Arnaud Gorin
email: arnaud.gorin@cerema.fr

Jérôme Gallet

Laurent Bigou

Recent advances of urban traffic management centers in French cities

Keywords: Traffic management, Urban ITS, Bus priority

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Introduction

Until recently, urban traffic management centers in France have been focused mainly on the improvement of the vehicle traffic flow and peak capacity [START_REF] Damas | La régulation du trafic urbain au service des politiques de déplacement[END_REF]. However, because of the political demand of encouraging modal shifts to decongest saturated cities, traffic management centers are now used to develop a better share of public space between vehicles and public transportations. In addition to a better share of public space, traffic management centers have to increase transportation efficiency, maximize travel time reliability, meet environmental requirements, reduce nuisances, promote economic activities and insure vulnerable person's safety. Urban traffic management centers have relied traditionally on signal light management and traffic light intersections synchronization. Signal light management consists to define the cycle time of traffic light according to a time-periodic pre-defined schedule. Traffic light intersections synchronization consists to coordinate a series of traffic lights to allow continuous traffic flow. However, though these methods yield to good results, they do not adapt in real-time to manage unpredictable events. In recent years, intelligent transport systems (ITS) such as artificial intelligence or automatic vehicle location system have become more attractive to manage traffic because they provide innovative solutions that can adjust in real-time to the traffic condition. Indeed, artificial intelligence involves the use of powerful algorithms to respond to real-time disruption, while automatic vehicle location (AVL) system involves the use of global positioning system (GPS) sensors that are able to collect real-time information about vehicle positioning. Despite this, they are few examples in the published literature of urban traffic management centers using ITS in France [START_REF] Acun | Les nouveaux outils de régulation du trafic urbain[END_REF][START_REF] Scemama | Claire-SITI: Système pour la supervision intelligente, intermodale et intégrée des réseaux de transport de surface l'expérience de la Ville de Toulouse[END_REF][START_REF] Lamboley | L'apport des nouvelles technologies dans la gestion du trafic urbain[END_REF]. This work proposes to review recent advances of traffic management centers in French cities, which are identified by their innovative projects. For each city, the operating system is described and feedback from traffic managers is discussed. The contribution of ITS to traffic management is also mentioned.

Systems description and feedback

City of Strasbourg: Preserving access to city center

Strasbourg is the capital region of Alsace in eastern France and has a metropolitan population estimated at 760,000 inhabitants. In the 70's, the city of Strasbourg realizes that the growing number of vehicles causes traffic related problems. Indeed the safety of pedestrians and bicycles is no longer assured. Buses get stuck in the traffic. Travel time in bus increases due to increasingly dense traffic. The noise, the stress and the pollution affect the quality of life. In 1976, the city decides to establish a traffic management center to reduce daily congestion. The traffic management center becomes operational in 1978. The principal objective is to give the bus priority at traffic light intersection. With the arrival of the tramway in 1994, the city of Strasbourg goes further. The intent is to reduce the traffic of vehicles in the city center. Trams have priority at all traffic signals. Transit of vehicles is prohibited in the historic center.

Pedestrian zones and bicycling routes are created.

Figure 1 -Retractable access control bollard system (credit: PC Sirac)

To enforce traffic rules, retractable bollards are installed in the heart of the city (figure 1).

They are used to permit only authorized vehicles to pass them, mainly taxis and garage owners. People, who live in the city center, have access too but only during the opening hours. 40 retractable bollards are now controlled directly from the traffic management center. They have succeeded in removing the traffic of vehicles in the city center.

The city aims also to change mobility behaviour by influencing people's choice of transport modes. One powerful tool is the use of mobile phone applications. Launched in 2011 by Strasbourg, Strasmap application enables people to organize their travel anywhere. All the real-time information content comes directly from the traffic management center. Strasmap informs in real-time about the traffic condition, the positioning of bus or tram station, the occupancy rate of car park, the air quality and the presence of road hazards. There is also a winter service, which allows users to know exactly which street is treated with ice melting chemicals. Up to now, the application has been downloaded more than 51,000 times.

To communicate, it is important to have a lot of traffic data. From 2015, the traffic management center is going to integrate vehicle probe data into the traffic management system. Vehicle probe data is defined as traffic flow information that is obtained from floating car data (FCD) or floating mobile data (FMD) [START_REF] Gorin | A probabilistic approach for data quality assessment of road hazard warnings in crowdsourced driving navigation systems[END_REF]. Indeed Strasbourg urban community consists actually of 1200 km streets and roads. From 2015, the traffic management center should be able to integrate more than 600 km of the Strasbourg highway network using FCD and FMD. Using these data, three goals will be pursued. First, by having traffic information on highway, it will be possible for the traffic management center to trigger real-time operations to control access of vehicles in the suburban area. This will desaturate surrounding municipalities and maintain traffic flow of public transportation. Secondly, these data will be used to improve mapping of traffic levels on the Strasmap application. Finally, it will help engineers to plan the transformation of the A35 highway into an urban boulevard.

City of Rennes: Bus priority based on a centralized architecture

Rennes is located on the north west of France. It is the capital region of Brittany and has 680,000 inhabitants in the town and suburban area. The public transportation is based primarily on an extensive bus network and a metro line. The bus network served around 72 million passengers per year. From 2005 to 2010, the bus priority was implemented into 60 traffic light intersections. The GPS/radio-based priority technology was selected and is described in figure 2. The technology is based on a decentralized architecture with short-range radio communication between bus on-board computer and traffic light intersection controller. The bus is localized by GPS and odometer measurements. The positioning is also corrected with road map information. The controller is equipped with a modem to exchange information with the bus. As traffic signal approaches, the priority request is transmitted automatically via the on-board computer to the controller. The system grants early or extended green signals to the bus via control commands issued by the controller. Though this method yields to good results, the cost of operating the bus priority system is expensive. Indeed, bus priority programming on each traffic light intersection is long and laborious. Furthermore, in the city center, due to the high number of traffic diversion, junction improvement and roadwork, the traffic condition can change and it is difficult in a short time to program again the bus priority.

Figure 2 -Functional description of bus priority

Hence in 2014, in order to improve the public transportation performance and to reduce the operating cost, the city of Rennes agreed to upgrade the traffic management center. The upgrade includes the development of a centralized architecture based on a priority algorithm. This time, the traffic management center receives information about the position of the bus via the controller. Using this information, the algorithm is able to take into account the predicted time to coordinate a series of traffic lights by gradually adapting the green signal window. The objective is to minimize the effect of a bus priority request on the operation of a traffic light intersection. The priority request is sent to the controller and contains the following information: the identification of the bus, the trajectory code, the priority code, its expected arrival time and a tolerance. The controller performs the priority request based on pre-programmed parameters, which are the green light delay before the arrival of the bus and the maximum/minimum green light lengths. The pre-programmed parameters can be directly downloaded from the traffic management center into the controller. The advantages of the centralized architecture are threefold: firstly it simplifies the programming of the controllers, secondly it can adapt to the traffic condition and thirdly it allows rapid actions. The implementation of the new algorithm is scheduled to deploy over the next eight years across all the controllers.

City of Nantes: Roundabouts with active priority request

Nantes is the eighth largest city in France in terms of urban area population with 880,000 inhabitants. Public transportation in the city comprises: 3 tram lines; 46 bus lines; 8 bus rapid transit lines (BRT) and 3 boat shuttles. These services provided around 537,000 trips per day in 2013.

Nantes policy is totally oriented towards the development of public transportation. The transportation policy of Nantes is based on 2 fundamental techniques -the lane reduction and the speed reduction. The lane reduction is that the number of lanes is reduced in favour of pedestrians, bicycles and public transportation. The speed reduction is that the road infrastructures are modified so that the vehicle speed can be reduced. Concretely, Nantes network consists of 375 km cycling routes, 132 km BRT and tramway lanes with 32 km lanes only dedicated to bus (ie. bus lane). Since the 2000s, the heart of the city is pedestrianized and in 2012, a limited traffic zone is implemented. Over the years, Nantes has been building roundabouts instead of the classical traffic light cross intersections to slow down drivers and to promote a continuous, circular flow of traffic. Typical roundabouts with bus priority in the city are represented in figure 3.

Figure 3 -Roundabout with bus priority and dedicated lane (credit: PC Nantes)

In parallel, policy measures are adopted to have BRT and tram priority at traffic light intersections. Due to the city road-building policies, the traffic management center has developed a specific strategy to implement the public transportation priority. At present, the traffic management center operates 346 traffic light intersections. These include 200 traffic light intersections (mostly roundabouts with active priority request) that are dedicated to the tramway and the BRT lines. Tramway priority at traffic signals uses selective loops, while BRT priority uses GPS/radio-based priority. The strategy is based on decentralized priority request communications. In the case of BRT priority (and similarly for tram), 4 positioning zones can be defined (figure 4):

1) Call zone: the BRT announces its arrival to the traffic light controller. The approach delay is periodically updated. It is to ensure a modification of the signal timing at the traffic light to allow the passing of the BRT. 2) Stop zone: If there is a bus station, the BRT door open/close status is used to update the approach delay and to release the green phase.

3) Crossing zone: Once the BRT passes the traffic signal, the red phase for the BRT is launched.

