Alexander Iglesias
email: aiglesias@ideko.es

Gorka Aguirre

Jokin Munoa

Zoltan Dombovari

Iñaki Laka

Gabor Stepan

Damping of in-process measuring system through variable stiffness tunable vibration absorber

Introduction

Growing demands on dimensional requirements in the manufacturing industry makes in-process control a fundamental tool for dimensional quality assurance. Nowadays, machine tool builders are increasingly implementing in-process measuring systems integrated in the own structure of the machine, with the purpose of cost-efficiently manufacturing products of complex geometries requiring a high degree of accuracy. The on-line measuring system allows an accurate determination of the dimensions of the workpiece, discarding the need of a final precision verification in an external measuring machine. Moreover, as these products usually require long manufacturing times, a single machining mistake can result in a faulty workpiece, which in turn brings along an enormous economic loss. Thus, the on-line measurement avoids the error or permits a fast correction, before it becomes too late. This integrated machining & measuring solution provides several benefits but it is also subject to the negative collateral effects coming from the machining process, such as vibration or thermal variations. As the available space inside the machine to locate the measuring system support structure is usually very limited, it often results in a rather flexible structure. Therefore, the sensitivity to vibrations is particularly compromising.

In grinding processes, typical vibrations from the wheel unbalance can be transmitted through the machine structure to the measuring probe, affecting a proper assessment of the measured dimension. Although grinding wheels are usually equipped with very effective wheel balance systems, there is always a slight remaining unbalance that, when the measurement structure is not stiff enough, could be capable of misleading the dimensional measure. This is especially critical when the grinding wheel is rotating in resonance (or close to the resonance) with some of the modes of the measuring system.

On the other hand, it must be taken into account that the wheel rotating speed changes as the wheel wears out, since the grinding cutting speed must be kept constant while the wheel diameter changes. Therefore, the wheel rotation speed varies from the rotation speed corresponding to the larger diameter (new wheel) to the smallest diameter (worn-out wheel). Thus, the excited frequency range is broad and the vibration becomes critical when the wheel rotating speed crosses a resonance of the measuring system. Different measuring structure designs have been implemented in grinding machines for on-line measurement. The typical arrangement consists of an additional carriage supporting the measuring unit and moving in parallel to the workpiece axis. The measurement can be performed either by a laser/optical measuring system or by a touch probe. Due to the classical architecture of external cylindrical grinding machines, the addition of this measuring structure must be carried out between the workpiece an the door of the machine, which can be detrimental for the operator to access the workpiece, especially when loading and unloading the workpiece. It is for this reason that in external cylindrical grinding machines, measuring systems are alternatively located on a gantry type frame (see Figure 1). Figure 1: HG72 grinding machine with in-process measuring system In this measuring system configuration, two bending modes and a rotation mode may be typically located within the working range of the grinding wheel.

Vibration mitigation strategies in measuring structures

There are different solutions to tackle vibrations on the measuring system. The most direct one would consist of redesigning the gantry-type structure using numerical models like Finite Element Method (FEM) models. The purpose would be to make it stiffer in order to shift the natural frequencies over the maximum excitation frequency (highest rotation speed of the wheel). However, this solution is many times unfeasible, due to the limited space to add new bracings and reinforcements to the gantry-type frame. The introduction of materials with high Young's modulus in zones where the strain energy of the modes is high can increase the stiffness of the critical mode without any increase in the required volume. Considering that steel and cast iron are the standard materials in machine structures, only costly materials like carbide and special metal alloys and composites can offer substantially larger Young's modulus [START_REF] Möhring | Materials in machine tool structures[END_REF], which makes the material variation not affordable for this kind of large gantry-type structures. Theoretically, the modifications to increase the stiffness could be combined with the reduction of weight in points with high critical modal displacement, with the purpose of increasing the natural frequencies of the system. However, the mass concentration at the critical locations is related to the own mass of the measurement device and, therefore, it is not easy to achieve important mass reductions. Moreover, the frequency range limited by the wheel rotating frequency range can be very broad in some cases, which makes the FEM based design to move modes out of this range extremely complex.

The introduction of flexible supports in the base of the gantry structure to isolate the vibrations coming from the machine can be also regarded as a possible solution. However, the grinding wheel rotation frequencies can be relatively low, which requires the introduction of very flexible supports for an effective passive isolation. The presence of moving masses and gravity centers of the measuring system out of the gantry-type frame makes the application of these compliant isolators on the base of the gantry not suitable.

For this reason, solutions seeking to add damping to the system reducing the impact of the resonance are more suitable to solve this problem. Active damping is usually an effective solution for any kind of broadband vibration problem like self-exited vibrations [START_REF] Zatarain | Estudio comparativo de los modelos matemáticos de chatter en fresado: monofrecuencia, multifrecuencia y simulación en el tiempo, XV Congreso de Máquinas-Herramienta y Tecnologías de Fabricación[END_REF][START_REF] Dombovari | General milling stability model for cylindrical tools[END_REF]. However, active damping strategy requires the integration of sensors, actuators and a control system, which usually makes the solution too costly [START_REF] Bilbao-Guillerna | Control laws for chatter suppression in milling using an inertial actuator[END_REF][START_REF] Bilbao-Guillerna | Low frequency chatter suppression using an inertial actuator[END_REF]. Thus, passive damping techniques, which are usually more affordable, are selected as more efficient solutions for this particular problem.

The introduction of special materials, coatings and foams with high internal damping has been proposed by several authors [START_REF] Vandeurzen | Additive damping treatments for mechanical structures[END_REF][START_REF] Takeyama | Improvement of dynamic rigidity of tool holder by appliying high-damping material[END_REF]. These materials should be placed at the locations where the modes have the highest strain energy. However, the main source of damping is at the interfaces and joints of the system, where damping is around one order higher than the structural damping of metallic parts [START_REF] Koenigsberger | Machine tool structures[END_REF]. This damping difference can be even higher in the case of low frequency modes. Some authors tried to increase the damping at joints by creating highly damped interfaces [START_REF] Rashid | Design & implementation of tuned viscoelastic dampers for vibration control in milling[END_REF][START_REF] Iglesias | Machining improvement on flexible fixture through viscoelastic damping layer[END_REF].

The introduction of passive auxiliary systems, such as Tuned Mass Dampers (TMD) [START_REF] Yang | Optimization of multiple tuned mass dampers to suppress machine tool chatter[END_REF], friction [START_REF] Edhi | Stabilization of high frequency chatter vibration in fine boring by friction damper[END_REF] and impact [START_REF] Ema | Suppression of chatter vibration of boring tools using impact dampers[END_REF] dampers, is another alternative to increase the damping by means of different physical mechanisms. Tuned mass dampers have been used for vibration control in many civil and industrial applications such as buildings [START_REF] Chang | Mass dampers and their optimal designs for building vibration control[END_REF], wind turbines [START_REF] Enevoldsen | Effects of a Vibration Mass Damper in a Wind Turbine Tower[END_REF] and machine tools [START_REF] Yang | Optimization of multiple tuned mass dampers to suppress machine tool chatter[END_REF], among many others. A TMD is an inertial mass added to the system to damp via a linear spring and a dashpot. These elements are tuned to damp the critical mode of the original system. By matching the natural frequency of the highly damped TMD with the critical frequency of the system, both modes can be coupled to increase the damping. Consequently, the original mode is split into two modes with higher dynamic stiffness. The TMDs have to be tuned accurately to the targeted frequency, and their positive effect is limited to a certain frequency range [START_REF] Den | Mechanical vibrations[END_REF][START_REF] Sims | Vibration absorbers for chatter suppression: a new analytical tuning methodology[END_REF]. Therefore, this system could only damp one mode of the gantry-type structure of the measurement device inside the grinding wheel rotation range. In this aspect the impact and friction dampers have the advantage to damp more than one mode, but they are usually applied in high frequency applications and are difficult to design and control. Some authors have tried to overcome the limitations of the TMD by implementing tunable TMDs. For instance, Slavicek and Bollinger [START_REF] Slavicek | Design and application of a self-optimizing damper for increasing machine tool performance[END_REF] proposed a damper for which the natural frequency and the damping of the system can be varied in process. These semi-active dampers require some energy to be tuned but after that the system behaves like a passive damper. The energy consumption is low compared with pure active devices and a small battery is providing sufficient energy for that purpose [START_REF] Munoa | Design of selftuneable mass damper for modular fixturing systems[END_REF]. The semi-active concepts have been proposed for chatter mitigation [START_REF] Munoa | Chatter suppression techniques in metal cutting[END_REF][START_REF] Slavicek | Design and application of a self-optimizing damper for increasing machine tool performance[END_REF][START_REF] Munoa | Design of selftuneable mass damper for modular fixturing systems[END_REF][START_REF] Aguirre | Self-tuning semi-active tuned-mass damper for machine tool chatter suppression[END_REF]. As chatter onset frequency strongly depends on the excited mode and the rotating speed, the capability of varying the stiffness of the tuned mass damper in order to tune the frequency at the desired value is highly useful in order to achieve an optimum performance [START_REF] Munoa | Design of selftuneable mass damper for modular fixturing systems[END_REF][START_REF] Aguirre | Self-tuning semi-active tuned-mass damper for machine tool chatter suppression[END_REF].

In this work, this tuning capability is applied for an effective vibration reduction in the measuring unit. Thus, instead of two or three different tuned mass dampers which damp each different mode, a variable stiffness vibration absorber which autonomously adapts its stiffness to tune according to the mode to be damped is used.

In order to tune the absorber to its optimum point, a tuning function has been derived in order to minimize the magnitude of the frequency response function (FRF) at the frequency coincident with the rotating speed of the wheel. Thus, an improved performance with respect to a fixed standard tuning criterion is achieved.

Finally, validation tests have been performed in a scale supporting structure and a real machine and the vibration reduction improvement comparison with respect to a standard fixed tuning strategy is evaluated.

Optimal tuning of the vibration absorber

In order to tune the absorber to its optimum point, an investigation about the optimum tuning has been carried out. As mentioned earlier, in an external cylindrical grinding machine, the main excitation is produced by the unbalance of the wheel, which has a frequency coincident with its rotation frequency. Therefore, the objective is to derive a tuning function that minimizes the dynamic response at the frequency coincident with the rotating speed of the wheel.

The advantage of the variable stiffness tunable absorber is that its stiffness and, therefore, its frequency can be varied along a specific range. Thus, an improved performance with respect to a fixed standard tuning criterion is achieved.

The vibration absorber must be tuned so that the lowest response is obtained in the measuring unit at each wheel rotating speed. Computing the achieved response reduction at every rotating speed, a pseudo FRF which describes the dynamic response at each individual frequency line can be obtained. This pseudo FRF is only valid to assess pure sinusoidal excitation.

Previously, Den Hartog [START_REF] Den | Mechanical vibrations[END_REF] defined an optimal tuning for a tuned mass damper with constant stiffness in order to minimize the amplitude and the real part respectively. In this case, as the tuning frequency will be modified for different wheel speeds, a new derivation will be sought.

A set of dimensionless parameters have been defined in order to simplify the subsequent equations:

Parameter Definition

Mass ratio The magnitude of the frequency response function of the system with the added vibration absorber can be expressed as [START_REF] Den | Mechanical vibrations[END_REF]:

, ) ( 1 ) ( 1 g M k g H  (1)
where k 1 is the modal stiffness of the original system and M(g) is the magnification function defined as:

2 2 1 3 2 2 2 2 2 1 2 2 1 2 2 2 2 2 2 2 2 )) 4 ( )) 1 ( 1 ( ( )) ( ( 4
4 ) ( ) (          g g g g g f g f g g g f g M               . ( 2
)
The optimum tuning is achieved with the minimization of M(g). Equation ( 2) is derived with respect to f and the following solution is obtained:

) 1 ) 1 ( ( 2
) ) 4 )) 1 ( 1 (( 16 16 ) 4 1 ( 2 ) 2 ( ( ) ( 2 2 2 2 1 2 2 2 2 1 4 2 6 2 1 3                        g g g g g g g g g f opt . (2)
If the damping of the original system is equal to zero c 1 =0, the magnification function yields:

2 2 2 2 2 2 2 2 4 2 2 2 2 2 2 2 )) 1 ( 1 ( 4 ))) 1 ( 1 ( ( 4 ) ( ) , (                 g g g f g g g g f g f M .
(3)

For both cases c 1 =0 and c 1 0, if the magnification function M(g) is derived with respect to  2 , it turns out that the minimization of the function is achieved when the damping of the added tuned mass damper c 2 tends toward 0. Therefore, in order to achieve the lowest possible response, the vibration absorber will need to have the lowest possible damping. This is the case of a vibration absorber, which is similar to a tuned mass damper but without adding damping to the moving mass (c 2 =0).

From equation (3), it is possible to derive that, when c 2 =0, the optimal tuning is achieved when f=g, where the magnification function would be M(g)= 0. This is not possible to achieve in practice due to the presence of damping in any mechanical system, but it will produce the lowest response.

g f opt  . (4)
Summarizing the optimum tuning of the vibration absorber for a measuring system in an external cylindrical grinding machine is achieved when c 2 =0 and f=g. This tuning is only appropriate for a pure single frequency sinusoidal excitation. If a broader excitation was present, this strategy would not be suitable.

Next, the comparison of a one critical mode system tuned according to the developed optimal tuning f opt , its approximation f=g and an optimal fixed tuning are presented through a theoretical simulation. As it can be observed in Figure 2, although the fixed tuning decreases significantly the response of the system around the critical mode, the variable tuning through the vibration absorber improves the behavior to a much higher extent. It is also noticeable that the "simplified tuning" f opt g is virtually as good as the exact tuning. Therefore, this approximate optimum tuning will be applied henceforth.

The variable stiffness vibration absorber is even more advantageous than an optimal fixed tuning when more than one critical mode is present. In order to assess the improvement achieved in systems with more than one dominant mode, a two-mode system with the characteristics described in Table 2 In Figure 3, the two critical mode system is compared when damped through optimal fixed tuning or the optimum tuning developed for the variable stiffness vibration absorber.

Figure 3: Comparison of optimal fixed tuning or through optimal variable stiffness vibration absorber in a two mode system.

Similarly to the one dominant mode case, the tuning trough the variable stiffness tunable vibration absorber outperforms the optimal fixed tuning.

Design of the vibration absorber

In this section, the variable stiffness DVA concept and tuning strategy developed in Section 3 will be implemented in a physical prototype capable of performing in practice the theoretical vibration reduction envisaged.

A vibration absorber prototype has been built. It consists of a unidirectional moving mass supported by a rotary spring located in the center of this moving mass. This rotary element, which is driven by an electric motor, is built according to a special shape that provides a variable stiffness as a function of its angular position (see Figure 4a). When the rotation angle is 0º, the rotary spring contributes to the system with its minimum stiffness, whereas when it is rotated at 90º, it provides the highest stiffness. An encoder coupled to the tuning motor controls the proper angular position in accordance with the rotating speed of the wheel. The design of the rotary spring must be fitted to cover the wheel rotating frequency range. The damper has a weight of 37kg with a damping ratio ranging from 0.24 to 0.09, depending on the tuning frequency. The tuning frequency ranges from 22Hz to 46Hz and varies according to the stiffness variation of the rotary spring, which fulfils the following equation:

  2 max 2 min sin cos k k k   , ( 5
)
where  is the rotation of the rotary spring from 0º to 90º.

The dynamic vibration absorber (DVA) is controlled through the CNC of the machine, which commands the rotary spring according to the wheel rotating speed through a tuning motor on real time. Therefore the developed optimal tuning f opt =g is achieved.

Experimental validation

Experimental setup

In order to validate the developed system, a scale prototype of the measuring unit mounted on the gantry type frame has been set up (see Figure 5). The system has been downsized, maintaining the proportion of the original system. The dynamic vibration absorber has been located on top of the gantry structure, at a corner, in the equivalent position where it will be later tested on the real structure. The DVA location and its orientation at 45º allows the operation against the main modes of the system.

Figure 6: Modes of the measuring system located on the wheel rotating frequency range According to Figure 6, there are two modes in the frequency range of the grinding wheel. The first one is a bending mode in z direction at 23.6Hz, whereas the second one is a torsion mode in the xy plane at 27.8Hz.

The excitation has been introduced through an unbalance motor located on the table where the gantry type structure is clamped, simulating the unbalance coming from a wheel in a real machine. A 56000gmm unbalance has been applied in order to achieve a high sensitivity in the performed tests, which turned out in a centrifugal force range of roughly 200-1000N, depending on the rotation speed. The vibration response has been measured at the location of the measuring probe, which is the point whose vibration directly affects the dimensional measurement.

Results

Figure 7 shows the response achieved when the dynamic vibration absorber is tuned at the frequency of these main modes, compared with an optimal fixed tuning for that mode: According to the figure, the improvement at the resonant frequency is remarkable with the fixed tuning with optimal damping and especially with the optimal tuning with no damping, achieving reductions of dynamic response of up to 97% and more than 99% respectively.

The scale measuring unit has been validated under different conditions: a) DVA detuned (constantly tuned at 46.5Hz) b) DVA tuned for the x mode with optimal damping at a fixed frequency of 23Hz c) DVA tuned for the y mode with optimal damping at a fixed frequency of 27Hz d) Self-tunable DVA tuned as f=g with minimum damping.

The purpose is to compare the strategy of tuning the DVA at a fixed frequency according to a single mode, which can be a sufficiently good approach when a single mode is affecting the operation range but can be highly unsuitable for multiple mode structures. When the DVA is not tuned, the vibration is magnified at the natural frequency regions (23Hz in z axis and 27Hz in x and y axes). The fixed tuning at 23Hz gives rise to amplification at higher frequencies, which can even result in higher amplitude than the original vibration value. The fixed tuning at 27Hz also leads to high vibration values below and over that particular frequency. The lowest vibration values are achieved with the self-tunable strategy. Therefore, the self-tunable system has proven its improved behavior with respect to traditional fixed damping systems, especially for multiple mode measuring structures. The vibration at the tip of the measuring probe is reduced more than 90% in every coordinate axis.

Conclusions

A tunable dynamic vibration absorber has been developed to minimize vibration in gantry type in-process measuring systems for external cylindrical grinding machines. The best tuning strategy has been defined, which consists of equating the wheel rotating frequency in order to minimize the amplification of the wheel unbalance vibration. This tuning is only suitable for pure single frequency sinusoidal excitation cases. The system is driven by a tuning motor which adds a self-tuning capability to minimize the amplification of the transmitted vibration throughout the whole wheel rotating speed range. The advantages of the system with respect to a conventional fixed tuning system have been demonstrated through theoretical simulations and experimental tests.

  Damping ratio of the vibration absorber with respect to the critical damping by Dan Hartog[START_REF] Den | Mechanical vibrations[END_REF]

Figure 2 :

 2 Figure 2: Comparison of tuning by optimal fixed tuning or through optimal variable stiffness vibration absorber. a)  1 =0.001, b)  1 =0.01, c)  1 =0.05, d)  1 =0.1

Figure 4 :

 4 Figure 4: a) Variation of the tuning frequency with the rotation of the rotary spring. b) Sketch of variable stiffness vibration absorber. Red shades represent the fixed parts whereas green shapes represent the moving mass

Figure 5 :

 5 Figure 5: Measuring unit scale prototype

Figure 7 :

 7 Figure 7: a) shows the comparison between the detuned and tuned (27.8Hz) frequency response functions in x direction; b) shows the comparison between the detuned and tuned (23.6Hz) frequency response functions in z direction

Figure 8

 8 Figure8shows the vibration result at different wheel rotation speeds (different frequencies) for the four previously defined DVA states.

Figure 8 :

 8 Figure 8: Vibration amplitude in the measuring probe location of the scale measuring system, a) in x axis, b) in y axis and c) in z axis

  has been analyzed:

		Dimensionless paramters		Modal parameters
		g		Frequency f	Mass m	Stiffness k	Damping c
	Mode I	1	0.01	f 1	2m 1	2k 1	c 1
	Mode II	1.73	0.0173	1.73f 1	2m 1	6k 1	3c 1

Table 2 :

 2 Dynamic parameters for the simulationThe tuning parameters used for this simulation are indicated in Table3:

	Parameter	Fixed tuning at Mode I	Fixed tuning at Mode II	Variable f=g tuning
	Mass ratio ()	0.2	0.2	0.2
	Damping of the tuned mass damper ( 2 )	Optimal (0.208)	Optimal (0.36)	0.01
	Tuning (f)	0.8	1.63	f=g

Table 3 :

 3 Tuning characteristics

Table 4 :

 4 The modal parameters of the original mode and the added dynamic vibration absorber are summarized in the table below: Modal parameters of the measuring system

	Parameter	Mode x Mode z
	Natural frequency 27.8Hz 23.6Hz
	Modal mass	194kg	233kg
	Damping ratio	0.0058 0.0076
	Mass ratio	0.19	0.16

Acknowledgements

The research leading to these results was partially supported by the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013)/ERC, advanced grant agreement no. 340889; the MC-SUITE project (grant agreement no. 680478) under the Horizon 2020 Programme (H2020-FoF-2015) and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences (BO/00589/13/6).


