ).

(4) Lo: forest cover differs by less than 50% of the watershed area between the compared watersheds, Hi: more than 50%.

(5) Period(s) of time selected for our analysis.

(6) NF: comparisons between natural forests and non-forest land uses, PF: comparison between planted forests and non-forest land uses. Syn: synchronic comparisons. Dia: diachronic comparisons. In parenthesis: description of watershed land cover ("w." = watershed). (ii) "small w." = watershed smaller than 1 km 2 , "large w." = watershed larger than 1 km 2 , "humid" = humid according to Table 1 definition, "dry" = dry or transition from humid to dry, "low diff."= forest cover differs by less than 50% of the watershed area between the compared watersheds, "high diff."= more than 50%.

  Number of studies and references in parenthesis (references are given with the first three letters of the first author's name and the 2-digit year).

Figure 1 .

 1 Figure 1. Location of the 20 studies selected for the meta-analyses.

  , only 16 are tropical watersheds. << INSERT Table 1. Characteristics of the selected studies. >> << INSERT Figure

Table 1 . Characteristics of the selected studies.

 1

	First	Country	Humidity	Watershed	Forest	Years (5)	Description (6)
	author and		class (2)	area (3)	cover
	year				difference
					(4)
	Bailly	Mada-	HU	S	Hi	64-71	NF Syn (4 w. forest, 2 w. agriculture/fallow)
	1974p(1)	gascar					PF Syn (1 w. Eucalyptus, 2 w. agriculture/fallow)
	Bailly	Mada-	HU	S	Hi	62-72	PF Syn (1 w. Eucalyptus, 2 w. grassland)
	1974m	gascar
	Bewket	Ethiopia HU	L	Lo	60-64 and	NF Dia (1 w. less forested in 1960 than in 1982).
	2005					80-84
	Blackie	Kenya	HU-DR	L	Hi	61-68	NF Syn (1 w. forest, 1 w. partially agriculture)
	1972
	Chandler	The	HU	S	Hi	95-96	NF Syn (1 w. forest, 2 w. grasslands)
	1998	Phili-
		ppines
	Costa	Brazil	HU	L	Lo	49-68 and	NF Dia (1 w. less forested in 49-68 than in 79-98).
	2003					79-98
	Dagg	Tanzania HU	S	Hi	57-63	NF Syn (1 w. forest, 1 w. agriculture)
	1965
	Fritsch	French	HU	S	Hi	77-81	NF Syn (2 w. forest, 6 w. deforested/pasture)
	1983	Guyana
	Fritsch	French	HU	S	Hi	77-87	NF Syn (2 w. forest, 2 w. pasture/slash-and-burn)
	1992	Guyana					PF Dia (2 w. before and after reforestation)
	Goujon	Mada-	HU	S	Hi	62-66	PF Syn (1 w. Pinus, 1 w. grassland)
	1968	gascar
	Lal 1997	Nigeria	HU-DR	S	Hi	74-75 and	NF Dia (1 w. deforested in 1979)
						79-84
	Mathur	India	DR	S	Hi	61-67 and	PF Syn (after 1969 1 w. Eucalyptus, 1 w. shrub)
	1976					69-73	PF Dia (1 w. under shrub before 1968, then
							Eucalyptus)
	Mungai	Sri	HU	L	Lo	51-61, 67-	NF Dia (1 w. under deforestation)
	2004	Lanka				77, 78-88
	Mwendera	Malawi	HU	L	Hi	61-65, 70-	PF Dia (1 w. under forestation with Pinus and
	1994					78	Eucalyptus)
	Raghunath	India	HU	L	Hi	Various	NF Syn (9 w. mainly under agriculture or pasture, 3
	1970					periods	w. mainly under forest)
	Samraj	India	HU	S	Hi	68-71, 73-	PF Syn (after 1972 1 w. Eucalyptus, 1 w. shrub)
	1988;					81, 82-91	PF Dia (1 w. under shrub before 1972, then
	Sharda						Eucalyptus)
	1998;
	Sikka
	2003
	Wilk 2001 Thailand HU	L	Hi	57-64, 87-	NF Dia (1 w. with decreasing forest cover)
						94
	Zhou	China	HU	S	Hi	81-90	PF Syn (1 w. Eucalyptus, 1 w. bare soil)
	2002

Table 2 . Results of the meta-analyses.

 2

	Results of the meta-	Selected studies (i)	Number of comparisons	Other significant results of the
	analysis with all data		and characteristics of the	meta-analyses with data subsets
			compared watersheds (ii)	(ii)

Differences in flows between natural forests vs. non-forest land uses

  Studies available only for lowland forests. No studies on cloud forests). Studies available only about Eucalyptus and Pinus plantations. No studies on other species, incl. native species).

	Total flow: No significant	11 (Bai74p Bew05 Bla72	15 (6 small w, 9 large	With small w. or high diff.: Less
	difference	Cha98 Cos03 Dag65 Fri83	w, 3 dry, 12 humid, 11	total flow in natural forest than
		Lal97 Mun04 Rag70 Wil01)	high diff., 4 low diff)	non-forest land uses
	Base flow: No significant	4 (Bai74p Bew05 Mun04	8 (1 small w, 7 large w,	In large w.: More base flow in
	difference	Wil01)	8 humid, 2 high diff., 6	natural forest than non-forest land
			low diff.)	uses
	Storm flow: No significant	8 (Bai74p Bew05 Cha98	16 (6 small w, 10 large
	difference	Cos03 Fri92 Mun04 Rag70	w, 16 humid, 9 high
		Wil01)	diff., 7 low diff.)
	Total flow: Less total flow	7 (Bai74m Bai74p Gou68	11 (11 small w, 1 dry,	With small w., humid or high diff.:
	in planted forest than non-	Mat76 Sam88 Sha98 Zho02)	10 humid, 11 high	Same result
	forest land uses		diff.)
	Base flow: Less base flow	5 (Bai74p Mwe94 Sam88	8 (7 small w, 1 large w,	With small w., humid or high diff.:
	in planted forest than non-	Sha98 Sik03)	8 humid, 8 high forest	Same result
	forest land uses		diff.)

(None Differences in flows between planted forests vs. non-forest cover (

Acknowledgements

The authors acknowledge Elke Verbeeten, Pablo Imbach, Carlos Perez, Daniel Murdiyarso, Thomas Koellner, Francisco Alpizar, and the anonymous reviewers for their useful comments. This document was produced within the framework of the project "Tropical Forests and Climate Change Adaptation" (TroFCCA), executed by CATIE and CIFOR and funded by the European Commission under contract EuropeAid/ENV/2004-81719. The contents of this document are the sole responsibility of the authors and can under no circumstances be regarded as reflecting the position of the European Union.


Thibault Mondary

Adeline Nazarenko

Haïfa Zargayouna

Sabine Barreaux

The Quaero Evaluation Initiative on Term Extraction

Keywords: Term extraction, Evaluation, Quaero, Pharmacology, Gradual relevance, Scalability

The Quaero program has organized a set of evaluations for terminology extraction systems in 2010 and 2011. Three objectives were targeted in this initiative: the first one was to evaluate the behavior and scalability of term extractors regarding the size of corpora, the second goal was to assess progress between different versions of the same systems, the last one was to measure the influence of corpus type. The protocol used during this initiative was a comparative analysis of 32 runs against a gold standard. Scores were computed using metrics that take into account gradual relevance. Systems produced by Quaero partners and publicly available systems were evaluated on pharmacology corpora composed of European Patents or abstracts of scientific articles, all in English. The gold standard was an unstructured version of the pharmacology thesaurus used by INIST-CNRS for indexing purposes. Most systems scaled with large corpora, contrasted differences were observed between different versions of the same systems and with better results on scientific articles than on patents. During the ongoing adjudication phase domain experts are enriching the thesaurus with terms found by several systems.

