C Lauvernet
email: claire.lauvernet@cemagref.fr

; Lefrancq

; Carluer

Development of an operational tool to simulate pesticide transfer through vegetated strips: the 2D runoff part

Development of an operational tool to simulate pesticide transfer through vegetated strips: the 2D runoff part

Claire Lauvernet, M. Lefrancq, N.

Introduction

Well-designed vegetative filter strips can be a solution to mitigate fluxes of pesticides from agricultural plots. Modelling can help in proportioning and positioning these structures. The aim of this work is then to provide to the French decision makers a modeling tool simple enough to be operationally used to assist them in the implementation of these buffer zones. This paper presents the first step, that is to say the runoff part on a grass strip topography in two dimensions.

Materiel and method

This tool uses Richards and Kinematic wave equations to describe respectively subsurface and surface processes and their interactions. Richards equations are solved on each soil column with the Ross method [START_REF] Ross | Modeling soil water and solute transport -fast, simplified numerical solutions[END_REF]. These columns will be coupled to take into account lateral flows in water tables. Kinematic wave calculations are solved in two-dimensional form, using a finite volume method and an uncentred Euler scheme. The requirement to simulate runoff in 2 dimensions is justified by many situations encountered in France: slopes in several directions on a same grass strip but also talwegs or tyre tracks which can concentrate the runoff and accelerate it, minimizing the efficiency of the buffer strip in function of their localization and direction. The model has been compared in one dimension to the VFSMOD model, which describes hydrology, sediment and pollutant transport through vegetative filter strips and is mainly validated [START_REF] Munoz-Carpena | Numerical approach to the overland flow process in vegetative filter strips[END_REF].

Results

Figure 1 shows the Iwagaki test, which represents the most famous test to check if a code solving the kinematic wave is robust against hard changes of slopes or roughness. It consists in simulating three rains in the same time in three different parts of the grass strip, with three values of Manning's coefficient for the roughness (0.015; 0.01; 0.02). This simulation is compared to data from [START_REF] Iwagaki | Fundamental studies on runoff analysis by characteristics[END_REF] and to VFSMOD simulations. Our finite volumes scheme gives good results, coherent with the data and quite good fit with VFSMOD simulation. The code proves to be robust against kinematic shocks. A simple sensitivity analysis was performed for several input parameter values to test the model robustness and coherence to input variation. Figure 2 shows the sensitivity of the output flow to the slope, which varies from 1% to 20%. Results show as expected that the most important is the slope, the more rapidly the outflow reaches the stationary state, but also that even for sharp slopes (20% is quite rare for grass strips), the code is still stable and ensures very good mass balances.

Finally, simulations of the runoff on grass strips have been performed in 2 dimensions. As no runoff data are available in 2 dimensions to validate the code, observations of the outflow and mass balance permit to check the logical behaviour of the model. Figure 3 shows a test where the grass strip contains a talweg from East to West, in the same direction as the incoming flow (from East). The flow begins to concentrate in the talweg close to the beginning of the simulation (left) and is more and more concentrated until the end (right), accelerating the flux and minimizing the possible effect of infiltration of the grass strip. Mass balance is very good (error of 10 - 13 m 3 /s). As a conclusion, the runoff module has been applied to several synthetic and real situations, and thereby validated. The most innovative aspect of this development lies in the 2 dimensional description of the topography, which will permit to simulate grass strips containing talwegs. It needs now to be coupled with the solution of Richards equation, as infiltration is the most important process of efficiency of grass strips. This model will then be able to be compared with other reliable complex models as a validation for the whole water part. Finally, the pesticides part will be added to this tool, which will be validated with experimental data including water tables which may, particularly in some French contexts, limit the efficiency of vegetated strips.

Figure 1 .

 1 Figure 1. Iwagaki test to check the kinematic wave solving code robustness: comparison of the code with data experiments (stars) and VFSMOD simulation (blue line).

Figure

  Figure 2. Sensitivity analysis of the output flow (m 3 /s) to the slope, from 1% to 20%.

Figure 3a .

 3a Figure 3a.-3.b. Runoff simulated in 2 dimensions on a grass strip containing a talweg with an input flux coming from right and concentrating in the talweg, at times 60s and 300s.


N Carluer
email: nadia.carluer@cemagref.fr

; Giannone

; Gouy

; Gril

Towards a tool to design vegetated strips for mitigation of pesticides transfers in surface runoff. Assessment of different scenarios

Introduction

Vegetated strips can be useful in order to limit the transport of pesticides from agricultural fields towards surface waters. However, as the efficiency of these structures is closely related to their ability to intercept incoming water, they need to be appropriately sited and of suitable size. In this paper, we assess the efficiency of vegetative strips to limit surface runoff in contrasting situations, representative of different parts of France.

Material and methods

The modelling tool is a version of HYDRUS-2D (Simunek et al,1999) coupled with a kinematic wave equation, which represents both surface runoff and variably saturated flow in soil. Three agro-pedo-climatic situations were investigated: (1) La Morcille, in a vineyard area close to Lyon, with very permeable soils and intensive runoff events, (2) Le Pays de Caux, in the North West of France, where soils are very susceptible to hortonian overland flow because of surface crusting, and (3) La Jaillière close to Brittany, where soils are often hydromorphic. For each site, scenarios tested several conditions for the vegetative strip: slope (2, 7 and 15%), width (1, 5, 10 and 20 m), surface roughness (Manning coefficient of 0.05; 0.1 and 0.4 s.m -1/3 , i.e. going from very low to high roughness). For each combination of these parameters, typical summer and winter events were simulated, considering initial water status (particularly depth of the water table, when present) and the incoming runoff (intensity, duration).

Results

Vegetative strip behaviour can be described using a "double cumulative" curve, which plots cumulative outcoming runoff versus cumulative incoming runoff. For example, Figure 1 illustrates results for strips of respectively 1, 5, 10 and 20 metres width, in La Morcille, during summer conditions (deep water table, incoming flow of 1600 L/h for a width of one metre, during one hour). The point for 0 m corresponds to incoming flow; we can see that the wider the vegetative strip, the greater the attenuation of the incoming flow (roughly 1000 L for a 10 m wide strip). The slope of the curve, which remains lower than 1.0 even when outcoming runoff has begun (incoming flow exceeds infiltration capacity), shows that infiltration is still occurring It is important to note that this is no more true when the buffer zone is totally saturated (for example for "La Jaillière" in winter): then, once outcoming runoff begins, it is equal to incoming runoff (slope of the curve = 1.0), i.e. the efficiency of the buffer is limited to an initial "buffer volume". Results show significant contrasts between areas, depending in a major way on the mechanism which leads to runoff on the buffer surface, namely hortonian overland flow or runoff by surface saturation (water table rises to the surface, and no more infiltration occurs). In the first case (Pays de Caux and La Morcille in summer), the reduction of flow is significant, often greater than 50 %, even if it depends closely on incoming flow intensity, such as in the case of Pays de Caux whose soils are crusted. In the second case, namely La Jaillière, the efficiency of the vegetated strip is very limited in winter. However, these findings have to be placed into context with the seasonal risk for a given area, taking into account the dominant crops and pesticide usage, before it is possible to make any firm conclusions on the actual efficiency of a vegetative buffer strip in a given context Though results show global trends for each agro-pedo-climatic set of scenarios, they also provide evidence that it is essential to design each vegetative strip individually, taking into account local conditions. Based on the results, the most influential variables were identified and correlation rules were established. Correlation rules allow us to determine the relative efficiency of a vegetative strip to limit surface runoff, without a complete simulation, given its specific characteristics and the characteristics of the incoming runoff. We can see in Figure 2 that the coefficient of determination is very good for "saturation" cases (R² = 0.9929). In this case, the regression is based on the incoming water volume[D1], the strip's width and the available volume of soil porosity [D2]at the beginning of the event. References Simunek, J., M. Sejna, et al. (1999). The HYDRUS-2D software package for simulating the two-dimensional movement of water, heat, and multiple solutes in variably-saturated media, International Groundwater Modeling Center, Riverside, California: 227 pp.

Figure

  Figure 1 : La Morcille, summer conditions, slope : 15% ; roughness : 0.2 s.m -1/3.

Figure 2 :

 2 Figure 2 : Outcoming volumes assessed by a correlation rule versus those simulated by Hydrus. Runoff arising from soil saturation (Jaillière ; Morcille in winter).


Cyril Cassagnes
email: cyril.cassagnes@gmail.com

Philippe Roose
email: philippe.roose@univ-pau.fr

Marc Dalmau
email: marc.dalmau@univ-pau.fr

KALIMUCHO -Software Architecture for Limited Mobile Devices

Keywords: D.2 [Software Engineering]: Requirements/Specifications, Design, Software Architectures, Interoperability, Adaptable Architecture, Reusable Software Design, Experimentation Heterogeneity, embedded devices, CDC/CLDC, platform

INTRODUCTION

Because of the massive growing of embedded mobile and limited devices uses in everyday life [START_REF] Lee | Context-Aware Service Composition for Mobile Network Environments[END_REF] (public transport, workplace, cars, domestic appliance, etc.), the study of distributed applications for such devices becomes a major thematic. Such tools do exist but are still too strongly relative to underlying technologies. Proposing design methods and software architectures models for the design, implementation and running of applications on mobile limited hardware is crucial. The objective of this paper is to present our platform able to manage dynamic reconfigurations of distributed applications deployed on various and heterogeneous embedded devices. The goal of the platform is also to ensure the dynamic management of software components in order to guarantee an adequate quality of service to the end user. The reminder of this paper is the following. We firstly present component and service-oriented component characteristics, and then we present related work our component model. In the third part, we introduce the Kalimucho platform: objectives, architecture, and functionalities. Finally, we present some perspectives before the conclusion.