Diego Julio Cirilo-Lombardo
email: diego77jcl@yahoo.com

Non-compact groups, Coherent States, Relativistic Wave equations and the Harmonic Oscillator

Keywords: 03.65.-w, 11.30.Pb, 42.50.-p Relativistic wave equation

Relativistic geometrical action for a quantum particle in the superspace is analyzed from theoretical group point of view. To this end an alternative technique of quantization outlined by the authors in a previous work and that is based in the correct interpretation of the square root Hamiltonian, is used. The obtained spectrum of physical states and the Fock construction consist of Squeezed States which correspond to the representations with the lowest weights λ = 1 4 and λ = 3 4 with four possible (non-trivial) fractional representations for the group decomposition of the spin structure. From the theory of semi-groups the analytical representation of the radical operator in the superspace is constructed, the conserved currents are computed and a new relativistic wave equation is proposed and explicitly solved for the time dependent case. The relation with the Relativistic Schrödinger equation and the Time-dependent Harmonic Oscillator is analyzed and discussed.

I. INTRODUCTION AND SUMMARY

The quantum behaviour of a relativistic particle in the superspace, besides to be a useful tool for certain studies and applications of Quantum Field Theory (QFT), is of notable importance in many physical contexts. Time-dependent Landau systems and the electronmonopole system are described naturally by the Super-Heisenberg-Weyl and OSP(1/2) algebras [1][2][3]. If several more or less well known physical systems are intrinsically supersymmetric in nature an obvious following question was: Can any supersymmetric toy model give us a good picture of not so well known physical systems? Part of the purpose of this paper is to demonstrate the positive answer to this question showing that a relativistic particle in the superspace can describe particles with fractionary spin for the which not concrete action is known.

On the other hand, the Time-Dependent Harmonic Oscillator (TDHO) was demonstrated to be a powerful tool to describe systems with more complicated dynamics in closed form.

From the famous reports of Ermakov [4] and Husimi [5] we can see that if any physical problem with a complicated or envolved dynamics can be represented faithfully or "mapped" to a TDHO system, this complicated dynamics admits a Coherent State (CS) or Squeezed States (SS) realization [START_REF] Klauder | Coherent States[END_REF]. It is clearly important that the model proposed here admits a Coherent State and Squeezed State realization. Most notably, squeezed states have been used in the context of quantum optics [START_REF] Shelby | [END_REF] and in the context of gravitational wave detection [8]. The correct choice for the realization of the physical states, however, will depend on the symmetry group that defines in some meaning the particular physical system under study.

Another part of this work will be devoted to discuss this point and what happens when different algebras can characterize the same physical problem.

In a previous paper we considered the simple model of superparticle of Volkov and Pashnev [START_REF] Pashnev | Supersymmetric lagrangian for particles in proper time[END_REF], that is type G4 in the description of Casalbuoni [START_REF] Casalbuoni | The Classical Mechanics for Bose-Fermi Systems[END_REF][START_REF] Casalbuoni | Relatively and supersymmetries[END_REF], in order to quantize it and to obtain the spectrum of physical states with the Hamiltonian remaining in the natural square root form. To this end, we used the Hamiltonian formulation described by Lanczos in [START_REF] Lanczos | Variational Principles in Mechanics[END_REF] and the inhomogeneous Lorentz group as a representation for the obtained physical states [START_REF] Yu | On massless fields and relativistic wave equations[END_REF][START_REF] Sannikov | Non-compact symmetry group of a quantum oscillator[END_REF][START_REF]Dirac: A positive-energy relativistic wave equation[END_REF]. The quantization of this model was performed completely and the obtained spectrum of physical states, with the Hamiltonian operator in its square root form, was compared with the spectrum obtained with the Hamiltonian in the standard form (i.e.: quadratic in momenta). We showed that the square root Hamiltonian can operate in a natural manner with the states with correspond to the representations with the lowest weights λ 1,2 = 1 4 and λ 1,2 = 3 4 and that there are four possible (non-trivial) fractional representations for the group decomposition of the spin structure from the square root Hamiltonian, instead of (1/2,0) and (0,1/2), as the case when the Hamiltonian is quadratic in momentum (e.g.

Ref. [START_REF] Pashnev | Supersymmetric lagrangian for particles in proper time[END_REF]). This result was a consequence of the geometrical Hamiltonian taken in its natural square root form and the Sannikov-Dirac oscillator representation for the generators of the Lorentz group SO (3,1). In this manner, we also showed that the superparticle relativistic actions as of Ref. [START_REF] Pashnev | Supersymmetric lagrangian for particles in proper time[END_REF] are a good geometrical and natural candidate to describe quartionic states [START_REF] Sorokin | Anti) commuting spinors and supersymmetric dynamics of semions[END_REF][START_REF] Sorokin | The Heisenberg algebra and spin[END_REF][START_REF] Volkov | Quartions in relativistic field theories[END_REF] (semions). In this paper we will complete the previous work giving now in more explicit form, how the states can be faithfully represented and realized from the geometrical point of view and from the dynamics of the group manifold.

In this work, strongly motivated for the several fundamental reasons described above, we considered the same simple model of superparticle of Volkov and Pashnev [START_REF] Pashnev | Supersymmetric lagrangian for particles in proper time[END_REF] to find the link with the TDHO problem, and for instance, with the CS and SS representations of the physical states obtained in [19] with the Hamiltonian remaining in its square root form. The plan of this paper is as follows: in order to make this work self-contained, in Sections 2, 3 and 4 we borrow from reference [19]

II. THE SUPERPARTICLE MODEL

In the superspace the coordinates are given not only by the space-time x µ coordinates, but also for anticommuting spinors θ α and θ . α . The resulting metric [START_REF] Pashnev | Supersymmetric lagrangian for particles in proper time[END_REF]19,[START_REF] Akulov | Is the neutrino a Goldstone particle?[END_REF] must be invariant to the action of the Poincare group, and also invariant to the supersymmetry transformations

x ′ µ = x µ + i θ α (σ) α . β ξ . β -ξ α (σ) α . β θ . β , θ ′α = θ α + ξ α , θ ′ . α = θ . α + ξ . α
The simplest super-interval that obeys the requirements of invariance given above, is the following (our choice for the metric tensor is g µν = (+ ---))

ds 2 = ω µ ω µ + aω α ω α -a * ω . α ω . α (1)
where (to simplify notation from here we avoid the contracted indexes between the spintensors (σ) α .

β and the anticommuting spinors θ α and θ .

α , as usual)

ω µ = dx µ -i dθ σ µ θ -θ σ µ dθ , ω α = dθ α , ω . α = dθ . α
are the Cartan forms of the group of supersymmetry [START_REF] Akulov | Is the neutrino a Goldstone particle?[END_REF].

The spinorial indexes are related as follows

θ α = ε αβ θ β , θ α = θ β ε βα , ε αβ = -ε βα , ε αβ = -ε βα , ε 12 = ε 12 = 1
and of analog manner for the spinors with punctuated indexes. The complex constants a and a * in the line element (1) are arbitrary. This arbitrarity for the choice of a and a * are constrained by the invariance and reality of the interval (1).

As we have extended our manifold to include fermionic coordinates, it is natural to extend also the concept of trajectory of point particle to the superspace. To do this we take the coordinates x (τ ), θ α (τ ) and θ

.

α (τ ) depending on the evolution parameter τ. Geometrically, the function action that will describe the world-line of the superparticle, is

S = -m τ 2 τ 1 dτ • ω µ • ω µ + a . θ α . θ α -a * . θ . α . θ . α = τ 2 τ 1 dτ L x, θ, θ (2)
where

• ω µ = . x µ -i . θ σ µ θ -θ σ µ .
θ and the upper point means derivative with respect to the parameter τ , as usual.

The momenta, canonically conjugated to the coordinates of the superparticle, are

P µ = ∂L/∂x µ = m 2 /L • ω µ P α = ∂L/ . ∂θ α = iP µ (σ µ ) α . β θ . β + m 2 a/L . θ α P . α = ∂L/ . ∂θ . α = iP µ θ α (σ µ ) α . α -m 2 a/L . θ . α (3)
It is difficult to study this system in the Hamiltonian formalism framework because of the constraints and the nullification of the Hamiltonian. As the action (2) is invariant under reparametrizations of the evolution parameter

τ → τ = f (τ )
one way to overcome this difficulty is to make the dynamic variable x 0 the time. For this, it is sufficient to introduce the concept of integration and derivation in supermanifolds as we introduce in Section 6 to write the action in the form

S = -m τ 2 τ 1 . x 0 dτ 1 -iW 0 ,0 2 -x i ,0 -W i ,0 2 + . x -2 0 a . θ α . θ α -a * . θ . α . θ . α
where the W µ ,0 was defined by

• ω 0 = .
x 0 1 -iW 0 ,0 [1] We take the Berezin convention for the Grassmannian derivatives:

δF (θ) = ∂F ∂θ δθ • ω i = . x 0 x i ,0 -iW i ,0
whence x 0 (τ ) turns out to be the evolution parameter

S = -m x 0 (τ 2 )
x 0 (τ 1 )

dx 0 1 -iW 0 ,0 2 -x i ,0 -W i ,0 2 + a . θ α . θ α -a * . θ . α . θ . α ≡ dx 0 L
Physically, this parameter (we call it the dynamical parameter) is the time measured by an observer's clock in the rest frame.