Introduction

Computational terminology is a twenty years old discipline that aims at building automatically or semiautomatically terminological resources from corpora. A lot of terminological tools have been developed, but despite the progress made, it remains difficult to get a clear idea of the maturity of this research field and to compare approaches. Some efforts have been made to set up an evaluation protocol adapted to the specificity of terminological tasks. The TEMREC task of the first NTCIR initiative [START_REF] Kando | The NTCIR workshop: the first evaluation workshop on japanese text retrieval and cross-lingual information retrieval[END_REF] was the first initiative dedicated to the evaluation of term extraction, but it was not reconducted due to its lack of popularity. CoRReCT [START_REF] Enguehard | Correct : Démarche coopérative pour l'évaluation de systèmes de reconnaissance de termes[END_REF] has proposed an interesting dataset and protocol to evaluate term recognition in corpora (a task close to controlled indexing). CE-SART [START_REF] Widad | Terminological resources acquisition tools: Toward a user-oriented evaluation model[END_REF] was the most comprehensive challenge: a gold standard list of terms and an acquisition corpus were chosen for a specific domain (medicine), the systems had to extract terms from the acquisition corpus and their results were compared against the gold standard using various relevance criteria. Quaero is a program promoting research and industrial innovation on technologies related to the analysis and classification of multimedia and multilingual documents1 . Partners technologies are yearly evaluated in internal or external initiatives. In this context, an internal Terminology Extraction evaluation was organized by LIPN in 2010 andreconducted in 2011. This paper presents the protocol and results obtained during these two evaluation initiatives.

The Task

Term extractors output lists of relevant terms from a domain specific corpus. The terms may be mono or polylexical units, i.e. made of one or several words. The goal of the evaluation was to measure the quality of the resulting lists of terms, through a comparison with a gold standard. We really focused on the specific task of term extraction without taking into account term ranking and variant clustering. This experiment has been done on the pharmacology domain: we proposed a set of corpora, every participant had to extract a list of relevant terms and submit it for evaluation. A participant could submit several runs and each run was evaluated against the gold standard. Three objectives were targeted in this initiative. The first one was to evaluate the behavior and scalability of systems regarding the size of corpora (the ability of extractors to deal with large corpora and the stability of scores regarding the size of the acquisition corpora). The second goal was to assess the progress between different versions of the same systems. The last one was to measure the influence of corpus type on the performance of term extractors.

Corpora and Gold Standard

Two different types of pharmacology corpora were used. A first set of European patents was supplied by Jouve, an industrial Quaero partner. From it, three selfincluding corpora of growing size (called C1, C2 and Peptide Extraction From Ionic Conjugates: [0042] A 50 mg sample of an ionic molecular conjugate was mixed into 20 mls of methylene chloride. The mixture was sequentially extracted with 50 ml, 20 ml and 20 ml portions of 2N acetic acid. The acetic acid extracts were combined and analyzed for peptide content by high performance liquid chromatography(HPLC). Peptide analysis by HPLC is as follows. HPLC analysis were performed using a Waters model M-45 solvent delivery pump and an EM Science MACS 700 detector at wavelength 220 nm and 1.0 AUFS. Peptides were run using a Lichrospher (EM separations) C18, 100A, 5um, 25cm x 4.6 mm column and 30% acetonitrile/0.1% TFA as an isocratic eluent buffer.

[0043] Following are details (Table VI) of. . . Voluntary exercise improves stress coping and lowers anxiety. Because of the role of GABA in these processes, we investigated changes in the central GABAergic system in rats with free access to a running wheel for 4 weeks. The control animals had no access to a running wheel. Using in-situ hybridisation histochemistry, we studied changes in gene expression of various GABA::(A) receptor subunits as well as the GABA-synthesising enzyme glutamic acid decarboxylase-67 (GAD67) in the forebrain. There were region-specific decreases in alpha 2, beta 3 and gamma 2 subunit mRNA expression and region-specific increases in beta 1 subunit expression. The alpha 5 and delta subunits, in the forebrain specifically associated with extrasynaptic GABA::(A) receptors in the hippocampus, showed differential increases in expression levels. . . . C3 in the following) were built, in order to test the systems' scalability. C1 contains 500,000 words, C2 1.5M words and C3 2.5M words. An excerpt of a patent is presented on Figure 1. In order to measure the systems' behavior against another type of corpora of the same domain, an additional corpus (called CA) composed of abstracts of scientific articles in pharmacology was extracted from the PASCAL2 database and provided by INIST-CNRS. CA weights 1.5M words (same size as C2), a typical abstract is presented on Figure 2. The gold standard was an unstructured version of the pharmacology thesaurus3 used by INIST-CNRS for indexing purposes. It consists of 76,466 English terms, mainly about general pharmacology, diseases and drugs. An excerpt is presented on Figure 3.

Metrics

Scoring was performed using terminological precision, recall and F-Measure, introduced in [START_REF] Nazarenko | Évaluation des outils terminologiques : enjeux, difficultés et propositions[END_REF]Zargayouna and Nazarenko, 2010) and presented on Equations 1-3. The terminological precision and recall metrics take into account a gradual relevance. The systems' outputs are tuned to find their maximal correspondence with the gold standard, which means that the outputs are adapted to the terminological type and granularity of the gold standard.

T P recision = i∈T (O) rel GS (i) |T (O)|
(1)

T Recall = i∈T (O)
rel GS (i)

|GS|

(2)

T F M easure = 2 × T P recision × T Recall T P recision + T Recall (3) |T (O)| is the size of tuned term extractor output T (O),
|GS| the size of the gold standard. rel GS (i) is the relevance of a term i with respect to the gold standard.

It is based on a terminological distance d t and on a threshold τ on this distance:

rel GS (i) = 1 -δ(i) if δ(i) < τ 0 otherwise (4)
where [START_REF] Drouin | Term extraction using nontechnical corpora as a point of leverage[END_REF][START_REF] Drouin | Termhood experiments: quantifying the relevance of candidate terms[END_REF] and YaTeA [START_REF] Aubin | Improving term extraction with terminological resources[END_REF]. During the second evaluation initiative, one system became unavailable (TermExtractor) and another underwent a major revision (TermoStatWeb). The participating systems are presented on Table 1 and detailed bellow.

δ(i) = min e∈GS (d t (i, e)
Acabit is a terminology extraction tool which takes as input a part-of-speech tagged corpus and proposes as output a ranked list of multi-word terms. Acabit is based on syntactical patterns and statistical filtering. It works on English and French, the tested version was 4.35 .

Fastr is a multi-lingual tool for automatic indexing.

In free indexing mode, it acts as a term extractor and extracts terms with their variations from a corpus. Fastr works on French and English, the tested version was 2.046 .

Jv1 is developed by Jouve for patent classification. It produces nomimal or verbal phrases based on shallow parsing and statistical metrics. This system works on English, German and French.

YaTeA aims at extracting noun phrases that look like terms from a corpus. It is based on simple syntactic patterns and endogenous disambiguation. Exogenous disambiguation using external resources is also possible, but it was not used here. YaTeA works on English and French, the tested version was 0.57 . The second filter merges terms sharing the same lemma. For this initiative, only one representative per cluster was output.