4) Clearance zone: Once the BRT reaches the clearance zone, the green phase for the vehicles is released. If for some reason, the BRT is blocked in the intersection, the phase for the vehicles is kept red. They cross traffic intersection using a special phase and they are the only ones permitted. To ease traffic flow, most traffic intersections work with two phases. The vehicles turning left have to wait on the platform. It is important to clear the platform before the arrival of the tram. A video detection system is used to do so. Up to four cameras monitor in real-time the most sensitive traffic intersections along the line. A video analyzer defines the platform occupancy area (figure 5). An alarm is transmitted to the controller if a predetermined threshold is exceeded. In that case, the controller can extend the duration of the red signal to clear the platform.

Figure 5 -Video detection system (credit: PC Lutèce)

Furthermore, it is possible to manage the priority request directly from the traffic management center via remote parameters adjustment. For example, the tram operator sends travel time data and can ask the traffic management center to adjust the approach delay on some intersections. It is also possible to change the priority request parameters in case of loops detection failure. Moreover, to manage special events (such as Congress at Porte de Versailles), stopping times in station can be changed in real-time.

Finally it should be noted that the traffic intersections are managed by 2 different systems, each having their own transmission network. The first system manages the traffic light timing plan of the whole city, while the second system manages specifically the priority request along the T3 tramline. Both systems communicate with each other to improve traffic flow when there is no tram.

Once the T3 tramline was completed, it was observed that the road traffic flow was divided by 2. However it seems that the tramline didn't create modal shifts, but rather that the drivers were diverted to other traffic routes (in particular the Boulevard Périphérique). A drop in traffic accidents was also observed.

City of Bobigny: Powerful algorithms to manage traffic flow

Bobigny is located in the Seine Saint Denis district of northeast Paris. The traffic management center of Bobigny (figure 6) is currently updating its system to operate traffic light intersections. The main purposes are to reduce nuisance and to improve the quality of traffic flow. Three methods are currently tested. The first method is based on a threshold algorithm that selects a timing plan in reaction to change in volume of traffic flow [START_REF] Abbas | Configuration methodology for traffic-responsive plan selection: a global perspective[END_REF]. The second method is based on a pattern-matching algorithm that uses pattern matching to choose among a set of predefined timing plans [START_REF] Abbas | Configuration methodology for traffic-responsive plan selection: a global perspective[END_REF]. The third method is based on an adaptive algorithm that applies pre-established-rules to respond to real-time disruption [START_REF] Acun | Les nouveaux outils de régulation du trafic urbain[END_REF]. To validate this system, a first zone was realized at the end of 2013. This zone includes the main access roads to Paris Air Show and a major road corridor. This zone is currently progressively extended to the whole Seine Saint Denis district. The objective is to operate 600 traffic light intersections in realtime at the beginning of 2017. The system can rely on a strong communication network made of 400 km optical fibre with a double ring structure. The optical fibre is able of transmitting up to 10 Gbits/sec. The double ring structure makes the network more secure and avoids the control loss of more than 1 or 2 traffic light intersections. Using traffic sensors, closed-circuit television (CCTV) and environmental sensors, the system is able to analyze permanently the traffic condition. The absolute priority is given to public transportation that is travelling with dedicated lane (tramway T1, T5, T8 and Mobilien bus). The bus priority in the overall traffic is going to be also implemented. Furthermore to manage special events, direct links have been established between the traffic management center and the police. Thus during the Paris Air Show and in 2017 at the Stade de France, the traffic management center is moved on the event site to allow a better collaboration with the police forces. Additionally the system was granted the European label "living lab" in 2013. The aim of this label is to create new services and to experiment innovative equipment within smart cities. Within this label, correlation between pollutants emission from vehicles and traffic condition was shown using sensors inside controllers. Finally the traffic management center is currently testing the information content quality provided by FMD and FCD. The purpose is to use these data to improve and optimize the algorithms. These data can also be used to inform road users about travel time or possible disruption through websites or variable message signs (VMS).

City of Troyes: Traffic monitoring using radio communication network

Troyes is located in north central France. There are about 61,000 inhabitants in the city and 190,000 inhabitants with the suburban area. Public transportation in the city comprises 15 bus lines and 13 school bus lines. The bus network serves around 8,800,000 passengers per year. Currently, several projects are in the planning phase within the transport city committee. The main project is to evacuate vehicles from the heart of the city while keeping an access to local shops and stores. Discussions are also under way to introduce lane reduction in favour of public transportation on the main road corridors.

The traffic management center of Troyes has upgraded its system in 2013. It is now based on a centralized architecture. The system takes fully in charge the commands of the traffic light intersections. It can change in real-time the cycle time of traffic signal. This implies a large number of traffic measurement points and a strong communication network to be efficient. Presently, the traffic management center operates 71 traffic light intersections in the city. They are connected to the traffic management center by copper wires or fibre. A further extension of the centralized system is planned to the suburban area of Troyes. Ultimately 160 traffic light intersections will be operated by the traffic management center. However there is no dedicated network in the suburban area (fibre or copper wires). Therefore the decision was taken to install an IP-based radio access network to communicate with the traffic management center and to install wireless magneto-resistive sensors to detect the presence and movement of vehicles. The basic operating system is described in figure 7.

Figure 7 -Basic operating system of Troyes centralized system

Magnetometers are installed in small holes cored in the roadway. They are usually positioned near the traffic light or 80 meters upstream to detect congestion. They transmit real-time data via low power radio communications to nearby access point. If the magnetometers are far, battery operated radio repeaters are installed to retransmit the information. The access point, the controller and the modem/radio equipment are connected with a wired (Ethernet) network.

The IP-based radio network uses elevated structures (such as water tower) to relay the information. This is to insure a permanent link between the traffic management center and the controllers. To obtain real-time information, high-speed network is also necessary. Transfer rates as high as 300 Mbits/sec are provided. Furthermore a radio system has been installed to give bus priority. Currently, 3 bus lines benefits from the bus priority. Each bus is automatically geolocated. The bus transmits its position and a priority request via radio communication directly to the traffic management center. The traffic management center processes the data and grants early green or green extension to the bus via direct communication with the controller. The system is continuously monitoring radio stream to be able to anticipate any problem or failure. To date, good results were obtained. Indeed, the use of radio system to grant priority request resulted in an average gain of 5 minutes on each bus ride.

City of Grenoble: A multimodal facility to inform and to coordinate actions

Grenoble is located in south-eastern France in the Rhône-Alpes region. The population of the Grenoble metropolitan area is about 675,000 inhabitants. Public transportation in the city comprises: 5 tram lines, 28 bus lines and 18 BRT lines. These services provide around 76,000,000 trips per year. The Grenoble agglomeration community has initiated an innovative approach to manage transportation. Indeed the Grenoble approach focuses on building a multimodal partnership to exchange information and to coordinate actions. This multimodal partnership consists of bringing together different traffic operation centers from the Grenoble metropolitan area. Concretely, in 2010, 6 operation centers were grouped together in one facility called StationMobile [START_REF] Dutartre | Avec StationMobile, se déplacer devient plus facile dans l'agglomération grenobloise[END_REF]. The StationMobile is a 2,500 square meters facility, which includes the following centers:

-the highways and national roads operation center of the Direction Interdépartementale des Routes Centre Est; -the departmental road network operation center of the Conseil Général de l'Isère; -the intercity bus service operation center of the Conseil Général de l'Isère; -the Grenoble traffic lights management center; -the public transportation urban operation center; -the operation center of the agglomeration community. The last one plays the role of coordinator. This multi-operators building provides direct and daily communications between the different centers. This creates relation between different transport operators to improve interaction and to propose solutions. Thus an interface between the different centers and a central computer has been put in place to centralize and to aggregate information. Information such as static data (network description) or dynamic data (traffic disruption, incident or roadwork) is supplied in real-time by the different centers. Each operator has also access to mapping of traffic levels, events and congestion in real-time. This enables better coordination to handle events and to take actions (use of VMS, reduction in bus delays etc.). Furthermore the StationMobile offers a range of free information services to the public: website, mobile application, voice server and disruption notifications (via email or text messages). Information about traffic condition or known disruptions is provided in real-time. A person can also inform the StationMobile about the presence of a road hazard via its mobile phone application. Moreover a multimodal mobility agency opened in September 2012. The objective is to gather in one place the sale of all transport offers (train, bus, bicycle, tram, car sharing). The agency provides information and advices. It raises citizen awareness to the use of alternative transport modes to the car. Finally the StationMobile is also a mobility and employment platform, a driving school, and an exhibition hall. This organization and these tools contribute to a better modal shift from road to public transportation or active transportation modes. They facilitate the sharing of information between managers and the public, the coordination of actions and response to crises.

Conclusion

Over the past four decades, French cities have implemented an array of measures aimed at tackling congestion and pollution. The first urban traffic management centers are launched in the 1970s. Bus priority strategy is applied in a number of cities to enhance service and attract more travellers. In city center, pedestrian zones are set-up, limited traffic zones are implemented and access control zones are created. City access alternatives are completed by the introduction of bicycle, BRT and tram dedicated lanes. Lane reduction or speed reduction techniques are used to plan and to build road infrastructures. The main goal is to encourage people to integrate multiple transportation modes, while maintaining an economic activity. By building sustainable environment and development, France's city policy seeks to improve quality of life of its citizens. Urban traffic management centers are at the front-line in responding to this challenge. With the arrival of ITS, they can rely on new technology tools to control and to operate traffic systems. This paper was able to identify some of them. They use AVL system to track vehicles and radio communication between on-board computer and controller to give bus or tram priority at traffic signal. Video detection systems can be used to identify and alert management about incidents. Wireless magnetometers are used to detect vehicles and to monitor traffic condition. Vehicle probe data will soon be integrated into traffic systems to optimize traffic flow in real-time using artificial intelligence. This technology can be used too to improve mapping of traffic levels. Traffic management centers advise, warn and guide citizen about possible traffic disruptions or travel time using VMS, websites or mobile phone applications. They monitor pollution with environmental sensors.

They participate to the construction of multimodal facilities to inform and to coordinate actions in real-time. Real-time operation would not be possible without strong communication networks and interfaces.

Figure 4 -

 4 Figure 4 -Positioning zones for BRT priority (credit: PC Nantes) This method provides bus priority by activating the green signal when necessary. It also ensures a maximum waiting time of 40 seconds for the vehicles at the traffic signal. Quality indicators are also sent back from the controller to the traffic management center for each traffic light intersection. These indicators inform about the proper functioning of the traffic light intersection. They allow the traffic management center to know if the road was blocked or if the detection system worked.

Figure 6 -

 6 Figure 6 -Traffic management center of Bobigny (credit: PC Gerfaut II)

Acknowledgements

This paper is based primarily on interviews. We would like to thanks in particular Yves Laugel at the Service SIRAC from the city of Strasbourg, Thierry Flaux at the Trafic-Sécurité from the city of Rennes, Yannick Le Guillou at the Service Régulation du Trafic from the city of Nantes, Justin Ledoux and Gérard Delthil at the PC Lutèce from the city of Paris, Jean-Yves Kergall at the Direction de la Voirie et des Déplacements from the city of Bobigny, Frédéric Guenin at the Direction Régulation du Trafic from the city of Troyes and Annick Dutartre at the Direction Générale Déléguée à la Mobilité et aux Transports from the Grenoble metropolitan area.


Guillaume Voisin
email: guillaume.voisin@obspm.fr

Silvano Bonazzola

Fabrice Mottez

Dirac states of an electron in a circular intense magnetic field

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

Electron and positron states with very low momentum perpendicular to the magnetic field have been of interest in the field of rotating neutron stars magnetospheres almost since their discovery in 1968 [1]. Indeed, the community soon realized that the extremely intense rotating magnetic fields of those magnetospheres, ranging from ∼ 10 4 Teslas at the surface of old millisecond pulsars to ∼ 10 11 Teslas at the surface of some magnetars with a typical ∼ 10 8 Teslas [2], could generate extremely large electric-potential gaps along the open magnetic-field lines ( see e.g. [START_REF] Arons | Neutron Stars and Pulsars[END_REF] for a review) which in turn accelerate charged particles to energies only limited by radiation reaction. It is believed that these magnetospheres are mostly filled with electron and positrons resulting from a cascade of pair creations : pairs are created by quantum-electrodynamics processes involving gamma rays, and in turn radiate their kinetic energy in gamma-rays that make other pairs. The process of radiation is that of an accelerated charge that inspirals around a curved magnetic field. Because the magnetic field B is so intense, the gyro-frequence ω = eB mγ of an electron of charge -e, mass m and Lorentz factor γ, is so large that the momentum perpendicular to the local field is dissipated to very low values almost instantaneously because of synchrotron radiation reaction. It follows that electrons and positrons are believed to remain mostly very close to the local field line, radiating mostly because of their motion along the curved field line rather than perpendicular to it. Such motion and radiation are described either by the synchro-curvature regime (see e.g. [4][5][6][7][8]) or the curvature regime [9], depending on whether the residual perpendicular motion is taken into account or neglected. With basic energetic arguments one then re-alizes that this can lead the particle to fall down in the quantified regime, both because radiation is efficient and because the energy levels are large in intense magnetic fields. This led the community to study transitions between low-lying Landau levels, see in particular the work of [10]. However, Landau levels are defined as the states of an electron in a constant uniform magnetic field and therefore are unable to produce transitions of momentum along the magnetic field, no more that they can explain a curved trajectory of the particle. Additionally, one will notice that the case of curvature radiation corresponds to an unphysical motion : a particle of charge e with a velocity v aligned with the local magnetic field B cannot undergo the Lorentz force e v ∧ B, and therefore cannot follow the magnetic field line.

Therefore, in this paper our purpose is to generalize the quantum motion of electrons and positrons to the motion in a locally uniform, circular and constant magnetic field, within the assumption that the radius of curvature is large compared to the Larmor radius. To this end we found a very precious help in the previous works about the motion of an electron in a constant uniform magnetic field by [11], [12] and particularly [START_REF] Sokolov | Synchrotron Radiation[END_REF]. Basing ourselves on the present paper, we will then be able to derive in an upcoming article the radiation of an electron on its lowest perpendicular levels, which could be called quantum synchro-curvature radiation.

We shall start by setting up the symmetries of the problem in section II, before deriving the solutions for a Klein-Gordon particle and more generally for the second order Dirac's equation in section III. Based on those results we derive the full set of Dirac's Hamiltonian proper states in section IV. Finally in section V we propose an interpretation of the obtained states.

II. SYMMETRIES

We consider a particle along a circular magnetic field line of radius ρ and of axis x, that we call in the following the main circle.

Further we assume that the characteristic extension ∆r of the wave function perpendicular to the magnetic field is very small compared to the radius of curvature. This, of course, must be checked a posteriori. In this case one may consider that the magnetic field is locally homogeneous upon an error of ∼ ∆r ρ B. Within these assumptions we have locally three symmetries of the system : one rotation around x, one rotation around the magnetic-field line and one radial translation from the magnetic-field line. This generates a solid torus around the field line. According to Noether's theorem there will be three corresponding conserved quantities, and so three quantum numbers characterizing the proper states of a particle in such a field, to which one has to add one for the spin symmetry :

• s that quantizes the orthogonal translation,

• l ⊥ that quantizes the rotation around the field line,

• l that quantizes the rotation around x,

• ζ that accounts for the spin orientation.

The only difference with the assumptions prevailing in the computation of regular Landau states is that the invariance by translation along the magnetic field is replaced by a rotation around the x axis.

III. SECOND ORDER AND KLEIN-GORDON SOLUTIONS

In this paper, except otherwise stated, we always assume summation over repeated indices, latin indices for space components and greek for space-time with a metric of signature (+ ---).

We start from Dirac's Hamiltonian for an electron of charge -e with minimal coupling to a classical magnetic field given by a potential A

Ĥ = α i Pi + βmc 2 , (1)
where i = {x, y, z} and the generalized impulsion is given by Pi

= -i ∂ i + i e A i . (2)
Dirac's matrices α i are given in standard representation (e.g. [START_REF] Berestetskii | Quantum electrodynamics[END_REF] where the σ i are the Pauli matrices :

§20) b β = 1 0 0 1 , α i = 0 σ i σ i 0 , (3)
σ x = 0 1 1 0 , σ y = 0 -i i 0 , σ z = 1 0 0 -1 . (4)
We need a coordinate system that makes explicit both the assumed rotation invariance around axis x and the part orthogonal to the magnetic field. Such a system is given by the "toroidal" coordinates, represented in figure 1. Toroidal coordinates are related to the cartesian system (x, y, z) by the homeomorphism

T : (r, θ, φ) →   x = r cos φ y = cos θ(ρ + r sin φ) z = sin θ(ρ + r sin φ)   , (5)
where θ represents the direct angle with respect to the y axis in the ( y, z) plane. φ represents the direct angle with respect to x in the plane ( x, y ) of the local frame ( x, y , u θ ) image of ( x, y, z) by a rotation of θ around x and r represents the distance to the main circle. In particular we will need

u θ = (0, -sin θ, cos θ) ( x, y, z) . (6)
In order to write Dirac's equation in this system of coordinates we need to use the Jacobian of T , J T , given in by formula (A2) in A.

Then the covariant quantities, and in particular the impulsion operators transform as follow, pj = (J -1 T ) ij pi (7) where i = {r, θ, φ}.

It is can be shown that Dirac's Hamiltonian keeps the same shape as in (1) if we express it with Dirac's matrices transformed in a "contravariant" way (see appendix A 3), namely

α i = α j (J -1 T ) i j . (8)
It follows that Dirac's Hamiltonian reads

Ĥ = α i Pi + βmc 2 . ( 9
)
We define a suited expression for the magnetic potential A in toroidal coordinates. Since the magnetic field is along ∂ θ , A r or A φ are its only non zero components. Since we impose rotation invariance A does not depend on θ and the local quasi-uniformity hypothesis implies the dependency in r and φ should be negligible as long as r ρ, and more precisely on the scale of the wave function ∆r. From the expression of the rotational in toroidal coordinates given by (A19) in A, a simple potential yielding a constant magnetic field to lowest order in r/ρ along ∂ θ is

A r = 0, A θ = 0, A φ = - 1 2 B. (10)
Using the metric g T (A10) we obtain the covariant component

A φ = 1 2 r 2 B. (11)
This corresponds to a magnetic field

B = B u θ + (r/ρ) . (12)
Let's remark that this is compatible with the uniform homogeneous field when ρ → ∞, as required, since the toroidal coordinates then tend to the cylindrical system.

Following the procedure described in [START_REF] Berestetskii | Quantum electrodynamics[END_REF], we seek a second order equation of which solutions include Dirac's equation solutions by the taking covariant form of Dirac's equation

DΨ = 0 ⇔ cγ µ Pµ -mc 2 Ψ = 0, (13)
and applying to it the operator Ĉ = cγ µ Pµ + mc 2 .

One obtains what we call here the second-order Dirac equation

Ĉ DΨ = 0 ⇔ 2 ∂ 2 t 2 Ψ = Ĥ2 Ψ. ( 14
)
Where Ĥ2 is the second-order "Hamiltonian" which for the magnetic potential given in (10) is explicitely given by Ĥ2

= ( c) 2 ∂ 2 r + 1 r ∂ r + cos φ ρ+r cos φ ∂ r + 1 r 2 ∂ 2 φ 2 - sin φ r(ρ+r cos φ) ∂ φ -2i e A φ ∂ φ + e 2 2 A φ A φ -ie r sin φ ρ+r cos φ A φ + 1 (ρ+r cos φ) 2 ∂ 2 θ 2 -e B • Σ -m 2 c 2 2 , (15)
where Σ = σ 0 0 σ are the spin 1 2 rotation generators in standard representation. The full derivation of ( 15) is given in B.

Now we notice that the interaction between the magnetic field and the state of the electron involves the characteristic magnetic length scale

λ = 2 eB 1/2 . ( 16
)
Anticipating on the result, we shall consider that λ defines the characteristic perpendicular extent of the wavefunction for low-perpendicular-momentum states. This is backed by the fact that the same magnetic length scale plays a similar role in the uniform-magnetic-field case (see e.g. [START_REF] Sokolov | Synchrotron Radiation[END_REF], where λ -2 is denoted γ).

We can define a dimensionless coordinate x ≡ r λ and the parameter ≡ λ ρ . Following our primary assumptions we shall consider that 1. One can check that this is particularly well verified in the case of a typical pulsar magnetic field of intensity B ∼ 10 8 Teslas and curvature radius ρ ∼ 10 4 meters

∼ 10 -16 B -1/2 8 ρ -1 4 , (17)
where ρ 4 = ρ/10 4 and B 8 = B/10 8 . Notice it could also be true in large particle accelerators, because of the soft (square-root) dependance on the magnetic-field intensity.

We can now give a quantitative meaning to the assumption of low perpendicular momentum, that is

∂ x ∼ ∂ φ -1 . ( 18
)
The longitudinal momentum can be larger. However we assume that

∂ θ -3/2 , (19)
and justify this approximation at the end of this section, where (46) translates in terms of maximum Lorentz factor the above approximation. We now rewrite Ĥ2 [START_REF] Olver | National Institute of Standards and Technology[END_REF] in terms of x, and λ keeping only the lowest order terms in

Ĥ2 = c λ 2 ∂ 2 x + 1 x ∂ x + 1 x 2 ∂ 2 φ 2 + 2i∂ φ -x 2 + 2 ∂ 2 θ 2 -2 u θ • Σ -m 2 c 2 λ 2 2 + ( ) . (20)
Compared to [START_REF] Olver | National Institute of Standards and Technology[END_REF] the rightmost term of the two first lines have vanished and we used the fact that

1 ρ+r sin φ = λ +
2 to simplify the others. The eigen problem of operator (20) is separable, which would not have been the case if we had considered ∂ θ ≥ -3/2 since we would have to take into account an additional 2 3 x cos φ∂ 2 θ 2 . It is also worth noticing that, as expected, this equation is very similar to the one found when solving the uniform field problem in cylindrical coordinates (see [START_REF] Sokolov | Synchrotron Radiation[END_REF]).

Our symmetry requirements impose that the sought states be proper states not only of the Dirac Hamiltonian but also of two rotation generators

Ĵθ = -i ∂ φ + 2 u θ • Σ, (21) Ĵx = -i ∂ θ + 2 Σ x , ( 22
)
where Ĵθ is the angular-momentum operator around the magnetic field centered on the main circle and J x is the angular-momentum around the axis of the main circle. These operators commute exactly, while commutation with the Dirac Hamiltonian is ensured to order ,

Ĵθ , Ĵx = 0, (23)
Ĥ, Ĵx = ( ) .

(

) 24
This leads to consider proper states of both operators which are of the form 2 , -1 2 , 1 2 , 3 2 .... We seek a solution to the eigen problem

χ l ,j ⊥ (θ, φ) = 1 2 e ij ⊥ φ+iθl         e -iθ 2 b 1 e iφ 2 + b 2 e -iφ 2 -e iθ 2 b 1 e iφ 2 -b 2 e -iφ 2 e -iθ 2 b 1 e iφ 2 + b 2 e -iφ 2 + e iθ 2 b 1 e iφ 2 -b 2 e -iφ 2 e -iθ 2 b 3 e iφ 2 + b 4 e -iφ 2 -e iθ 2 b 3 e iφ 2 -b 4 e -iφ 2 e -iθ 2 b 3 e iφ 2 + b 4 e -iφ 2 + e iθ 2 b 3 e iφ 2 -b 4 e -iφ 2         , (25)
Ĥ2 Ψ = E 2 Ψ, ( 26
)
where one can show that ±E are also a proper energies of the Dirac Hamiltonian Ĥ. We seek a solution of the problem (26) with the help of the ansatz

Ψ(x, θ, φ) = χ(θ, φ)Φ(x). (27)
Inserting this ansatz into (26) one gets for χ an eigen problem of the form

2 ∂ 2 θ 2 -u θ • Σ χ(θ, φ) = -C θ χ(θ), (28)
where C θ is a constant. As expected, the operator on the left hand side of equation ( 28) commutes with Ĵx and Ĵθ . Therefore χ is necessarily a combination of χ l ,j ⊥ .

Moreover, it is ca be shown that the operator in (28) is degenerate with respect to j ⊥ but not to l . Therefore we have that

χ(θ, φ) = j ⊥ χ l ,j ⊥ (θ, φ), (29)
where each χ l ,j ⊥ has a different set of unknowns (b i ) = (b j ⊥ i ), i ranging from 1 to 4.

However, we notice that a solution of (28) can be found with a combination of only two states,

χ(θ, φ) = χ l ,j ⊥ -1 + χ l ,j ⊥ , if for a given j ⊥ one takes b j ⊥ 1 = b j ⊥ 3 = b j ⊥ -1 2 = b j ⊥ -1 4 = 0, giving χ(θ, φ) = e il θ e il ⊥ φ         e -i θ 2 b j ⊥ -1 1 + b j ⊥ 2 -e i θ 2 b j ⊥ -1 1 -b j ⊥ 2 e -i θ 2 b j ⊥ -1 1 + b j ⊥ 2 + e i θ 2 b j ⊥ -1 1 -b j ⊥ 2 e -i θ 2 b j ⊥ -1 3 + b j ⊥ 4 -e i θ 2 b j ⊥ -1 3 -b j ⊥ 4 e -i θ 2 b j ⊥ -1 3 + b j ⊥ 4 + e i θ 2 b j ⊥ -1 3 -b j ⊥ 4         , (30)
where we define l ⊥ ≡ j ⊥ - 1 2 . With this choice, l ⊥ is no longer the exact angular momentum, it will however simplify upcoming calculations and allow a direct comparison with the uniform case as presented in [START_REF] Sokolov | Synchrotron Radiation[END_REF]. We will give its exact meaning in section (V).

Then, χ is a solution provided that the remaining free coefficients satisfy the following systems, obtained after inserting (30) back into (28) ,

M θ b j ⊥ 2,4 b j ⊥ -1 1,3 = 0, (31)
where M θ is defined by the matrix coefficients

M θ 11 = -2 (2l + 1) 2 -4(2 -C θ ), M θ 12 = (2l + 1) 2 2 -4(2 + C θ ), M θ 21 = -(1 -2l ) 2 2 -4(2 -C θ ), M θ 22 = -(1 -2l ) 2 2 + 4(C θ + 2).
(32

)
This system has a none trivial solution only if

C σ θ = 1 4 4l 2 2 + 2 + σ4 l 2 4 + 4 , (33)
which leads to the solution coefficients

b j ⊥ 2,4 σ = c σ 1,2 2 -l 2 + σ l 2 4 + 4 , (34)
b j ⊥ -1 1,3 σ = c σ 1,2 -2 -l 2 + σ l 2 4 + 4 , (35)
where c 1 and c 2 are for now arbitrary constants describing the two proper spaces found for b

j ⊥ 2 , b j ⊥ -1 1 (two first lines of χ) and b j ⊥ 4 , b j ⊥ -1 3
(two last lines of χ) respectively. The number σ = ±1 distinguishes two classes of solutions that we shall denote ↑= +1 and ↓= -1 for reasons that will become obvious when we see its physical meaning in section V.