Therefore, the invariance of a theory with respect to the invariance of the coordinate evolution parameter means that one of the dynamic variables of the theory (x 0 (τ ) in this case) becomes the observed time with the corresponding non-zero Hamiltonian

H = P µ . x µ + Π α . θ α + Π . α . θ . α -L = m 2 -P i P i + 1 a Π α Π α - 1 a * Π . α Π . α (4)
where

Π α = P α + i P µ (σ µ ) α . β θ . β Π . α = P . α -iP µ θ α (σ µ ) α .
α That gives the well known mass shell condition and losing, from the quantum point of view, the operatibility of the Hamiltonian.

In the paper [START_REF] Pashnev | Supersymmetric lagrangian for particles in proper time[END_REF], where this type of superparticle action was explicitly presented, the problem of nullification of Hamiltonian was avoided in the standard form. This means that the analog to a mass shell condition (4) in superspace was introduced by means of a multiplier (einbein) to obtain a new Hamiltonian

H = κ 2 m 2 -P 0 P 0 -P i P i + 1 a Π α Π α - 1 a * Π . α Π . α (5)
(here κ, as in Ref. [START_REF] Pashnev | Supersymmetric lagrangian for particles in proper time[END_REF], is a constant with the dimensions of the square of a length).With this Hamiltonian it is clear that in order to perform the quantization of the superparticle the problems disappear: P 0 is restored into the new Hamiltonian, and the square root is eliminated. The full spectrum from this Hamiltonian was obtained in [START_REF] Pashnev | Supersymmetric lagrangian for particles in proper time[END_REF] where the quantum Hamiltonian referred to the center of mass was

H cm = κ 2 m 2 -M 2 + 2 3/2 M |a| 1 -(σ 0 ) α . β s . β s α (6a)
with the mass distribution of the physical states being the following : two scalar supermul-

tiplets M 1s = 2 1/2 |a| + 2 |a| + m 2 and M 2s = 2 |a| + m 2 -2 1/2
|a| ; and one vector supermultiplet M v = m. The Fock's construction in the center of mass for the eq.(6a) ( Hamiltonian quadratic in momenta) consists of the following vectors:

S 1 = |0 e iM t Ξ 1α = d • α |0 e iM t P 1 = d • β d • β |0 e iM t Ξ 2α = s • α |0 e iM t V αβ = s • α d • β |0 e iM t Ξ 3α = s • α d • β d • β |0 e iM t P 2 = s • α s • α |0 e iM t Ξ 4α = d • α s • β s • β |0 e iM t S 2 = d • β d • β s • α s • α |0 e iM t (6b)
where operators s α and d α acting on the vacuum give zero:

s α |0 = d α |0 = 0.
We will show in this report that it is possible, in order to quantize the superparticle action, to remain the Hamiltonian in the square root form. As it is very obvious, in the form of square root the Hamiltonian operator is not linearly proportional with the operator n s = s

. β s α . The Fock construction for the Hamiltonian into the square root form agrees formally with the description given above for reference [START_REF] Pashnev | Supersymmetric lagrangian for particles in proper time[END_REF], but the operability of this Hamiltonian is over basic states with lowest helicities λ = 1/4, 3/4. This means that the superparticle Hamiltonian preserving the square root form operates over physical states of particles with fractional quantum statistics and fractional spin (quartions).

III. HAMILTONIAN TREATMENT IN LANCZO'S FORMULATION

In order to solve our problem from the dynamical and quantum mechanical point of view, we will use the formulation given in [START_REF] Lanczos | Variational Principles in Mechanics[END_REF][START_REF] Bakai | Adiabatic Invariants[END_REF]. This Hamiltonian formulation for dynamical systems was proposed by C. Lanczos and allows us to preserve the square root form in the new Hamiltonian. We start from expression (4)

H = m 2 -P i P i + 1 a Π α Π α - 1 a * Π . α Π . α if dt dτ ≡ dx 0 dτ = g P 0 , P i , Π α , Π . α , x 0 , x i , θ α , θ .
α with the arbitrary function g given by

g = m 2 -P i P i + 1 a Π α Π α -1 a * Π . α Π . α -P 0 m 2 -P 0 P 0 -P i P i + 1 a Π α Π α -1 a * Π . α Π . α m 2 -P i P i + 1 a Π α Π α -1 a * Π . α Π . α -P 2 0 (7)
the new Hamiltonian H takes the required "square root" form

H ≡ g (H + P 0 ) = m 2 -P 0 P 0 -P i P i + 1 a Π α Π α - 1 a * Π . α Π . α (8)
where we shall set H = 0 (now depending on 2n + 2 canonical variables), and the variable P 0 is clearly identificated by the dynamical expression

dP 0 dτ = -g ∂H ∂x 0 or dP 0 dτ = - ∂H ∂t (9)
This means that P 0 = -H + const. Concerning the equations of motion , the integration and derivatives are consistent with the geometrical treatment of supermanifolds that we will describe with some detail in Section 6 .

In order to make an analysis of the dynamics of our problem, we can compute the Poisson brackets between all the canonical variables and their conjugate momentum [START_REF] Pashnev | Supersymmetric lagrangian for particles in proper time[END_REF][START_REF] Casalbuoni | The Classical Mechanics for Bose-Fermi Systems[END_REF][START_REF] Casalbuoni | Relatively and supersymmetries[END_REF] •

P µ = {P µ , H} pb = 0 (10)
.

θ α = {θ α , H} pb = 1 a Π α H (11)
.

θ • α = θ • α , H pb = - 1 a * Π • α H (12)
•

x µ = {x µ , H} pb = 1 H P µ + i a Π α (σ µ ) α . β θ . β + i a * θ α (σ µ ) α . β Π • β (13)
•

Π α = {Π α , H} pb = 2i a * H P α . β Π • β ( 14
) • Π • α = Π • α , H pb = -2i aH Π β P β . α (15)
where P α .

β ≡ P µ (σ µ ) α .
β . From the above expressions the set of classical equations to solve is easily seen

•• Π α = - 4P 2 |a| 2 H 2 • Π α (16) •• Π • α = - 4P 2 |a| 2 H 2 • Π . α (17) Assigning 4P 2
|a| 2 H 2 ≡ ω 2 , and having account for Π + α = -Π • α , the solution to the equations ( 16) and ( 17) takes the form

Π α = ξ α e iωτ + η α e -iωτ Π • α = -η • α e iωτ -ξ • α e -iωτ (18)
By means of the substitution of above solutions into ( 14) and ( 15), we find the relation between ξ α and η α

η α = 2 a * Hω P α . β ξ . β
From eqs. ( 18) and above we obtain

Π α = ξ α e iωτ + 2 a * Hω P α . β ξ . β e -iωτ (19)
Π • α = - 2 aHω ξ β P β . α e iωτ -ξ • α e -iωτ (20)
where we used the fact that the constant two-component spinors ξ α verify ξ • α = ξ + α . Integrating expressions [START_REF] Casalbuoni | Relatively and supersymmetries[END_REF] and [START_REF] Lanczos | Variational Principles in Mechanics[END_REF], we obtain explicitly the following

θ α = ζ α - i aHω ξ α e iωτ - 2 a * Hω P α . β ξ . β e -iωτ (21)
θ . α = ζ • α + i a * Hω - 2 aHω ξ β P β . α e iωτ + ξ • α e -iωτ (22)
where ζ α and ζ • α = ζ + α are two-component constant spinors. Analogically, from expression (13), we obtain x µ in explicit form

x µ = q µ - 1 H P µ - ωH P 2 ξσ µ ξ τ + 1 Hω 1 a e iωτ ξσ µ ζ + 1 a * e -iωτ ζσ µ ξ + (23)
+ P µ 2P 2 ζ α ξ α e iωτ -ζ • α ξ • α e -iωτ

IV. QUANTIZATION

Because of the correspondence between classical and quantum dynamics, the Poisson brackets between coordinates and canonical impulses are transformed into quantum commutators and anti-commutators

[x µ , P µ ] = i {x µ , P µ } pb = -ig µν {θ α , P β } = i {θ α , P β } pb = -iδ α β θ • α , P • β = i θ • α , P • β pb = -iδ • α • β (24)
and the new Hamiltonian ( 8) operates quantically as follows

m 2 -P 0 P 0 -P i P i + 1 a Π α Π α - 1 a * Π . α Π . α |Ψ = 0 ( 25
)
where |Ψ are the physical states. From the (anti)commutation relations [START_REF] Sudarshan | Dirac positive energy wave equation with para-Bose internal variables[END_REF] it is possible to obtain easily the commutators between the variables

ξ α , ξ • α , ζ α , ζ • α , q µ , P µ ξ α , ξ • α = -P α . α ζ α , ζ • α = -1 2P 2 P α . α [q µ , P µ ] = -ig µν (26)
To obtain the physical spectrum we use the relations given by ( 26) into ( 25) and the Hamiltonian H takes the following form

H = m 2 -P 0 P 0 -P i P i - 2 3/2 (P µ ) 2 |a| - 2 3/2 |a| (P µ ) 2 ξ α P α . β ξ . β (27)
Passing to the center of mass of the system, and defining new operators

s α = (1/ √ M)ξ α , s . α = (1/ √ M )ξ . α , d α = √ 2Mζ α , d . α = √ 2M ζ .
α (where M = P 0 ), the H cm is

H cm = m 2 -M 2 + 2 3/2 M |a| 1 -(σ 0 ) α . β s . β s α (28)
being the anti-commutation relations of the operators s α , s . α , d α , d . α :

s α , s • α = -(σ 0 ) α . α d α , d • α = -(σ 0 ) α . α (29)
Now the question is: how does the square-root H Hamiltonian given by expression (28) operate on a given physical state? The problem of locality and interpretation of the operator like ( 25) is very well known. Several attempts to avoid these problems were given in the literature [START_REF] Sucher | Relativistic invariance and the Square-Root Klein-Gordon equation[END_REF][START_REF] Schweber | An Introduction to Relativistic Quantum Field Theory[END_REF]. The main characteristic of all these attempts is to eliminate the square root of the Hamiltonian: e.g.imposing constraints. In this manner, the set of operators into the square root operates freely on the physical states, paying the price to lose the concept of locality and quantum interpretation of the spectrum of a well possed field theory.

Our plan is "to take " the square root to a bispinor in order to introduce the physical state into the square root Hamiltonian. In the next section we will perform the square root of a bispinor and obtain the mass spectrum given by the Hamiltonian H.