MIG-

TermExtractor is an online term extractor designed to build ontologies. It is based on two entropy measures. The first one is used to select the terms which are consensually referred throughout the corpus documents, the second one is used to select only the terms which are relevant to the domain of interest. TermExtractor8 works on English.

TermoStatWeb is an online term extractor based on TermoStat [START_REF] Drouin | Term extraction using nontechnical corpora as a point of leverage[END_REF], which basic principle is to compare the distribution of words between a specialized document (the corpus to be processed) and a large corpus of "general language", using different statistical measures, e.g. log likelihood, specifity or chi2. It works on English, Spanish, French and Italian. This version has been superseded by TermoStatWebV3 at the end of 2010.

TermoStatWebV3 is the new version of Ter-moStatWeb 9 . Different runs were produced, some with only simple terms (TermoStat-WebV3-nomulti) and other with both simple and multi-terms (TermoStatWebV3-multi).

TS3 is an experimental version of TermoStatWebV3 enriched by INRIA/TexMex with post-processing and filtering. Different runs were produced, some with only simple terms (TS3-nomulti) and other with both simple and multi-terms (TS3-multi).

Evaluation Results

Terminological scores have been computed for a total of 32 runs. Table 2 presents the number of returned terms. Huge differences can be observed. Although the corpus of abstracts (CA) contains 1.5MWords as C2, term extractors produce much more term candidates from CA than from C2.

Overall results are presented on Table 3, with best scores highlighted in bold font for each corpus. Two families of systems can be observed from the last two tables: those who favor precision by returning a small amount of correct terms (Jv1, TermExtractor, TermoStatWebV3-multi and TermoStatWebV3-nomulti) and those who favor recall (Acabit, Fastr, MIG-YaTeA, TS3-multi, TS3-nomulti and YaTeA). TermoStatWeb cannot be clearly categorized. This difference between systems can be explained by their design choices.

Scalability

Almost all systems scale with large corpora, i.e. provide results without crashing. However, one system has not been tested on C1 and C2 because it was designed to work with very large corpora. For a first group of systems, precision decreases as the corpus size increases (it is an expected behavior, as more terms are able to be found, more terms might be erroneous). For the systems of a second group (TermoStatWebV3 and TS3), however, the results (amount of terms, TPrecision, TRecall) do not increase monotonically with the corpus size. This is probably due to the statistical underlying filtering strategy.

Systems evolution

Evolution can be observed between the different versions of the same systems. As described before, TS3 and TermoStatWebV3 both derived from Progression between TS3-multi and TermoStat-WebV3-multi is presented on Figure 6. Significant improvements can be observed in terms of TPrecision, mostly for C2. However, TRecall notably decreases because TermoStatWebV3 filters terms much more aggressively, focusing on precision.

The most significant improvement is obtained between YaTeA and MIG-YaTeA, as presented on Figure 7. TPrecision, TRecall and TFMeasure increase as the corpus grows.

Sensitivity to the corpus type

Regarding the sensitivity to the corpus type, results show that CA almost always provides better scores than C2 (although they both weight 1.5M words), both in term of TPrecision and in term of TRecall. The contrast between C2 and CA can be seen on Figure 8. These improvements can be explained because CA, composed of 7,030 abstracts where C2 contains only 106 patents, can potentially cover a larger field of knowledge. esting keys to better characterize term extractors in their design choices, their scalability regarding corpus size and their robustness with respect to corpus type. As the evaluation organizer, LIPN set up an adjudication phase after 2011 initiative, in order to get more reliable precision results. We actually noticed that relevant terms are missing in the gold standard, which potentially penalizes the systems. Tested term extractors have provided a total of more than one hundred thousand of distinct terms. Some of them are found by at least two systems and are not already present in the gold standard. Some experts from INIST-CNRS are asked to check whether these terms are relevant or not. To ease this process, LIPN has developed a web based validation interface, which allows experts to view terms in context (the sentences where they appear). A screenshot of this interface is presented on Figure 9. For each term, experts have to decide if it is GOOD, BAD or ?, a generic category for problematic terms. A free comment zone is available to indicate the correct form when needed.

These evaluations were also helpful for developers and for the gold standard provider. INIST-CNRS plans to exploit the results of the ongoing adjudication work to extend their thesaurus.

Acknowledgement

This work has been partially founded by OSEO under the Quaero program.

Figure 1 :

 1 Figure 1: Excerpt of a patent

Figure 2 :

 2 Figure 2: Excerpt of a typical abstract from CA

Figure 3 :

 3 Figure 3: Excerpt of the gold standard

  YaTeA is an enriched version of YaTeA developed by INRA/MIG. It uses two post-processing filters for cleaning and merging results. The first filter gets rid of extraction errors, incomplete terms, spelling errors and overgeneralized terms.

Figure 4 :

 4 Figure 4: Increase of TPrecision, TRecall and TFMeasure from TermoStatWeb to TS3-multi

Figure 6 :Figure 7 :Figure 8 :

 678 Figure 5: Increase of TPrecision, decrease of TRecall and TFMeasure from TermoStatWeb to TermoStatWebV3-multi

Figure 9 :

 9 Figure 9: Screenshot of the validation interface

TABLE VI

 VI

		IN-VITRO ASSAY DATA
	DAY OF ASSAY PERCENT OF TOTAL PEPTIDE RELEASED
		Example #8 Example #9	Example #10
	1	5.5%	12.5%	11%
	7	26.9%	21.3%	53%
	14	55.2%	47.3%	55%
	. . .

Table 2 :

 2 Number of returned terms

	9 http://olst.ling.umontreal.ca/
	˜drouinp/termostat_web

http://www.quaero.org

The multidisciplinary bibliographical database produced by INIST-CNRS. PASCAL page on the official INIST sitehttp://inist.fr/spip.php?article11.

3 Available on TermSciences, the multidisciplinary terminological portal developed by INIST-CNRShttp://www.termsciences.fr

http://sourceforge.net/projects/ termometerxd

http://www.bdaille.fr/index. php?option=com_content&task= blogcategory&id=5&Itemid=5

http://perso.limsi.fr/Individu/ jacquemi/FASTR

http://search.cpan.org/ ˜thhamon/ Lingua-YaTeA-0.5

http://lcl2.di.uniroma1.it/ termextractor, available until early 2011


Michele Arzano
email: marzano@uu.nl

Jerzy Kowalski-Glikman
email: jkowalskiglikman@ift.uni.wroc.pl

Kinematics of a relativistic particle with de Sitter momentum space

We discuss kinematical properties of a free relativistic particle with deformed phase space in which momentum space is given by (a submanifold of) de Sitter space. We provide a detailed derivation of the action, Hamiltonian structure and equations of motion for such free particle. We study the action of deformed relativistic symmetries on the phase space and derive explicit formulas for the action of the deformed Poincaré group. Finally we provide a discussion on parametrization of the particle worldlines stressing analogies and differences with ordinary relativistic kinematics.

I. INTRODUCTION

Field theories with curved momentum space have a long history which dates back to the early 60s when they were studied as possible examples of divergence free theories [START_REF] Yu | On the introduction of an "elementary length" in the relativistic theory of elementary particles[END_REF]. The idea of curving momentum space had been actually already introduced by Snyder in his classic paper on non-commutative space [START_REF] Snyder | Quantized space-time[END_REF]. In the past decade the topic has received a boost of new interest for different reasons. On one side certain classes of non-commutative field theories covariant under deformed relativistic symmetries have been shown to posses Fourier transformed counterparts which are functions of group-valued momenta. On the other hand "group field theories", in the path integral approach, have been studied as powerful tools to generate amplitudes in the spin foam approach to quantum gravity .

