Geometric Structures and Gauge Symmetries for Various Fermionic Fields from a Faddeev–Jackiw–Dirac Argument

Abstract

This paper focuses on the identification of presymplectic/symplectic structures, as well as gauge symmetries, for various fermionic fields in four spacetime dimensions, using a combined Faddeev–Jackiw–Dirac approach. The intrinsic first-order dynamics of fermionic fields allow a straightforward Faddeev–Jackiw analysis, which avoids unnecessary primary second-class constraints, introduces no artificial hierarchies, and, in general, constitutes a foolproof strategy when combined with the Dirac method. The Majorana spinors and Majorana spinor-vectors, with dynamics of various orders, as fundamental constituents of all SUSY and SUGRA paradigms, are here taken into consideration. Their Faddeev–Jackiw–Dirac analysis exhibits several symplectic/presymplectic structures, which represent the main novelty, together with the reconfirmation of their gauge symmetries.

1. Motivation

In the effervescence of the late nineteen-eighties and the beginning of the nineteen-nineties, produced by the cohomological approaches to quantization issues for gauge systems, an economical method for extracting the pure dynamics of a first-order system was proposed [1]. The procedure, devised for first-order systems linear in the time derivatives, shortcuts in some sense the well-known Dirac algorithm [2,3] by identifying the canonical structure and constraints via some transformations that bring the ‘kinetic term’, i.e., the linear term in the generalized velocities, to a ‘standard’ form, an expression that is guaranteed by the well-known Darboux theorem for 1-forms [4]. Geometrically, these transformations (i) isolate the symplectic leaves corresponding to the original, presymplectic structure associated with the ‘kinetic term’; (ii) exhibit a surface where the dynamics take place, and (iii) factorize the pure gauge modes, i.e., variables that do not enter into the action functional. As any local variational dynamics can, with the help of some appropriate auxiliary variables, be reformulated as a first-order one, it follows that, in principle, the Faddeev–Jackiw approach might be implemented for any theory, whenever one does not account for locality. It is worth mentioning that, motivated by the Faddeev–Jackiw Hamiltonian reduction, a deep analysis of the generic first-order systems has been performed [5]. Additionally, related approaches that emphasize [6,7] and solve [8] the restriction of evolution to the constraint surface have been implemented. In all these approaches, one isolates the physical degrees of freedom for generic first-order systems with finitely many degrees of freedom . Hence, the quantization is straightforward [9], but the gauge symmetries of the gauge unfixed action are not available, except for the shift symmetries in the pure gauge variables. Additionally, in field theories, essential properties such as locality or fundamental symmetries can be lost. As Faddeev and Jackiw remarked [1], a combined approach, involving Dirac’s classification of constraints and the elimination of second-class ones, would be recommended, especially in field theory.

This paper focuses on a combined Faddeev–Jackiw–Dirac analysis of various Majorana spinor-based first-order field theories. The approach furnishes the main ingredients for the BRST- or BRST-based quantization schemes [10] of the models under consideration, i.e., the symplectic structure and the first-class constraints. The motivation for the scrutinized field theories is threefold: (i) by excellence, they are linear, first-order theories; (ii) they are fundamental pieces in all SUSY and SUGRA models [11,12,13], describing various superpartners of spins $1/2$ and $3/2$; and (iii) their Faddeev–Jackiw approach asks for the Dirac formalism, both for eliminating some second-class constraints and for identifying the gauge symmetries. To the best of the authors’ knowledge, this is the first time that Rarita–Schwinger fields, both massive and massless, and the ‘conformal’ gravitino field are analyzed by means of a combined Faddeev–Jackiw–Dirac procedure. The advantage of blending the two approaches consists of (a) the economy in identifying the gauge and symplectic structure, by skipping steps in the Dirac method, especially by avoiding the primary second-class constraints, (b) the preservation of essential symmetries and properties, such as locality in field theory, aspects not taken into consideration in the standard Faddeev–Jackiw approach, and (c) the offering of all pieces necessary for completing a BRST- or BRST-based quantization scheme.

This paper is structured, besides the introductory and conclusive parts, into three main sections as follows. Section 2 discusses the Faddeev–Jackiw Hamiltonian reduction procedure and its related geometric aspects. Here, one considers a generic first-order system, with finite degrees of freedom, and, by means of a suitable reparameterization of the configuration manifold, one factorizes the pure gauge variables and displays a symplectic manifold, a time-evolution generator, and a set of constraints, casting the Lagrangian 1-form in a total action-like expression. At this point, as the procedure is to be applied to field theories, where locality is a fundamental requirement, the Dirac prescriptions are to be fulfilled, depending on the nature of constraints with respect to symplectic structure. The need for Dirac algorithm elements stems mainly from the impossibility, in field theory, of satisfying various constraints while maintaining locality or fundamental symmetries such as Poincaré or Lorentz invariance. Section 3 is devoted to the Faddeev–Jackiw–Dirac analysis of spin-$1/2$ and -$3/2$ Majorana fields with dynamics linear in first-order derivatives. This begins with a ‘warm-up’ on the chargeless spin $1/2$ fields, where only the $1+3$ decomposition of the Lagrangian density displays both the symplectic structure and the generator of evolution, i.e., the Hamiltonian density. Then, one passes to chargeless spin $3/2$ fields, described by Majorana spin-vectors. For completeness, the massive case, $\mathrm{m}\ne 0$, is considered, the massless case peculiarities being drawn in the limit of the vanishing mass parameter, $\mathrm{m}\to 0$. The reason for choosing this strategy lies in the phase-space and symplectic structure, which are common to both situations. For the massive instance, it is found that the dynamics are confined to a submanifold that is symplectic with respect to the induced 2-form. In the complementary case, the previous submanifold becomes co-isotropic with respect to the same induced 2-form. In this situation, the system displays local symmetries that are found to be the standard gauge invariances of the massless Rarita–Schwinger field [14]. This is the first instance where the standard Faddeev–Jackiw approach is adjusted with Dirac’s algorithm ingredients, as the geometric meaning of constraints should be considered with respect to the symplectic structure. This is not an embellishment but a necessity, for the constraints cannot be solved without spoiling the locality of the field model. In Section 4 one analyzes a spin-$3/2$ field with linear dynamics of order three in the spacetime derivatives. This is the main ingredient in various conformal supergavity theories [12], with its quanta known as Q-gravitino. The dynamics of the considered field model are initially reformulated, via the introduction of auxiliary variables, as a first-order system, which is then analyzed using the Faddeev–Jackiw method. It exhibits no pure gauge field but three sets of constraints, among which two are of first class and the third is of second class. This situation also asks for a Dirac treatment, which is implemented in two ways: (i) by maintaining the second-class constraints and constructing the generating set of gauge transformations and (ii) by restricting to the second-class submanifold and determining the gauge symmetries of the corresponding theory. Finally, a conclusive section ends this paper.