V. MASS SPECTRUM AND SQUARE ROOT OF A BISPINOR

The square root from a spinor was extracted in 1965 by the soviet scientist S. S. Sannikov from Kharkov (Ukraine) [START_REF] Sannikov | Non-compact symmetry group of a quantum oscillator[END_REF] and the analysis of the structure of the Hilbert space containing such "square root " states was worked out by E. C. G. Sudarshan, N. Mukunda and C. C.

Chiang in 1981 [START_REF] Sudarshan | Dirac positive energy wave equation with para-Bose internal variables[END_REF]. Taking the square root from a spinor was performed also by P.A.M.

Dirac [START_REF]Dirac: A positive-energy relativistic wave equation[END_REF] in 1971.

We know that the group SL(2, C) is locally isomorph to SO(3,1), and SL(2,R) is locally isomorph to SO(2,1). For instance, the generators of the group SO(3,1) for our case can be constructed from the usual operators a, a + (or q and p) in the following manner: we start from an irreducible unitary infinite dimensional representation of the Heisenberg-Weyl group, which is realized in the Fock space of states of one-dimensional quantum oscillator [START_REF] Yu | On massless fields and relativistic wave equations[END_REF][START_REF] Sorokin | The Heisenberg algebra and spin[END_REF][START_REF] Volkov | Quartions in relativistic field theories[END_REF].

Creation operators and annihilation operators of these states obey the conventional commutation relations [a, a + ] = 1 [a, a] = [a + , a + ] = 0 . To describe this representation to the Lorentz group one may also use the coordinate-momentum realization (q, p = -i ∂ ∂q ) of the Heisenberg algebra, which relates to the a, a + realization by the formulas

a = q+ip √ 2 a + = q-ip √ 2 (30)
as usual. Let us introduce the spinors

L α =   a 1 a + 1   L • α =   a 2 a + 2   (31)
The commutation relations take the form

[L α , L β ] = iε αβ ; L • α , L • β = iε • α • β ; L • α , L β = 0 ( 32
)
The generators of SL(2, C) are easily constructed [START_REF] Volkov | Quartions in relativistic field theories[END_REF] from L α and L

• α S αβ ≡ iS 1i (σ i ) αβ = 1 4 {L α , L β } S • α • β ≡ iS 2i (σ i ) • α • β = 1 4 L • α , L • β ( 33
)
where the index i = 1, 2, 3 and the six vectors S ai (a, b = 1, 2; a = b), characteristics of the representation of SL(2, C) ≈ SO(3, 1), satisfy the commutation relations

[S ai , S aj ] = -iε ijk S k a , [S bi , S bj ] = -iε ijk S k b , [S ai , S bj ] = 0 ( 34
)
Notice that the above construction obeys the described decomposition of SL(2, C) ≈

SO(3, 1)
Then the quantities

Φ α ≡ Ψ| L α |Ψ , Φ • α ≡ Ψ L • α Ψ , (35)
are the two-components of a bispinor

Φ ≡ Ψ L Ψ =   Φ α Φ • α   where we define Ψ ≡   |Ψ Ψ   .
Notice that |Ψ and Ψ are the square root of each component of this bispinor and can have the same form (given the isomorphism between the generators L α and L • α ), that is very easy to verify. In terms of q the basic vectors of the representation can be written as [START_REF] Yu | On massless fields and relativistic wave equations[END_REF][START_REF] Sannikov | Non-compact symmetry group of a quantum oscillator[END_REF][START_REF] Sorokin | The Heisenberg algebra and spin[END_REF]

q |n = ϕ n (q) = π -1/4 (2 n n!) -1/2 H n (q) e -q 2 /2 (36) dqϕ * m (q) ϕ n (q) = δ mn (37)
(where H n (q) are the Hermite polynomials) and form a unitary representation of SO (3, 1), and

|n = (n!) -1/2 a + n |0 (38)
the normalized basic states where the vacuum vector is annihilated by a . The Casimir operator, that is S ai S i a , has the eigenvalue λ(λ -1) = -3 16 (for each subgroup ISO(2,1) given by eqs. [START_REF] Majorana | [END_REF])and indeed corresponds to the representations with the lowest weights λ = 1 4 and λ = 3 4 . The wave functions which transform as linear irreducible representation of ISO(2, 1) , subgroup of ISO (3,1) generated by operators [START_REF] Majorana | [END_REF] are

Ψ 1/4 (x, θ, q) = +∞ k=0 f 2k (x, θ) ϕ 2k (q) (39) Ψ 3/4 (x, θ, q) = +∞ k=0 f 2k+1 (x, θ) ϕ 2k+1 (q) (40)
(analogically for the Ψ

d α → Φ 1/4 α ≡ Ψ 1/4 L α Ψ 1/4 d • α → Φ 1/4 • α ≡ Ψ 1/4 L • α Ψ 1/4 (41)
and, analogically, the spinors s α and s • α with

s α → Φ 3/4 α ≡ Ψ 3/4 L α Ψ 3/4 s • α → Φ 3/4 • α ≡ Ψ 3/4 L • α Ψ 3/4 (42)
where the new spinors L α L • α are defined as

L α =   a 1 a 1 a + 1 a + 1   (42a) L . α =   a 2 a 2 a + 2 a + 2  
The reason to this choice is the following: as it was shown in Ref. [25] the Hilbert space for each subgroup ISO(2,1)≈ SU(1, 1) [26] can be decomposed as direct sum of two independent subspaces characterized for the states of helicity λ = 1 4 and λ = 3 4 respectively. Each subspace is composed by the even λ = 1 4 and odd states λ = 3 4 given by eqs. (39)(40). These "cat" states admit (after a convenient choice for the functions f 2k (x, θ) and f 2k+1 (x, θ)) a coherent state realization being eigenvectors not of the ladder operator a of the Heisenberg-Weyl algebra , but those of the quadratic ladder operator aa of the SU(1, 1) algebra defined in general by

K + = 1 2 a + a + , K -= 1 2 aa , K 0 = 1 4 (a + a + aa + ) , (42b)
That means that when we are in the full Hilbert space the algebra is Heisenberg-Weyl and the states |Ψ = 1 √ 2

Ψ 1/4 + Ψ 3/4 are eigenvectors of the operator a, and when we pass to the decomposed space (by means of a suitable unitary transformation) the algebra becomes the SU(1, 1) algebra with the quadratic ladder operators given by expression (42b).

The relations [START_REF] Lachieze-Rey | On three quantization methods for a particle on hyperboloid[END_REF] and [START_REF] Delbourgo | A square root of the harmonic oscillator[END_REF] give a natural link between the spinors ξ α ξ • α and ζ α ζ • α , solutions of the dynamical problem, with the only physical states that can operate freely with the Hamiltonian H : the "square root" states |Ψ , Ψ from the bispinor Φ. Notice that there are four (non-trivial) representations for the group decomposition of the bispinor Φ,as follows

Φ 1 =   Φ 1/4 Φ 3/4   → (1/4, 0) ⊕ (0, 3/4) Φ 2 =   Φ 3/4 Φ 1/4   → (3/4, 0) ⊕ (0, 1/4) Φ 3 =   Φ 1/4 Φ 1/4   → (1/4, 0) ⊕ (0, 1/4) Φ 4 =   Φ 3/4 Φ 3/4   → (3/4, 0) ⊕ (0, 3/4)
This result is a consequence of the geometrical Hamiltonian taken in its natural square root form and the Sannikov-Dirac oscillator representation for the generators of the Lorentz group SO(3,1).

Commutation relations (29) obey the Clifford's algebra for spinorial creation-annihilation operators. In this manner, the square root of the operators s α and d α in the representation given by the associations ( 41) and ( 42) acting on the vacuum give zero, symbolically :

√ s α → Φ 3/4 α |0 = L rs α Ψ r 3/4 Ψ s 3/4 |0 = 0 d α → Φ 1/4 α |0 = L rs α Ψ r 1/4 Ψ s 3/4 |0 = 0
where here we introduce r, s, t... latin indexes to design the fractional spin states. The Fock's construction in the center of mass of the system consists now, in contrast to the construction (6b), of the following vectors:

S 1 = |0 e iM t 2 , Ξ 1r = Ψ r 1/4 |0 e iM t 2 , P 1 = Ψ r 1/4 Ψ r 1/4 |0 e iM t 2 , Ξ 2r = Ψ r 3/4 |0 e iM t 2 , V rs = Ψ r 3/4 Ψ s 1/4 |0 e iM t 2 , Ξ 3r = Ψ r 3/4 Ψ s 1/4 Ψ s 1/4 |0 e iM t 2 , (43)
P 2 = Ψ r 3/4 Ψ r 3/4 |0 e iM t 2 , Ξ 4r = Ψ r 1/4 Ψ s 3/4 Ψ s 3/4 |0 e iM t 2 , S 2 = Ψ r 1/4 Ψ r 1/4 Ψ s 3/4 Ψ s 3/4 |0 e iM t 2
Notice that the vectors given above are the only states that can operate into the square root operator given by expression (28), and not that constructed directly with the operators s α and d α . Schematically we have, e.g. for Ξ 4r , the following operability :

m 2 -M 2 + 2 3/2 M |a| 1 -(σ 0 ) α . β s . β s α Ψ r 1/4 Ψ s 3/4 Ψ s 3/4 |0 e iM t 2 ≡ ≡ m 2 -M 2 + 2 3/2 M |a| 1 -(σ 0 ) α . β s . β s α d • γ s • β s • β e iM t |0
From expression [START_REF] Bars | [END_REF] and taking into account that the number operator is s

.

β s α ≡ n s , because s .
β and s α work as creation-annihilation operators, we can easily obtain the mass for the different "square root" or fractional supermultiplets :

i) n s = 0 → M I = -2 1/2 |a| + 2 |a| 2 + m 2 ; Fractional supermultiplet I: S 1 , Ξ 1r , P 1 ii) n s = 1 → M II = √ m ; Fractional supermultiplet II: (Ξ 2r , V rs , Ξ 3r ). iii) n s = 2 → M III = 2 |a| 2 + m 2 + 2 1/2
|a| ; Fractional supermultiplet III: P 2 , Ξ 4r , S 2 . We emphasize now that the computations and algebraic manipulations given above were Another important point is that the link between the new Hamiltonian H given by expression (8) and the relativistic Schrödinger equation (e.g. ref. 27 ) can be given through the relation between the conserved currents of the fermionic "square" states and the para-states.

with d • α → Φ 1/4 • α and s • α → Φ 3/4 • α (
This important issue will be analyzed in Section VII.