Interestingly enough group-valued momenta emerge even in more "canonical" settings namely when analyzing the behaviour of a relativistic particle coupled to gravity. Indeed it is well known that the phase space of a standard relativistic particle is geometrically described by the cotangent bundle of Minkowski spacetime. In this case the cotangent bundle is just a product of Minkowski spacetime, the position space, and the dual of its tangent space, the momentum space, which is again isomorphic to R 4 . In the case when gravity is present this simple picture becomes more complex. First, gravity usually curves space, resulting in the fact that elements of momentum space become non-commutative, with degree of non-commutativity provided by spacetime curvature. It turns out however that even in the case when local degrees of freedom of gravity are not present and spacetime is flat gravity may still influence the form of the phase space of the particle in a nontrivial way. This happens, for example, for gravity in 2 + 1 dimensions. In this case it follows from Einstein equations that the Riemann tensor vanishes thus spacetime is flat [START_REF] Staruszkiewicz | Gravitation Theory In Three-Dimensional Space[END_REF], [START_REF] Deser | Three-Dimensional Einstein Gravity: Dynamics Of Flat Space[END_REF], [START_REF] Witten | 2+1)-Dimensional Gravity as an Exactly Soluble System[END_REF] and the only effect of gravity is in the possibility of non-trivial topology of the space-time manifold. If we however couple 2 + 1 gravity to point particles the behaviour of the latter becomes highly nontrivial: one can show that taking (topological) degrees of freedom of gravity into account results in effective relativistic particle kinematics with curved momentum space, the picture somehow dual to the particle moving in curved spacetime (see e.g., [START_REF] Matschull | Quantum mechanics of a point particle in 2+1 dimensional gravity[END_REF] and [START_REF] Schroers | Lessons from (2+1)-dimensional quantum gravity[END_REF] for a recent compact review.).

In 3 + 1 dimensions the situation is much more complex, and only some partial results have been obtained so far. First of all, contrary to 2 + 1 dimensions, four-dimensional gravity is not described by a topological field theory, simply because it clearly possesses local degrees of freedom. However it is possible to formulate gravity as a constrained topological field theory [START_REF] Smolin | General relativity with a topological phase: An action principle[END_REF], [START_REF] Freidel | Quantum gravity in terms of topological observables[END_REF], with well defined topological limit, in which the local degrees of freedom are switched off and the spacetime becomes locally the flat Minkowski (in the vanishing cosmological constant limit) with its standard group of spacetime symmetries. It is also possible to couple point particles to gravity formulated in this guise [START_REF] Freidel | Particles as Wilson lines of gravitational field[END_REF], [START_REF] Kowalski-Glikman | Can we see gravitational collapse in (quantum) gravity perturbation theory?[END_REF]. One expects that when the topological limit is taken, the theory is going to exhibit the same kind of phenomena as in 2 + 1 dimensional case: the particle action is deformed with the particle's momentum space becoming curved, the curvature provided by the mass scale of the theory, the Planck mass κ.

In this paper we will investigate properties of a relativistic particle, whose momentum space is the four-dimensional de Sitter space of curvature κ, which, in terms of the standard coordinates on five dimensional Minkowski space P A , A = 0, . . . , 4 is defined as a four-dimensional surface

-P 2 0 + P 2 1 + P 2 2 + P 2 3 + P 2 4 = κ 2 . (1.1)
In what follows we will make use of the coordinates k µ , µ = 0, . . . , 3 on this space defined by

P 0 (k 0 , k) = κ sinh k 0 /κ + k 2 2κ e k0/κ , P i (k 0 , k) = k i e k0/κ , (1.2)
P 4 (k 0 , k) = κ cosh k 0 /κ - k 2 2κ e k0/κ .
It should be noticed that the coordinates k are not global and cover only a half of de Sitter space defined by the inequality P 0 (k) + P 4 (k) > 0, which turns out to be a group manifold of the group AN(3). More details can be found in [START_REF] Freidel | Field theory on κ-Minkowski space revisited: Noether charges and breaking of Lorentz symmetry[END_REF]. As explained in [START_REF] Arzano | Lorentz invariant field theory on kappa-Minkowski space[END_REF] there are pathologies in the construction of theories if we do not restrict the momentum manifold to AN(3) which we are going to do here. The Casimir for Lorentz transformations acting on the first four coordinates (forming the SO(3, 1) subgroup of the SO(4, 1) group that leaves the quadratic form (1.1) invariant) is given by

C = P 2 0 -P 2 -m 2 , (1.3)
and it follows from (1.2) that in k coordinates it has the form

C κ ≡ 4κ 2 sinh 2 k 0 2κ -k 2 e k0/κ = m 2 κ = 2κ 2 1 + m 2 κ 2 -1 . (1.4)
These are the prerequisites for our construction of the particle's with de Sitter momentum space action which we carry on in the next Section. After deriving the action we will briefly describe the construction of the phase space and the Hamtonian analysis of the model. Then in Section III we turn to the description of the action of deformed symmetries on the phase space and discuss the possible physical interpretation. Such a discussion, we hope, could be of interest in view of recent debate concerning some alleged paradoxes [START_REF] Hossenfelder | Bounds on an energy-dependent and observer-independent speed of light from violations of locality[END_REF] 1 , from which, as it is claimed, the whole class theories with deformations provided by an observer-independent energy scale [START_REF] Amelino-Camelia | Testable scenario for relativity with minimum-length[END_REF], [START_REF] Amelino-Camelia | Relativity in space-times with short-distance structure governed by an observer-independent (Planckian) length scale[END_REF], [START_REF] Kowalski-Glikman | Observer independent quantum of mass[END_REF], [START_REF] Bruno | Deformed boost transformations that saturate at the Planck scale[END_REF], [21] [22] suffers. We hope that the analysis presented in this paper will help to further clarify the issue of internal consistency of this class of theories.

II. RELATIVISTIC PARTICLE WITH CURVED MOMENTUM SPACE

Let us turn to the construction of the action principle for a single free particle whose momentum space is half de Sitter space covered by the coordinates (1.2) [START_REF] Girelli | The free particle in deformed special relativity[END_REF]. Noticing that de Sitter space is most simply described by equation (1.1) it seems convenient to start our investigations in 5 dimensions and then reduce the model one dimension down. Thus the lagrangian of a relativistic particle with de Sitter valued momenta should contain three terms: the standard kinetic term L = ẊA P A , A = 0, . . . , 4 with dot denoting derivative over the evolution parameter τ , which makes the unconstrained phase space of the model 5 + 5 dimensional, and two constraints: one forcing the momentum space to be de Sitter

S ≡ P

A P A -κ 2 ≈ 0 , (2.1)
and the second enforcing the mass-shell conditions2 (1.3)

C ≡ -P µ P µ -m 2 ≈ 0 . (2.
2)

The action we start with has then the form

S = dτ P A ẊA + Λ S + λ C , ẊA = dX A dτ . (2.
3)

It follows that the Poisson brackets of the basic phase space variables X A and P A are canonical, i.e.,

X A , P B = δ A B (2.4)
We will solve explicitly the S constraint to obtain the action in terms of the 4 + 4 phase space variables with momenta belonging to the AN(3) group manifold. As a first step notice that since S is a first class constraint we must gauge fix it, by adding to the model another constraint G whose Poisson bracket with S does not vanish {S, G} = 0 .