2. Faddeev–Jackiw Method: A Brief Review

Adopting the language of systems with finitely many degrees of freedom, the theories addressed by this method have dynamics governed, via the variational principle, by a Lagrangian density, which is a smooth function defined on the total space of a first-order jet bundle that, in addition, is linear with respect to ‘vertical’ coordinates

$$\lambda =a_A\left(z\right)\mathrm{d}{z}^A-V\left(z\right)\mathrm{d}t,\lambda \in {\Lambda }^1(P\times \mathbb{R}),$$

(1)

with P being the configuration (super)manifold. The ‘kinetic term’ in (1) displays the potential 1-form

$$\alpha =a_A\left(z\right)\mathrm{d}{z}^A\alpha \in {\Lambda }^1\left(P\right),$$

(2)

which plays a central role in the method under discussion. Dynamics of the theory (1) come from the variational principle and take the concrete form

$${\omega }^{♯}\dot{z}=\mathrm{d}V,$$

(3)

where

$$\omega =\mathrm{d}\alpha ,\omega \in {\Lambda }^2\left(P\right),$$

(4)

and

$${\omega }^{♯}:\mathfrak{X}\left(P\right)⟶{\Omega }^1\left(P\right),{\omega }^{♯}\left(X\right)\left(Y\right):=-\omega (X,Y).$$

(5)

For an arbitrary (super)manifold P, the coefficients of the presymplectic 2-form (4) read

$${\omega }_{AB}={\partial }_A{a}_B+{(-)}^{({ϵ}_A+1)({ϵ}_B+1)}{\partial }_B{a}_A,$$

(6)

with ${\partial }_A$ being the left derivatives with respect to the configuration manifold coordinates $z^A$ and ${ϵ}_A$ the corresponding Grassmann parities.

Throughout this paper, the notations from [4] are preferred; i.e., ${\Lambda }^{•}\left(P\right)$ denotes the exterior algebra associated with the smooth manifold P, while ${\mathfrak{X}}^{•}\left(P\right)$ represents the dual of ${\Lambda }^{•}\left(P\right)$, i.e., the Gerstenhaber algebra of multi-vector fields.

With these preparations at hand, if

$$\mathrm{rank}\omega =r\le \dim P$$

(7)

then, by using Darboux’s theorem [15], one can choose a set of local coordinates on P

$$z⟷(q^i,{\pi }_a,u^{\Delta }),|i|=r,,|a|=g,|\Delta |=\dim P-r-g$$

(8)

with respect to which the dynamical 1-form reads

$$\lambda =a_i\left(q\right)\mathrm{d}{q}^i-\left(h\left(q\right)+u^{\Delta }{\Phi }_{\Delta }\left(q\right)\right)\mathrm{d}t.$$

(9)

Previously, the standard notation $\left|I\right|$ for the range of a discrete index I was employed, and the rank condition

$$\mathrm{rank}(\omega \left(q\right)=\mathrm{d}\left(a_i\left(q\right)d{q}^i\right))=r$$

(10)

was understood.

It is worth noticing that transformation (8) can be implemented directly, or step by step [1], until reaching the rank condition (10). Envisaging the relation with Dirac’s approach of singular Lagrangian systems, it has been shown [16] that (9) and Dirac’s Hamiltonian 1-form are merely related, so they are equivalent by a canonical transformation. From the expression of the dynamical 1-form (9), one identifies a parameterization of the reduced phase-space, q, and the corresponding inherent symplectic structure, $\omega$. Invoking the same geometric object (9), one can extract the generator of the infinitesimal time evolution, i.e., the Hamiltonian

$$q\mapsto h\left(q\right),$$

(11)

and also some constraints

$${\Phi }_{\Delta }\approx 0.$$

(12)

At this point, according to the standard Faddeev–Jackiw approach, one restricts the dynamics on the surface (12) by solving the constraints, determines the induced ‘kinetic’ 1-form, and isolates the ‘true’ degrees of freedom. In the case of field theories, because implementing these steps destroys essential symmetries, such as Poincaré invariance, and, more critically, spoils locality, a fundamental aspect of the quantization of field theory, it appears natural that, at stages (11) and (12), the analysis be concluded via the Dirac approach.

From the perspective of symplectic structure (10), the set of constraints (12) could be of first or second class or mixed [2,3]. Also, regarding the function (11), it could be a first- or second-class one. The presence of constraints results in a smaller phase-space accessible for the time evolution of the dynamical system, a manifold that should be preserved by the Hamiltonian vector field associated with the time evolution generator. With this information at hand, it is natural to solve the dynamics a la Dirac [2,3], i.e., by extending the Hamiltonian off the constraint surface to obtain the associated first-class extension, followed by removing the second-class constraints and constructing the induced symplectic 2-form, the well-known Dirac bracket. At this point, with all the ingredients constructed, one can proceed as in [17,18,19], by adopting the Dirac conjecture, to identify the gauge transformations and classical observables of the initial theory (9) or (1).

In the remainder of this paper, we will analyze fundamental fermionic field theories using the economical Faddeev–Jackiw method, supplemented, whenever needed, with the Dirac algorithm.

3. Chargeless Spin-1/2 and -3/2 Fields with First-Order Dynamics

This part focuses on the Faddeev-Jackiw analysis of fermionic field theories of spins $1/2$ and $3/2$ in four spacetime dimensions [14], fundamental ingredients in all SUSY- and SUGRA-type paradigms [11,13]. All these theories exhibit Lagrangian densities of the type (1), which means that the Faddeev–Jackiw approach can be properly initiated.

The arena of the considered field theories consists of a four-dimensional Minkowski spacetime of mostly minus signature, ${\mathbb{R}}^{1|3}$, and the Majorana representation of the Clifford algebra $\mathcal{C}(1,3)$

$${\gamma }_{\mu }{\gamma }_{\nu }+{\gamma }_{\nu }{\gamma }_{\mu }=2{\sigma }_{\mu \nu },$$

i.e., all the representation generators are purely imaginary, with ${\gamma }^0$ skew-symmetric and ${\gamma }^j$ symmetric. Also, one denotes by

$${\gamma }_{{\mu }_1\cdots {\mu }_k}={\displaystyle \frac{1}{k!}}\sum _{\sigma \in S_4}{(-)}^{\sigma }{\gamma }_{{\mu }_{\sigma \left(1\right)}}\cdots \gamma {\mu }_{\sigma \left(k\right)}$$

the elements of the Clifford algebra representation standard basis, and one uses the Fierz relations

$${\gamma }_{\mu \nu }{\gamma }^{\lambda \rho }=-{\delta }_{[\mu }^{\lambda }{\delta }_{\nu ]}^{\rho }-{\delta }_{[\mu }^{[\lambda }{\gamma }_{\nu ]}^{\phantom{\nu }\rho ]}+{\gamma }_{\mu \nu }^{\phantom{\mu \nu }\lambda \rho },$$

(13)

with the spacial components simply reading

$${\gamma }_{ij}\left({\gamma }^{jk}-{\eta }^{jk}\right)=2{\delta }_i^k.$$

(14)

Also, as Majorana spinors play the role of the protagonist fields, one considers the charge conjugation matrix

$$\mathcal{C}:=-{\gamma }^0,$$

that defines the real (Majorana) spinors as being those that enjoy the relation

$$\overline{\psi }:={\psi }^{†}{\gamma }^0={\psi }^{\mathcal{C}}:={\left(\mathcal{C}\psi \right)}^{\top }.$$

(15)

With this framework at hand, the Faddeev–Jackiw procedure is applied to the common fermionic fields of spins $1/2$ and $3/2$.