Let's now solve the equation for Φ(x). After inserting our ansatz (27) including the previously found expression for χ, we get the equation

∂ 2 x 2 + 1 x ∂ x - l 2 ⊥ x 2 -x 2 -2l ⊥ Φ(x) = -C x Φ(x), ( 36
)
where C x is a constant to be determined. We give the detailed resolution of this equation in C, where we find that

Φ(x) = x l ⊥ e -x 2 2 L l ⊥ s (x 2 ) (37) C x = 4 n + 1 2 , (38)
where L l s is a generalized Laguerre polynomial of degree s as defined in [START_REF] Olver | National Institute of Standards and Technology[END_REF] ( §18.5) by

L l s (x) = s i=0 (l + i + 1) s-i (s -i)!i! (-x) i , (39)
where (a

) n = Γ(a + n)/Γ(a) is a Pochhammer's symbol.
Besides, n = s + l ⊥ is the primary perpendicular quantum number, s is a positive integer and the perpendicular angular momentum must be positive or null l ⊥ ≥ 0 to ensure that the wave function vanishes at infinity and is square-integrable. We will come back later to the interpretation of these quantum numbers. Notice that we use the same notations as in [START_REF] Sokolov | Synchrotron Radiation[END_REF] for the uniform-magneticfield case, where the radial dependency has exactly the same form but is expressed with a different coordinate system. Notice as well that (37) is proportional to a normalization constant that we dropped here for simplicity. Normalization will be determined farther on.

Putting the whole Ψ back into the main equation ( 26), we get the proper energies

E 2 = m 2 c 4 + 2 c 2 λ 2 (C x + C σ θ ) , (40)
which develops as

E = ±mc 2 1 + 2 ω c mc 2 n + 1 2 + σ ω c mc 2 1 + ( 2 l ) 2 + Ω mc 2 2 1 4 + l 2 1 2 , (41)
where ω c = eB m is the cyclotron pulsation and Ω = c ρ is the pulsation of the circular trajectory.

Proper functions Ψ and proper values E 2 are the exact solutions of eigen problem of the approximated operator Ĥ2 (20). However, our approximations do not allow us to take meaningfully into account terms of order and higher. At this order, the complete solutions of the second order eigen problem (26) is explicitly given by

Ψ 2 (x, θ, φ) = 4x l ⊥ e -x 2 2 L l ⊥ s (x 2 )e il θ e il ⊥ φ         c σ 1 σe -i θ 2 + e i θ 2 c σ 1 σe -i θ 2 -e i θ 2 c σ 2 σe -i θ 2 + e i θ 2 c σ 2 σe -i θ 2 -e i θ 2         + ( ) , (42)
E = ±mc 2 1 + 2 ω c mc 2 n + 1 + σ 2 + Ω mc 2 2 l 2 , (43)
where in such a development, one has to remind that l can be of order -1 . In this limit we obtain degenerate states : indeed states with n, σ = +1 have the same energy as states with numbers n + 1, σ = -1. The only exception is for what we will from now on call the perpendicular fundamental state : n = 0, σ = -1, which is non-degenerate.

Before going farther, let's notice that we already obtained the solution of the Klein-Gordon equation for an electron in a circular magnetic field. Indeed, Ĥ2 corresponds to the Klein-Gordon "Hamiltonian" plus a spin term u θ • Σ . Neglecting this term it comes that χ(θ, φ) = e il θ e il ⊥ φ and the proper states of energy (43) are given by

Ψ KG (x, θ, φ) = e il θ e il ⊥ φ x l ⊥ e -x 2 2 L l ⊥ s (x 2 ). ( 44
)
We now justify a posteriori approximation (19). Assuming as in typical pulsar magnetospheres that the motion is dominated by the momentum along the field and that particles are ultra-relativistic with a classical Lorentz factor γ 1 we obtain, using equation ( 43),

E = γmc 2 = Ωl + 1 l . (45)
This allows to translate approximation (19) in terms of a limit on the Lorentz factor

γ 6 • 10 6 ρ 1/2 4 B 3/4 8 , (46)
compatible with a variety of pulsar magnetospheres situations. We briefly come back to the interpretation in terms of possible drifts at the end of section V.

IV. DIRAC'S EQUATION SOLUTIONS A. General solution

It is can be shown from the derivation of the second order equation ( 14) that from any second-order solution a first-order solution can be obtained by applying the Ĉ operator to it like

Ψ(x, θ, φ) = Ĉe -i E t Ψ 2 (x, θ, φ). ( 47
)
This is the approach suggested in [START_REF] Berestetskii | Quantum electrodynamics[END_REF]. However, following naively this procedure leads to obtain as many first order solutions as second order solutions while there should be half as many. One can check that we now have 4 independent second-order solutions for each triplet (n, l ⊥ , l ), two for each value of σ, as shown in equations (34) and ( 35). Moreover, one can check that the obtained second-order solutions are neither directly solutions of Ĉ or of D which implies that proper states for a given energy must be linear combinations of the second order solutions.

We are going to show that such solutions can be obtained using the combination

Ψ = e -i E t Ψ 2l ⊥ -1,σ=+1 + Ψ 2l ⊥ ,σ=-1 , (48)
E = ±mc 2 1 + 2 ω c mc 2 n + Ω mc 2 2 l 2 . ( 49
)
The state Ψ above is thus defined by the superposition of two states having the same quantum number s. The proper energy (49) can be equivalently defined as

E 2 l ⊥ -1,σ=+1 or E 2 l ⊥ ,σ=-1 .
We chose the second option in (49). Remark that as such Ψ is undefined for l ⊥ = 0. Prescribing that Ψ 2l ⊥ =-1 = 0, we find the perpendicular fundamental state as the particular case Ψ s=0,l ⊥ =0 , One can show that solving the equation DΨ = 0 amounts to solve the linear problem

M D     c - 1 c - 2 c + 1 c + 2     = 0, ( 50
)
Calculations to obtain the matrix M D are lengthy but appeal to relatively simple operations for which a formal calculation engine can be helpful. We consider the details of it would be of little interest for the reader, for this reason we give here only the main steps. It goes as follow :

• Divide by the following common factor to isolate the "spinor part" of the equation

S ≡ DΨ/(e il θ e i(l ⊥ -1)φ x l ⊥ e -x 2 2 ) = 0. (51)
• The remaining function can be expanded on the basis of the four orthogonal functions : e ai θ 2 e biφ , where a = ±1 and b = {0, 1}. Taking into account the four spinor components, labeled by j hereafter, this gives us 16 coefficients depending on the four unknowns

(c ↓ 1 , c ↓ 2 , c ↑ 1 , c ↑
2 ) that we call s j,a,b . It follows that the equation DΨ = 0 reduces to a linear system of 16 equations ∀j, ∀a, ∀b, s j,a,b = 0.

(52)

Notice that, at this stage, the coefficients still depend on functions of x .

• A lot of these equations are actually equivalent.

The coefficients with a = +1 are proportional to coefficients with a = -1 for any given doublet (j, b). Also notice that the components of the spinor are related two by two : s 1,a,b ∝ s 2,a,b and s 3,a,b ∝ s 4,a,b for all a and b. Finally, there are only four a priori independent equations. To fix ideas, we will go on with the system

     s 1,1,1 = 0 s 3,1,1 = 0 s 1,1,0 = 0 s 3,1,0 = 0 . ( 53
)
• Using the two relations

L l ⊥ +1 s (x 2 ) = L l ⊥ +1 s-1 (x 2 ) + L l ⊥ s (x 2 ), (54)
L l ⊥ s (x 2 ) = (1 + l ⊥ )L l ⊥ +1 s (x 2 ) -x 2 , L l ⊥ +2 s-1 (x 2 ) (s + l ⊥ + 1) (55)
which can be derived from the Laguerre-polynomial recurrence relations given in [START_REF] Olver | National Institute of Standards and Technology[END_REF] §18.9, one shows that s 1,1,1 and s 3,1,1 are proportional to xL l ⊥ +1 s while s 1,1,0 and s 3,1,0 are proportional to L l ⊥ s . • It follows from the previous point that after dividing each equation by its respective x polynomial as well as 2 c/λ (to make it dimensionless), the system (53) gives (50) with

M D =       E-mc 2 c/λ l 0 2i -l -E+mc 2 c/λ -2i 0 0 -2i(1 + n) E-mc 2 c/λ -l 2i(1 + n) 0 l -E+mc 2 c/λ       . ( 56
)
The determinant of M D is null which means, as expected, that the kernel of M D is not empty. One finds two independent solutions given by

c ↓ 1 c ↓ 2 = E+mc 2 c/λ -l and c ↑ 1 c ↑ 2 = 2n 0 i , (57)
c ↓ 1 c ↓ 2 = -l E-mc 2 c/λ and c ↑ 1 c ↑ 2 = 2n i 0 . (58)
Notice that the perpendicular fundamental state comes out naturally from the two solutions (57) and (58). Indeed, for n = 0 the σ =↑ coefficients vanish, and the en-

ergy becomes E = mc 2 1 + Ω mc 2 2 l 2 .
Then the two solutions are proportional, as expected from the non degeneracy of the perpendicular fundamental, since one finds (58) by simply multiplying (57) by -E-mc 2 c/λ . The two solutions (57) and (58) correspond to two spin states that we shall respectively label by ζ = -1 and ζ = +1. Some more details will be given in section V.

B. Normalization

We now have obtained the three parts of the wave function. We still need to impose normalization with

d 3 x 4 i=1 Ψ * i Ψ i = 1. ( 59
)
We need the Jacobian determinant of the toroidal coordinates d 3 x = |r(ρ + r sin φ)| dr dθ dφ .