It is interesting to note that the arbitrary c-parameters a and a * generate a deformation of the usual line element for a superparticle in proper time, and this deformation is responsible, in any meaning, for the multiplets given above. This is not a casuality: one can easily see how the quantum Hamiltonian ( 28) is modified in the center of mass of the system by the c-parameters a and a * . The implications of this type of superparticle actions with deformations of the quantization will be analyzed in a future paper [28].

VI. SQUARE ROOT HAMILTONIAN AND THE THEORY OF SEMIGROUPS

From the Hamiltonian eq.( 8)

H = m 2 -P 0 P 0 -P i P i + 1 a Π α Π α - 1 a * Π . α Π . α
where

Π α = P α + i P µ (σ µ ) α . β θ . β Π . α = P . α -iP µ θ α (σ µ ) α .
α (P α and P µ was defined from the Lagrangian, as usual), we start with the equation

S [Ψ] = H s Ψ = m 2 -P 0 P 0 -P i P i + 1 a Π α Π α - 1 a * Π . α Π . α Ψ (44)
In order to construct a general analytic representation for above equation, let us set ω 2 = m 2 and -P 0 P 0 -P

i P i + 1 a Π α Π α -1 a * Π . α Π . α = -G.
Because we are treating it now with a supermanifold instead of the simple flat space, it is necessary to give a consistent definition of integration on it, in order to perform successfully the construction of the general analytical representation of eq.( 44). To do this we make firstly some remarks on superspace and supermanifolds. to B L will be taken to be G ∞ , i.e. infinitely differentiable with respect to all arguments which in turn implies that the function admits a finite Taylor series expansion in the odd arguments, with infinitely differentiable functions of the even arguments as coefficients [START_REF] Rogers | A global theory of supermanifolds[END_REF]. A (D 0 + D 1 )-dimensional supermanifold M (D 0 ,D 1 ) is constructed from B If the overlap U i ∩ U j is non-empty we require the transition function

ϕ j • ϕ -1 i to be G ∞ . A chart map ϕ induces coordinates ϕ M (m) = z M = {x µ , θ α } (M = 1, ..., D 0 + D 1 ; µ = 1, ...D 0 , α = 1, ..., D 1
). If we change to other coordinates we require this change to respect evenness/oddness in the following sense: if (X) denotes a grading of an Grassmann element X, i.e. (X) = 0 if X is even and (X) = 1 if X is odd, then under a coordinate change z M → z M we require z M = z M .

Equipped with the notion of differentiability in B

(D 0 ,D 1 ) L one can construct the tangent bundle T M (D 0 ,D 1 ) . At a point m ∈ U ⊂ M (D 0 ,D 1 ) the tangent space T (m) M (D 0 ,D 1 ) is spanned (on a coordinate basis) by M ∂ = → ∂
∂z M (we use the de Witt conventions [START_REF] Witt | Supermanifolds[END_REF] for index manipulation in order to avoid factors of (-1)). The dual space to T (m)

M (D 0 ,D 1 ) denoted T * (m)
M (D 0 ,D 1 ) is spanned by z M where N ∂ z M = N δ M (The Kronecker delta). This in turn gives rise to the cotangent bundle T * M. In general, field tensors of type (p, r) are elements of ⊗ p T * M (D 0 ,D 1 ) ⊗ ⊗ r T M (D 0 ,D 1 ) , then the components of a tensor of type (p, r) are displayed on a coordinate basis as

dz N 1 ⊗ ... ⊗ dz Np Np...N 1 T M 1 ....Mr ⊗ Mr ∂ ⊗ ... ⊗ M 1 ∂
We will need tensors with special symmetry properties that generalize the differential forms of the ordinary differential geometry. This space of (p, r) tensors is denoted Λ r p M (D 0 ,D 1 ) and spanned by

dz N 1 ∧ ... ∧ dz Np ⊗ Mr ∂ ∨ ... ∨ M 1 ∂
where ∧ is the graded antisymmetric wedge product

dz N ∧ dz M = dz (N ∧ dz M ] = -(-1) (N )(M ) dz M ∧ dz N ,
where (N) = dz N , and ∨ is the graded symmetric product

N ∂ ∨ M ∂ = (N ∂ ∨ M ] ∂ = + (-1) (N )(M ) N ∂ ∨ M ∂ An element of Λ r p M (D 0 ,D 1
) is denominated a hyperform [START_REF] Picken | Integration on Supermanifolds and a Generalized Cartan Calculus[END_REF]. For p = 0 a hyperform is also called a derivative r-form; for r = 0 a hyperform is a differential p-form.

We are now in the position to introduce the following definition that precises the relationship between the supermanifold integration and the Berezin/Riemann integration: Definition. Let M (D 0 ,D 1 ) be the total space of a fibre bundle E = (M (D 0 ,D 1 ) , π, B 0 , F 1 ), and let p ∈ M (D 0 ,D 1 ) . Let (π -1 (U) , ϕ) be a chart on the G ∞ supermanifold, where U ⊂ B 0 and p ∈ U. Because of the local trivialization property we take π

-1 (U) ∼ = U × F 1 . Let A be a D H 0 , D V 1 hyperform with support compact in ϕ (U) ⊂ O,
where O is open in B D 0 L,0 . In natural coordinates z M = ϕ M (p), and with respect to the canonical basis

A = a (z) Ω (D 0 ,D 1 ) 1 D 0 ! 1 D 1 ! dx µ 1 ∧ ... ∧ dx µ D 0 µ D 0 ... µ 1 A α 1 ....α D 1 α D 1 ∂ ∨ ... ∨ α 1 ∂ Then A π -1 (u) := Ber O×B D 1 L,1 a (x, θ) dx 1 ...dx D 0 dθ 1 ...dθ D 1 (45)
Notice that it makes no sense to demand that A has compact support in the " θ direction", as e.g. for B

{x µ , θ α } → x µ (x) , θ α (x, θ) (46)
that is nothing more than a bundle morphism from one set of natural coordinates to another.

This corresponds to the choice of a different section and a change of basis in the fibres. From the practical point of view we can easily see that, following the definition (45), the procedure consists of replacing the one-forms dx µ by the integration symbols "dx µ ", the derivative one-forms α ∂ by the integration symbols "dθ α ", and deleting the ∧ and ∨ products. This procedure is justified by the fact that both sides transform identically under the coordinate transformations (46).

We assume that -G + ω 2 satisfies the conditions required to be a generator of a unitary group (self-adjoint); we can write (44) as :

S [Ψ] = √ -G + ω 2 Ψ (47)
Using the analytic theory of fractional powers of closed linear operators (e.g. Refs. [27,[START_REF] Yosida | Functional Analysis[END_REF]), it can be shown that (for generators of unitary groups) we can write (47) as

S [Ψ] = 1 π ∞ 0 -G + λ + ω 2 -1 Ψ dλ √ λ (48)
where [-G + (λ + ω 2 )] -1 is the resolvent associated with the operator (-G + ω 2 ) . The resolvent can be computed directly if we can find the fundamental solution to the equation

∂Q ∂t (z 1 , z 2 ; t) + G -ω 2 Q(z 1 , z 2 ; t) = δ(z 1 -z 2 ) (49)
It is shown in [27] that the equation

i ∂Q ∂t (z 1 , z 2 ; t) + 1 2M G -V Q(z 1 , z 2 ; t) = δ(z 1 -z 2 ) ( 50
)
has the general (infinitesimal) solution

Q(z 1 , z 2 ; t) = M 2πi t 3/2 e it M 2 ( z 1 -z 2 t ) 2 -V (z 2 ) + ie c (z 1 -z 2 )A[ 1 2 (z 1 -z 2 )] (51)
provided that A and M are time independent. We used the midpoint evaluation in the last part for equation (51) A 1 2 (z 1z 2 ) and from the supersymmetric Hamiltonian ( 44) that

A ≡ 0, -i P µ (σ µ ) α . β θ . β , i P µ (σ µ ) α . β θ . β in superspace components. If we set ω 2 i = V and M = i 2 then Q(z 1 , t; z 2 , 0) = z(t)=z 1 z(0)= z 2 DW z,t [z (s)] e { t 0 V [z(s)]ds+ ie c x y A[z(s)]dz(s)}
solves (50), where

z(t)=z 1 z(0)= z 2 DW z,t [z (s)] = z(t)=z 1 z(0)= z 2 D[z (s)] e -1 4 t 0 | dz(s) ds | 2 ds = lim N →∞ 1 4πε (N) n N 2 M N j=1 dz j e - N j=1 [ 1 4ε(N) (z j -z j-1 ) 2 ]
and ε (N) = t/N. Now we construct the solution of equation (47) for the constant A case. The solution for other examples will be discussed in a future paper applying explicitly these results to the Hamiltonian formulation in supermanifolds. First, rewrite equation (51) as

Q(z 1 , z 2 ; t) = 1 4πt 3/2 exp (z 1 -z 2 ) 2 4s -µ 2 t + ie c (z 1 -z 2 ) .A
Finally, using the theory of fractional powers, we note if T [t, 0] is the semigroup associated with -G + ω 2 , then the semigroup associated with [-G + ω 2 ] 1/2 is given by 27,32

T 1/2 [t, 0] ϕ (x) = ∞ 0 M 1 4πt 3/2 exp (z 1 -z 2 ) 2 4t -µ 2 t + ie c (z 1 -z 2 ) .A ϕ (z 2 ) dz 2 × × ct √ 4π 1 s 3/2 exp - (ct) 2
4s ds

Laplace transforming we get

T 1/2 [t, 0] ϕ (x) = ct 4π 2 M 1 4πt 3/2 exp ie 2 c (z 1 -z 2 ) .A 2µ 2 K 2 µ (z 1 -z 2 ) 2 + c 2 t 2 (z 1 -z 2 ) 2 + c 2 t 2 ϕ (z 2 ) dz 2 (52)
where the order of integration was interchanged. We can easily see from expression (52) that the solution is non-local(MacDonald's function of second order) and does not coincide with the solution to the same problem where the square root has been eliminated by reparametrization and introducing constraints. It is important to note here the following:

i) from the semi-group representation of the radical operator we see that is not the same to operate with the square root Hamiltonian as that with its square or other power of this operator: the states under which the Hamiltonian operates are sensible to the power of such Hamiltonian .

ii) from the practical point of view the explicit determination of the functions (states) ϕ (z) can carry several troubles in any specific physical problems. In a future paper [28] we can give a detailed study of this problem in different physical contexts.