3.1. Chargeless Spin-1/2 Field

The dynamics of these fields are generated, via the variational principle, from the Lagrangian density

$${\mathcal{L}}_{\mathrm{m}}^{(1/2)}\left(\left[\psi \right]\right)={\displaystyle \frac{\mathrm{i}}{2}}\overline{\psi }\widehat{\partial }\psi -{\displaystyle \frac{\mathrm{m}}{2}}\overline{\psi }\psi ={\displaystyle \frac{\mathrm{i}}{2}}{\psi }^{\top }\dot{\psi }+{\displaystyle \frac{\mathrm{i}}{2}}{\psi }^{\top }{\gamma }^0(\tilde{\partial }+\mathrm{i}\mathrm{m})\psi ,$$

(16)

where one employs the notations

$$\widehat{\partial }:={\gamma }^{\mu }{\partial }_{\mu },\tilde{\partial }:={\gamma }^k{\partial }_k.$$

By detailing the kinetic term in (16),

$${\displaystyle \frac{\mathrm{i}}{2}}{\psi }^{\top }\dot{\psi }={\displaystyle \frac{\mathrm{i}}{2}}{\psi }^A{\delta }_{AB}{\dot{\psi }}^B,$$

(17)

it results that the considered spin-$1/2$ field displays a phase-space labeled at each point in space by

$${\mathcal{Q}}_{1/2}:\psi =\left({\psi }_A\right).$$

Furthermore, using definition (6) in the context of the 1-form associated with (17), one immediately identifies the 2-form

$${\omega }_{AB}=\mathrm{i}{\delta }_{AB},$$

which is invertible; i.e., ${\mathcal{Q}}_{1/2}$ is symplectic, with the coefficients of the non-degenerated Poisson 2-vector

$$[{\psi }^A,{\psi }^B]=-\mathrm{i}{\delta }^{AB}.$$

Finally, according to the general prescriptions in the previous section, the second term on the right-hand side of the nonintegrated density (16) is, up to a global factor, the Hamiltonian density

$${\mathcal{H}}_{\mathrm{m}}^{(1/2)}=-{\displaystyle \frac{\mathrm{i}}{2}}{\psi }^{\top }{\gamma }^0(\tilde{\partial }+\mathrm{i}\mathrm{m})\psi .$$

Simple counting reveals that the chargeless spin-$1/2$ field has two physical degrees of freedom at each space point. This analysis demonstrates that, in this case, the Dirac algorithm is unnecessary, as the Faddeev–Jackiw procedure skips the primary second-class constraints generated by the definition of canonical momenta in the Dirac algorithm.

3.2. Chargeless Spin-3/2 Field

The chargeless spin-$3/2$ field considered here, which also exhibits first-order dynamics, is a fundamental ingredient in all SUSY-type models [11,13]. It is mathematically expressed by a Majorana spin-vector ${\psi }_{\mu }$, with the evolution generated, via the variational principle, by the well-known Rarita–Schwinger Lagrangian action [14] with the non-integrated density

$${\mathcal{L}}_{\mathrm{m}}^{(3/2)}\left(\left[\psi \right]\right)={\displaystyle \frac{\mathrm{i}}{2}}{\overline{\psi }}_{\mu }{\gamma }^{\mu \nu \rho }{\partial }_{\nu }{\psi }_{\rho }-{\displaystyle \frac{\mathrm{m}}{2}}{\overline{\psi }}_{\mu }{\gamma }^{\mu \nu }{\psi }_{\nu }.$$

(18)

As the local function (18) pertains to the space of real smooth functions defined on a first-order jet bundle, it allows the initiation of the Faddeev–Jackiw approach, which is naturally achieved by performing the $1+3$ decomposition of the density (18)

$${\mathcal{L}}_{\mathrm{m}}^{(3/2)}=-{\displaystyle \frac{\mathrm{i}}{2}}{\psi }_i^{\top }{\gamma }^{ij}{\dot{\psi }}_j+{\displaystyle \frac{\mathrm{i}}{2}}{\overline{\psi }}_i{\gamma }^{ijk}{\partial }_j{\psi }_k-{\displaystyle \frac{\mathrm{m}}{2}}{\overline{\psi }}_k{\gamma }^{kl}{\psi }_l+\mathrm{i}{\psi }_0^{\top }({\gamma }^{ij}{\partial }_i{\psi }_j+\mathrm{i}\mathrm{m}{\gamma }^k{\psi }_k).$$

(19)

The previous expression exhibits a phase-space parameterized by

$${\mathcal{Q}}_{3/2}:{\psi }_k=\left({\psi }_{Ak}\right),$$

(20)

and a first-order dynamics confined on the surface

$${\Phi }_A^{\left(\mathrm{m}\right)}:={({\gamma }^{ij}{\partial }_i{\psi }_j+\mathrm{i}\mathrm{m}{\gamma }^k{\psi }_k)}_A\approx 0.$$

(21)

Also, from the same expression, one can read the time-evolution generator

$$H_{\mathrm{m}}^{(3/2)}:=\int {\mathrm{d}}^3\mathbf{x}{\mathcal{H}}_{\mathrm{m}}^{(3/2)}$$

with the Hamiltonian density

$${\mathcal{H}}_{\mathrm{m}}^{(3/2)}=-{\displaystyle \frac{\mathrm{i}}{2}}{\overline{\psi }}_i{\gamma }^{ijk}{\partial }_j{\psi }_k+{\displaystyle \frac{\mathrm{m}}{2}}{\overline{\psi }}_k{\gamma }^{kl}{\psi }_l$$

(22)

In addition, the temporal components of the spin-vector, ${\psi }_0=\left({\psi }_{A0}\right)$, here play the role Lagrange multipliers in (9).

Invoking again the ‘kinetic term’ in (19), one identifies the coefficients of the potential 1-form

$$a_{Bj}=-{\displaystyle \frac{\mathrm{i}}{2}}{\left({\gamma }_{ij}\right)}_{AB}{\psi }^{Ai},$$

that, by direct computations performed by means of (6), produce those of the canonical 2-form

$${\omega }_{Ai;Bj}=-\mathrm{i}{\left({\gamma }_{ij}\right)}_{AB}.$$

(23)

Using the Fierz relations (14), one concludes that the phase-space (20) is symplectic. The coefficients of the non-degenerated Poisson 2-vector, which give the fundamental Poisson brackets, read

$$\left[{\psi }^{Ai},{\psi }^{Bj}\right]:={\omega }^{Ai;Bj}={\displaystyle \frac{\mathrm{i}}{2}}{({\gamma }^{ij}-{\eta }^{ij})}^{AB}.$$

(24)

At this stage, one has identified the geometry of the phase-space, together with the region where the dynamics are confined, namely the surface (21). From the concrete expressions (21), it becomes transparent that the constraints can not be solved unless one gives up the locality. This is the point at which the Dirac formalism appears as the natural solution to complete the analysis of the field model under consideration. By direct computation, one obtains the Poisson brackets between the constraints

$$\left[{\Phi }_A^{\left(\mathrm{m}\right)},{\Phi }_B^{\left(\mathrm{m}\right)}\right]=-{\displaystyle \frac{3\mathrm{i}{\mathrm{m}}^2}{2}}{\delta }_{AB}$$

(25)

as well as those between the constraints and Hamiltonian

$$\left[{\Phi }_A^{\left(\mathrm{m}\right)},H_{\mathrm{m}}^{(3/2)}\right]={\displaystyle \frac{\mathrm{i}\mathrm{m}}{2}}{\left({\gamma }^0{\Phi }^{\left(\mathrm{m}\right)}\right)}_A+{\displaystyle \frac{3\mathrm{i}{\mathrm{m}}^2}{2}}{\left({\gamma }^0{\gamma }^k{\psi }_k\right)}_A,$$

(26)

that display a strong dependence of the Hamiltonian gauge algebra on the mass parameter $\mathrm{m}$.