(60)

Expressing it as a function of the dimensionless variable x ,

d 3 x = λ 2 ρ |x(1 + x sin φ)| dx dθ dφ , (61)
it becomes obvious that the sin φ term can be removed at lowest order in .

For the integration over x the following integral [START_REF] Olver | National Institute of Standards and Technology[END_REF] +∞

x=0 x 2l e -x 2 L l s x 2 2 xdx = (s + l)! 2s! ( 62
)
is useful. We get the normalization

N = 4π ρλ 2 (n-1)! E+ζmc 2 c/λ 2 +( l ) 2 n+2n 2 2s! . (63)

C. Complete proper states

Eventually the proper states of a particle of energy E (49) in a toroidal magnetic field can be explicitly given by

Ψ = 1 N e -x 2 2 e il θ e i(l ⊥ -1 2 )φ e -i φ 2 x l ⊥ -1 L l ⊥ -1 s x 2 χ ↑ ζ (θ) +e i φ 2 x l ⊥ L l ⊥ s x 2 χ ↓ ζ (θ)
.

(64) The two χ σ ζ spinors are explicitly given by

χ ↓ ζ =         1+ζ 2 E+mc 2 c/λ -1-ζ 2 l -e -i θ 2 + e i θ 2 1+ζ 2 E+mc 2 c/λ -1-ζ 2 l -e -i θ 2 -e i θ 2 -1+ζ 2 l + 1-ζ 2 E-mc 2 c/λ -e -i θ 2 + e i θ 2 -1+ζ 2 l + 1-ζ 2 E-mc 2 c/λ -e -i θ 2 -e i θ 2         , ( 65
)
χ ↑ ζ = 2ni         1-ζ 2 e -i θ 2 + e i θ 2 1-ζ 2 e -i θ 2 -e i θ 2 1+ζ 2 e -i θ 2 + e i θ 2 1+ζ 2 e -i θ 2 -e i θ 2         . ( 66
)
One can see that the constant uniform-magnetic-field case can be recovered by taking the limit ρ → ∞ in (64) after having performed the replacements : θ → z/ρ, l → ρk z where z is the axis along the magnetic field and k z is the associated wave number. FIG. 2. Left panel : probability density of detecting an electron in a state s = 1, l ⊥ = 1 in a plane orthogonal to the magnetic-field line of radius ρ. The color goes from blue (inner part of each ring), spin aligned with the magnetic field, to red (outer part of each ring), spin anti-aligned. Here the parallel motion is small, l -1 such that both spin components are almost equally important. In the case of a relativistic parallel motion, l > -1 , only the anti-aligned (red) component plays a significant role as one can see from equation ( 64), ( 65) and (66). Right panel : representation of a family of off-centered classical trajectories defined by a = 1.5 where a is defined in (D22).

V. INTERPRETATION OF THE QUANTUM NUMBERS

In this section we always consider an electron state (positive energy) to simplify the discussion without any loss of generality.

The parallel quantum number l quantifies, by construction, the angular momentum around the x axis. From the expression of the proper energy we can also interpret Ωl as the "component" of the energy corresponding to the motion along the magnetic field.

We move on to interpreting the perpendicular motion. Our treatment is similar to that of [START_REF] Sokolov | Synchrotron Radiation[END_REF]. The energy of the motion perpendicular to the magnetic field is quantified by the quantum number n = s + l ⊥ . It can be interpreted as the quantification of the square of the radius of the trajectory of the electron since the classical gyroradius can be expressed as r g = p ⊥ /(mω c ) with p ⊥ the perpendicular momentum and, in case of a purely perpendicular motion, E 2 = p 2 ⊥ + m 2 c 4 . We see below that this assertion can be very quickly proven in the classical limit in the particular case n = l ⊥ .

As we saw in the previous section the wave function is not defined for a strictly negative l ⊥ . From a classical point of view this is easily understandable since l ⊥ quantifies the angular momentum around the local axis of the magnetic field. Therefore, l ⊥ > 0 corresponds to a rotation in the direct sense, which is the orientation that an electron takes under the action of the classical Lorentz force v ∧ B, where v is the speed of the electron.

Going a little bit deeper, one can show that the solution (64) is a proper state of the angular momentum around the magnetic field Ĵθ of proper value (l ⊥ -1/2). This means that the perpendicular fundamental has a negative angular momentum of -/2. However, it does not mean that the electron classically turns backwards around the magnetic field, but rather that the spin is oriented backward, while the orbital angular momentum is zero. Indeed, one can show that the spinors χ σ ζ are proper states of the operator of projection of the spin onto the main circle of the magnetic field, 2 u θ • Σ (the spin part of Ĵθ ), with proper values σ/2. We here justify the notation ↑ or ↓ for σ = ±1 as meaning that the spin is aligned or anti-aligned with the magnetic field. The perpendicular fundamental state is thus the only purely anti-aligned state, as we will see. Since it has no orbital momentum, one cannot interpret the trajectory of the particle following the magnetic field as the result of the classical Lorentz force but rather as a strictly quantum phenomenon of the interaction between spin and magnetic field. States with n > 0 (i.e. l ⊥ > 0 or s > 0 ) are in a superposed spin state, both aligned and anti-aligned with the magnetic field which results into a degeneracy into two states parametrized by ζ. It is in theory possible to find measurable quantities, hermitian operators that commute with the Hamiltonian, such that this degeneracy would be lifted and the spin orientation fixed (see e.g. [START_REF] Sokolov | Synchrotron Radiation[END_REF] or [START_REF] Melrose | [END_REF] for possibilities in the case of a uniform homogeneous magnetic field). However, it is usually impossible to determine the state of the spin, in particular in astrophysics. We therefore prefer to consider the most general case in which a state of energy E is the superposition of the two ζ states of (64) combined through a mixing angle η , Ψ = cos(η)Ψ ζ=-1 + sin(η)Ψ ζ=+1 .

(67)

Parametrization by such an angle takes into account the constrain of having a norm of the final state that is still one. Notice that it is impossible to form a purely aligned or anti-aligned state for any value of the mixing angle, as we previously stated.

We now explain the role of the quantum numbers s and l ⊥ and why their role in the energy is degenerated. First consider an electron with s = 0, then the radial part of the wave function ( 64) is merely ∝ e -x 2 x 2p where p = l ⊥ -1 for the anti-aligned term and p = l ⊥ for the aligned term. Now, this function is peaked at x p = √ p with an amplitude at the peak of p p . This means that, apart for the perpendicular fundamental, the electron always has a double orbit : one of aligned spin and, a bit further, one of anti-aligned spin, as shown in figure 2.

Considering a high value of the perpendicular angular momentum one can quickly recover classical results analogous to the uniform magnetic field case. For simplicity we will consider that momentum along the field is zero, l = 0. From the previous discussion the particle orbits at a distance r p λ √ l ⊥ . Expressing l ⊥ as a function of the energy one gets

l ⊥ = E 2 -m 2 c 4 2mc 2 ω c . ( 68
)
In the classical limit the numerator simply identifies with the square of the perpendicular momentum of the particle p 2 ⊥ . Inserting (68) into r p , we obtain the classical (relativistic) Larmor radius :

r p = r g = p ⊥ eB (69)
This is in agreement with the more general result given above. Moreover, it confirms that the typical extent of the wave-function can be taken to be the gyro-radius, at least for high enough quantum numbers, and approximation ( 18) can we written in a more intuitive way

r g ρ (70)
Similarly, one finds that the group velocity of a wavepacket, v g = ∂ω ∂k , can be found after identifying ω = E/ and k = l ⊥ /r p : v g = r p ω c /γ with γ = E/(mc 2 ). We here recognize the classical relativistic gyro-frequency ω c /γ of an electron in a uniform magnetic field. Now, for a same energy we may as well have states of lower l ⊥ and higher s. This degeneracy also appears, to some extent, in the classical treatment of this problem. Since the radial part of the motion is mostly identical to the uniform-magnetic-field case, we can use the later to better understand the former. We developed in D the Newtonian solution of the uniform problem based on the Hamilton-Jacobi formalism which, because of its parenting with quantum mechanics, allows a formulation in similar terms. In particular, it is found that there are two terms in the perpendicular energy : one related with the angular momentum (noted p θ in the classical case) and the other to a shift of distance r 0 of the center of the trajectory with respect to the origin of the coordinate. This is summarized in formula (D33) recalled here,

E = p θ ω c + 1 2 mr 2 0 ω 2 c . (71)
We will follow that guide. If l ⊥ obviously corresponds to the angular-momentum, p θ term, we can show that s corresponds to the second term. The position of the center of the trajectory given by equation (D31) can be generalized as the operator

ˆ x 0 = ˆ x + ˆ J ∧ ˆ p mω c ˆ J (72)
Where ˆ

x is the position operator, ˆ p the impulsion and ˆ J the angular momentum with respect to the coordinate origin. For our set of solutions ˆ J = Ĵθ and the previous operator simplifies to the two components

x0 = x - py mω c , (73)
ŷ 0 = ŷ + px mω c , (74)
where (x, y ) are the coordinates locally perpendicular to the magnetic field as defined in figure 1. From that, an operator r2 0 = x2 0 + ŷ 2 0 is readily obtained. Using the dimensionless coordinate x = r/λ

r2 0 = λ 2 x 2 -∂ 2 x 2 - 1 x ∂ x - 1 x 2 ∂ 2 φ 2 + 2i∂ φ , (75)
where one recognizes the radial part of the second order equation previously solved (see (36) or (C3) ) except for the ∂ φ term which sign is reversed. From that observation it is straightforward to see that the proper values of r2

0 are r 2 0 = 4λ 2 (s + 1 2 ). ( 76
)
The 1 2 term comes from the spin interaction that broadens the orbits as we saw previously. Now interpreting the trajectory as an off-centered circle obviously breaks the assumed rotation invariance around the coordinate center. This apparent paradoxe is solved by considering that a proper wave-function is analogous not to a single classical trajectory but to the set of all the trajectories corresponding to the invariants of motion defining the proper state : n or the perpendicular energy and l ⊥ or the perpendicular angular momentum. We see from the expression of the trajectory (D20) that this set is classically parametrized by the constant of integration θ 0 . This constant is defined by the initial conditions of the motion, and sets the position of the center of trajectory on the circle of radius r 0 centered on the main circle. Then it is obvious that this set is invariant by rotation, as shown in figure 2. Thus, we recover the interpretation of s as characterizing the radial symmetry assumed in section II.