DISTRIBITED APPLICATIONS

An application for heterogeneous mobile embedded and limited (low bandwidth, power consumption, etc.) device has to firstly prevent hardware and mobility limitation. We decided to use dynamic reconfiguration to bypass this problem. Intervention on the architecture of the application needs to have a specific software component model. Such model permits to act on the lifecycle of the components instances and to provide information on their execution state. Because we implemented the platform upon the OSGi service based platform, let's briefly make an introduction to OSGi.

Open Service Gateway Initiative (OSGI)

The OSGi (Open Service Gateway Interface) Framework implements a complete and dynamic service model [START_REF] Cremene | Miron -Adaptation dynamiques de services -DECOR'04 Déploiement et (Re) Configurations de Logiciels[END_REF]. Projects such as Apache Felix, Oscar or Concierge implement OSGi specifications. We use the power of OSGi with a component model over it.

Heterogeneity

The hardware heterogeneity leads an important limit for a full OSGi development. The platform is not available on all devices, and particularly on sensors or smart sensors. So, we cannot deploy the OSGI framework to the Sun SPOTs because the Squawk VM hosted on Sun Spot Sensors implements CLDC 1.1 specifications whereas OSGI requires CDC due to its need of the java.lang.ClassLoader mechanism. On such devices we shall thus use the isolate (JSR 121) mechanism intead. The component framework that we have designed and implemented take into account such limitations to bypass the heterogeneity limit.

Service oriented software components

A classic component model provides a separation of concerns (functional/non-functional preoccupations) but do not provide a tool for dynamic management. Service oriented approaches proposed new models called service oriented components. Several service oriented models do exist [START_REF] Ketfi | Donsez -Composants adaptable au dessus d'OSGi -Journée Systèmes à composants adaptables et extensibles -17 et 18 octobre[END_REF] and particularly over the OSGi one on which we will base the Osagaia model.

Certainly the most used by professionals is the Spring Framework (Spring Dynamic Modules) [START_REF] Escoffier | iPOJO : Un modèle à composant à service flexible pour les systèmes dynamiques[END_REF]. Nevertheless, it has two unacceptable restrictions: it does not permit substitution of service providers (into a composition) and doesnt support reconfiguration [START_REF] Escoffier | Dynamically Adaptable Applications with iPOJO Service Components -6th International Symposium on Software Composition (SC 2007)[END_REF]. Moreover, Spring is more oriented for application server development. It exists other models, more opened and extendable as iPOJO [15]. This model, proposed by the Apache foundation is based on OSGi. It mainly proposed customization by handlers. A handler is an object who can be plugged on an instance container. It is possible to extend the component model by creating a specific handler. A handler manages one non-functional requirement of the instances [START_REF] Escoffier | Dynamically Adaptable Applications with iPOJO Service Components -6th International Symposium on Software Composition (SC 2007)[END_REF][17] [START_REF] Diaconescu | Autonomic iPOJO: Towards Self-Managing Middleware for Ubiquitous Systems[END_REF]. Nevertheless, it depends on the following OSGi platforms: Apache Felix, Eclipse Equinox, Knoplerfish and cannot implemented without OSGi platform (or on other OSGi platforms), this is a problem for the heterogeneity, because it can not manage hosts as Sun Spots. Nevertheless, the handler mechanism is very interesting and the Osagaia model widely draws with: InputUnit, OutputUnit and ControlUnit, able to manage input/output data flows as well as control flows and commands. We also (as others models) will use component description [13] [START_REF] Louberry | Architectures Logicielles pour des Applications Hétérogènes Distribuées et Reconfigurables -NOTERE[END_REF] in order to describe the architecture of the application. So, the non-functional part of the Osagaia model is the same for all components composing the application.

Related Works

In [START_REF] Escoffier | Dynamically Adaptable Applications with iPOJO Service Components -6th International Symposium on Software Composition (SC 2007)[END_REF] authors explain how interesting it is to provide dynamic reconfiguration with sensor networks. In [START_REF] Desertot | Un modèle à composants pour les services basés capteurs -Proceeding in Journées Composants JC'05[END_REF] we can find a software component model for sensors with connectors dedicated to measures. Our approach is close to this philosophy. We propose to goes on and uniform the deployment of components for dynamic reconfiguration of application of heterogeneous hosts.

Building such applications has been the topic of several works [START_REF] Cremene | Miron -Adaptation dynamiques de services -DECOR'04 Déploiement et (Re) Configurations de Logiciels[END_REF][5] [17][19] [13]. Nevertheless, none of such design method integrates strongly limited hosts. Some of them use additional software layers highly consumer of resources, unacceptable for very limited hosts.

The goal of out software component model is to facilitate the work of the software developer. He focuses on the applicative/functional part and dont care about mobility, dynamic nor heterogeneity. So, we propose a Java uniform framework. Software architectures are often coupled to specific hardware [13] and highly power consumer. We propose a lightened architecture able to be deployed on (very) limited hosts. We do not use middleware but a set of services integrated to the Java platform with specific component and connectors containers, called Osagaia (means component in Basque language) and Korrontea (means connector in Basque language). The following part details each of these containers and how they interact with the platform.

COMPONENT MODEL

Functional properties implemented into the components are the core of the application. Non-functional ones deal with mechanisms implementing the dynamic of the architecture and the supervision of the application by the platform which manages its QoS (user and hardware oriented). The Osagaia is a first class component container (Fig. 1) divided into three non-functional entities: InputUnit (IU), OutputUnit (OU) and ControlUnit (CU). The functional part of the component is called the Business Component (BC). The CU exports the interface for managing the life cycle (creation, configuration, validity, destruction), initializes configurable values of component, receives commands and sends states. The configuration deals with the management of QoS events. These events are then captured and evaluated by the platform which triggers dynamic reconfiguration [START_REF] Laplace | Conception d'Architectures Logicielles pour intégrer la qualité de service dans les applications multimédias réparties[END_REF]. The IU and OU allow the container to be connected to the corresponding Korrontea connectors according to the connection commands send by the platform. The platform receives such QoS data; compute them in order to decide if a reconfiguration is needed.

Osagaia Component Model

KORRONTEA CONNECTOF MODEL

KALIMUCHO platform

The objective of the distributed platform, called Kalimucho (means "lot of quality") is to supervise the Osagaia software components and Korrontea connectors as well as to reconfigure the application acting on life cycle of components/connectors or moving them on other hosts. A reconfiguration is done in order to ensure an adequate QoS [START_REF] Laplace | Conception d'Architectures Logicielles pour intégrer la qualité de service dans les applications multimédias réparties[END_REF].

Table. 1. JVM with corresponding JSR

As previously mentioned, the hardware heterogeneity is hidden with the use of the Java Virtual Machines (JVM). This language is quit natural because of the wide variety of devices/hardware supported, and with the wide possibility to get light JVM for (strongly) limited (mobile) devices. The Java environment is also suitable for devices without OS as Sun Spot Sensors. It is made of several platforms (J2EE, J2SE, J2ME), Java API, and a JVM specific to each device.

Services

The Kalimucho platform is based on services. As the platform will be deployed on each device, including limited ones, the question is how to distribute those services (it is a utopia to deploy the whole platform on too limited devices) according to limitations of the host. So, all services cannot be deployed verbatim on all devices. Consequently, the platform has various architectures, which is totaly compatible with the Kalimucho service based platform. Kalimucho proposes the following functionalities:

1. Creation of Osagaia containers encapsulating the business component (service factory).

Création of Korrontea connectors establishing communications between two components (service factory).

3. Capture of QoS events from components/connectors and determine when they imply to reconfigure the application. 4. Provide new configurations (ie. new assembling) with addition/suppression/moving/substitution of components and links between them. 5. Ensure coherency between services distributed over devices (discovering, unicity, addition, suppression, etc.). 6. Manage the mobility of the business code, ie, BC.

Container and Connectors factories are a solution for the first and second point. They create entities specifically adapted to the host onto which they are deployed (it exists a container for each kind of host). The Supervision Service and Configuration Generator correspond to the third point. When an event occurs, the supervision needs the evaluation of the current QoS. Then the Configuration Generator provides another configuration, corresponding to the fourth point. The fifth point is solved with the Component Register having the knowledge and maintaining the list of available components for mobile and/or limited devices which are not able to run the OSGi platform. So the Kalimucho platform has a global vision of the deployed application.

Each service needs more or less resources. For example, the configuration generator needs numerous calculations. So, the selection of services to be deployed on limited hosts is done according to resources available on devices. That is why all services are not deployed on each one. The objective is to maximize the distribution of the platform in order to maximize the autonomy of each device according to its potential (memory, energy, power, etc.).

We present here the deployment of the platform according to resources available of the host. On mobile devices (see. Fig. 3), development is adapted to the CDC API. For example, the Configuration service is not implemented, so the Supervision service does not exactly run as on a fixed device with no limitations (desktop PC) because it can't directly access to this service. Consequently, the configuration evaluation will be done by the nearest non limited host. Factories still have the same role but they are adapted to the CLDC implementation interface. The container factory service allows building Osagaia containers. This service is available on each device family because instantiation of a software component is strongly coupled to the hosting device. This service interacts with the supervision service in ordre to get the ID Card (a component description as used with COTS [START_REF] Roose | COTS: a help to COTS Products Integration -Third International Off-The-Shelf-Based Development Methods Workshop[END_REF]) of the business component from a component repository available on the network. When using Squawk JVM on very limited devices (CLDC), the repository is embedded onto the device. This choice is done because such devices do not provide class loader mechanisms (the number of BC has to be limited due to storage limits on such devices). ID Cards contain static properties (name, version, etc.) as well as dynamic information (input/output connectors, target host, etc.).

Connector Container factory service Figure 5. Connector Container Factory

Connector Container Factory as well as Component Container Factory receives commands from the Supervision Service. Such factory creates links between component instances. The connector encapsulates a business component implementing the communication mode/politic. If the connector is distributed on two devices, each supervision service builds one part of the connector. The supervision service provides information necessary to build the connection hosts (IP address, port, etc.). For a given communication politic, the designer has to provide as much connectors as device types. In order to modify the QoS, we must have several components able to provide the same service with various qualities, or we must provide various components or components combinations ensuring this service. The choice of the adequate combination is done by the platform.