From (25), it results that in the presence of mass,

$$\mathrm{m}\ne 0$$

constraints (21) are second-class and irreducible. As discussed previously, they cannot be removed from the theory by means of the Dirac bracket unless one gives up on locality. In this case, simple counting reveals four degrees of freedom for the chargeless spin-$3/2$ field, while the dynamics are confined on the second-class (with respect to Poisson structure (24)) surface (21).

In the complementary situation, invoking the same arguments, it results that, in the absence of a mass parameter

$$\mathrm{m}=0,$$

constraints (21) are Abelian and irreducible, while the Hamiltonian, in light of (26), is a first-class function; i.e., the Hamiltonian gauge algebra reads

$$\left[{\Phi }_A^{(\mathrm{m}=0)},{\Phi }_B^{(\mathrm{m}=0)}\right]=0,\left[{\Phi }_A^{(\mathrm{m}=0)},H_{\mathrm{m}=0}^{(3/2)}\right]=0.$$

(27)

As the first-class constraints, geometrically co-isotropic submanifolds of the manifold equipped with the symplectic 2-form (23) generate gauge transformations [2,3,17,18,19],

$${\delta }_{ϵ}{\psi }_k:=\left[{\psi }_k,2\mathrm{i}{\Phi }_A^{(\mathrm{m}=0)}{ϵ}^A\right],$$

by direct computation one identifies

$${\delta }_{ϵ}{\psi }_{\mu }={\partial }_{\mu }ϵ$$

(28)

as an irreducible and Abelian generating set of gauge transformations [14] for (18) in the massless case, $\mathrm{m}=0$. Also, the standard counting of the physical degrees of freedom exhibits only two degrees of freedom for the chargeless spin-$3/2$ field. Physically, these correspond to $\pm 3/2$ polarization states of the gravitino [11], which allow non-trivial couplings with the Einstein–Hilbert graviton, described in the free limit by the Pauli–Fierz action, leading to the well-known simple supergravity [13].

It is worth noticing that the Abelian expression of the Hamiltonian gauge algebra (27), together with the irreducible character of the first-class constraints ${\Phi }_A^{(\mathrm{m}=0)}\approx 0$, are incorporated into the Hamiltonian BRST formalism as a BRST charge and BRST-invariant Hamiltonian that reduce only to antighost number zero components.

4. Chargeless Spin-3/2 Field with Higher-Derivative Dynamics

This part is devoted to the fermionic field that enters conformal SUGRA theories [12], well-known as the Q-gravitino field, a higher-derivative spin-vector with the evolution generated, via the variational principle, from the Lagrangian density

$${\mathcal{L}}^{\left(\mathrm{Q}\right)}={\displaystyle \frac{\mathrm{i}}{2}}{\overline{\Phi }}_{\mu }{\gamma }^{\mu \nu \rho }{\partial }_{\nu }{\Phi }_{\rho },$$

(29)

with

$${\Phi }_{\mu }:=\mathrm{i}{\gamma }^{\nu }{\partial }_{[\nu }{\psi }_{\mu ]}+{\displaystyle \frac{\mathrm{i}}{2}}{\gamma }_{\mu \nu \rho }{\partial }^{\nu }{\psi }^{\rho }.$$

(30)

As in the previous part, here, ${\psi }_{\mu }$ is also a Majorana spin-vector, which, in light of definition (30), implies that ${\Phi }_{\mu }$ is also a real spin-vector.

The concrete expression of Lagrangian density (29) shows that it pertains to the space of smooth functions defined on the second-order jet space associated with the manifold parameterized by the coordinate ${\psi }_{\mu }$. Moreover, the variational principle displays for Lagrangian density (29) equations of motion with three spacetime derivatives. To apply the Faddeev–Jackiw procedure, one first builds the first-order version of (29). This is defined on the total space of a first-order jet space associated with a larger manifold, parameterized by three real spin-vectors

$$P^{\left(\mathrm{Q}\right)}:\left({\psi }_{\mu },{\varphi }_{\mu },{\pi }^{\mu }\right),$$

and is described by the Lagrangian density

$${\mathcal{L}}_1^{\left(\mathrm{Q}\right)}:={\displaystyle \frac{\mathrm{i}}{2}}{\overline{\varphi }}_{\mu }{\gamma }^{\mu \nu \rho }{\partial }_{\nu }{\varphi }_{\rho }+{\overline{\pi }}^{\mu }(-{\varphi }_{\mu }+{\Phi }_{\mu }).$$

(31)

Invoking the concept of an auxiliary variable [17], it becomes transparent that the field models (31) and (29) are equivalent.

The first-order formulation of a massless Q-gravitino allows its analysis according to the Faddeev–Jackiw method, completed, as previously, with Dirac’s approach of constrained Hamiltonian systems. This begins with the $1+3$ decomposition of density (31), but, for a convenient expression of the ‘kinetic term,’ it is useful to initially perform the reparameterization of the configuration manifold

$$({\pi }^0,{\pi }^k)⟷(p,\Pi )$$

(32)

with

$$p:=-\mathrm{i}{\gamma }^k{\pi }_k,{\Pi }_k:=-{\gamma }_0{\gamma }_k{\pi }^0-{\displaystyle \frac{1}{2}}({\gamma }_{kl}-2{\eta }_{kl}){\pi }^l.$$

(33)

By direct computation, it can be checked that transformation (33) is invertible,

$${\pi }^0={\displaystyle \frac{1}{3}}{\gamma }^0{\gamma }^k{\Pi }_k,{\pi }_k={\displaystyle \frac{1}{3}}\left[\mathrm{i}{\gamma }_{k}p-{\displaystyle \frac{2}{3}}({\gamma }_{kl}-2{\eta }_{kl}){\Pi }^l\right]$$

(34)

which proves that (32) is indeed one-to-one.