One notices that in the present solution, the particle remains localized around the magnetic-field line, and therefore there is no drift perpendicular to the line as in the classical theory where the drift is due to the centrifugal force (see e.g. [8]). This is justified by the fact that we considered only the lowest perpendicular states and a "moderate" longitudinal momentum (19) that allows us to neglect coupling terms between longitudinal motion (∂ θ terms) and perpendicular motion (x, ∂ x and ∂ φ terms). We notice that several works on the classical theory of synchro-curvature radiation ( for example [4][5][6][7]) did not take this drift into account either, and this approximation is widely used for lepton trajectories in pulsar magnetospheres even with Lorentz factors largely above the limit given in (46). Besides, in [8] the authors show that the effect on radiation of the drift classically results in an effective radius of curvature.

VI. CONCLUSION

In this paper we were able to generalize the relativistic Landau states to the case of circular magnetic field (64), in the approximation that the curvature radius is large compared to the Larmor radius of the particle while the momentum along the field is not excessively large (( 18) or ( 70) and ( 19) ). Our main interest is for applications to the very intense magnetic fields around rotating neutron stars, pulsars and magnetars, in which radiation from very low perpendicular momentum electrons and positrons is believed to be ubiquitous. In an upcoming paper, we will address the problem of radiations from transitions between the states derived in the present paper. We could call them quantum curvature radiation for transitions involving only the ground perpendicular state and more generally quantum synchro-curvature radiation (see e.g. [4]).

Appendix A: Toroidal coordinates toolbox

The toroidal coordinates are defined by the following diffeomorphism T T :

  r θ φ   →   x y z   =   r cos φ cos θ(ρ + r sin φ) sin θ(ρ + r sin φ)   , (A1)
such that surfaces of constant r are torii centered on the circle of radius ρ > 0.

The primed quantities denote quantities in the basis ∂T ∂r , ∂T ∂θ , ∂T ∂φ .

Jacobian

The Jacobian of this coordinate system is (A3)

J T =   cos φ 0 -r

Transformation of covariant quantities

Covariant quantities transform like

A i → A i A = t J T A ⇔ A i = (J T ) ji A j , (A4)
A = t (J -1 T )A ⇔ A i = (J -1 T ) ji A j . (A5)
Developing the kinetic part one finds

(γ µ D µ ) (γ ν D ν ) = ∂ µ ∂ µ + (ie) 2 A µ A µ +ie{γ µ A µ , γ ν ∂ ν }, (B4) where {γ µ A µ , γ ν ∂ ν } = {γ µ , γ ν }A µ ∂ ν + γ ν γ µ ∂ ν (A µ ) , (B5)
and where

γ ν γ µ ∂ ν (A µ ) = 1 2 [γ µ γ ν ∂ µ (A ν ) + γ ν γ µ ∂ ν (A µ )] (B6) = 1 2 [γ µ γ ν (∂ µ A ν -∂ ν A µ ) + 2η µν ∂ ν A µ ] .
Using the identities

{γ µ , γ ν } = 2η µν 1, (B7) γ i γ j = -δ ij -i ijk Σ k , ( B8
)
γ 0 γ i = α i , (B9)
and recognizing the electromagnetic field tensor,

F µν = ∂ µ A ν -∂ ν A µ , (B10)
from which we get the contravariant components of the electric and magnetic fields (see e.g. [START_REF] Gourgoulhon | Special Relativity in General Frames[END_REF])

E i = η ij F 0j = -F 0i , (B11)
B i = ijk F jk , (B12)
we get

γ ν γ µ ∂ ν (A µ ) = -α • E -iΣ k ijk ∂ i A j curl(A) k =B k +2η µν ∂ ν A µ .
(B13) The anti-commutator (B5) then becomes

{γ µ A µ , γ ν ∂ ν } = 2(A µ ∂ µ + 1 2 ∂ µ A µ )-α• E -i B • Σ. (B14)
Inserting (B14) back into (B4), and reorganizing the terms a little we obtain

(γ µ D µ ) (γ ν D ν ) = ∂ µ ∂ µ + 2i e (A µ ∂ µ + 1 2 ∂ µ A µ ) - e 2 2 A µ A µ (∂µ+i e Aµ) 2 - ie α • E + e B • Σ , (B15)
from what we get the same expression for the secondorder Dirac equation as in [START_REF] Berestetskii | Quantum electrodynamics[END_REF] (One will pay attention that in Cartesian coordinates A x = -A x and that the usual magnetic potential A is defined as a contravariant quantity. With the metric signature used here :

A 0 = Φ/c = A 0 .), ( c) 2 1 c ∂ t + i e c Φ 2 -x∈{x,y,z} ∂ x + i e A x 2 - i e c α • E + e B • Σ + m 2 c 2 2 Ψ = 0 (B16)
We now switch to another spatial coordinate system denoted by primes, with the only assumption that this system is orthogonal. The Jacobian of the transformation is given by (J ij ). In (B17) we separate the time components from the space components,

(γ µ D µ ) (γ ν D ν ) = (∂ t + ieΦ) 2 - ∆ -2i e A i ∂ i + 1 2 ∇ • A + e 2 2 A i A i - ie α • E + e B • Σ . (B17)
Where A i = J ij A i and ∆ , ∇ • represent the Laplacian and the divergence in the primed system of coordinates. We have used the orthogonality of J ij to eliminate cross terms. If only spatial coordinates change the electric field E here transforms like a covariant vector and α like a contravariant quantity (as shown in the text) such that α • E = α • E . The rules of transformation of the magnetic field are less straighforward and it might be simpler to just express it as a function of the primed variables without changing its basis. That is the choice of this paper.

In the case proposed in this paper, we use the toroidal coordinates defined in A, with a Laplacian and a divergence respectively given by (A16) and (A17). The magnetic potential is assumed to be only along the third direction : A = (0, 0, A φ ). All replacements made we obtain

-( c) 2 -1 c 2 ∂ 2 t 2 + ∂ 2 r + 1 r ∂ r + 1 (ρ+r cos φ) 2 ∂ 2 θ 2 + 1 r 2 ∂ 2 φ 2 - 2i e A φ ∂ φ + e 2 2 A φ A φ -e B • Σ -m 2 c 2 2 + cos φ ρ+r cos φ ∂ r - sin φ r(ρ+r cos φ) ∂ φ -ie r sin φ ρ+r cos φ A φ Ψ = 0 . ( B18
) The non-negligible (see the text) Laplacian terms are on the second line. The terms involving the magnetic potential are on the third line. All the terms on the fourth line are neglected in this paper, the two leftmost terms coming from the Laplacian and the rightmost term being the divergence.

Appendix C: Resolution of the radial differential equation

In this appendix, we develop the detailed solution of the differential equation (36) giving the radial dependency of the proper states of Dirac's equation. Here we recall the equation

∂ 2 x 2 + 1 x ∂ x + 1 x 2 ∂ 2 φ 2 + 2i∂ φ -x 2 + C f (x, φ) = 0, ( C1
) Where -C is the proper value of the equation, to be determined.