Component Register Service Figure 6. Component Register Service

The main task of the Component Register Service is to provide the two following interfaces: ReachSensorComponent and UpdateSensorComponent. Its role is to replace the lack of the OSGi platform, and especially the Service Register on CLDC devices. With this service, the platform is aware of components deployments of each CLDC compliant sensor. When reconfiguring, the Component Register Service sends the new deployment schema to the Supervision of the nearest non limited host. Then, the register is updated if needed and components hosted on sensors are reconfigured with the new local deployment schema.

The Supervision Service

The Supervision Service uses the set of previously presented interfaces. This service supervises all the components of the application using their Control Unit (CU) interface. Communication between the application and the platform are only done between CU and the Supervision Service hosted on the device onto which components are hosted. The Supervision Service controls the execution of the application using QoS events raised by each container and monitors the hardware resources (battery, memory, CPU, number of current connections, etc.). Because all services are not hosted on all devices, when QoS information is captured, according to the device type, it has to be relayed towards a Supervision service able to access to all services. Such organization implicitly expresses a hierarchy between Supervision Services (Supervision, SupervisionCDC, SupervisionCLDC) as well as it exists between devices (more or less limited). The organization is detailed in the figure 3: When a local Supervision Service express the lack of a hardware resource (memory, energy, etc.), it raises a QoS event. The platform decides to act according to its configuration and thresholds values defined by the designer. If the device is mobile, the reconfiguration has to follow the Supervision hierarchy. When re-deploying, the Supervision service send query to the Configuration Generator with the QoS states. According to the values, the evaluation can be positive or negative. If so, according to the device, the Supervision Service gets the list of components causing the failure. This list comes from a description file sent by a distant device or by the Component Register and is transmitted to the Configuration Generator. It makes evaluation and starts the reconfiguration process. The new generated configuration is provided to the Supervision Service which broadcast the new deployment to corresponding hosts and starts its deployment.

Concerning the functional parts, the CU of the container detects that the QoS level is no more ensured. It then raises an event to the local Supervision Service ant the process goes on. A reconfiguration can have consequences on the architectures. A local or distributed reconfiguration can trigger other reconfigurations of the architecture. The evaluation is firstly based on parameters given by the user, secondly, the Configuration Generator obtained from the description file of the application, the list of related components. Consequently, the configuration generator provides the list of component to suppress, to migrate and to deploy on each corresponding device.

Configuration Generator Service

Conclusion and Future work

The design of ubiquitous applications needs to integrate the management of the hardware hosts heterogeneity as well as functional aspects. This implies to provide solutions in order to reconfigure applications (software deployment) in such mobile and versatile environment.

The service based Kalimucho platform has been implemented in order to adapt its own structure to application hosts possibilities as well as functional adaptations. These works are supported by Sun Microsystems (for the use of SunSpot sensors), "Conseil Général des Pyrénées Atlantiques" with a technological transfer with Dev 1.0. Software components, connectors and the Kalimucho platform are available on PC (OSGi/Felix platform), PDA (iPaq/OSGi/concierge) and SunSpor smart sensors (Squawk Java Virtual Machine).

Figure 1 .

 1 Figure 1. Osagaia Component

{Figure 2 .

 2 Figure 2. Korrontea ComponentBecause the assembling of Osagaia components may need various communications paradigms, we decided to design a connector model. The Korrontea container is based on the same architecture as Osagia (first class component). The main difference is that the Business Component implements communications modes. Each one is made of a triplet: Communication (local, distant), Mode (Synchronous/ Asynchronous/Real-Time/With-Without loss), Interface (socket, RPC, RMI)}. Thanks to the CU, the platform also supervises connectors in order to detect transmission problems (bandwidth, loss of connection, errors, etc.) and to change communication and interface characteristics. A distributed application is deployed with Osagaia components instantiations linked by Korrontea connectors. Our component model allows creating a dynamic architecture as well as giving QoS information through the CU.

Figure 3 .

 3 Figure 3. Deployment example of a limited device 4.2 Component Container factory service

Figure 4 .

 4 Figure 4. Component Container Factory

Figure 7 .

 7 Figure 7. Supervision on a non limited device.

Figure 8 .

 8 Figure 8. Configuration Generator Service This service provides EvaluateConfiguration and ComputeReconfiguration interfaces. It uses an enhancement criterion and a configurable user defined criterion [2]. The final user can configure the platform by modifying critical range values (energy level, CPU charge, memory, execution time, etc.).


Luigi Accardi
email: accardi@volterra.uniroma2.itameur@volterra.uniroma2.it

Ameur Dhahri

Polynomial extensions of the Weyl C * -algebra

.

Introduction

The program to develop a theory of renormalized higher (≥ 2) powers of white noise [START_REF] Accardi | Quantum probability, renormalization and infinite dimensional * -Lie algebras[END_REF] has led to investigate the connections between central extensions and renormalization. This has, in its turn, led to the discovery of the unique (up to isomorphisms) non trivial central extension of the 1dimensional Heisenberg algebra heis(1) and of its identification with the Galilei algebra (cf [START_REF] Accardi | Central extensions and stochastic processes associated with the Lie algebra of the renormalized higher powers of white noise, Noncommutative Harmonic Analysis with Applications to Probability II[END_REF], [START_REF] Accardi | Boukas Analytic central extensions of infinite dimensional white noise *-Lie algebras[END_REF]). In the present paper we will work in the 1dimensional Schroedinger representation where heis(1) can be realized as the real linear span of {1, p, q} (central element is the identity 1) and the operators (qϕ)(x) := xϕ(x); pϕ(x)

:= 1 i ∂ ∂x ϕ(x) ; ϕ ∈ L 2 (R)
are defined on suitable domains in L 2 (R) and satisfy the Heisenberg commutation relations [q, p] = i and all other commutators are zero. In the same representation Ceheis(1) can be realized as the real linear span of {1, p, q, q 2 } It has been recently proved that the current algebra over R of this algebra admits a non-trivial (suitably defined) Fock representation: this is equivalent to prove the infinite divisibility of the vacuum distributions of the operators of the form αq 2 + βq + γp ; α, β, γ ∈ R

For each N ∈ N, the real Lie algebra with generators {1, p, q, . . . , q N } in the following denoted heis(1, 1, N) is a natural generalization of the Galilei algebra. The analogy with the usual Heisenberg algebra, i.e heis(1, 1, 1, ) naturally suggests the following problems:

1) To describe the Lie group generated in the Schroedinger representation by heis(1, 1, N).

2) To find an analogue of the Weyl commutation relations for the group in item 1). This is equivalent to describe the (1, N)-polynomial extensions of the Heisenberg group.

3) To determine the vacuum distribution of the nontrivial elements of heis(1, 1, N) and their moments. These elements have the form up + N j=0 a j q j with u = 0 and some a j = 0 whith j ≥ 2.

4) To extend the constructions of items 1), 2) above to the continuous case, i.e. to describe the current * -Lie algebra of heis(1, 1, N) over R.

The solutions of these problems are discussed in the following. In particular we show that the group structure emerging from problem 2) is non trivial and yet controllable (see Theorem 1) thus allowing a solution of problem 3) (see Proposition 2). Problem 4) was solved in [START_REF] Accardi | Renormalization and central extensions of Lie algebras[END_REF] for the Galilei algebra (N = 2) but the technique used there cannot be applied to the polynomial case. In section 4 we introduce the q-projection-method in order to overcome this difficulty. This reduces the problem to the calculation of the expectation value of functionals of the form exp(P (X) + icX) where P is a polynomial with real coefficients, c is a real number, and X is a standard Gaussian random variable. If P has degree 2, this gives a different proof of result obtained in [START_REF] Accardi | Central extensions and stochastic processes associated with the Lie algebra of the renormalized higher powers of white noise, Noncommutative Harmonic Analysis with Applications to Probability II[END_REF]. The calculation of the moment in subsection 4.3 and the continuous second quantization versions of the polynomial extensions of the Weyl C * -algebra are new even in the quadratic case.

2 Polynomial extensions of the Heisenberg algebra: Discrete case The associated position and momentum operators are defined respectively by

q = b + + b √ 2 , p = b -b + i √ 2
Proposition 1 For all u, w ∈ C (w = 0) and a polynomial P in one determinate, we have e iwp+iuP ′ (q) = e iu P (q+w)-P (q)

w e iwp (2)
Proof. Let P a polynomial in one determinate and w, z ∈ C such that w = 0. Then from the identity

[P (q), p] = iP ′ (q)
It follows that [P (q), [P (q), p]] = 0

Hence [START_REF] Accardi | Central extensions and stochastic processes associated with the Lie algebra of the renormalized higher powers of white noise, Noncommutative Harmonic Analysis with Applications to Probability II[END_REF] implies that e izP (q) e iwp e -izP (q) = e iw(p-zP ′ (q)) (

But, one has also e iwp e -izP (q) = (e iwp e -izP (q) e -iwp )e iwp = e -izP (q+w) e iwp (4)

The identities ( 3) and ( 4) imply e iw(p-zP ′ (q)) = e izP (q) e -izP (q+w) e iwp

With the change of variable u := -wz, (5) becomes (2). Since the real vector space of polynomials of degree ≤ N with coefficients in R, denoted in the following

R N [x] := {P ′ ∈ R[x] : degree (P ′ ) ≤ N} has dimension N + 1,
in the following we will use the identification

P ′ (x) = N k=0 a k x k ≡ (a 0 , a 1 , • • • , a N ) ∈ R N +1
Now, for all w ∈ R, we define the linear map T w by T 0 := id if w = 0 and if w = 0

T w : P ′ ∈ R N [x] → 1 w w 0 P ′ (. + y)dy ∈ R N [x] (6)
so that

T w P ′ = P (• + w) -P w ( 7
)
where P is any primitive of P ′ .

Properties of the maps T w

Lemma 1 For each w ∈ R \ {0} the linear map T w is invertible. Moreover, we have

T -1 w = N k=0 (-1) k (T (w)) k
where T (w) is a lower triangular nilpotent matrix whose coefficients, with respect to the mononial real basis {1, x, . . . , x N } of R N [x], are given by

t mn (w) = χ {m<n} n! (n + 1 -m)!m! w n-m , 0 ≤ m, n ≤ N ( 8
)
where for all set I

χ {m<n} = 0, if m ≥ n 1, if m < n Proof. Let k ∈ {0, 1, • • • , N}. If we take P ′ (x) := x k
then, in [START_REF] Accardi | The quadratic Fock functor[END_REF] we can choose

P (x) = x k+1 k + 1 so that P (x + w) -P (x) w = 1 w(k + 1) [(x + w) k+1 -x k+1 ] Since (x + w) k+1 = k+1 h=0 k + 1 h x h w k+1-h = x k+1 + k h=0 k + 1 h x h w k+1-h it follows that (x + w) k+1 -x k+1 = k h=0 k + 1 h x h w k+1-h Hence T w 1 = 1 (9)
and, for all k ≥ 1

T w x k = 1 w(k + 1) k h=0 k + 1 h w k+1-h x h = k h=0 k! (k + 1 -h)!h! w k-h x h = k-1 h=0 k! (k + 1 -h)!h! w k-h x h + x k (10)
Therefore the matrix of T w in the basis {1, . . . , x N } of R N [x], we continue to denote it by T w , has the form I +T (w) where T (w) is a lower triangular nilpotent matrix whose coefficients, with respect to the basis {1, x, x 2 , . . . , x N } of R N [x], are given by

t mn (w) = χ {m<n} n! (n + 1 -m)!m! w n-m , 0 ≤ m, n ≤ N (11)
Hence T w is invertible because det(T w ) = 1. Moreover, one has

T -1 w = (I + T (w)) -1 = N k=0 (-1) k (T (w)) k (12) Let S denote the action of R on R N [x] by translations (S u P ′ )(x) := P ′ (x + u), x, u ∈ R Then T u P ′ = 1 u (S u -id)P
where P is a primitive of P ′ . The shift S u is a linear homogeneous map of

R N [x] into itself. Moreover N k=0 a k (x + u) k = N k=0 k h=0 a k k h x h u k-h = N h=0 N k=h a k k h u k-h x h = N h=0 (S u P ′ ) h x h
where

(S u P ′ ) h = N k=h a k k h u k-h
Thus, in the monomial basis {1, . . . , x N } of R N [x], the matrix associated to S u is

(S u ) hk = χ {h≤k} k h u k-h (13)
Note that S u is invertible with inverse S -u and

S u S v = S u+v Now denote ∂ the action of derivation operator on R N [x], i.e ∂ : N k=0 a k x k → N k=1 ka k x k-1 (14)
Then clearly

∂S u = S u ∂, ∀u ∈ R (15) Let R N [x] 0 the subalgebra of R N [x] of all polynomials vanishing in zero R N [x] 0 := {P ′ ∈ R N [x] : P ′ (0) = 0} It is clear that R N [x] = R N [x] 0 ⊕ R1
so that, thanks to (9), T w is uniquely determined by its restriction to R N [x] 0 . Denote

T 0 w := T w | R N [x] 0 ∂ 0 := ∂| R N [x] 0
Then (14) implies that ∂ 0 is invertible and (15) implies that

∂ -1 0 S u = S u ∂ -1 0 , ∀u ∈ R (16) Lemma 2
The following identities hold:

1) T w S u = S u T w , ∀u, w ∈ R 2) T w T u = T u T w , ∀u, w ∈ R 3) T u S v = (1 + v u )T u+v -v u T v , ∀u, v ∈ R such that u = 0 4) T u S -u = T -u , ∀u ∈ R 5) T u T v = ( 1 v + 1 u )T u+v -1 u T v -1 v T u , ∀u, v ∈ R \ {0} Proof.
In the above notations

T 0 w = 1 u (S u -id)∂ -1 0
and (16) implies that T 0 w S u = S u T 0 w , ∀u, w ∈ R Since the constant function 1 is fixed both for S u and for T w , it follows that

T w S u = S u T w , ∀u, w ∈ R
and therefore also

T w T u = T u T w , ∀u, w ∈ R
which implies that assertions 1) and 2) are satisfied. Now notice that translations commute with the operation of taking primitives, i.e. if P is a primitive of P ′ , then ∀v ∈ R, S v P is a primitive of S v P ′ , that is

T u S v P ′ = 1 u (S u -id)S v P = 1 u (S u+v -S v )P = 1 u (S u+v -id)P + 1 u (id -S v )P = u + v u T u+v - v u T v P ′
This proves 3). In particular, choosing v = -u, one finds T u S -u = T -u and 4) holds. Finally, from the identity

T u T v = 1 v T u S v - 1 v T u
and from 3), the assertion 5) follows.

Polynomial extensions of the Heisenberg algebra

Recall that the 1-dimensional Heisenberg group, denoted Heis(1), is the nilpotent Lie group whose underlying manifold is R × R 2 and whose group law is given by

(t, z)•(t ′ , z ′ ) = (t+t ′ -2σ(z, z ′ ), z+z ′ ); t, t ′ ∈ R ; z := (α, β), z ′ := (α ′ , β ′ ) ∈ R 2
where σ( • , • ) the symplectic form on R 2 given by

σ(z, z ′ ) := αβ ′ -α ′ β; z := (α, β), z ′ := (α ′ , β ′ ) ∈ R 2
In the Schroedinger representation of the real Lie algebra Heis(1) one can define the centrally extended Weyl operators through the map

(t, z) ∈ R × R 2 → W (t, z) := e i( √ 2(βq-αp)+t1) = W (z)e it (17)
where the Weyl operators are defined by

W (z) := e i √ 2(βq-αp) ; z := (α, β) ∈ R 2
From the Weyl relations

W (u)W (z) = e -iσ(u,z) W (z + u) (18)
one then deduces that

W (t, z)W (t ′ , z ′ ) = W ((t, z) • (t ′ , z ′ )) (19)
i.e. that the map (19) gives a unitary representation of the group Heis(1).

The generators of the centrally extended Weyl operators have the form

up + a 1 q + a 1 1 =: up + P ′ (q) ( 20
)
where u ∈ R and P is a polynomial in one indeterminate with real coefficients of degree at most 1. Thus they are exactly the elements of the one dimensional Heisenberg algebra heis(1).

Replacing P ′ , in (20), by a generic polynomual of degree at most N (in one indeterminate with real coefficients ), one still obtains a real Lie algebra because of the identity

[up + P ′ (q), vp + Q ′ (q)] = u[p, Q ′ (q)] -v[p, P ′ (q)] = -iuQ ′′ (q) + ivP ′′ (q)
and, since R N [x] has dimension N + 1, this real Lie algebra has dimension N + 2 on R.

Definition 1 For N ∈ N, the real Lie algebra

heis(1, 1, N) := {up + P ′ (q) : u ∈ R, P ′ ∈ R N [x]} is called the (1, N)-polynomial extensions of the 1-dimensional Heisenberg algebra.
The notation (1, N) emphasizes that the polynomials involved have degree 1 in the momentum operator and degree ≤ N in the position operator. With these notations the (1, N)-polynomial extensions of the centrally extended Weyl operators are the operators of the form

W (u, P ′ ) := e i(up+P ′ (q)) ; u ∈ R ; P ′ ∈ R N [x] (21)
By analogy with the 1-dimensional Heisenberg group, we expect that the pairs (u,

P ′ ) ∈ R × R N [x]
form a group for an appropriately defined composition law. The following theorem shows that this is indeed the case.

Theorem 1 For any (u, P ′ ), (v, Q ′ ) ∈ R × R N [x] one has W (u, P ′ )W (v, Q ′ ) = W ((u, P ′ ) • (v, Q ′ )) (22)
where

(u, P ′ ) • (v, Q ′ ) := (u + v, T -1 u+v (T u P ′ + T v S u Q ′ )) (23)
Proof. From Proposition 1, one has

W (u, P ′ ) = e iup+iP ′ (q) = e iTuP ′ (q) e iup Therefore W (u, P ′ )W (v, Q ′ ) = e iup+iP ′ (q) e ivp+iQ ′ (q) = e iTuP ′ (q) e iup e iTv Q ′ (q) e ivp = e iTuP ′ (q) e iTvQ ′ (q+u) e i(u+v)p = e i[TuP ′ (q)+Tv Q ′ (q+u)] e i(u+v)p = e i(u+v)p+iT -1 u+v (TuP ′ (q)+Tv SuQ ′ (q)) = W u + v, T -1 u+v (T u P ′ + T v S u Q ′ )
This proves ( 22) and (23).

Remark 1 The associativity of the group law ( 23) can be directly verified using Lemma 2.

Our next goal is to determine the (1, N)-polynomial extensions of the Weyl commutation relations. To this goal define the ideal

R N [x] 0 := {P 0 ∈ R N [x] : P 0 (0) = 0} (24)
Thus the projection map from

R N [x] onto R N [x] 0 is given by P ′ (x) ∈ R N [x] → (P ′ ) 0 (x) := P ′ (x) -P ′ (0) ∈ R N [x] 0 (25)
and, in the notation (25), the (1, N)-polynomial extensions of the centrally extended Weyl operators (21) take the form

W (u, P ′ ) = e i(up+P ′ (q)) = e i(up+(P ′ ) 0 (q)) e iP ′ (0) ; u ∈ R ; P ′ ∈ R N [x] (26)
The analogy between ( 26) and ( 19) naturally suggests the following definition.

Definition 2

The unitary operators

W (u, P ′ 0 ) := e i(up+P ′ 0 (q)) ; u ∈ R ; P ′ 0 ∈ R N [x] 0 ( 27
)
will be called (1, N)-polynomially extended Weyl operators.

From ( 22) and ( 23) we see that, if

P ′ 0 , Q ′ 0 ∈ R N [x] 0 , then in the notation (25) one has W (u, P ′ 0 )W (v, Q ′ 0 ) = W u + v, T -1 u+v (T u P ′ 0 + T v S u Q ′ 0 = (28) = W u + v, (T -1 u+v (T u P ′ 0 + T v S u Q ′ 0 )) 0 e i(T -1 u+v (TuP ′ 0 +TvSuQ ′ 0 ))(0)
By analogy with the usual Weyl relations we introduce the notation

σ((u, P ′ 0 ), (v, Q ′ 0 )) := T -1 u+v (T u P ′ 0 + T v S u Q ′ 0 ))(0) (29)
i.e. σ((u,

P ′ 0 ), (v, Q ′ 0 )) is the degree zero coefficient of the polynomial T -1 u+v (T u P ′ 1 + T v S u P ′ 2 ) ∈ R N [x]. Notice that the map ((u, P ′ 0 ), (v, Q ′ 0 )) ≡ ((u, v), (Q ′ 0 , P ′ 0 )) → σ((u, P ′ 0 ), (v, Q ′ 0 ))
is linear in the pair (Q ′ 0 , P ′ 0 ) but polynomial in the pair (u, v). This is an effect of the duality between p, which appears at the first power, and q, which appears in polynomial expressions. In order to prove the (1, N)-polynomial analogue of the classical Weyl relations (18) one has to compute the scalar factor (34). We will compute more generally all the coefficients of T -1 u+v (T u P ′ 1 + T v S u P ′ 2 ). In view of (12) this leads to compute the powers of the matrices T (w) given by [START_REF] Bourbaki | Fonctions d'une variable Réelle[END_REF].

Lemma 3 Let k ∈ {2, . . . , N} and w ∈ R\{0}. Then, the matrix of (T (w)) k , in the monomial basis {1, x, . . . ,

x N } of R N [x], is given by (T [k] ij (w)) 0≤i,j≤N , where T [k] ij (w) = t ij (w)C [k] i,j (30)
the coefficients t ij (w) are given by ( 8) and the C

[k] i,j are inductively defined by

C [k] i,j = N h=0 C (h) i,j C [k-1] i,h (31)
where

C (h) i,j = (j + 1 -i)! (h + 1 -i)!(j + 1 -h)! χ {i<h<j} (h)
and C [START_REF] Accardi | Central extensions and stochastic processes associated with the Lie algebra of the renormalized higher powers of white noise, Noncommutative Harmonic Analysis with Applications to Probability II[END_REF] i,j = χ {i<j} for all i, j = 0, . . . , N.

Proof. We prove the lemma by induction. For k = 2, one has

t ih (w)t hj (w) = χ {i<h} χ {h<j} 1 h + 1 h + 1 i w h-i 1 j + 1 j + 1 h w j-h = χ {i<h<j} j! (j + 1 -i)!i! w j-i (j + 1 -i)! (h + 1 -i)!(j + 1 -h)! = t ij (w)C (h) i,j (32)
where

C (h) i,j = (j + 1 -i)! (h + 1 -i)!(j + 1 -h)! χ {i<h<j}
It follows that

T [2] ij (w) = t ij (w) N h=0 C (h) i,j = t ij (w)C [2] i,j
Now, let 2 ≤ k ≤ N -1 and suppose that (30) holds true. Then, one has

T [k+1] ij (w) = N h=0 T [k] ih (w)t hj (w)
Using induction assumption, one gets

T [k+1] ij (w) = N h=0 t ih (w)t hj (w)C [k] i,h (33)
Therefore the identities (31), ( 32) and (33) imply that

T [k+1] ij (w) = t ij (w) N h=0 C (h) i,j C [k] i,h = t ij (w)C [k+1] i,j
As a consequence of Theorem 1 and Lemma 3, we are able to explicitly compute the scalar factor in the (1, N)-polynomial extensions of the Weyl relations (28).

Proposition 2 In the notation (34), for any u, v ∈ R and for any

P ′ 1 , P ′ 2 ∈ R N [x] 0 given by P ′ 1 (x) = α 1 x + . . . + α N x N , P ′ 2 (x) = β 1 x + . . . + β N x N one has σ((u, P ′ 0 ), (v, Q ′ 0 )) = N j=0 1 j + 1 (u + v) j N m=0 (-1) m C [m] 0,j (34)
j≤h≤N t jh (u)α h + t jh (v) h≤k≤N (S u ) hk β k
where the coefficients (S u ) hk are given by ( 13) and with the convention

C [0] 0,j = δ j,0 , α 0 = β 0 = 0, t jj (w) = 1
, for all j = 0, . . . , N.

Proof. Let u, v ∈ R and P ′ 1 , P ′ 2 ∈ R N [x] 0 be as in the statement. Then, using together Lemmas 1, 3 and identity (13), we show that

T -1 u+v (T u P ′ 1 + T v S u P ′ 2 )(x) = (T -1 u+v (T u P ′ 1 + T v S u P ′ 2 )) 0 (x) + N j=0 1 j + 1 (u + v) j N m=0 (-1) m C [m] 0,j j≤h≤N t jh (u)α h + t jh (v) h≤k≤N (S u ) hk β k Definition 3 For N ∈ N, the real Lie group with manifold R × R × R N [x] 0 ≡ R × R N [x] ( 35
)
and with composition law given by

(u, P ′ ) • (v, Q ′ ) := (u + v, T -1 u+v (T u P ′ + T v S u Q ′ )
) is called the (1, N)-polynomial extensions of the 1-dimensional Heisenberg group and denoted Heis(1, 1, N).

Remark 2

The left hand side of (35) emphasiyes that the coordinates (t, u, a 1 , . . . , a N ) ∈ R N +2 of an element of Heis(1, 1, N) are intuitively interpreted as time t, momentum u and coordinates of the first N powers of position (a 1 , . . . , a N ). Putting t = a 0 is equivalent to the identification

R × R N [x] 0 ≡ R N [x].

The discrete Heisenberg algebra {1, p, q}

From Definition 1, it follows that {1, p, q} is a set of generators of heis(1, 1, 1) so that heis(1, 1, 1) ≡ heis(1)

Moreover the composition law associated to Heis(1) is given by the following.

Proposition 3 Let P ′ 1 (x) = α 0 + α 1 x, P ′ 2 (x) = β 0 + β 1 x and let u, v ∈ R. Then, we have (u, P ′ 1 ) • (v, P ′ 2 ) = (u + v, Q ′ )
where

Q ′ (x) = (α 0 + β 0 + 1 2 uβ 1 - 1 2 vα 1 ) + (α 1 + β 1 )x
Proof. Let u, w ∈ R. Then, one has

T w = 1 1 2 w 0 1 , T -1 w = 1 -1 2 w 0 1 S u = 1 u 0 1 Let P ′ 1 (x) = α 0 + α 1 x, P ′ 2 (x) = β 0 + β 1 x.
Then, from Thereom 1, the composition law is given by

(u, P ′ 1 ) • (v, P ′ 2 ) = (u + v, Q ′ ) where Q ′ (x) = T -1 u+v (T u P ′ 1 + T v S u P ′ 2 )(x) = (α 0 + β 0 + 1 2 uβ 1 - 1 2 vα 1 ) + (α 1 + β 1 )x
In particular recall that for all z = α + iβ, the Weyl operator W (z) is defined by

W (z) = W (α, β) := e i √ 2(βq-αp) Put P ′ 1 (x) = β √ 2x, P ′ 2 (x) = β ′ √ 2x
Then from the above proposition, one has

(- √ 2α, P ′ 1 ) • (- √ 2α ′ , P ′ 2 ) = (- √ 2(α + α ′ ), Q ′ )
where

Q ′ (x) = (α ′ β -αβ ′ ) + (β + β ′ ) √ 2x
Therefore, for all complex numbers z = α + iβ, z ′ = α ′ + iβ ′ , it follows that

W (z)W (z ′ ) = W (α, β)W (α ′ , β ′ ) = e i(α ′ β-αβ ′ ) e i( √ 2(β+β ′ )p- √ 2(α+α ′ )p = e -iℑ(zz ′ ) W (z + z ′ ) Remark 3
The above proposition proves that the composition law given in Definition 3 is indeed a generalization of the composition law of the Heisenberg group.

2.4

The discrete Galilei algebra {1, p, q, q 2 } In the case of N = 2, the composition law is given by the following.

Proposition 4 Let P ′ 1 , P ′ 2 ∈ R 2 [x], such that P ′ 1 (x) = α 0 + α 1 x + α 2 x 2 and P ′ 2 (x) = β 0 + β 1 x + β 2 x 2 .
Then, for all u, v ∈ R, we have

(u, P ′ 1 ) • (v, P ′ 2 ) = (u + v, Q ′ ) where Q ′ (x) = γ + βx + αx 2 with γ = α 0 + β 0 - 1 2 vα 1 + 1 2 uβ 1 + 1 6 (v -u)vα 2 + 1 6 (u -v)uβ 2 β = α 1 + β 1 -vα 2 + uβ 2 α = α 2 + β 2
Proof. From Lemma 1, the matrices of T w and T -1 w , in the monomial basis {1, x, x 2 } of R 2 [x], are given by

T w =   1 1 2 w 1 3 w 2 0 1 w 0 0 1   , T -1 w =   1 -1 2 w 1 6 w 2 0 1 -w 0 0 1  
Moreover, identity (13) implies that

S u =   1 u u 2 0 1 2u 0 0 1   Let P ′ 1 (x) = α 0 + α 1 x + α 2 x 2 , P ′ 2 (x) = β 0 + β 1 x + β 2 x 2 and u, v ∈ R.
Then, using the explicit form of T -1 u+v , T u and S u , it is straightforward to show that T -1 u+v (T u P ′ 1 + T v S u P ′ 2 )(x) = γ + βx + αx 2 where α, β, γ are given in the above proposition. Finally, from Theorem 1 we can conclude.

q t = b t + b + t √ 2 ; p t = b t -b + t i √ 2 Then [q t , p s ] = i δ(t -s) ; [q k t , p s ] = ik q k-1 t δ(t -s), k ≥ 1 [q t , q s ] = [p t , p s ] = [q k t , q k s ] = [q k t , q s ] = 0 and (q s ) * = q s ; (q k s ) * = q k s ; (p s ) * = p s Now for all real function f ∈ L 2 (R ∩ L ∞ (R), we define p(f ) = R f (t)p t dt, q k (f ) = R f (t)q k t dt, k ∈ N Proposition 5 For all real functions g, f 1 , . . . , f n ∈ L 2 (R)∩L ∞ (R),
we have e i(p(g)+q n (fn)+...+q(f

1 )) = e i n k=1 k h=0 k! (k+1-h)!h! q h (g k-h f k ) e ip(g)
Proof. Consider real functions g, f 1 , . . . , f n ∈ L 2 (R) ∩ L ∞ (R). Put U t = e it(p(g)+q n (fn)+...+q(f 1 )) e -itp(g)

Then one has ∂ t U t = ie it(p(g)+q n (fn)+...+q(f 1 )) (p(g) + q n (f n ) + . . . + q(f 1 ))e -itp(g) -ie it(p(g)+q n (fn)+...+q(f 1 )) p(g)e -itp(g) = ie it(p(g)+q n (fn)+...+q(f 1 )) (q n (f n ) + . . . + q(f 1 ))e -itp(g) = ie it(p(g)+q n (fn)+...+q(f 1 )) e -itp(g) [e itp(g) (q n (f n ) + . . .

+ q(f 1 ))e -itp(g) ] = iU t [e itp(g) (q n (f n ) + . . . + q(f 1 ))e -itp(g) ] (36)
Note that e itp(g) q k (f )e -itp(g) = R e itp(g) q k s e -itp(g) f (s)ds Put V t,s = e itp(g) q k s e -itp(g) Then, one has

∂ t V t,s = ie itp(g) [p(g), q s ]e -itp(g) = ie itp(g) R g(u)[p u , q s ]e -itp(g) = R g(u)δ(s -u)du = g(s)
This gives V t,s = q s + tg(s)

It follows that e itp(g) q k s e -itp(g) = (q s + tg(s)) k Therefore, one gets

e itp(g) q k (f )e -itp(g) = R e itp(g) q k s e -itp(g) f (s)ds = R (q s + tg(s)) k f (s)ds = k h=0 k h t k-h R q h s (g(s)) k-h f (s)ds = k h=0 k h t k-h q h (g k-h f ) (37)
Using identities (36) and (37), one gets

∂ t U t = U t n k=1 k h=0 k h t k-h q h (g k-h f k )
This implies that

U t = e i n k=1 k h=0 k!t k-h+1 (k+1-h)!h! q h (g k-h f k )
Finally, one gets e it(p(g)+q n (fn)+...+q(f

1 )) = e i n k=1 k h=0 k!t k-h+1 (k+1-h)!h! q h (g k-h f k ) e itp(g)
As a consequence of the above proposition, we prove the following.

Lemma 4 For all real functions g, f 1 , . . . , f n ∈ L 2 (R) ∩ L ∞ (R), we have e i(p(g)+q(f 1 )+...+q n (fn)) = e iTg (q(f 1 )+...+q n (fn)) e ip(g)

where

T g q k (f k ) = k h=0 k! (k + 1 -h)!h! q h (g k-h f k ) = R T g(s) (q k s )f k (s)ds ( 38
)
with T g(s) = I + T (g(s)), where, for all s ∈ R, T (g(s)) is given by Lemma 1.

Proof. From Proposition 5, one has e i(p(g)+q(f 1 )+...+q n (fn

)) = e i n k=1 k h=0 k! (k+1-h)!h! q h (g k-h f k ) e ip(g) = e i n k=1 Tg q k (f k ) e ip(g)
Moreover, one has

T g q k (f k ) = R k h=0 k! (k + 1 -h)!h! q h s g k-h (s) f k (s)ds
Now using identity (10), it follows that

k h=0 k! (k + 1 -h)!h! q h s g k-h (s) = T g(s) (q k s )
This ends the proof.

Lemma 5 For all real functions g,

f k ∈ L 2 (R) ∩ L ∞ (R), we have T -1 g q k (f k ) = R T -1 g(s) (q k s )f k (s)ds Proof. Let g, f k ∈ L 2 (R) ∩ L ∞ (
R) be real functions. Define F g as follows

F g q k (f k ) = R T -1 g(s) (q k s )f k (s)ds
Then one has

F g (T g q k (f k )) = k h=0 k! (k + 1 -h)!h! F g q h (g k-h f k ) = k h=0 k! (k + 1 -h)!h! R T -1 g(s) (q h s )g k-h (s)f k (s)ds = R T -1 g(s) k h=0 k! (k + 1 -h)!h! q h s g k-h (s) f k (s)ds = R (T -1 g(s) T g(s) (q k s ))f k (s)ds = q k (f k ) = T g (F g q k (f k ))
which proves the required result.

Theorem 2 Let g, f 1 , . . . , f n , G, F 1 , . . . , F n ∈ L 2 (R) ∩ L ∞ (R) be real func- tions.
Then, the composition law associated to the continuous polynomial extensions of the Heisenberg algebra is given as follows

(g, q(f 1 ) + . . . + q n (f n )) • (G, q(F 1 ) + . . . + q n (F n )) = (g + G, T -1 g+G (T g (q(f 1 ) + . . . + q n (f n )) + T G S g (q(F 1 ) + . . . + q n (F n )))
where for all real function f ∈ L 2 (R) ∩ L ∞ (R), we have

S g q k (f ) = R (q s + g(s)) k f (s)ds = R S g(s) (q k s )f (s)ds (39)
with S g(s) is the translation operator.

Proof. Consider real functions g, f 1 , . . . , f n , G, F 1 , . . . , F n ∈ L 2 (R) ∩ L ∞ (R).
Then, from Lemma 4, one has e i(p(g)+q(f 1 )+...+q n (fn)) e i(p(G)+q(F 1 )+...+q n (Fn)) = e i(q(f 1 )+...+q n (fn)) e ip(g) e i(q(F 1 )+...+q n (Fn)) e ip(G) = e i(q(f 1 )+...+q n (fn)) e ip(g) e i(q(F 1 )+...+q n (Fn)) e -ip(g) e ip(g+G)

= e i(q(f 1 )+...+q n (fn)) e i(e ip(g) q(F 1 )e -ip(g) +...+e ip(g) q n (Fn)e -ip(g) ) e ip(g+G)

But, from identity (37), one has

e ip(g) q k (F k )e -ip(g) = R (q s + g(s)) k F k (s)ds = S g q k (F k )
This gives e i(p(g)+q(f 1 )+...+q n (fn)) e i(p(G)+q(F 1 )+...+q n (Fn)) = e i(Tg(q(f 1 )+...+q n (fn))+T G Sg(q(F 1 )+...+q n (Fn))) e ip(g+G) = e i(p(g+G)+T -1 g+G (Tg(q(f 1 )+...+q n (fn))+T G Sg (q(F 1 )+...+q n (Fn))))

which ends the proof of the above theorem.

The continuous Heisenberg algebra

{1, p(f ), q(f )} For all f ∈ L 2 (R), define B(f ) = R b s f (s)ds, B + (f ) = R b + s f (s)ds Let f = f 1 + if 2 , g = g 1 + ig 2 ∈ L 2 (R) such that f i , g i , i = 1, 2, are real functions.
Then, a straightforward computation shows that

B(f ) + B + (f ) = √ 2(q(f 1 ) + p(f 2 )) = q(f ′ 1 ) + p(f ′ 2 )
where for all i = 1, 2,

f ′ i = √ 2f i .
Recall that the Weyl operator, associated to an element

f = f 1 + if 2 ∈ L 2 (R), is defined by W (f ) = e i(B(f )+B + (f )) = e i √ 2(q(f 1 )+p(f 2 )) = e i(q(f ′ 1 )+p(f ′ 2 ))
Note that from Theorem 2, one has

e i(q(f ′ 1 )+p(f ′ 2 )) e i(q(g ′ 1 )+p(g ′ 2 )) = e i(p(f ′ 2 +g ′ 2 )+T -1 f ′ 2 +g ′ 2 (T f ′ 2 q(f ′ 1 )+T g ′ 2 S f ′ 2 q(g ′ 1 ))) (40)
But, one has

T f ′ 2 q(f ′ 1 ) = 1 2 R f ′ 1 (s)f ′ 2 (s)ds + q(f ′ 1 ) (41)
and

T g ′ 2 S f ′ 2 q(g ′ 1 ) = R g ′ 1 (s)f ′ 2 (s)ds + 1 2 R g ′ 1 (s)g ′ 2 (s)ds + q(g ′ 1 ) (42)
Moreover, recall that the matrix of T -1

f ′ 2 (s)+g ′ 2 (s) in the canonical basis {1, x} of R 2 [x] is given by 1 -1 2 (f ′ 2 (s) + g ′ 2 (s)) 0 1 This proves that T -1 f ′ 2 +g ′ 2 q(f ′ 1 ) = - 1 2 R f ′ 1 (s)f ′ 2 (s)ds - 1 2 R f ′ 1 (s)g ′ 2 (s)ds + q(f ′ 1 ) (43)
T -1 f ′ 2 +g ′ 2 q(g ′ 1 ) = - 1 2 R g ′ 1 (s)f ′ 2 (s)ds - 1 2 R g ′ 1 (s)g ′ 2 (s)ds + q(g ′ 1 ) (44)
Therefore, using identities (40)-(44), one gets

e i(q(f ′ 1 )+p(f ′ 2 )) e i(q(g ′ 1 )+p(g ′ 2 )) = e i 2 ( g ′ 1 ,f ′ 2 -f ′ 1 ,g ′ 2 ) e i(q(f ′ 1 +g ′ 1 )+p(f ′ 2 +g ′ 2 )) (45)
Finally, by taking

f ′ i = √ 2f i , g ′ i = √ 2g i , i = 1, 2, in (45)
, one obtains the well known Weyl commutation relation

W (f )W (g) = e -iℑ f,g W (f + g) 3.2 The continuous Galilei algebra {1, p(f ), q(f ), q 2 (f )}
The composition law associated to the continuous Galilei algebra

{1, p(f ), q(f ), q 2 (f ), f = f ∈ L 2 (R) ∩ L ∞ (R)} is giving by the following. Proposition 6 For all real functions g, G, f 1 , f 2 , F 1 , F 2 ∈ L 2 (R) ∩ L ∞ (R), we have g, q(f 1 ) + q 2 (f 2 ) • G, q(F 1 ) + q 2 (F 2 ) = g + G, 1 2 g, F 1 - 1 2 G, f 1 + 1 6 g, (g -G)F 2 + 1 6 G, (G -g)f 2 +q(f 1 ) + q(F 1 ) + q(gF 2 ) -q(Gf 2 ) + q 2 (f 2 ) + q 2 (F 2 )
Proof. Recall that the matrices of T g(s) , S g(s) and T -1 g(s)+G(s) in the monomial basis {1, x, x 2 } of R 2 [x] are given by

S g(s) =   1 g(s) g(s) 2 0 1 2g(s) 0 0 1   , T g(s) =   1 1 2 g(s) 1 3 g(s) 2 0 1 g(s) 0 0 1   , T -1 g(s)+G(s) =   1 -1 2 (g(s) + G(s)) 1 6 (g(s) + G(s)) 2 0 1 -(g(s) + G(s)) 0 0 1  
Using all togethers the explicit form of the above matrices, identities (38), (39) and Lemma 5, one gets

S g (q(F 1 ) + q 2 (F 2 )) = g, F 1 + g 2 , F 2 + q(F 1 ) + 2q(gF 2 ) + q 2 (F 2 ) T G S g (q(F 1 ) + q 2 (F 2 )) = g, F 1 + g 2 , F 2 + 1 2 G, F 1 + G, gF 2 + 1 3 G 2 , F 2 +q(F 1 ) + 2q(gF 2 ) + q(GF 2 ) + q 2 (F 2 ) T g (q(f 1 ) + q 2 (f 2 )) = 1 2 g, f 1 + 1 3 g 2 , f 2 + q(f 1 ) + q(gf 2 ) + q 2 (f 2 ) T -1 g+G (T g (q(f 1 ) + q 2 (f 2 )) + T G S g (q(F 1 ) + q 2 (F 2 ))) = 1 2 g, F 1 - 1 2 G, f 1 + 1 6 g, (g -G)F 2 + 1 6 G, (G -g)f 2 + q(f 1 ) + q(F 1 ) + q(gF 2 ) -q(Gf 2 ) +q 2 (f 2 ) + q 2 (F 2 )
Finally from Theorem 2 we can conclude.

4

The q-projection method

the Weyl algebra coincides with the complex linear span of the products e iβ( √ 2q) e iα( 2p) . Therefore a state on the Weyl algebra is completely determined by the expectation values of these products. In particular the Fock state Φ is characterized by the property that the vacuum distribution of the position operator is the standard Gaussian √ 2q ∼ N (0, 1) together with the identity

√

e iα( √ 2p) Φ = e -α 2 e - √ 2αq Φ ( 47
)
which follows from i √ 2αp = -√ 2αq + 2αb. The q-projection method consists in using ( 47) and ( 4) to reduce the problem to compute vacuum expectation values of products of the form e -izP (q) e -iwp to the calculation of a single Gaussian integral. In the following sub-sections we illustrate this method starting from the simplest examples.

Vacuum characteristic functions of observables in

Heis(1, 1, 1)

In this section we show that the q-projection method, applied to Heis(1), gives the standard result for the spectral measure of the Weyl operators. From (47) and the CCR it follows that

e i(α( √ 2p)+β( √ 2q
)) Φ = e iαβ e -α 2 e (iβ-α)( √ 2q) Φ from which one obtains Φ, e i(α(

√ 2p)+β( √ 2q)) Φ = e iαβ e -α 2 Φ, e (iβ-α)( √ 2q) Φ = e -α 2 e iαβ √ 2π R e iβx-αx e -x 2 2 dx = e -α 2 2 e iαβ √ 2π R e iβx e -(x+α) 2 2 dx = e -α 2 2 1 √ 2π R e iβx e -x 2 2 dx = e -α 2 2 e -β 2 2
In particular, for all z = α + iβ, z ′ = α ′ + iβ ′ , one has

W (z)Φ, W (z ′ )Φ = Φ, W (-z)W (z ′ )Φ = e -iℑ(z z′ ) Φ, e i √ 2((β ′ -β)q-(α ′ -α)p) Φ = e -iℑ(z z′ ) e -z-z ′ 2 2

Vacuum characteristic functions of observables in

Heis(1, 1, 2)

In this section we use the q-projection method to give a derivation of the expression of the vacuum characteristic functions of observables in {1, p, q, q 2 } different from the one discussed in [] [x.].

Proposition 7 For all, α, β, γ ∈ R, one has

Φ, e iα( √ 2q) 2 +iβ( √ 2q)+γ( √ 2q) Φ = (1 -2iα) -1 2 e γ 2 2(1-2iα) e -β 2 2(1-2iα) e i βγ 1-2iα Proof. Put Ψ 1 (β) := Φ, e iα( √ 2q) 2 +iβ( √ 2q)+γ( √ 2q) Φ = E(e iαX 2 +iβX+γX )
where X is a normal gaussian random variable. Then, one has

Ψ ′ 1 (β) = iE(Xe iαX 2 +iβX+γX ) (48) = 1 √ 2π R xe iαx 2 +iβx+γx e -x 2 2 dx
Taking the changes of variables

u(x) = e iαx 2 +iβx+γx , v ′ (x) = xe -x 2 2
Then, one gets

u ′ (x) = (2iαx + iβ + γ)e iαx 2 +iβx+γx , v(x) = -e -x 2 2
This gives

E(Xe iαX 2 +iβX+γX ) = 2iαE(Xe iαX 2 +iβX+γX ) + (iβ + γ)E(e iαX 2 +iβX+γX ) (49)
Therefore, identities (48) and (49) imply that

Ψ ′ 1 (β) = iγ -β 1 -2iα Ψ 1 (β) which yields Ψ 1 (β) = C(α, γ)e i βγ 1-2iα e -β 2 2(1-2iα) (50)
where

C(α, γ) = Ψ 1 (0) := Ψ 2 (γ) = E(e iαX 2 +γX )
Now, one has

Ψ ′ 2 (γ) = E(Xe iαX 2 +γX ) = 1 √ 2π R xe iαx 2 +γx e -x 2 2 dx
Taking the changes of variables

h(x) = e iαx 2 +γx , l ′ (x) = xe -x 2
It follows that

h ′ (x) = (2iαx + γ)e iαx 2 +γx , l(x) = -e -x 2 2
Then, one obtains

Ψ ′ 2 (γ) = 2iαΨ ′ 2 (γ) + γΨ 2 (γ)
This implies that

Ψ 2 (γ) = C(α)e γ 2 2(1-2iα) (51)
where

C(α) = Ψ 2 (0) = E(e iαX 2 ) = (1 -2iα) -1 2 (52)
Finally, using identities (50), ( 51) and ( 52), one obtains

Φ, e iα( √ 2q) 2 +iβ( √ 2q)+γ( √ 2q) Φ = (1 -2iα) -1 2 e γ 2 2(1-2iα) e -β 2
2(1-2iα) e i βγ

1-2iα

Theorem 3 For all A, B, C ∈ R, one has

Φ, e it(A( √ 2q) 2 +B( √ 2q)+C( √ 2p)) Φ = (1 -2itA) -1 2 e 4C 2 (A 2 t 4 +2iAt 3 )-3|M | 2 t 2 6(1-2iAt)
where M = B + iC.

Proof. We have

it(Aq 2 + Bq + Cp) = itCp + itP ′ (q)
where P (X) = 1 3 AX 3 + 1 2 BX 2 . Then, Proposition 1 implies that e it(Aq 2 +Bq+Cp) = e it P (q+tC)-P (q) tC e itCp (53)

But, one has

P (q + tC) -P (q) tC = Aq 2 + (tAC + B)q + 1 2 tBC + 1 3 A(tC) 2 (54)
Using ( 53) and (54) for getting e it(Aq 2 +Bq+Cp) = e it( 13 A(tC) 2 + 1 2 tBC) e it(Aq 2 +(tAC+B)q) e itCp

It follows that

e it(A( √ 2q) 2 +B( √ 2q)+C( √ 2p)) = e it( 4 3 A(tC) 2 +tBC) e it(A( √ 2q) 2 +(2tAC+B)( √ 2q
))

e itC( √ 2p) (55)
Therefore from (55), ( 47) and ( 47), one has

e it(A( √ 2q) 2 +B( √ 2q)+C( √ 2p)) Φ = e it( 4 3 AC 2 t 2 +t(B+iC)C) e it(A( √ 2q) 2 +(2ACt+(B+iC))( √ 2q)) Φ = e it( 4 3 AC 2 t 2 +tM C) e it(A( √ 2q) 2 +(2ACt+M )( √ 2q
√ 2q) 2 +(2ACt+M )( √ 2q)) Φ = (1 -2iAt) -1 2 e C 2 t 2 2(1-2iAt) e -(2At 2 +Bt) 2 2(1-2iAt) e -i Ct(2ACt 2 +Bt) 1-2iAt (57)
Finally, identities (56) and (57) imply that Φ, e it(A(

√ 2q) 2 +B( √ 2q)+C( √ 2p)) Φ = (1 -2itA) -1 2 e 4C 2 (A 2 t 4 +2iAt 3 )-3|M | 2 t 2 6(1-2iAt)
This ends the proof. Using together Proposition 4 and Theorem 3, we prove the following theorem.

Theorem 4 For all α i , β i , γ i ∈ R, i = 1, 2, we have

e it(α 1 ( √ 2q) 2 +β 1 ( √ 2q)+γ 1 ( √ 2p)) Φ, e it(α 2 ( √ 2q) 2 +β 2 ( √ 2q)+γ 2 ( √ 2p)) Φ = (1 -2i(α 2 -α 1 )t) -1 2 e Lt 4 +Z 1 t 3 +Z 2 t 2 6(1-2i(α 2 -α 1 )t)
where

L = -4(α 1 γ 2 -α 2 γ 1 )[3(α 1 γ 2 -α 2 γ 1 ) + 2(α 2 -α 1 )(γ 1 + γ 2 )] +4(γ 2 -γ 1 ) 2 (α 2 -α 1 ) 2 Z 1 = 12[(α 2 -α 1 )(β 1 γ 2 -γ 1 β 2 ) -(β 2 -β 1 )(α 1 γ 2 -α 2 γ 1 )] +4i[2(α 2 -α 1 )(γ 2 -γ 1 ) 2 + (γ 1 + γ 2 )(α 2 γ 1 -α 1 γ 2 )] Z 2 = -3(γ 2 -γ 1 ) 2 + 6i(β 1 γ 2 -γ 1 β 2 )
Proof. We have

e it(α 1 ( √ 2q) 2 +β 1 ( √ 2q)+γ 1 ( √ 2p)) Φ, e it(α 2 ( √ 2q) 2 +β 2 ( √ 2q)+γ 2 ( √ 2p)) Φ = Φ, e -it(α 1 ( √ 2q) 2 +β 1 ( √ 2q)+γ 1 ( √ 2p)) e it(α 2 ( √ 2q) 2 +β 2 ( √ 2q)+γ 2 ( √ 2p)) Φ (58) Put P ′ 1 (x) = -t √ 2β 1 x -2tα 1 x 2 P ′ 2 (x) = t √ 2β 2 x + 2tα 2 x 2
Then, Proposition 4 implies that

(-t √ 2γ 1 , P ′ 1 )(t √ 2γ 2 , P ′ 2 ) = (t √ 2(γ 2 -γ 1 ), Q ′ ) ( 59
)
where

Q ′ (x) = C + Bx + Ax 2 with C = (β 1 γ 2 -γ 1 β 2 )t 2 + 2 3 (γ 1 + γ 2 )(α 2 γ 1 -α 1 γ 2 )t 3 B = √ 2(β 2 -β 1 )t + 2 √ 2(α 1 γ 2 -α 2 γ 1 )t 2 A = 2(α 2 -α 1 )t
Therefore, from identities (58) and (59), one obtains

e it(α 1 ( √ 2q) 2 +β 1 ( √ 2q)+γ 1 ( √ 2p)) Φ, e it(α 2 ( √ 2q) 2 +β 2 ( √ 2q)+γ 2 ( √ 2p)) Φ = e i((β 1 γ 2 -γ 1 β 2 )t 2 + 2 3 (γ 1 +γ 2 )(α 2 γ 1 -α 1 γ 2 )t 3 ) Φ, e it (α 2 -α 1 )( √ 2q) 2 +((β 2 -β 1 )+2(α 1 γ 2 -α 2 γ 1 )t)( √ 2q)+(γ 2 -γ 1 )( √ 2p)
Φ Now, from Theorem 3, one gets

e it(α 1 ( √ 2q) 2 +β 1 ( √ 2q)+γ 1 ( √ 2p)) Φ, e it(α 2 ( √ 2q) 2 +β 2 ( √ 2q)+γ 2 ( √ 2p)) Φ = e i((β 1 γ 2 -γ 1 β 2 )t 2 + 2 3 (γ 1 +γ 2 )(α 2 γ 1 -α 1 γ 2 )t 3 ) (1 -2i(α 2 -α 1 )t) -1 2 e 4(γ 2 -γ 1 ) 2 ((α 2 -α 1 ) 2 t 4 +2i(α 2 -α 1 )t 3 )-3|M| 2 t 2 6(1-2i(α 2 -α 1 )t) (60)
where

M = (β 2 -β 1 ) + 2(α 1 γ 2 -α 2 γ 1 )t + i(γ 2 -γ 1 )
Finally, using the identity (60), it is easy to show that e it(α where L, Z 1 , Z 2 are given above.

4.3 Vacuum moments of observables in {1, p, q, q 2 } In this subsection we use the results of the preceeding section to deduce the expression of the vacuum moments of observables in {1, p, q, q 2 }. The form of these moments was not known and may be used to throw some light on the still open problem of finding the explicit expression of the probability distributions of these observables. Then, one gets

g (k) (t) = k h=0 k h ϕ (h) 1 (t)ϕ (k-h) 2 (t)
Because ϕ (h) 1 (t) = 0 for all h ≥ 5 and ϕ 1 (0) = ϕ ′ 1 (0) = 0, one gets

g (k) (0) = k! (k -2)! γϕ (k-2) 2 (0) + i k! (k -3)! βϕ (k-3) 2 (0) + k! (k -4)! αϕ (k-4) 2 (0) (64)
where by convention ϕ (i-j) 2

(0) = 0 if i < j. Therefore, identities (63) and (64) imply that

g (k) (0) = k!(2iA) k-2 γχ {k≥2} + ik!(2iA) k-3 βχ {k≥3} + k!(2iA) k-4 αχ {k≥4}
Then, for all k ≥ 1, one obtains Finally, by using identity (61) the result of the above theorem holds true.

ϕ (k) (0) k! = ϕ (k) 3 (0) + g (k) (0) = (2i) k A k 2k - A k-2

Vacuum characteristic functions of observables in

Heis(1, 1, N )

From ( 2) and (47) we deduce that Φ, e iwp+iuP ′ (q) Φ = Φ, e iu P (q+w)-P (q) w e iwp Φ = Φ, e iu P (q+w)-P (q) w e iwp Φ = e -w 2 /2 Φ, e iu P (q+w)-P (q) The q-projection method reduces the problem to an integral of the form Φ, e iQ(q) Φ where Q is a polynomial.

  The one-mode Hesienberg algebra is the * -Lie algebra generated by b, b + , 1 with commutation relations [b, b + ] = 1, [b, 1] = [b + , 1] = 0 and involution (b) * = b +

  We will use the known fact that, if L is a Lie algebra and a, b ∈ L are such that [a, [a, b]] = 0 then, for all w, z ∈ C e za e wb e -za = e w(b+z[a,b])

3

  Polynomial extensions of the Heisenberg algebra: Continuous case Let b + t and b t satisfy the Boson commutation and duality relations [b t , b + s ] = δ(ts) ; [b + t , b + s ] = [b t , b s ] = 0 ; (b s ) * = b + s where t, s ∈ R and δ is the Dirac delta function. Define

Because of the relation (see Proposition 1 )

 1

  )) Φ (56) where M = B + iC. Now, by taking α = At, β = 2ACt 2 + Bt, γ = -Ct and using Proposition 7, one gets Φ, e it(A(

  ) = (2iA) k k!(1 -2iAt) -k-1 , ϕ k (k -1)!(1 -2iAt) -k (63)

  = e -α 2 e - √ 2αq Φ

  Theorem 5 Define, for A, B, C ∈ R 1 +2i 2 +...+ki k =n 2 3n n! i 1 ! . . . i k ! w i 1 1 . . . w i k + iβt 3 + γt 2 1 -2iAtwith α, β and γ are given above. Hence, one has 1 +2i 2 +...+ki k =n2 2n n! i 1 ! . . . i k ! (t) = αt 4 + iβt 3 + γt 2 , ϕ 2 (t) = (1 -2iAt) -1 ,

	Note that
	ϕ
					X := A(	√	2q) 2 + B(	√	2q) + C(	√	2p)
	Then
			Φ, X n Φ =			k
	where
		w k =	A k 2k	-	A k-2 4	γχ {k≥2} -	A k-3 8	βχ {k≥3} +	A k-4 16	αχ {k≥4}
	with	α =	2 3	A 2 C 2 , β =		4 3	AC 2 , γ = -	1 2	|M| 2 = -	1 2	(B 2 + C 2 )
	Proof. Recall that
											E(e itX ) = e ϕ(t)
	where			ϕ(t) = -αt 4 E(X n ) = 1 2 ln(1 -2iAt) + 1 i n d dt n t=0	e ϕ(t)	(61)
	Now we introduce the following formula (cf [8])
	d n dt n e ϕ(t) =									ϕ (1) (t) 1!	i 1	. . .	ϕ (k) (t) k!	i k	e ϕ(t)	(62)
	Put
			ϕ 1 ϕ 3 (t) = -	1 2	ln(1 -2iAt), g(t) = ϕ 1 (t)ϕ 2 (t)
											29

i i

  1 +2i 2 +...+ki k =n 2 2n n! i 1 ! . . . i k ! (2i) i 1 +...+ki k w i 1 1 . . . w i k k = i n i 1 +2i 2 +...+ki k =n 2 2n 2 n n! i 1 ! . . . i k ! w i 1 1 . . . w i k

								4	γχ {k≥2} -	A k-3 8	βχ {k≥3} +	A k-4 16	αχ {k≥4}
		= (2i) k w k
	where
		w k =	A k 2k	-	A k-2 4	γχ {k≥2} -	A k-3 8	βχ {k≥3} +	A k-4 16	αχ {k≥4}
	Thus, from identity (62), one has
	d dt	n	t=0	e ϕ(t) =
										k

i