(2.5)

After imposing G we are dealing with a system with two second class constraints, which therefore describes 4 + 4dimensional phase space. It is convenient to parametrize this reduced phase space with variables x µ , k µ that have vanishing Poisson bracket with the constraints

{x µ , S} = {x µ , G} ≈ 0 , (2.6) {k µ , S} = {k µ , G} ≈ 0 , (2.7)
where the last equalities mean that the Poisson brackets are weakly zero, i.e., equal zero on the surface S = G = 0. It follows that the Dirac brackets of these variables and arbitrary functions of them is equal to their original Poisson bracket, because they differ just by terms proportional to the brackets (2.6), (2.7). One can interpret these equations along with (2.5) as providing the parametrization of the original phase space in terms of (S, G, x µ , k µ ) with the reduced phase space of interest being defined by condition S = G = 0. In order to find the reduced action describing the resulting dynamics of the system with curved momentum space we must choose the gauge fixing function G, find the expressions for X A , P A in terms of the variables x µ , k µ satisfying (2.6), (2.7) in such a way that both S(X, P ) and G(X, P ), when expressed in terms of the new variables, are identically zero, and plug them back to the original action (2.3). The first step of the construction is therefore to choose the explicit form of the gauge fixing G. Here we must recall that we are not interested in the whole de Sitter space as a space of momenta, but only in its AN(3) group submanifold defined in addition to the condition S = 0 by the requirement P 0 + P 4 > 0. Thus we have a natural choice for G if we impose its Poisson bracket with S to be fixed by such condition, namely

{G, S} = P 0 + P 4 , (2.8)
which leads to the following expression

G = 1 2 X 4 -X 0 -T (2.9)
with T being an arbitrary number.

The momentum sector of the reduced phase space can be straightforwardly described by k µ given by the coordinates on AN(3) given in (1.2). Indeed any function of the momenta P A has vanishing Poisson bracket with S; as for G it has vanishing Poisson bracket with P i , i = 1, 2, 3 and with P 0 + P 4 = κ e k0/κ and thus with k 0 which can be seen as a function of P 0 + P 4 .

The position part of the reduced phase space can be constructed as follows. Notice that de Sitter space has a ten parameter group of symmetries, whose action is generated, through Poisson bracket by five dimensional Lorentz transformations, with generators J AB = 1 κ (X A P B -X B P A ), where we add the prefactor to make J AB dimensionless. One easily checks that indeed

{J AB , S} = 0 .
(2.10)

We can therefore define coordinates using linear combinations of J AB which leave G invariant. Since these combinations must give us four coordinates x 0 , x i one of whose is a rotation scalar and the remaining three form a three vector we choose

x 0 ≡ J 04 , x i ≡ J 0i + J 4i ≈ - 1 κ (P 0 + P 4 ) X i = -X i e k0/κ , (2.11)
which indeed both have weakly vanishing Poisson bracket with G. One finds (keeping in mind the changes in sign in raising and lowering indices)

x 0 , x i = 1 κ x i , x i , x j = 0 , (2.12)
i.e. the same algebra structure of the so called κ-Minkowski space [START_REF] Majid | Bicrossproduct Structure Of Kappa Poincare Group And Noncommutative Geometry[END_REF].

To derive the reduced action it remains to plug the expressions for X A as functions of x µ and k µ along with (1.2) into (2.3). One finds (disregarding the total derivatives and changing the sign)

S red = dτ k µ ẋµ - 1 κ k i x i k0 + λ P µ (k)P µ (k) + m 2 (2.13)
We see that the deformation appears in two places: first we have the term 1 κ k i x i k0 which deforms the symplectic structure, and second, we have to do with the deformed on-shell condition, enforced by the Lagrange multiplier λ, P µ (k)P µ (k) + m 2 κ = 0. In the following we will take 3 λ = 1 remembering that we must add the on-shell condition to the equations of motion.

Let us point out here that the kinetic term in the action (2.13) can be derived also in the following, more straightforward way. Namely, since the momentum manifold is a group manifold of the group AN(3), we can make use of the Kirillov symplectic form [START_REF] Kirillov | Elements of the Theory of Representations[END_REF] to define the kinetic term to be

x, g -1 d dτ g (2.14)
where

AN(3) ∈ g = exp(ik 0 X 0 ) exp(ik i X i ) ( 2 . 1 5 ) and x = x µ Y µ (2.16)
with X µ are generators of the an(3) Lie algebra and Y µ are elements of the dual algebra i.e.,

[X 0 , X i ] = - i κ X i , [X i , X j ] = 0 , Y µ , X ν = iδ ν µ (2.17)
Substituting (2.15), (2.16) to (2.14) one easily reproduces the kinetic term of (2.13).

It is worth mentioning also that there is nothing strange with the fact that time derivatives of momenta appear in the action. This just reflects the fact that we have constructed our action on the cotangent bundle to the curved momentum space. In fact, one can get an exact analogue of the kinetic term of the action (2.13), with the role of momentum and coordinates exchanged in the κ dependent term, considering an ordinary relativistic particle moving in de Sitter spacetime, in flat coordinates.

Varying S red with respect to x µ we get the condition that momenta are conserved kµ = 0 .

(2.18)

Varying with respect to momenta we obtain relations between components of four velocity and momenta

ẋ0 = 1 κ k i ẋi + ∂ ∂k 0 (P µ (k)P µ (k)) , ẋi = ∂ ∂k i (P µ (k)P µ (k)) . (2.19)
3 Notice that λ is a Lagrange multiplier that is needed first to enforce the on-shell condition, and second to make the action invariant under τ local reparametrizations (one dimensional general coordinate transformations.) Indeed, if we take τ → τ = f (τ ) then assuming that both x(τ ) and k(τ ) are scalars with respect to this transformation, the kinetic term in S red is invariant. To make the mass-shell term invariant as well we must assume that λ(τ ) transforms so that dτ λ (τ ) = dτ λ(τ ) i.e.,

λ (τ ) = " df (τ ) dτ « -1 λ(τ ) ,
which means that λ is a one dimensional vielbein determinant. This means that we can always use this transformation to make λ constant. Although one cannot make it equal 1 in general because the length of the interval

Z τ 2 τ 1 dτ λ is invariant
, in what follows we will ignore this difficulty and use the gauge λ = 1.

We will return to these equations in the following section.

Let us now finish this section calculating the Poisson brackets of the system described by the action S red , (2.13). To this end instead of doing the full canonical analysis we can use the following trick. The kinetic term of S red can be rewritten as follows

-ki x i i -x 0 + 1 κ k i x i k0
from which it follows that the Poisson brackets are

x i , k j = δ i j , x 0 + 1 κ k i x i , k 0 = 1
with vanishing all other brackets between these variables. Since x i , k 0 = {k i , k 0 } = 0 we deduce that x 0 , k 0 = 1.

Similarly we compute the brackets of x 0 with k i and x i to obtain the following set of the Poisson brackets of the variables x µ and k µ

x 0 , x i = 1 κ x i , (2.20)
x 0 , k 0 = 1 , (2.21)

x i , k j = δ i j , (2.22)
x 0 , k j = -

1 κ k j , (2.23)
with all other brackets being zero. This is the desired classical counterpart of the κ-deformed phase space [START_REF] Ghosh | [END_REF][START_REF] Lukierski | Classical Quantum Mechanics Of Free Kappa Relativistic Systems[END_REF]. At this point it will be useful to pause for a moment to discuss the structures we have encountered so far from an algebraic viewpoint. By construction the "momentum" variables {k µ } are coordinate functions on the group AN (3). They are also generators of translations and as such can be viewed as a (trivial) Lie algebra with Lie bracket given by their Poisson brackets. Being coordinate functions on the group AN (3) they have a natural non-abelian composition rule which turns their space into a group. The space-time coordinates we chose above, due to their non-trivial Poisson brackets, can be be seen as generators of such group. At the same time due to the "pairing" {x µ , k ν } = δ µ ν the co-ordinates {x µ } span the (non-trivial) Lie algebra dual to the one of translation generators. In both cases the non-trivial Poisson and Lie brackets are characteristic signature left by the curvature of momentum space on the "dual" space spanned by the x µ .