Assuming the following form for f :

f (x, φ) = e iξφ g(x) (C2)
And inserting it into (C3) we obtain

∂ 2 x 2 + 1 x ∂ x - ξ 2 x 2 -2ξ -x 2 + C g(x) = 0. (C3)
We notice that

∂ 2 x 2 + 1 x ∂ x g(x) = 1 √ x ∂ 2 x 2 √ xg(x) + 1 4x 2 g(x), (C4)
which once put into (C3) gives the following form

∂ 2 x 2 √ xg(x) + 1 4 -ξ 2 x 2 -2ξ -x 2 + C √ xg(x) = 0.
(C5) Here we recognize the differential equation giving generalized Laguerre functions given in [START_REF] Olver | National Institute of Standards and Technology[END_REF], table 18.8.1,

∂ 2 x 2 + 1 4 -α 2 x 2 -x 2 + 4s + 2α + 2 h(x) = 0, (C6)
where α is a real parameter stritly larger than -1, s a positive integer and the solution h is

h(x) = e -x 2 2 x α+1/2 L (α) s (x 2 ). ( C7
)
Identifying α and C in (C5) we find

α = ± |ξ| if |ξ| < 1 |ξ| if |ξ| ≥ 1 , (C8)
and

C = 4s + 2(ξ ± |ξ|) + 2 if |ξ| < 1 4s + 2(ξ + |ξ|) + 2 if |ξ| ≥ 1 . (C9)
Finally the solutions of equation (C3) are

f (x, φ) = e iξφ e -x 2 2 x α L (α) s (x 2 ). (C10)
In the specific case of this paper we have ξ = l ⊥ . Therefore we obtain

α = ± 1 2 if l ⊥ = 1 2 |l ⊥ | otherwise , (C11)
and

C = 4s + 2 if l ⊥ ≤ 0 4(s + l ⊥ ) + 2 if l ⊥ ≥ 0 . ( C12
)
Appendix D: Resolution of the Newtonian Hamilton-Jacobi problem of an electron in a constant uniform magnetic field

We work out the general solution, without assuming the center of motion, of the motion of an electron in a uniform constant magnetic field in polar coordinates using the Hamilton-Jacobi formalism of Newtonian mechanics. Although heavy, this way of obtaining a common result is interesting in view of the comparison with the quantum mechanical result, given the parenting between the Hamilton-Jacobi formalism and Hamiltonian quantum mechanics.

Without loss of generality we restrict ourselves to a plane motion with polar coordinates (r, θ). The position vector r and the velocity ˙ r are then expressed in the polar basis ( u r , u θ ) by r = r u r , (D1) ˙ r = ṙ u r + r θ u θ .

(D2)

The link with cartesian coordinates comes with u r = (cos θ, sin θ) and u θ = (-sin θ, cos θ).

We choose to write the magnetic potential giving a field of flux intensity B orthogonal to the plane of motion in a symmetric gauge with

A = 1 2 rB u θ . (D3)
Then the Lagrangian of an electron of charge -e is given by

L = 1 2 m( ṙ2 + (r θ) 2 ) - 1 2 er 2 θB. ( D4
)
We readily see that θ is a cyclic coordinate as only its derivative participates in the Lagrangian. Therefore, its conjugate momentum is a constant of motion

p θ = ∂L ∂ θ = mr 2 ( θ - ω c 2 ). (D5)
Notice that p θ is actually the angular momentum of the particle.

We may now define the typical time scale T and length scale λ

(ω c /2) -1 = eB 2m -1 , (D6) λ = 2p θ mω c , (D7)
which define an energy scale = m(λω c /2) 2 . (D8)

We switch now to dimensionless coordinates

r → x = r/λ, (D9) t → τ = ω c 2 t, (D10)
and to a dimensionless Lagrangian

L = L/ = 1 2 x 2 + θ (1 + x 2 ) -1 -x 2 , (D11)
where denotes the derivation with respect to τ while ẇas with respect to t. The momentum conjugated to x is simply

p x = ∂ L ∂x = x . (D12)
The Legendre transform of L gives us the corresponding Hamiltonian H = p x x + p θ θ -L = 1 2 p 2 x + 1

x 2 + 2 + x 2 , (D13)

where we made obvious that θ was an ignorable coordinate by using (D5) to get that θ = 1/x 2 + 1 . Let's now introduce the Hamilton characteristic function W (see e.g. [START_REF] Goldstein | Classical mechanics (Reading (Mass.)[END_REF]), of which we consider only the x dependence. We get the following Hamilton-Jacobi equation

∂W ∂x 2 + 1 x 2 + 2 + x 2 = 2 Ẽ, ( D14
)
where Ẽ = E/ is the dimensionless energy of the system. It follows that one gets the following Hamilton function

W = ± dx Ẽ - 1 x 2 -2 -x 2 . ( D15
)
Choosing Ẽ as the new momentum we get that its conjugate coordinate is :

Q Ẽ = ∂W ∂ Ẽ = ± dx 2 Ẽ -1 x 2 -2 -x 2 (D16)
which integrates as

Q Ẽ = ± 1 2 arctan a -x 2 √ 2ax 2 -x 4 -1 (D17)
where a = Ẽ -1.

The denominator of the arctan argument in (D17) is necessarily positive since it is proportional to ∂W In order to keep a continuous trajectory, one will switch from the + to the -solution whenever the sin function switches as well.

Finally, we use the conservation of angular momentum (D5) to obtain the equation for θ and switch back to the international unit system r = λ a + √ a 2 -1 sin (ω c t + θ 0 ), θ = ωc 2 t + arctan a tan ωc Thus, we obtained a solution depending only on the invariants of motion, E and p θ , and a constant of integration θ 0 depending on the initial conditions.

2 t + θ0 2 + √ a 2 -1 + nπ, (
Let's interpret the different trajectories. First, if we assume that r or θ is constant, from the expression of p θ (D5) we see that the other one is constant as well. From (D20) we see that this is allowed only for a = 1 . In this case we find readily that r = λ, (D24) θ = ω c t.

(D25)

Using the relation between the conjugate momentum p θ and the orthogonal "kinetic" momentum p ⊥ = mr θ

p θ = rp ⊥ -mr 2 ω c 2 . ( D26
)
Replacing r by λ in the above expression one obtains that the radius of the trajectory is the usual Larmor radius

r = λ = p ⊥ mω c (D27)
In this configuration the radial momentum is zero, and

E = p θ ω c . (D28)
In this particular configuration we see that p ⊥ is constant as well. We see that by construction : a ≥ 1. Now, we consider the solutions for higher energies i.e. higher values of a. We see that they are all circles after computing the curvature radius using the formula We obtain, after a lengthy but straightforward calculation, the general formula for the radius of curvature

ρ = λ a + 1 2 = 2E mω 2 c . ( D30
)
Reminding that the magnetic field does not work, we find that the kinetic energy is constant and therefore these circles are circulated at constant speed ω c ρ. This allow us to find the center of each circle r 0 by simple geometric considerations

r 0 = r - 1 ω c d r dt ∧ p θ p θ , (D31)
where p θ is the vectorial angular momentum, which here is orthogonal to the plane of motion. The distance of the center of motion to the origin of coordinates takes a simple form

r 0 = ρ 2 -λ 2 = 2 E -p θ ω c mω 2 c . ( D32
)
Put differently, this gives a simple expression for the energy in terms of

E = p θ ω c + 1 2 mr 2 0 ω 2 c . (D33)
Thus we found the very intuitive result that, when the trajectory is centered on the origin of coordinates all the energy is stored in the angular momentum. However, we also see that the energy is "degenerate" with another invariant of motion, r 2 0 , as in the quantum case. In any case, according to equation (D30) the radius of the trajectory is proportional to the square-root of the full energy.

FIG. 1 .

 1 FIG.1. Representation of a circular magnetic field line (green) of radius ρ, that we call in this paper the main circle. The blue shadow around the line represents the wave function of a ground orthogonal level with a characteristic extent λ. The relation between the toroidal coordinates (r, θ, φ) and the cartesian coordinates (x, y, z) is also shown.

where b 1

 1 , b 2 , b 3 , b 4 are constants of φ and θ, and j ⊥ and l are the proper values of Ĵθ and Ĵx respectively. l and j ⊥ are half-integers ... -3

cos φ cos θ sin φ sin θ sin φ 0

 0 sin φ cos θ sin φ -sin θ(ρ + r sin φ) r cos θ cos φ sin θ sin φ cos θ(ρ + r sin φ) r sin θ cos φ   , (A2) with determinant det J T = -r(ρ + r sin φ), and inverse

∂x 2 .

 2 By construction we now haveQ Ẽ = ∂ H ∂ Ẽ = 1.Integrating and equating to (D17) one obtains :a -x 2 √ 2ax 2 -x 4 -1 = ± tan (2τ + θ 0 ) (D18)Solving for x 2 in (D18), one obtainsx 2 = a ± a 2 -1 |sin(2τ + θ 0 )| . (D19)

2 + r 2 -r d 2 r dθ 2 .

 222

Here is an example with the derivation operators ,

∂ y = cos θ sin φ∂ r -sin θ ρ + r sin φ ∂ θ + cos θ cos φ r ∂ φ , (A7)

The Minkowski metric η = (1, -1, -1, -1) transforms according to

which gives

(A10)

Transformation of contravariant quantities

Contravariant quantities transform like

This is the case for example of the magnetic potential or of the Dirac matrices α (which are not really contravariant but we use this type of transformation i the text). In particular, α r = cos φα x + cos θ sin φα y + sin θ sin φα z , (A13)

The Laplacian is needed for the kinetic part of the second order Dirac equation,

Practically, we always have ρ+r sin φ > 0 in this paper.

b. Divergence

The divergence can be used to derive the second order Dirac equation and is given by

c. Rotational of covariant components

We need the rotational of the magnetic covariant vector which gives the magnetic field

x , (A18) which explicitly reads

Appendix B: Second-order Dirac equation in toroidal coordinates

Greek indices are used for Minskowski space-time of metric signature (+ ---) while latin indices are used for the spatial part only. η µν represents the Minkowski metric, ijk the fully antisymetric (Ricci) pseudo-tensor, 1 represents the identity.

We start with the derivation of the second-order Dirac equation in Cartesian coordinates and then turn it into toroidal coordinates. We take into account the coupling of an electron of charge -e to a classical electromagnetic field defined by a four-potential (A µ ) = (Φ/c, A) through the covariant derivative defined as