Spacetime structure of massive gravitino
Abstract
We present reasons as to why an ab initio analysis of the spacetime structure of massive gravitino is necessary. Afterwards, we construct the relevant representation space, and finally, give a new physical interpretation of massive gravitino.
keywords:
Massive gravitino, Spacetime symmetries, RaritaSchwinger framework.,
Journalref: Phys. Lett. B 529 (2002) 124131.
1 Motivation
A massive gravitino is described by . As far as its spacetime properties are concerned, it transforms as a finite dimensional nonunitary representation of the Lorentz group,
(1) 
The unitarily transforming physical states are built upon this structure [1].
We enumerate two circumstances that motivate us to take an ab initio look at this representation space.

For the vector sector, it has recently been called to attention that the Proca description of the representation space is incomplete [2]. An ab initio construction of this sector reveals that the Stückelberg contribution to the propagator, so important for the renormalization of the gauge theories with massive bosons [3], is found to naturally reside in the representation space.

At the same time, the properties of the , along with that of the , representation space determine the structure of . In order to impose a singlespin, i.e., spin , interpretation on the latter, the lower spin and spin components of are considered as redundant, unphysical, states that are claimed to be excluded from consideration by means of the two supplementary conditions: , and , respectively. However, this timehonored framework was questioned by a recent empirical observation regarding the and resonances [4]. The available data on highspin resonances reveal an unexpected and systematic clustering in terms of the representations with and without imposition of the supplementary conditions. For the and resonances these results are summarized in Fig. 1. For example, in the standard theoretical framework , , and should have been absent. Experimental data shows them to be present at statistically significant level.^{1}^{1}1The , , carry four star status, while at present simply has a one star significance [6].
In regard to the latter of the two enumerated circumstances, we take the position that any solution of the QCD Lagrangian for particle resonances must carry welldefined transformation properties when looked upon from different inertial frames. This forces these resonances to belong to one, or the other, of various representation spaces of the Lorentz group. For this reason the data on particle resonances may furnish hints on physical interpretation of various Lorentz group representations that one needs in gauge theories, or theories of supergravity.
For exploring the spacetime structure of massive gravitinos the charge conjugation properties play an important role. Under the operation of charge conjugation, one may choose the spinor sector to behave as a Dirac object, and implement the Majorana nature of the massive gravitino at the level of the Fock space. This is standard, see, e.g., Ref. [7]. Or, from the very beginning choose the spinor sector to behave as a Majorana object. Since we wish to stress certain nontrivial aspects of massive gravitino that do not – at least qualitatively – depend on this choice, we shall here treat the spinor sector to be of Dirac type.
Very nature of our ab initio look at the representation space defined in Eq. (1), obliges us to present sufficient pedagogic details so that by the end of the paper much that is needed to form an opinion on the arrived results is readily available. At the same time, Letter nature of this manuscript would prevent us from delving into subtle details which are, for present, of secondary importance (but have been studied and are planned to be presented elsewhere).
We shall work in the momentum space. The notation will be essentially that introduced in Ref. [2].
2 Construction of the Spinor and Vector Sectors
We now wish to construct the primitive objects that span the representation space defined in Eq. (1) for an arbitrary . We first construct the spinor and vector sectors in the rest frame, and then boost them using the following boosts:
(2) 
with
(3)  
(4) 
where stands for an identity matrix, while the remaining symbols carry their usual contextual meaning. We define the spin helicity operator: where , and
(5) 
2.1 Representation space
In this subsection we present the kinematic structure of the representation space in such a manner that its extension to the representation space becomes transparent. The familiarity of the equation presented shall not, we hope, mar the procedure adopted. The entire construct, in essence, relies on nothing more than the boost operators.
The restframe spinors are then chosen to be:
(6) 
The choice of the phases made in writing down these spinors has been determined by the demand of parity covariance.
The boosted spinors, and are obtained by applying the boost operator to the above spinors, yielding:
(7) 
In the standard notation, these satisfy the orthonormality and completeness relations, along with the Dirac equation:
(8)  
(9) 
The wave equation satisfied by the and spinors follows if we note:^{2}^{2}2This part of the calculation is best done if one starts with the momenta in the Cartesian coordinates, and takes the “axis of spin quantization” to be the zaxis.
(10) 
where we defined,
(11) 
The are null matrices. Adding/subtracting Eqs. (9) and (10), yields:
(12)  
(13) 
Multiplying Eq. (12) from the right by , and Eq. (13) by , and using Eqs. (8), immediately yield the momentumspace wave equation for the representation space,
(14) 
In Eq. (14), the minus sign is to be taken for, , and the plus sign for, .
The essential element to note in regard to Eq. (14) is that it follows directly from the explicit expressions for the and .
2.2 Representation space – An ab initio construct
Next, we introduce the restframe vectors for the representation space,
(15)  
(16)  
(17)  
(18) 
The boosted vectors are thus:
(19) 
In the notation of Ref. [2],^{3}^{3}3In the cited work we presented the representation space in its parity realization. Here, the presentation is in terms of helicity realization. The two descriptions have mathematically similar but physically distinct structures, which, e.g., show up in their different behavior under the operation of Parity. these satisfy the orthonormality and completeness relations, along with a new wave equation. The orthonormality and completeness relations are:
(20) 
where
(21) 
with
(22) 
The parity operator for the representation space is:
(23)  
while the helicity operator for this space is, , with given by:
(24) 
We now must take a small definitional detour towards the notion of the dragged Casimirs for spacetime symmetries. It arises in the following fashion. The second Casimir operator, , of the Poincaré group is defined as the square of the PauliLubanski pseudovector:
(25) 
where is the standard LeviCivita symbol in four dimensions, while denote generators of the Lorentz group,
(26) 
where each of the runs over . The are generators of the spacetime translations. In general, these have nonvanishing commutators with ,
(27) 
On using Eq. (27), we rewrite as
(28)  
The squared brackets vanishes due to antisymmetry of the LeviCivita symbol. As such, the spacetime translation operators entering the definition of can be moved to the very right.
This observation allows for introducing the dragged Casimir as an operator with the same form as – the difference being that the commutator in Eq. (27) is now set to zero [as is appropriate for finite dimensional representations]. Consequently, while and carry same invariant eigenvalues when acting upon momentum eigenstates, their commutators with the Lorentz group generators are no longer identical.^{4}^{4}4To avoid confusion, note that is defined in the ; while is defined in the Poincaré group. For the representation space, vanishes. For the representation space, does not vanish (except when acting upon rest states), and equals . This leads to the fact that while the former representation space is endowed with a welldefined spin, the latter is not:
As an immediate application, for the representation space bifurcates this space into two sectors. The three states with are associated with the eigenvalue, ; while the, , corresponds to eigenvalue zero.
Thus, all the , except for the rest frame, cease to be eigenstates of the ’s and do not carry definite spins. This contrasts with the situation for the representation space, where the are eigenstates of the corresponding .
Now in order that the carry the standard contravariant Lorentz index, we introduce a rotation in the representation space via [2]:
(29) 
Then, the representation space is spanned by four Lorentz vectors:
(30) 
and the superscript is the standard Lorentz index. Following the procedure established in Sec. 2.1, they can be shown to satisfy a new wave equation [2],
(31) 
where the plus sign is to be taken for, , while the minus sign belongs to, . The matrices are: , , , , and
(32) 
The remaining are obtained from the above expressions by noting: . Parenthetically, we note that the Stransformed equals and is nothing but the standard spacetime metric (for flat spacetime).
It can also be seen that , for , coincide with the solutions of Proca framework (and are divergenceless); whereas , that gives the Stückelberg contribution to the propagator, lies outside the Proca framework:^{5}^{5}5We define the dragged PauliLubanski pseudovector, , in a manner parallel to the introduction of the dragged second Casimir operator.
Remarks  