III. SYMMETRIES AND NOETHER CHARGES

The action S red , (2.13) is, by construction, invariant under action of deformed, infinitesimal, global spacetime symmetries: translations and Lorentz transformations, which together form a ten-parameters algebra. Here we will find an explicit form of such transformations and the associated Noether charges. Consider first time translation generated by k 0 . Since k 0 has a nontrivial bracket with x 0 only we find that δx 0 = a 0 , δx i = 0 , δk µ = 0 .

(3.1)

For an arbitrary spacial translation generated by a i k i we have instead

δx 0 = 1 κ k i a i , δx i = a i , δk µ = 0 (3.2)
so the action will be invariant under the following transformations

δx 0 = a 0 + 1 κ k i a i , δx i = a i , δk µ = 0 . (3.3)
To find Noether conserved charges associated with these transformations we use the usual trick: we first make the parameters local, i.e., τ dependent and then read off the Noether charges Q trans from the general expression4 δL = ȧµ Q µ . One checks that

Q trans µ = k µ , (3.4)
which are obviously time independent by virtue of equations of motion, (2.18).

Since the action S red is manifestly invariant under space rotations it is straightforward to write down the infinitesimal rotations with parameter θ i

δx i = θ j ji l x l , δk i = θ j j i l k l , δx 0 = δk 0 = 0 . (3.5)
The associated Noether charges are just the components of angular momentum

M i = ij l x l k j (3.6)
and again by using equations of motion (2.18), (2.19) one easily checks that they are conserved.

Let us finally turn to the invariance of the action S red under infinitesimal boosts parametrized by ξ i . To do that we must first find the transformation rules for components of the momenta k µ . To this end it is sufficient to notice that it follows from the form of the original action (2.3) that the spacetime components of the momenta P transform under boosts as four-vectors, to wit

δ ξ P 0 = ξ i P i , δ ξ P i = ξ i P 0 . (3.7)
Using this and the explicit expressions for P as functions of k (1.2) by straightforward calculation we find

δ ξ k i = ξ i κ 2 1 -e -2k0/κ + k 2 2κ - k i ξ • k κ , δ ξ k 0 = ξ • k , (3.8)
with ξ • k = ξ i k i . Imposing invariance of the action S red one finds that positions must transform like

δ ξ x i = -ξ i x 0 + 1 κ x i ξ • k - 1 κ k i ξ • x (3.9)
and

δ ξ x 0 = -ξ • x + 1 κ x 0 ξ • k + ξ • x 1 2 1 -e -2k0/κ + k 2 2κ 2 .
(3.10)

The conserved charges associated with boosts are

N i = -k i x 0 -x i κ 2 1 -e -2k0/κ + k 2 2κ . (3.11)
Notice that the transformation laws for positions (3.9) and (3.10) can be rewritten in the following simple form

δ ξ x i = -ξ i x 0 - 1 κ ijk ξ j M k (3.12)
and

δ ξ x 0 = -ξ • x + 1 κ ξ • N . (3.13)
It can be checked by explicit calculations that the charge (3.11) generates Lorentz transformations through Poisson bracket.

δS = Z τ 2 τ 1 dτ " ∂L ∂φ δφ + ∂L ∂ φ d dτ δφ « = Z τ 2 τ 1 dτ d dτ " ∂L ∂ φ δφ « ,
where in the last equality we use equations of motion. Since τ 1 and τ 2 are arbitrary, and δφ = af (φ) with some f and τ -independent a, the Noether charge Q = ∂L ∂ φ f (φ) is time independent, i.e., conserved. But from inspection of the middle term in the formula above it follows that if we allow a to be τ -dependent the only term containing ȧ is ȧ Q.

IV. FINITE DEFORMED POINCAR É TRANSFORMATIONS AND κ-POINCAR É GROUP

In this section we address the problem of integration of infinitesimal Lorentz transformations derived in the previous section. In this way we will find κ-Poincaré group action5 , which describes the finite transformations corresponding to Poisson bracket representation of κ-Poincaré algebra.

Let us start with Lorentz transformations of momenta. For simplicity we choose our coordinates in such a way that the boosts act along the first axis, ξ i = (ξ, 0, 0). Instead of calculating the finite boosts k µ (ξ) directly it is more convenient to start with the P A variables that transform as components of four-vector and a scalar. Thus we have

P 0 (ξ) = P (0) 0 cosh ξ + P (0) 1 sinh ξ , P 1 (ξ) = P (0) 1 cosh ξ + P (0) 0 sinh ξ , (4.1)
P 2 (ξ) = P (0) 2 , P 3 (ξ) = P (0) 3 , P 4 (ξ) = P (0) 4 = κ 2 + m 2 , (4.2)
where P (0)

A denotes the initial values of the components. Using these equations and relations (1.2) one can calculate the expressions for k µ (ξ), which have been first obtained in [START_REF] Bruno | Deformed boost transformations that saturate at the Planck scale[END_REF]. In what follows we will not need these explicit formulas.

Our main goal is to derive the action of finite boosts and translations on position variables. Let us start with boosts. The transformation laws (3.12) and (3.13) can be rewritten as differential equations for ξ dependence

d dξ x 0 (ξ) = -x 1 (ξ) + 1 κ N 1 (ξ) d dξ x 1 (ξ) = -x 0 (ξ) d dξ x 2 (ξ) = 1 κ M 3 (ξ) , d dξ x 3 (ξ) = - 1 κ M 2 (ξ) ( 4 . 3 )
To solve these equations we must first find out what is the ξ dependence of the relevant components of rotation and boost charges. Since the generator of boost is

N 1 from d/dξN 1 ≡ {N 1 , N 1 } = 0 we see that N 1 = n 1 is just a constant.
Similarly, the first component of the rotation charge M 1 (ξ) = m 1 is constant. As for the other components we have the equations

d dξ M 2 (ξ) = -N 3 (ξ) , d dξ M 3 (ξ) = N 2 (ξ) , d dξ N 2 (ξ) = -M 3 (ξ) , d dξ N 3 (ξ) = M 2 (ξ) .
Denoting the initial values with lowercase letters we have

M 2 (ξ) = m 2 cosh ξ + n 3 sinh ξ M 3 (ξ) = m 3 cosh ξ -n 2 sinh ξ N 2 (ξ) = n 2 cosh ξ -m 3 sinh ξ N 3 (ξ) = n 3 cosh ξ + m 2 sinh ξ. (4.4)
Substituting these expressions in (4.3) we find

x 0 (ξ) = x 0 (0) cosh ξ + x 1 (0) sinh ξ - n 1 κ sinh ξ, x 1 (ξ) = x 0 (0) sinh ξ + x 1 (0) cosh ξ - n 1 κ cosh ξ + n 1 κ x 2 (ξ) = x 2 (0) - n 2 κ + n 2 κ cosh ξ - m 3 κ sinh ξ x 3 (ξ) = x 3 (0) - n 3 κ + n 3 κ cosh ξ + m 2 κ sinh ξ. (4.5)
Using (3) these equations can be rewritten in a simpler form, similar to the standard Lorentz transformations, to wit

x 0 (ξ) = x 0 (0) cosh ξ + x 1 - n 1 κ (0) sinh ξ, x 1 - n 1 κ (ξ) = x 0 (0) sinh ξ + x 1 - n 1 κ (0) cosh ξ x 2 - n 2 κ (ξ) = x 2 - n 2 κ (0) x 3 - n 3 κ (ξ) = x 3 - n 3 κ (0) . (4.6)
It follows that for a free particle with given values of boost charge n i the set of four numbers (x 0 , X i ), X i = x i -n i /κ behaves as a set of components of a standard Lorentz four-vector. Notice also that with the help of (3.6) and (3.11) we can express the n i and m i constants in terms of the initial positions and components of (conserved) momentum k. It should be stressed that for the coordinates (x 0 , x i ), contrary to the action of the standard Lorentz group, there is a nontrivial action in transverse direction, which is suppressed by the deformation scale κ.