With these preparations at hand, the $1+3$ splitting of Lagrangian density (31) becomes

$$\begin{array}{ccc}\hfill {\mathcal{L}}_1^{\left(\mathrm{Q}\right)}& =& -{\displaystyle \frac{\mathrm{i}}{2}}{\varphi }_k^{\top }{\gamma }^{kl}{\dot{\varphi }}_l+\mathrm{i}{\Pi }_k^{\top }{\dot{\psi }}^k\hfill \\ & +& {\displaystyle \frac{\mathrm{i}}{2}}{\overline{\varphi }}_j{\gamma }^{jkl}{\partial }_k{\varphi }_l+{\displaystyle \frac{2}{9}}{\overline{\varphi }}^k({\gamma }_{kl}-2{\eta }_{kl}){\Pi }^l+{\displaystyle \frac{\mathrm{i}}{12}}\partial {}^{[k}\overline{\psi }^{l]}(5{\gamma }_{klm}+{\eta }_{m\left[k\right.}{\gamma }_{\left.l\right]}){\Pi }^m\hfill \\ & +& \mathrm{i}{\varphi }_0^{\top }({\gamma }^{kl}{\partial }_k{\varphi }_l+{\displaystyle \frac{\mathrm{i}}{3}}{\gamma }^k{\Pi }_k)-\mathrm{i}{\psi }_0^{\top }\left({\partial }^k{\Pi }_k\right)+{\displaystyle \frac{1}{3}}\overline{p}(\mathrm{i}{\gamma }^k{\varphi }_k-{\displaystyle \frac{3}{2}}{\gamma }^{kl}{\partial }_k{\psi }_l),\hfill \end{array}$$

(35)

which, in light of the general procedure briefly synthesized in Section 2 implies the following: (i) The first-order formulation of the model under discussion exhibits a phase-space parameterized by the fields

$${\mathcal{Q}}^{\left(\mathrm{Q}\right)}:({\varphi }_{Ak},{\Pi }_{Ak},{\psi }^{Ak}),$$

(36)

while the remainder Majorana spinor-fields

$$({\varphi }_0,p,{\psi }^0)$$

(37)

play the role of Lagrange multipliers. (ii) The phase-space ${\mathcal{Q}}^{\left(\mathrm{Q}\right)}$ is equipped with the 2-form

$$\left[\omega \right]=\left[\begin{array}{ccc}-\mathrm{i}{\left({\gamma }_{ij}\right)}_{AB}& 0& 0\\ 0& 0& \mathrm{i}{\delta }_i^k{\delta }_A^C\\ 0& \mathrm{i}{\delta }_k^i{\delta }_C^A& 0\end{array}\right]$$

(38)

which is non-degenerate; i.e., it organizes the phase-space (36) as a symplectic manifold. (iii) The time-evolution is generated by

$$H^{\left(\mathrm{Q}\right)}:=\int {\mathrm{d}}^3\mathbf{x}{\mathcal{H}}^{\left(\mathrm{Q}\right)}$$

with the Hamiltonian density

$${\mathcal{H}}^{\left(\mathrm{Q}\right)}=-{\displaystyle \frac{\mathrm{i}}{2}}{\overline{\varphi }}_j{\gamma }^{jkl}{\partial }_k{\varphi }_l-{\displaystyle \frac{2}{9}}{\overline{\varphi }}^k({\gamma }_{kl}-2{\eta }_{kl}){\Pi }^l-{\displaystyle \frac{\mathrm{i}}{12}}\partial {}^{[k}\overline{\psi }^{l]}(5{\gamma }_{klm}+{\eta }_{m\left[k\right.}{\gamma }_{\left.l\right]}){\Pi }^m.$$

(39)

(iv) The time-evolution is confined on the constraints surface

$$\begin{array}{ccc}\hfill G^{\left(1\right)}& :=& {\gamma }^{kl}{\partial }_k{\varphi }_l+{\displaystyle \frac{\mathrm{i}}{3}}{\gamma }^k{\Pi }_k\approx 0,\hfill \end{array}$$

(40)

$$\begin{array}{ccc}\hfill G^{\left(2\right)}& :=& {\partial }^k{\Pi }_k\approx 0,\hfill \end{array}$$

(41)

$$\begin{array}{ccc}\hfill \chi & :=& {\displaystyle \frac{\mathrm{i}}{3}}{\gamma }^k{\varphi }_k-{\displaystyle \frac{1}{2}}{\gamma }^{kl}{\partial }_k{\psi }_l\approx 0,\hfill \end{array}$$

(42)

where the spinor indices have been omitted.

The previous results impose that the analysis has to be completed using the Dirac formalism. To fulfill this objective, one computes the non-degenerate Poisson structure associated with the symplectic one (38). Invoking again the Fierz relations (13) and using the expression of the symplectic 2-form coefficients (38), it results that the only fundamental non-trivial Poisson brackets read

$$\left[{\varphi }_{Ak},{\varphi }_{Bl}\right]={\displaystyle \frac{\mathrm{i}}{2}}{({\gamma }_{kl}-{\eta }_{kl})}_{AB},\left[{\Pi }_{Ak},{\psi }^{Bl}\right]=-\mathrm{i}{\delta }_A^B{\delta }_k^l.$$

(43)

Investigating the nature of the constraint functions (40)–(42) with respect to Poisson structure (43), by direct computation, it results that among the constraints (40)–(42), (40) and (41) are Abelian,

$$\left[G^{\left(a\right)},G^{\left(b\right)}\right]=0=\left[G^{\left(a\right)},\chi \right],a,b=1,2$$

(44)

while the remaining ones enjoy

$$\left[\chi ,\chi \right]=-{\displaystyle \frac{\mathrm{i}}{6}}.$$

(45)

Including the generator of time evolution in the computations, one further completes the Hamiltonian gauge algebra with

$$\left[G^{\left(1\right)},H^{\left(\mathrm{Q}\right)}\right]={\displaystyle \frac{\mathrm{i}}{3}}{\gamma }^0{G}^{\left(2\right)},\left[G^{\left(2\right)},H^{\left(\mathrm{Q}\right)}\right]=0$$

(46)

and

$$\left[\chi ,H^{\left(\mathrm{Q}\right)}\right]=-{\displaystyle \frac{2}{3}}{\gamma }^0\tilde{\partial }\chi +{\displaystyle \frac{\mathrm{i}}{6}}{\gamma }^0{G}^{\left(1\right)}+{\displaystyle \frac{1}{18}}{\gamma }^k{\Pi }_k.$$

(47)

The analysis of the Hamiltonian gauge algebra (40)–(47) enforces the following conclusions: (a) the constraints (40) and (41) are of first class, which means they generate gauge symmetries; (b) the constraints (42) are of second class, which results in a smaller accessible phase-space; (c) the number of physical degrees of freedom for the model under investigation is found to be equal to eight; and (d) the canonical Hamiltonian (39) is not a first-class function.