VII. RELATION WITH THE RELATIVISTIC SCHR ÖDINGER EQUATION: COMPATIBILITY CONDITIONS AND PROBABILITY CURRENTS

Looking at formula (25) the new Hamiltonian operates as (g µν = (+ ---))

m 2 -P 0 P 0 -P i P i + 1 a Π α Π α - 1 a * Π . α Π . α |Ψ = 0
for instance, the action of the radical operator is

m 2 -P 0 P 0 -P i P i + 1 a Π α Π α - 1 a * Π . α Π . α γ β (ΨL γ ) Ψ 1/2 = 0 (53)
that seems as a parabosonic supersymmetric version of the relativistic Schrödinger-De Broglie equation. In the next paragraph we will see that this equation corresponds to the family of equations given firstly by E. Majorana [START_REF] Majorana | [END_REF] and P. A. M. Dirac [START_REF]Dirac: A positive-energy relativistic wave equation[END_REF], and in its para-bosonic version by Sudarshan, N. Mukunda and C. C. Chiang in 1981 [START_REF] Sudarshan | Dirac positive energy wave equation with para-Bose internal variables[END_REF].

As we have been obtain from expression (58) j 0 (x) for the square states Φ α is positively definite because the energy E appears squared. Now in order to find the current vector for the para-Bose states Ψ we proceed analogically as above for the states Φ α arriving to

Ψ † Ψ -Ψ Ψ † = 0
as we expected because these square root states obey the square root operator equation and, for instance, also obey the equation for the squared operator (the inverse is not true in general). In fact, in some references in the literature the authors don't take care of the fact we can pass to the equation with the square root Klein-Gordon operator to its squared traditional version operating both on the same state but not the inverse (see e.g. ref. [34]).

The correct form to do this is as follows: if we start with

(-∆ + m 2 )Ψ = i∂ t Ψ (59)
the relation with any pseudo-differential operator A is

AΨ = (-∆ + m 2 )Ψ = i∂ t Ψ AAΨ = iA∂ t Ψ = A (-∆ + m 2 )Ψ = -∆ + m 2 Ψ
This happens clearly because Ψ obeys (59). Finally the current for the quartionic states that we were looking for is

∂ µ Ψ∂ µ Ψ † -Ψ † ∂ µ Ψ = 0 = -∂ µ j µ j µ (x) ≡ -i Ψ∂ µ Ψ † -Ψ † ∂ µ Ψ (60)
Is not difficult to see that in this case from expression (60) the zero component of the current is not positively definite given explicitly by

j 0 (x) = 2EΨ † Ψ
The compatibility condition, as usual, is given by the follow expression

[τ α , τ β ] Ψ = 0 (61)
where we defined

τ β ≡ ( + m 2 ) α β (ΨL α ) 1/2
. After a little algebra and using expression (61) we arrive to

+ m 2 δ α + m 2 γ β ǫ δγ 1/2 Ψ = 0 (62)
It is good to remember here that eq.( 53) describes a free particle in a N=1 superspace and the term of interaction appears from the supersymmetry between the bosonic and fermionic fields. The last expression shows that our equation ( 53) is absolutely compatible and consistent because its character fermionic coming from the supersymmetric part, and for instance not necessary to introduce any extra term in order to include spin. It is well known, that it terms (put "by hand" in equations containing a second order derivatives) destroy the compatibility condition leading to the impossibility of including interactions [35].

VIII. RELATIVISTIC WAVE EQUATION

Following the arguments given in the precedent paragraphs, it is natural to propose the following form for a square root of the second order supersymmetric wave equation

m 2 -P 0 P 0 -P i P i + 1 a Π α Π α - 1 a * Π . α Π . α α β (ΨL α ) Ψ 1/2 = 0 (63)
In order to reduce the expression (63) to the simplest form it is necessary pass to the center of mass of the system and redefining the variables as:

t → (aa * ) -1/2 t, θ → a -1/2 θ, θ → a -1/2 θ
we obtain the following expression

|a| 2 ∂ 2 0 + 1 4 ∂ η -∂ ξ + i ∂ 0 σ 0 ξ 2 - - 1 4 ∂ η + ∂ ξ + i ∂ 0 σ 0 ξ 2 + m 2 α β Φ α 1/2 = 0 ( 64
)
where

η ≡ θ + θ , ξ ≡ θ -θ , and ∂ 0 (σ 0 ) α . β θ . β -θ α ≡ ∂ 0 (σ 0 ) ξ (65)
The trick that we used above [START_REF] Bakai | Adiabatic Invariants[END_REF]36,[START_REF] Akhiezer | Quantum Electrodynamics[END_REF] is based on the observation that the expression (63) has similar form that the equation for an electron in constant electromagnetic field (with

P µ (σ µ ) α .
β as the constant electric field). Imposing the condition ∂ η Φ α = 0 ⇒ Φ α (ξ); the "square" of the solution eigenfunction of eq.( 63) takes the form Φ γ (t) = e A(t)+ξ̺(t) Φ γ (0) (66)

with ̺ (t) = φ α + χ . α (i.e.chiral plus anti-chiral parts). The system of equations for A (t) and ̺ (t) that we are looking for, is easily obtained inserting the expression (66) in the eq.( 64)

|a| 2 Ä + m 2 = 0 .. χ . α -i ω 2 σ 0 α . α φ α = 0 - .. φ α + i ω 2 σ 0 . β α χ . β = 0
The above system can be solved given us the following result

A = - m |a| 2 t 2 + c 1 t + c 2 ; c 1 , c 2 ∈ C (67)
and

φ α = • φ α αe iωt/2 + βe -iωt/2 + 2i ω σ 0 . β α Z . β (68)
χ . α = σ 0 α . α • φ α αe iωt/2 -βe -iωt/2 + 2i ω σ 0 α . α Z α (69)
where • φ α , Z α and Z .

β are constant spinors. The superfield solution for the square states that we are looking for, have the following form

Φ γ (t) = e -( m |a| ) 2 t 2 +c 1 t+c 2 e ξ̺(t) Φ γ (0) (70)
with

̺ (t) = • φ α αe iωt/2 + βe -iωt/2 -σ 0 α . α αe iωt/2 -βe -iωt/2 + 2i ω σ 0 . β α Z . β + σ 0 α . α Z α (71)
and

Φ γ (0) = Ψ (0)| L γ |Ψ (0) (72)
that is nothing more that the mean value of L γ between the states |Ψ in the initial time, where the subalgebra is the Heisenberg-Weyl algebra (with generators a, a + and n + 1 2 ). As we have pointed out in Section V, the states |Ψ span all the Hilbert space and for instance, we can not obtain useful information from the point of view of the topology of the group manifold then, about the spin.

The dynamics of the square root fields, in the representation that we are interested in, can be simplified considering these fields as coherent states in the sense that are eigenstates of a 2

Ψ 1/4 (0, ξ, q) = +∞ k=0 f 2k (0, ξ) |2k = +∞ k=0 f 2k (0, ξ) a † 2k (2k)! |0 (73)
Ψ 3/4 (0, ξ, q) = +∞ k=0 f 2k+1 (0, ξ) |2k + 1 = +∞ k=0 f 2k+1 (0, ξ) a † 2k+1 (2k + 1)! |0
From a technical point of view these states are a one-mode squeezed states constructed by the action of the generators of the SU(1,1) group over the vacuum. For simplicity, we will take all normalization and fermionic dependence or possible CS fermionic realization, into the functions f (ξ). Explicitly at t=0

Ψ 1/4 (0, ξ, q) = f (ξ) |α + Ψ 3/4 (0, ξ, q) = f (ξ) |α - (74)
where |α ± are the CS basic states in the subspaces λ = 1 4 and λ = 3 4 of the full Hilbert space. From expression (70) and expressions [START_REF] Delbourgo | A square root of the harmonic oscillator[END_REF] we obtain

Φ α (t, λ) = Ψ λ (t)| L α |Ψ λ (t) = e -( m |a| ) 2 t 2 +c 1 t+c 2 e ξ̺(t) Ψ λ (0)|   a 2 (a + ) 2   α |Ψ λ (0) (75) Φ α (t, λ) = e -( m |a| ) 2 t 2 +c 1 t+c 2 e ξ̺(t) |f (ξ)| 2   α 2 λ α * 2 λ   α (76)
where λ label the helicity or the spanned subspace (e.g. ±). The "square root" states solution of the expression (63) take the following form

Ψ λ = e -1 2 ( m |a| ) 2 t 2 +c 1 t+c 2 e ξ̺(t) 2 |f (ξ)|   α α *   λ ( 77
)
where λ = 1/4, 3/4. Notice the difference with the case in which we used the HW realization for the states Ψ

|Ψ = f (ξ) 2 (|α + + |α -) = f (ξ) |α (78)
where, however, the linear combination of the states |α + and |α -span now the full Hilbert space being the corresponding λ to this CS basis λ = 1 2 .The "square" state at t=0 are

Φ α (0) = Ψ (0)| L α |Ψ (0) = Ψ (0)|   a a +   α |Ψ (0) (79) = f * (ξ) f (ξ)   α α *   α
The square state at time t

Φ γ (t) = e -( m |a| ) 2 t 2 +c ′ 1 t+c ′ 2 e ξ̺(t) |f (ξ)| 2   α α *   α (80)
And the "square root" solution becomes now to

Ψ (t) = e -1 2 ( m |a| ) 2 t 2 +c ′ 1 t+c ′ 2 e ξ̺(t) 2 |f (ξ)|   α 1/2 α * 1/2   (81)
We can see the change in the solutions from the choice in the representation of the Hilbert space. The algebra (topological information of the group manifold) is "mapped" over the spinors solutions through the eigenvalues α and α * . Notice that the constants c ′ 1 c ′ 2 in the exponential functions in expressions (80) and (81) differ from the c 1 and c 2 in (76) and (77), because these exponential functions of the Gaussian type come from the action of a unitary operator over the respective CS basic states in each representation (h 3 or HW).