V. PARTICLE WORLDLINES AND NON-LOCALITY

Let us now turn to the discussion of some consequences of the deformed Poincaré transformations derived in the preceding section. The question we would like to address is the following: suppose we have two free particles whose worldlines cross for one observer, say Alice. Do they cross for another Lorentz boosted observer, Bob? This problem has been discussed first (in a slightly different guise) in [START_REF] Schutzhold | Problems of doubly special relativity with variable speed of light[END_REF], and the argument of this paper has been questioned in [START_REF] Arzano | Comment on 'Large-scale non-locality in doubly special relativity with an energy-dependent speed of light[END_REF]. Recently discussion on this issue has been sparked by the paper [START_REF] Hossenfelder | Bounds on an energy-dependent and observer-independent speed of light from violations of locality[END_REF] and the extensive debate on this issue has appeared in the follow-up papers [START_REF] Amelino-Camelia | Taming nonlocality in theories with deformed Poincare symmetry[END_REF], [START_REF] Smolin | Classical paradoxes of locality and their possible quantum resolutions in deformed special relativity[END_REF], [START_REF] Smolin | On limitations of the extent of inertial frames in non-commutative relativistic spacetimes[END_REF].

Following [START_REF] Amelino-Camelia | Taming nonlocality in theories with deformed Poincare symmetry[END_REF] we find it convenient to describe the worldline of a particle as a set of points in spacetime for which the boost charge N i is constant. Take X i and T the coordinates of the point where the two wordlines cross (for Alice) and denote t = x 0 . Then the spacetime points belonging to the particle worldline satisfy

k i (t(τ ) -T ) + E (x i (τ ) -X i ) = 0 , (5.1)
where

E ≡ κ 2 1 -e -2k0/κ + k 2 2κ . (5.2)
Since E = e -k0/κ P 0 (cf. (1.2)), we see that for massless particles for which P 0 = |P| the velocity is

v = x i (τ ) -X i t(τ ) -T = |k| E = |P| P 0 = 1 , (5.3)
(see [START_REF] Daszkiewicz | Velocity of particles in doubly special relativity[END_REF] and [START_REF] Girelli | The free particle in deformed special relativity[END_REF].). Notice in passing that in terms of k and E the mass shell relation reads

E 2 -k 2 = m 2 e -2k0/κ . (5.4)
Let us now turn to the non-locality issue raised in [START_REF] Hossenfelder | Bounds on an energy-dependent and observer-independent speed of light from violations of locality[END_REF] and further discussed in [START_REF] Smolin | Classical paradoxes of locality and their possible quantum resolutions in deformed special relativity[END_REF] and [START_REF] Amelino-Camelia | Taming nonlocality in theories with deformed Poincare symmetry[END_REF]. Take two worldlines of massless particles that cross for an observer, Alice, at the spacetime event with coordinates X i and T . Thus the worldlines are described by the equations

k (1) i (t (1) (τ ) -T ) + E (1) (x (1) i (τ ) -X i ) = 0 , k (2) i (t (2) (τ ) -T ) + E (2) (x (2) i (τ ) -X i ) = 0 . (5.5)
where E (1/2) = |k (1/2) | because both particles are massless by assumption, and it is clear that they cross at T , X. These worldlines are seen by Bob as

k (1) i (t (1) (τ ) -T (1) ) + E (1) (x (1) i (τ ) -X (1) i ) = 0 , k (2)
i (t (2) (τ ) -T (2) ) + E (2) (x (2) i (τ ) -X (2) i ) = 0 , (5.6)
and in general they do not cross, because the point T (1) = T (2) , X

(2) i as a result of the fact that the points belonging to the two worldlines do not transform in the same way. To see this explicitly, let in the Alice's frame the coordinates of the crossing point be (T, X) = (0, X, 0, 0) and let the particle (1) moves along the second axis k (1) = (0, k (1) , 0), while the particle (2) along the third k (2) = (0, 0, k (2) ). The components of the boost and angular momentum vectors for the particles are (since these are the conserved quantities i.e., independent of the worldline point, it is sufficient to calculate them at the crossing point) n (1) = (|k (1) | X, k (1) T, 0) n (2) = (|k (2) | X, 0, k (2) T ) ,

m (1) = (0, 0, k (1) X) m (2) = (0, -k (2) X, 0) .

(5.8)

Writing the worldline equation as seen by Alice in the form

E (I) x (I) i + k (I) i t (I) = n (I) i , (5.9)
we find that in Bob's boosted frame they have the form

E (I) x (I) i + k (I) i t (I) = n (I) i , (5.10)
with (see (4.4))

n (1) = (|k (1) | X, k (1) (T cosh ξ -X sinh ξ), 0) , n (2) = (|k (2) | X, 0, k (2) (T cosh ξ -X sinh ξ)) .

(5.11)

We still need explicit expressions for transformed energy and momentum, which can be found straightforwardly using the fact that the P 0 , P i transform as components of a Lorentz four-vector (4.1), (4.2), to wit

E = κ E cosh ξ + k 1 sinh ξ E cosh ξ + k 1 sinh ξ + e -k0/κ √ κ 2 + m 2 , k 1 = κ k 1 cosh ξ + E sinh ξ E cosh ξ + k 1 sinh ξ + e -k0/κ √ κ 2 + m 2 , k 2/3 = κ k 2/3 E cosh ξ + k 1 sinh ξ + e -k0/κ √ κ 2 + m 2 , (5.12)
where on-shell, for massless particles e -k0/κ = 1 -|k| κ .

(5.13) Substituting these into (5.10) and denoting

D = |k| κ (cosh ξ -1) + 1 ,
we find that the worldlines in Bob's frame take the form t (1) sinh ξ + x (1) cosh ξ = XD (1) t (2) sinh ξ + x (2) cosh ξ = XD (2) t (1) + y (1) cosh ξ = (T cosh ξ -X sinh ξ)D (1) y (2) = 0 (5.14)

z (1) = 0 t (2) + z (1) cosh ξ = (T cosh ξ -X sinh ξ)D (2) .

It is clear that if D (1) = D (2) i.e., if k (1) = k (2) the worldlines do not cross anymore.

For completeness consider now the case, in which the second particle moves along the first axis, towards the origin. We have n (2) = (|k (2) |(X -T ), 0, 0) , m (2) = (0, 0, 0) (5.15) and n (2) = (|k (2) |(X -T ), 0, 0) (5.16) so that

x (2) -t (2) = (X -T ) e -ξ + |k (2) | κ (1 -e -ξ ) , y (2) = z (2) = 0 .