In light of these outputs, it results that the Q-gravitino field analysis can be performed by using a mixing of the Faddeev–Jackiw method, which avoids supplementary constraints generated by the Legendre transformation, and the Dirac algorithm, which eliminates second-class constraints. This combined approach [1] is specific to field theory, where important symmetries and essential properties, such as spacetime locality, have to be maintained. With these arguments in mind, one constructs the off-shell extension, with respect to the constraint set (40)–(42), of the function (39) to identify the first-class Hamiltonian density, the classical observable generating the time evolution

$${{\mathcal{H}}^{\prime }}^{\left(\mathrm{Q}\right)}:={\mathcal{H}}^{\left(\mathrm{Q}\right)}-{\displaystyle \frac{\mathrm{i}}{3}}{\chi }^{\top }{\gamma }^0{\gamma }^k{\Pi }_k.$$

(48)

Taking into account the previously built classical observable, the completion of the Hamiltonian gauge algebra (44) and (45) reads

$$\left[G^{\left(1\right)},{H^{\prime }}^{\left(\mathrm{Q}\right)}\right]={\displaystyle \frac{\mathrm{i}}{3}}{\gamma }^0{G}^{\left(2\right)},\left[G^{\left(2\right)},{H^{\prime }}^{\left(\mathrm{Q}\right)}\right]=0$$

(49)

and

$$\left[\chi ,{H^{\prime }}^{\left(\mathrm{Q}\right)}\right]=-{\gamma }^0\tilde{\partial }\chi +{\displaystyle \frac{\mathrm{i}}{6}}{\gamma }^0{G}^{\left(1\right)}.$$

(50)

It is worth noticing that the incorporation of the first-class Hamiltonian (48) into the first-order Lagrangian density (35) is, as usual, a matter of redefinition of the Lagrange multipliers associated with the constraints $\chi \approx 0$.

At this stage, the analysis of the model under investigation is completed with the identification of gauge symmetries of the original theory (29), in two equivalent manners: (i) by maintaining the second-class constraints and constructing the generating set of gauge transformations for

$${{\mathcal{L}}_1^{\prime }}^{\left(\mathrm{Q}\right)}=-{\displaystyle \frac{\mathrm{i}}{2}}{\varphi }_k^{\top }{\gamma }^{kl}{\dot{\varphi }}_l+\mathrm{i}{\Pi }_k^{\top }{\dot{\psi }}^k-{{\mathcal{H}}^{\prime }}^{\left(\mathrm{Q}\right)}+\mathrm{i}{\varphi }_0^{\top }{G}^{\left(1\right)}-\mathrm{i}{\psi }_0^{\top }{G}^{\left(2\right)}+{\displaystyle \frac{1}{3}}\overline{p}\chi ,$$

(51)

and (ii) by solving the second-class constraints in (51) and using the corresponding induced symplectic structure, i.e., the Dirac bracket. It is worth mentioning that Lagrangian models (29), (35), and (51) are equivalent. Precisely, the equivalence of the first two dynamics rests on the elimination of the auxiliary variables used for linearization of the starting model (29). In comparison, that of the last two dynamics is obtained just by a mere redefinition of the Lagrange multiplier p in (35).

In the last part of this section, we apply both strategies for deriving the desired generating set of gauge transformations.

When one takes into account all the constraints (40)–(42), according to general prescription [2]

$${\delta }_{ϵ}F:=\left[F,G^{\left(1\right)\top }{ϵ}^{\left(1\right)}+G^{\left(2\right)\top }{ϵ}^{\left(2\right)}\right]$$

for the gauge transformation of a local function defined on the phase-space (here a smooth function depending on the variables (36)), by direct computation, it results that

$${\delta }_{ϵ}{\varphi }_k=-\mathrm{i}{\partial }_k{ϵ}^{\left(1\right)},{\delta }_{ϵ}{\Pi }_k=0,{\delta }_{ϵ}{\psi }_k={\displaystyle \frac{1}{3}}{\gamma }_k{ϵ}^{\left(1\right)}+\mathrm{i}{\partial }_k{ϵ}^{\left(2\right)}.$$

(52)

Now, by considering the previous infinitesimal transformations and taking the gauge invariance of the action associated with (31), i.e.,

$${\delta }_{ϵ}{S}_1^{\left(\mathrm{Q}\right)}=0,S_1^{\left(\mathrm{Q}\right)}:=\int {\mathrm{d}}^{4}x{\mathcal{L}}_1^{\left(\mathrm{Q}\right)},$$

one determines the gauge transformations for the Lagrange multipliers

$${\delta }_{ϵ}{\varphi }_0=-\mathrm{i}{\dot{ϵ}}^{\left(1\right)},{\delta }_{ϵ}p=0,{\delta }_{ϵ}{\psi }_0={\displaystyle \frac{1}{3}}{\gamma }_0{ϵ}^{\left(1\right)}+\mathrm{i}{\dot{ϵ}}^{\left(2\right)}.$$

(53)

Eliminating the auxiliary variables from (52) and (53), one identifies a generating set of gauge transformations for the starting model (29)

$${\delta }_{ϵ}{\psi }_{\mu }={\displaystyle \frac{1}{3}}{\gamma }_{\mu }{ϵ}^{\left(1\right)}+\mathrm{i}{\partial }_{\mu }{ϵ}^{\left(2\right)}$$

(54)

which, modulo some homogeneity redefinitions of the gauge parameters, coincides with that given in [12] and used in simple conformal SUGRA in four spacetime dimensions. The same arguments further lead to the infinitesimal gauge transformations

$${\delta }_{ϵ}{\Phi }_{\mu }=-\mathrm{i}{\partial }_{\mu }{ϵ}^{\left(1\right)},$$

with ${\Phi }_{\mu }$ introduced in (30).

To implement the second strategy, i.e., to solve the second-class constraints, initially reparameterize the manifold (36) such that the second-class surface (42) is solved directly. In view of this, instead of variables ${\varphi }_k$, one chooses

$$\left({\varphi }_k\right)\to ({\phi }_k:=({\gamma }_{kl}-2{\eta }_{kl}){\varphi }^l,\phi :={\gamma }^k{\varphi }_k)$$

(55)

with the inverse

$$({\phi }_k,\phi )\to \left({\varphi }_k={\displaystyle \frac{1}{3}}({\gamma }_k\phi -{\phi }_k)\right).$$

(56)

The pairing between old and new field variables is performed well, as the $\phi$-coordinates are overcomplete in the sense that they are $\gamma$-traceless,

$${\gamma }^k{\phi }_k=0.$$

To find the symplectic structure in the new parameterization of the phase-space

$${\mathcal{Q}}^{\left(\mathrm{Q}\right)}:\left(\phi ,{\phi }_k,{\psi }_k,{\Pi }^k\right)$$

(57)

one uses the expression

$$\begin{array}{ccc}\hfill -{\displaystyle \frac{\mathrm{i}}{2}}{\varphi }_k^{\top }{\gamma }^{kl}{\dot{\varphi }}_l& =& -{\displaystyle \frac{\mathrm{i}}{3}}{\varphi }_k^{\top }{\gamma }^k{\gamma }^l{\dot{\varphi }}_l-{\displaystyle \frac{\mathrm{i}}{6}}{\varphi }_k^{\top }\left({\gamma }^{kl}-2{\eta }^{kl}\right){\dot{\varphi }}_l\hfill \\ & =& -{\displaystyle \frac{\mathrm{i}}{3}}{\varphi }_k^{\top }{\gamma }^k{\gamma }^l{\dot{\varphi }}_l+{\displaystyle \frac{\mathrm{i}}{18}}{\varphi }_k^{\top }\left({\gamma }^{kl}-2{\eta }^{kl}\right)\left({\gamma }_{lm}-2{\eta }^{lm}\right){\dot{\varphi }}^m\hfill \end{array}$$

which projects the ‘kinetic term’ in the sector $\left({\varphi }_k\right)$ onto its $\gamma$-traceless part and onto one of its algebraic complements. This further gives the potential 1-form in the parameterization (57)

$${\alpha }^{\left(\mathrm{Q}\right)}=-{\displaystyle \frac{\mathrm{i}}{3}}{\phi }^{\top }\mathrm{d}\phi +{\displaystyle \frac{\mathrm{i}}{18}}{\phi }_k^{\top }\mathrm{d}{\phi }^k+\mathrm{i}{\Pi }_k\mathrm{d}{\psi }^k,$$