These constants can be easily determined as functions of the frequency ω as in ref. [5] for the Schrodinger equation. A detailed analysis of this point and the other type of solutions will be given elsewhere [28]. About the possible algebras that contain an SU(1,1) as subgroup that can lead or explain the fermionic factors of type e ξ̺(t) 2

|f (ξ)| in the solutions are 2 subgroups that are strong candidates [START_REF] Bars | [END_REF]: the supergroup OSP (2, 2) [2] and the supergroup OSP (1/2, R) [39]. In the case of the OSP (2, 2) we have bosonic and fermionic realizations and the CS and SS can be constructed from the general procedure given by M. Nieto et al. in Refs. [1][2][3]. On the other hand, the OSP (1/2, R) realization is more "economic", the number of generators is minor than in the OSP (2, 2) case and the realization is bosonic: the K ± and K 0 generators operate over the bose states and the HW algebra given by a and a + operate over the fermionic part.

In this case the CS and the SS that can be constructed are eigenstates of the displacement and squeezed operators respectively but they cannot minimize the dispersion of the quadratic Casimir operator, so that they are not minimum uncertainty states.

The important point to remark here is that when we describe from the mostly geometrical grounds any physical system through SU(1,1) CS or SS, the orbits will appear as the intersections of curves that represent constant-energy surfaces, with one sheet of a two sheeted hyperboloid-the curved phase space of SU(1,1) or Lobachevsky plane-in the space of averaged algebra generators. In the specific case treated in this paper, the group containing the SU(

IX. CONCLUDING REMARKS

In this work the problem of the physical interpretation of the square root quantum operators and possible relation with the TDHO and coherent and squeezed states was analyzed considering the simple model of superparticle of Volkov and Pashnev [START_REF] Pashnev | Supersymmetric lagrangian for particles in proper time[END_REF]. Besides the extension and clarification of the results of our previous works [19], of which we have already made mention in the Introduction, we can summarize as follows:

i) the Fock construction for these fractional or "square root states" was proposed, explicitly constructed and compared with the Fock construction given in the reference [START_REF] Pashnev | Supersymmetric lagrangian for particles in proper time[END_REF] for the superparticle model with the Hamiltonian in standard form;

ii) we have shown that, in contrast to [START_REF] Pashnev | Supersymmetric lagrangian for particles in proper time[END_REF], the only states that the square root Hamiltonian can operate with correspond to the representations with the lowest weights λ = 1 4 and λ = 3 vii) As for the Klein-Gordon equation, the conserved currents for the "square-root" states (para-fields) and for the square states were explicitly computed and analyzed. The component zero of the current is linearly dependent on the energy E in the para-field case and for the "square" state the dependence on the energy is quadratic .

viii) The compatibility conditions were analyzed and the consistency of the proposed equation was established. The explanation of this consistency and the relation with the free dynamics and the supersymmetry of the model was given.

ix) New wave equation is proposed and explicitly solved for the time-dependent case. As for the TDHO the physical states are realized on the CS and SS basis, and the link between the topology of the (super)-group manifold and the obtained solution from the algebraic and group theoretical point of view was discussed and analyzed.

It is interesting to see that the results presented here for the superparticle are in complete agreement with the results, symmetry group and discussions for non-supersymmetric examples given in references [START_REF] Lachieze-Rey | On three quantization methods for a particle on hyperboloid[END_REF][START_REF] Delbourgo | A square root of the harmonic oscillator[END_REF][START_REF] Elizalde | On the concept of determinant for the differential operators of quantum physics[END_REF], where group and geometrical quantization was used.

This fact gives a high degree of reliability of our method of quantization and the correct interpretation of the radical Hamiltonian operator. It is clear that the ordinary Canonical method of quantization fails when the reparametrization procedure affects the power of the starting Hamiltonian modifying inexorably the obtained spectrum of the physical states [see e.g. [START_REF] Delbourgo | A square root of the harmonic oscillator[END_REF][START_REF] Elizalde | On the concept of determinant for the differential operators of quantum physics[END_REF]]. For instance, we conclude that quantically it is not the same to operate with the square root Hamiltonian as that with its square or other power of this operator because the obtained states (mass spectrum, spin) under which the Hamiltonian operates are sensible to the power of such Hamiltonian.; and seeing that the lowest weights of the states under the square root Hamiltonian can operate, and because not concrete action is known to describe particles with fractional statistics, superparticle relativistic actions as of [START_REF] Pashnev | Supersymmetric lagrangian for particles in proper time[END_REF] can be good geometrical and natural candidates to describe quartionic states [START_REF] Yu | On massless fields and relativistic wave equations[END_REF][START_REF] Sorokin | Anti) commuting spinors and supersymmetric dynamics of semions[END_REF][START_REF] Sorokin | The Heisenberg algebra and spin[END_REF][START_REF] Volkov | Quartions in relativistic field theories[END_REF] (semions). S. Gavrilov put our attention on references [44] and [45] where the canonical quantization was performed in non-commutative formulation and in the ordinary space respectively We are very acknowledge to M. Plyushcay and S. Gavrilov for your very valuable references.

  the geometrical description, the Hamiltonian treatment and quantization of the superparticle model. Section 5 is devoted to describe the process of quantization and the obtaining of the mass spectrum of the superparticle model under consideration emphasizing the relation between the group representation of the physical states and their CS or SS realizations. In Section 6 from the theory of semigroups we construct a general analytic representation of the square root Hamiltonian. In Sections 7 and 8 the relation of the model with the relativistic Schrödinger equation is discussed and a new relativistic wave equation is proposed. Finally, some conclusions and remarks are given in Section 9.

  the square of the true states) into the square root Hamiltonian. That means that the physical states are constrained by the the explicit form of the Hamiltonian operator. Notice from expressions(35), (41) and (42) that the physical states for the Hamiltonian in the square root form are one half the number of physical states for the Hamiltonian quadratic in momenta.

  The supermanifolds we are dealing with are modelled over flat superspace B (D 0 ,D 1 ) L , the cartesian product of D 0 copies of B L,0 and D 1 copies of B L,1 , where B L,0 and B L,1 are the even, respectively odd, subspaces of a real Grassmann algebra B L (with L anticommuting generators). Functions from B (D 0 ,D 1 ) L

(D 0

 0 ,D 1 ) L in the usual way by means of an atlas of charts ∪ i∈I (U i , ϕ i ) with U i an open cover of M (D 0 ,D 1 ) and a homeomorphism ϕ i of U i onto an open subset of B (D 0 ,D 1 ) L .

(0, 1 )

 1 L the only G ∞ function of θ which has compact support is ,, f (θ) = 0. By construction, the left-hand side of the definition (45) transform with the Berezinian under the following change of coordinates

;

  iii) there are four possible (non-trivial) fractional representations for the group decomposition of the spin structure from the square root Hamiltonian, instead of (1/2,0) and (0,1/2) as the case when the Hamiltonian is quadratic in momentum (e.g. Ref.[START_REF] Pashnev | Supersymmetric lagrangian for particles in proper time[END_REF]) as a consequence of the geometrical Hamiltonian taken in its natural square root form and the Sannikov-Dirac oscillator representation for the generators of the Lorentz group SO(3,1); now we make this research complete with the following new results:iv) The relation between the structure of the Hilbert space of the states, the spin content of the sub-Hilbert spaces and the CS and SS realization of the physical states was established for the particular model presented here. iv) We construct explicitly from the theory of semi-group the analytical representation of the radical operator in the N=1 superspace and we see that it is not the same to operate with the square root Hamiltonian as that with its square or other power of this operator from the point of view of the spectrum of the physical states: the states under which the Hamiltonian operates are sensible to the power of such Hamiltonian . v) If expression constructed in iv) gives a closed representation for the radical operator, from the practical point of view the explicit determination of the functions (states) ϕ (z) can carry several troubles in any specific physical problems. vi) The relation between the relativistic Schrödinger equation and other type of equations that involve variables with fractional spin and the model analyzed here was established and discussed.

  thankful to Professors John Klauder for his advisements, E. C. G. Sudarshan for his interest demonstrated in this work in a private communication and in particular to N. Mukunda for his interest to put in this research and give me several remarks and references on the problem of consistency of the positive energy wave equations with para-fields variables. I appreciate deeply Professor Tepper Gill's efforts to clarify to me many concepts on the correct description of the quantum systems, Hamiltonian formulation and the relation with the proper energy of these systems. I am very grateful to the Directorate of JINR, in particular of the Bogoliubov Laboratory of Theoretical Physics, for their hospitality and support. Note added: when this paper was alredy finished, the Professors M. Plyushcay and

  1/4 and Ψ 3/4 states with contrary helicity). We can easily see that the Hamiltonian H (28) operates over the states Ψ , which become into H as its square Φ α and Φ .

α . It is natural to associate, up to a proportional factor, the spinors d α and d • α with

  1,1) as subgroup linear and bilinear functions of the algebra generators can factorize operators as the Hamiltonian or the Casimir operator (when averaged with respect to group CS or SS), defining corresponding curves in the averaged algebra space. If we notice that the validity of the Ehrenfest's theorem for CS (SS) implies that, if the exact dynamics is confined to the SU(1,1) hyperboloid, it necessarily coincides with the variational motion, the variational motion that comes from the Euler-Lagrange equations for the lagrangian

L = z| i ∂ ∂t -H |z will be different if |z = |α or z = |α ± , as is evident to see.

It is interesting to note also that similar picture holds in the context of the pseudospin SU(1,1) dynamics in the frame of the mean field approximation induced by the variational principle on nonlinear Hamiltonians

[40]

.

We can see from the above expression that if we put the (super)momenta together in the operator, we obtain a more suitable equation in order to compute the currents as in the Fock-Klein-Gordon case

now eliminating the exponent 1/2 and taking the Hermitian conjugation to equation we have

Following the same procedure as Dirac in Ref. [START_REF]Dirac: A positive-energy relativistic wave equation[END_REF] we multiply the square of expression (54) from the left side by Φ † α and multiply the square of expression (55) from the left side by Φ α , integrating and subtracting the final expressions we obtain

Using the relations: 56) the current for the square states Φ α is

If we suppose that a link between the relativistic Schrödinger equation (53) and our new Hamiltonian H holds, the relation with the quartionic states is the following i .

Ψ = EΨ

Squaring the above expression and having account as H operates over Ψ and Ψ † , we can easily obtain

that into the explicit expression for j 0 (x) permits us to analyze the positivity of this component of the current for the square states Φ α j 0 (x) = 2E 2 Φ α † Φ α (58)