(5.17)

Again, for Bob, the worldlines are missing each other. It can be also shown directly that an analogous result holds if Bob is a translated observer. We conclude that if we use the κ-Minkowski coordinates x to parametrize events in spacetime, or in the geometrical language, to provide coordinates, if two worldlines cross at (T, X i ) for Alice, for the Lorentz boosted Bob, in general, the coordinates of all points on the worldlines do not coincide. In other words it would seem that while Alice sees the worldline cross, Bob does not.

In the next Section we argue that, according to our analysis, this conclusion (taken as a strong point in [START_REF] Hossenfelder | Bounds on an energy-dependent and observer-independent speed of light from violations of locality[END_REF] and implicitly accepted in [START_REF] Amelino-Camelia | Taming nonlocality in theories with deformed Poincare symmetry[END_REF], [START_REF] Smolin | Classical paradoxes of locality and their possible quantum resolutions in deformed special relativity[END_REF], [START_REF] Smolin | On limitations of the extent of inertial frames in non-commutative relativistic spacetimes[END_REF]) does not seem to be the correct description of the state of affairs.

VI. DISCUSSION

In ordinary classical mechanics there are two equivalent ways to describe the phase space of a free relativistic particle. The first, more common, approach is simply to take as the configuration space the "range" space of the coordinates of a particle (Minkowski space) and define the (unreduced) phase space as the cotangent bundle of such configuration space. The physical (on-shell) phase space will be given by a six-dimensional sub-manifold of the unreduced phase space whose coordinates parametrize geodesics in Minkowski space. For the alternative description one starts from an unreduced phase space given by the dual space of the Poincaré algebra. For a spinless free particle the physical phase space is then given by the six-dimensional submanifold defined by the orbits of the Poincaré group6 on such space (for details see e.g. [START_REF] Carinena | Relativistic quantum kinematics in the Moyal representation[END_REF] and references therein) defined by hypersurfaces with vanishing Pauli-Lubanski vector and given mass. In this case the (physical) momenta are naturally associated to the (linear) momenta and the physical positions are defined using the energy of the particle and the boost charges x i = n i /p 0 (for a pedagogical discussion see [START_REF] Bacry | The Notions Of Localizability And Space: From Eugene Wigner To Alain Connes[END_REF]).

In a deformed symmetry setting it is by no mean obvious (as the ongoing debate in the literature shows [START_REF] Hossenfelder | Bounds on an energy-dependent and observer-independent speed of light from violations of locality[END_REF][START_REF] Smolin | Classical paradoxes of locality and their possible quantum resolutions in deformed special relativity[END_REF][START_REF] Amelino-Camelia | Taming nonlocality in theories with deformed Poincare symmetry[END_REF][START_REF] Smolin | On limitations of the extent of inertial frames in non-commutative relativistic spacetimes[END_REF]) what natural definition of the configuration space of a free relativistic particle one should adopt. Indeed there is no a priori reason why one should pick the particular set of κ-Minkowski coordinates as "physical" coordinates of a particle since (as discussed above) they just happen to describe a dual space to the translation sector of deformed algebra of symmetry but have no other compelling role that makes them desirable candidates to describe a particle's wordline. Indeed since the equivalent of "picture one" above for the phase space is now missing it seems sensible to resort to the second option and define a particle's worldline using a combination of Poincaré momenta. Looking at the rules governing transformations of positions x under boosts (3.12), (3.13) one easily sees that the source of the problems is the fact that positions belonging to different worldlines transform in different way because the outcome of transformation depends on the momentum that the particle carries. Intuitively we want to construct new positions, linear combinations of x, such that their Lorentz transformations do not depend on momentum. Since the boost generators have the standard Poisson bracket it is convenient to parametrize momentum with the help of P µ coordinates (1.2), which transform as components of a standard Lorentz vector. This is precisely what one would expect following the more general description of phase space in terms of Poincaré momenta as described above.

More concretely we characterize every worldline by six parametrs: the components of boosts n i and a tangent vector whose components can be identified with those of the momentum P µ (notice that since the particles are on shell the P 0 component can be computed from P 2 0 -P 2 i = m 2 ). Thus the set of worldlines (physical phase space) is, as expected, six dimensional. We can now use coordinates in such space to describe coordinates of events in spacetime. Let us first define three dimensional space by mean of the phase space coordinates ζ i = n i /P 0 . Intuitively these coordinates span the space slice at time ζ 0 = 0. Now take a subset of worldlines which are characterized by P i = 0, P 0 = M , which are just worldlines corresponding to particles at rest with respect to the origin. Now take any worldline with n i = 0, i.e. that passes through the "origin" at the initial time. We can define a time coordinate ζ 0 as follows. Let at some event our worldline with n i = 0 cross the worldline of a static particle characterized by a given n stat i = Mζ i . Then the time is given be the relation P 0 ζ i + P i ζ 0 = 0, where ζ i is known from the boost components of the static particle. Clearly this procedure provides spacetime coordinates analogous to the standard ones, with the usual special relativistic transformation rules. Thus at the classical one particle level the kinematics of particle with group valued momenta can be described with usual special relativistic tools.

One may wonder where the deformation is gone if the particle kinematics is just that of special relativity? To answer this question we must recall the discussion at the end of Section II. Even though classically a description in terms of undeformed covariant coordinates is available the curvature of momentum space is still "hiding" in the dual space of translation generators under the form of non-trivial brackets. Such non-trivial bracket is ultimately responsible 7 for a series of non-symmetric behaviours of the space-time symmetry generators which affect quite crucially the definition of particle, antiparticle and multi-particle states of the corresponding quantum theory [START_REF] Arzano | Lorentz invariant field theory on kappa-Minkowski space[END_REF][START_REF] Arzano | Fun from none: deformed symmetries and Fock space[END_REF]. Thus the physical effects of deformation seem to show up fully only in the quantum theory and it is a challenge for future studies 8 to establish what, if any, kind of physical effects of the deformation could be revealed at the classical level.

For discussion of these claims see the recent analyses reported in[START_REF] Smolin | Classical paradoxes of locality and their possible quantum resolutions in deformed special relativity[END_REF],[START_REF] Amelino-Camelia | Taming nonlocality in theories with deformed Poincare symmetry[END_REF].

Note in passing that there is another natural candidate for the deformed mass-shell condition, namely P 4m ≈ 0, m2 = κ 2 + m 2 .

The proof goes as follows. Let the infinitesimal transformation with parameter a leave the action S invariant, δS = 0. Let us denote by φ(τ ) collectively all variables of the model. Then

We warn the reader, to avoid confusion, that the terminology "κ-Poincare group" has been used sometime to denote the Hopf algebra dual to the κ-Poincare algebra[START_REF] Kosinski | The Duality between kappa Poincare algebra and kappa Poincare group[END_REF].

Such approach offers also the most general formulation of a relativistic particle's phase space since encompasses the case of spinning particle and naturally leads to a description of the quantized system and its coupling to gauge fields.

Acknowledgments

We would like to thank Lee Smolin for many discussions and encouragement in carrying out this project. JKG is supported in part by research projects NN202318534 and Polish Ministry of Science and Higher Education grant 182/N-QGG/2008/0. He thanks Prof. Renate Loll for her hospitality during his visit to University of Utrecht, where this work started. MA is supported by a Marie Curie Intra-European Fellowship. MA would also like to thank the Perimeter Institute for Theoretical Physics for hospitality while this work was being completed. Research at Perimeter Institute for Theoretical Physics is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MRI.