(58)

whose differentiation displays the symplectic structure of the phase-space ${\mathcal{Q}}^{\left(\mathrm{Q}\right)}$

$${\omega }^{\left(\mathrm{Q}\right)}=-{\displaystyle \frac{\mathrm{i}}{3}}\mathrm{d}{\phi }^{\top }\mathrm{d}\phi +{\displaystyle \frac{\mathrm{i}}{18}}\mathrm{d}{\phi }_k^{\top }\mathrm{d}{\phi }^k+\mathrm{i}\mathrm{d}{\Pi }_k\mathrm{d}{\psi }^k.$$

(59)

Previously, due to the nature of Grassmann-odd coordinates, the wedge product, which means a symmetric one, has been understood. Also, by computing the inverse of the symplectic 2-form (59), one obtains the Poisson 2-vector that displays the fundamental non-trivial Poisson brackets

$$\left[\phi ,\phi \right]={\displaystyle \frac{3\mathrm{i}}{2}},\left[{\phi }_k,{\phi }_l\right]=3\mathrm{i}\left({\gamma }^{kl}-2{\eta }^{kl}\right),\left[{\Pi }_k,{\psi }^l\right]=-\mathrm{i}{\delta }_k^l,$$

where the spinor indices have been omitted.

In this natural parameterization, the constraints become

$$\begin{array}{ccc}\hfill G^{\left(1\right)}& :=& {\displaystyle \frac{2}{3}}\tilde{\partial }\phi +{\displaystyle \frac{1}{3}}{\partial }^k{\phi }_k+{\displaystyle \frac{\mathrm{i}}{3}}{\gamma }^k{\Pi }_k\approx 0,\hfill \end{array}$$

(60)

$$\begin{array}{ccc}\hfill G^{\left(2\right)}& :=& {\partial }^k{\Pi }_k\approx 0,\hfill \end{array}$$

(61)

$$\begin{array}{ccc}\hfill \chi & :=& {\displaystyle \frac{\mathrm{i}}{3}}\phi -{\displaystyle \frac{1}{2}}{\gamma }^{kl}{\partial }_k{\psi }_l\approx 0,\hfill \end{array}$$

(62)

while the first-class Hamiltonian density reads

$$\begin{array}{ccc}\hfill {{\mathcal{H}}^{\prime }}^{\left(\mathrm{Q}\right)}& =& {\displaystyle \frac{\mathrm{i}}{9}}\overline{\phi }\tilde{\partial }\phi +{\displaystyle \frac{\mathrm{i}}{9}}\overline{\phi }{\partial }^k{\phi }_k-{\displaystyle \frac{\mathrm{i}}{18}}{\overline{\phi }}_k{\gamma }^{klm}{\partial }_k{\phi }_l\hfill \\ & -& {\displaystyle \frac{2}{9}}{\overline{\Pi }}_k{\phi }^k-{\displaystyle \frac{\mathrm{i}}{12}}{\overline{\Pi }}_k\left(5{\gamma }^{klm}-{\eta }^{k[l}{\gamma }^{m]}\right){\partial }_{[l}{\psi }_{m]}+{\displaystyle \frac{\mathrm{i}}{3}}{\overline{\Pi }}_k{\gamma }^k\chi .\hfill \end{array}$$

(63)

Restricting the phase-space ${\mathcal{Q}}^{\left(\mathrm{Q}\right)}$ to the second-class surface

$${\tilde{\mathcal{Q}}}^{\left(\mathrm{Q}\right)}\ni \left({\phi }_k,{\overline{\Pi }}_k,{\psi }^k\right)\mapsto \left(\phi =-{\displaystyle \frac{3\mathrm{i}}{2}}{\gamma }^{kl}{\partial }_k{\psi }_l,{\phi }_k,{\overline{\Pi }}_k,{\psi }^k\right)\in {\mathcal{Q}}^{\left(\mathrm{Q}\right)}$$

and inducing on the reduced phase-space the symplectic 2-form (59) results in the Poisson 2-vector components, most easily computed as Dirac brackets among the coordinates on ${\tilde{\mathcal{Q}}}^{\left(\mathrm{Q}\right)}$,

$$\left[{\tilde{\omega }}^{-1}\right]=\left[\begin{array}{ccc}3\mathrm{i}{\left({\gamma }^{kl}-2{\eta }^{kl}\right)}_{AB}& 0& 0\\ 0& -{\displaystyle \frac{3\mathrm{i}}{2}}{\left({\gamma }_{ki}{\gamma }_{jl}\right)}_{AB}{\partial }^{ij}& -\mathrm{i}{\delta }_i^k{\delta }_A^B\\ 0& -\mathrm{i}{\delta }_k^i{\delta }_B^A& 0\end{array}\right]$$

(64)

The dynamics on the reduced phase-space take place on the constraints

$$\begin{array}{ccc}\hfill {\tilde{G}}^{\left(1\right)}& =& -\mathrm{i}\tilde{\partial }{\gamma }^{kl}{\partial }_k{\psi }_l+{\displaystyle \frac{1}{3}}{\partial }^k{\phi }_k+{\displaystyle \frac{\mathrm{i}}{3}}{\gamma }^k{\Pi }_k\approx 0,\hfill \end{array}$$

(65)

$$\begin{array}{ccc}\hfill {\tilde{G}}^{\left(2\right)}& =& {\partial }^k{\Pi }_k\approx 0,\hfill \end{array}$$

(66)

which are Abelian and irreducible, and are generated by the first-class Hamiltonian density

$$\begin{array}{ccc}\hfill {{\tilde{\mathcal{H}}}^{\prime }}^{\left(\mathrm{Q}\right)}& =& {\displaystyle \frac{\mathrm{i}}{4}}{\partial }_k{\overline{\psi }}_l{\gamma }^{kl}\tilde{\partial }{\gamma }^{mn}{\partial }_m{\psi }_n-{\displaystyle \frac{1}{6}}{\partial }_k{\overline{\psi }}_l{\gamma }^{kl}{\partial }^m{\phi }_m-{\displaystyle \frac{\mathrm{i}}{18}}{\overline{\phi }}_k{\gamma }^{klm}{\partial }_k{\phi }_l\hfill \\ & -& {\displaystyle \frac{2}{9}}{\overline{\Pi }}_k{\phi }^k-{\displaystyle \frac{\mathrm{i}}{12}}{\overline{\Pi }}_k\left(5{\gamma }^{klm}-{\eta }^{k[l}{\gamma }^{m]}\right){\partial }_{[l}{\psi }_{m]}.\hfill \end{array}$$

(67)

The remaining part of the Hamiltonian gauge algebra in the presence of the reduced phase-space reads

$${\left[{\tilde{G}}^{\left(1\right)},{{\tilde{H}}^{\prime }}^{\left(\mathrm{Q}\right)}\right]}^{*}={\displaystyle \frac{\mathrm{i}}{3}}{\gamma }^0{\tilde{G}}^{\left(2\right)},{\left[{\tilde{G}}^{\left(2\right)},{{\tilde{H}}^{\prime }}^{\left(\mathrm{Q}\right)}\right]}^{*}=0,$$

(68)

where the obvious notation was employed,

$${{\tilde{H}}^{\prime }}^{\left(\mathrm{Q}\right)}:=\int {\mathrm{d}}^3\mathbf{x}{{\tilde{\mathcal{H}}}^{\prime }}^{\left(\mathrm{Q}\right)}.$$

The second strategy ends with the identification of a generating set of gauge transformations for the first-order, in the evolution parameter, density

$$\begin{array}{ccc}\hfill {{\tilde{\mathcal{L}}}_1^{\prime }}^{\left(\mathrm{Q}\right)}& =& -{\displaystyle \frac{3\mathrm{i}}{4}}{\partial }_k{\psi }_l^{\top }{\gamma }^{kl}{\gamma }^{mn}{\partial }_m{\dot{\psi }}_n-{\displaystyle \frac{\mathrm{i}}{54}}{\phi }_k^{\top }\left({\gamma }^{kl}-2{\eta }^{kl}\right){\dot{\phi }}_l+\mathrm{i}{\Pi }_k^{\top }{\dot{\psi }}^k\hfill \\ & -& {{\tilde{\mathcal{H}}}^{\prime }}^{\left(\mathrm{Q}\right)}+\mathrm{i}{\varphi }_0^{\top }{\tilde{G}}^{\left(1\right)}-\mathrm{i}{\psi }_0^{\top }{\tilde{G}}^{\left(2\right)}.\hfill \end{array}$$

(69)

According to Dirac’s conjecture, the gauge invariances of (69) are obtained from

$${\delta }_{ϵ}\tilde{F}:={\left[\tilde{F},{\tilde{G}}^{\left(1\right)\top }{ϵ}^{\left(1\right)}+{\tilde{G}}^{\left(2\right)\top }{ϵ}^{\left(2\right)}\right]}^{*},$$

(70)

for any smooth function on the reduced phase-space $\tilde{F}$ and completed with Lagrange multipliers transformations that leave (69) invariant. From the prescription (70) one obtains

$${\delta }_{ϵ}{\phi }_k=-\mathrm{i}({\gamma }_{jk}-2{\eta }_{jk}){\partial }^k{ϵ}^{\left(1\right)},{\delta }_{ϵ}{\psi }_k={\displaystyle \frac{1}{3}}{\gamma }_k{ϵ}^{\left(1\right)}+\mathrm{i}{\partial }_k{ϵ}^{\left(2\right)},{\delta }_{ϵ}{\Pi }_k=0,$$

which is completed with

$${\delta }_{ϵ}{\varphi }_0={\dot{ϵ}}^{\left(1\right)},{\delta }_{ϵ}{\psi }_0={\displaystyle \frac{1}{3}}{\gamma }_0{ϵ}^{\left(1\right)}+\mathrm{i}{\dot{ϵ}}^{\left(2\right)}$$

obtained from the invariance of (69) requirement.

As expected, both strategies yield the same generating set of gauge transformations, with the same number of independent degrees of freedom. The Abelian generating set of gauge transformations of the Q-gravitino field allows its non-trivial couplings with the 1-form and the Weyl-graviton, leading to simple conformal supergravity in four spacetime dimensions [12].

Finally, even though the second strategy requires extra computations, it furnishes the main ingredients for the BRST quantization of the analyzed higher-derivative field theory. Concerning this quantization procedure, the irreducibility of Abelian constraints (65) and (66) displays a BRST charge that pertains to the antighost number zero subspace in the Hamiltonian BRST complex, while the remaining part of the Hamiltonian gauge algebra (68) exhibits a BRST-invariant Hamiltonian that stops at antighost number one terms, contrary to the case of the massless Rarita–Schwinger field, as one concluded in the preceding section.

5. Conclusions

In this paper, the Faddeev–Jackiw approach to first-order systems has been revisited in the context of some fermionic field theories widely used in SUSY/SUGRA-type paradigms. Three types of first-order models were analyzed: (a) a chargeless spin-$1/2$ field, described by a Majorana spinor, that could be entirely analyzed on using the Faddeev–Jackiw approach; (b) a Rarita–Schwinger spin-$3/2$ field, described by a Majorana spinor-vector, that, to maintain the spacetime locality, could be treated in the Faddeev–Jackiw fashion, only by supplementing with prescriptions in the Dirac formalism; and (c) a chargeless spin-$3/2$ field with a third-order Lagrangian description, also treated in a combined Faddeev–Jackiw–Dirac manner.

Concretely, in the case of a spin-$1/2$ field, the interpretation performed only by means of the Faddeev–Jackiw approach isolated the independent degrees of freedom, the symplectic structure, and the time-evolution generator. For the spin-$3/2$ Rarita–Schwinger field, one preferred a unitary, massive/massless implementation of the Faddeev–Jackiw procedure. In this case, arguments concerning the preservation of locality require adjusting the analysis with Dirac’s algorithm arguments. In the massive case, the model displays only one second-class spinor constraint and four physical degrees of freedom. In contrast, in the massless case, it exhibits one Abelian spinor constraint and two physical degrees of freedom. Here, the Hamiltonian gauge algebra has been computed, providing the ingredients for the BRST quantization of the model. The last spinor field considered was reformulated by reducing, via auxiliary variables, the evolution, initially with three spacetime derivatives, to a first-order one. Then, arguments for locality preservation favored a blended Faddeev–Jackiw–Dirac approach that imposed three spinor constraints, two of which were Abelian and one second-class, and yielded eight physical degrees of freedom for the field model. At that point, the analysis completion was performed in two equivalent manners: (i) by keeping in the formalism the second-class constraint and (ii) by restricting the evolution to the second-class constraint surface, supplemented with the induced symplectic 2-form. Also, the Hamiltonian gauge algebra has been derived here.

To conclude, in field theories, where locality and Poincaré invariance are mandatory, the Faddeev–Jackiw method eases the canonical analysis, but, in most cases, it should be supplemented with Dirac’s approach to constrained dynamical systems. On the other hand, field models that do not exhibit gauge symmetries, i.e., matter field theories, can be primarily analyzed using only the Faddeev–Jackiw approach.

The blended Faddeev–Jackiw and Dirac approach is expected to work, with no additional extra variables, as enforced in Section 4, for all fermionic higher-spin models, especially in the non-constrained spinor Christoffel-type construction [20,21], where the first-order dynamics come naturally as generalizations of those given in Section 3 of the present paper. Such an endeavor provides the tools for the BRST quantization of free-fermionic higher-spin fields.