Proca Sector  
Stückelberg Sector 
In the above table we have introduced the via the equation:
(33) 
3 Construction of spinor vector: The massive gravitino
We now wish to present the basis vectors for the representation space defined by Eq. (1) in a language which is widely used [7]. This would allow the present analysis to be more readily available, and also bring out the relevant similarities and differences with the framework of Rarita and Schwinger [8].
In writing down the basis spinorvectors, we will use the fact that in the representation space the charge conjugation is implemented by
(34) 
In the representation space the charge conjugation operator is where complex conjugates the spinor to its right. Then, with the uniform choice,
(35) 
we obtain:
(36) 
In the spirit outlined, the massive gravitino lives in a space spanned by sixteen spinorvectors defined in items A, B, C below:

Of these, eight spinorvectors have – but not – eigenvalues, . These can be further subdivided into particle,
and antiparticle sectors:
Here, , .

Four spinorvectors have – but not – eigenvalues, :

Another set of four spinorvectors with – but not – eigenvalues, :
We have evaluated , , and , for all of the above sixteen spinor vectors. The , when transformed to the configuration space, tests the divergence of .
For , identically vanishes. Requiring it to vanish for results in:
The sectors, if (wrongly) imposed with the vanishing of, and , results in kinematically acausal dispersion relation (i.e., in ). This could be the source of the wellknown problems of the RaritaSchwinger framework as noted in works of Johnson and Sudarshan [9], and those of Velo and Zwanziger [10]. In this context one may wish to recall that interactions can induce transitions between different sectors.
The analysis for all the sectors of the can be summarized in the following table:
Remarks  

RaritaSchwinger Sector  
The table clearly illustrates that there is no particular reason – except (the unjustified) insistence that each particle of nature be associated with a definite spin – to favor one sector over the other. Each of the sectors is endowed with specific properties. The RaritaSchwinger sector has no more, or no less, physical significance than the other two sectors. While, for instance, the RaritaSchwinger sector can be characterized by vanishing of the and ; the sector is uniquely characterized by vanishing of . The sector allows for vanishing of only.
Except for the rest frame, the , in general, are not eigenstates of the for representation space (1). Instead, the three sectors of the representation space under consideration correspond to the following inertialframe independent values of the associated dragged second Casimir invariant: