Spinning particle interacting with electromagnetic and antisymmetric gauge fields in anti-de Sitter space

Massless spinning particle model that interacts with electromagnetic and antisymmetric gauge fields in anti-de Sitter space-time is considered as a constrained Hamiltonian system. d-dimensional anti-de Sitter space-time is realized as a real projective manifold parametrized by the homogeneous coordinates. Classical constraints that generate in the presence of interactions minimal world-line supersymmetry algebra extended by the dilatations of the ambient-space homogeneous coordinates are found. Various representations of the Lagrangian of the spinning particle are obtained. Dirac quantization is shown to produce first- and second-order equations for the wave function of the spinning particle that are presented in the homogeneous, inhomogeneous and intrinsic coordinates of AdSd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$AdS_d$$\end{document}.


Introduction
Spinning particle models [1][2][3][4] are known to provide classical realization of the spin 1/2 field equations in Minkowski space-time as odd generators of the minimal world-line supersymmetry algebra that is the finite-dimensional subalgebra of the infinite-dimensional superVirasoro algebra of the superstring. Since the world-line supersymmetry is less restrictive than the world-sheet one, spinning particle models admit a wide variety of generalizations. In particular, it is possible to include interactions with background electromagnetic [1,[5][6][7][8], Yang-Mills [9,10], gravitational [5,11] and antisymmetric gauge fields [12] in a way consistent with minimal world-line supersymmetry. Such models upon quantization yield Dirac equation for spin 1/2 field interacting with background fields. Apart from Minkowski space-time of the special interest are maximally symmetric spaces such as anti-de Sitter space-time. There the interplay between the space-time geometry and world-line supersymmetry appears to be quite non-trivial [13,14]. a e-mail: d_uvarov@hotmail.com Anti-de Sitter space can be described as a manifold embedded into flat space-time with extra dimension(s) and it is possible to consider respective spinning particle models [15][16][17][18]. These provide pseudoclassical realization of the idea of formulating field dynamics in (anti-)de Sitter (as well as Minkowski) space in a way exhibiting (conformal) isometries that dates back to the seminal works of Dirac [19,20]. It was also applied to examine conformal field theories in 4-dimensional Minkowski (Euclidean) space-time [21][22][23] and to formulate dynamical equations for the gauge fields in 4-dimensional anti-de Sitter space [24,25]. More recently embedding (or ambient) space description was applied to study correlation functions in d-dimensional conformal field theories taking advantage of the AdS/CFT inspired techniques [26][27][28] and to study higher-spin field equations in Ad S d and its conformal boundary [29][30][31]. In Ref. [32] there was considered the possibility of applying twistor methods to the Ad S/C FT duality based on the projective-space description of the bulk anti-de Sitter space parametrized by the homogeneous coordinates that naturally combines linear realization of SO(2, d − 1) isometry and the projective light-cone description of the (d − 1)-dimensional conformal boundary space-time. Shortly after that two-twistor formulation of the spinning particle in Ad S d for d = 4, 5, 7 was proposed in [33]. It is based on the generalization [34] of the two-twistor formulation of the massive bosonic particle in Ad S 5 [35]. 1 Utility of the projective-space realization of anti-de Sitter space from the viewpoint of canonical description of massless particle (tensionless string) models can be justi-fied as follows. Description of Ad S d as an embedded hyperboloid assumes imposition of the constraint y 2 + 1 ≈ 0 on the ambient-space inhomogeneous coordinates y m , where the SO(2, d − 1)-invariant scalar product y 2 = (y · y) = y m η mn y n is taken w.r.t. Minkowski metric η mn = diag(−, −, +, · · · , +) and Ad S d radius is set to unity. In the canonical approach the mass-shell constraint for the massless particle (tensionless string zero modes) in its simplest form is p 2 ≈ 0 and its Poisson bracket (PB) relations with the above constraint imply that (y · p) ≈ 0 is also a constraint forming with y 2 + 1 ≈ 0 the pair of the second-class constraints. The presence of the second-class constraints necessitates introduction of the Dirac brackets (DB) that in general essentially complicates analysis of the Hamiltonian dynamics (see, e.g. [40]) so it is convenient to treat the constraint y 2 +1 ≈ 0 as a gauge-fixing condition for the first-class constraint (y· p) ≈ 0 that generates dilatations of the embeddingspace coordinates [41]. Gauged dilatations implement the projective-space realization of Ad S d , so the set of the two first-class constraints (y · p) ≈ 0 and p 2 ≈ 0 can be taken as the starting point for description of the massless particle (tensionless string zero modes) models in such an approach.
In the Lagrangian approach important feature of the parametrization of Ad S d by the homogeneous coordinates x m : y m = |x| −1 x m , |x| = √ −x 2 , is that the object that can be naturally identified with the metric tensor, taking into account the form of the line element, is degenerate det θ = 0. So one is led to consider particle (string, brane) mechanics in the space with degenerate metric [42]. Tensor θ mn and associated differential operator θ mn ∂/∂ x n also enter dynamical equations for the Ad S d higher-spin fields in the ambient-space formulation [24,25,[43][44][45]. In Ref. [46] there was proposed massless spinning particle model in Ad S d realized as the projective space parametrized by the homogeneous coordinates. Three first-class constraints of the model (one odd and two even) span minimal world-line supersymmetry algebra extended by the gauged space-time dilatations. Dirac quantization of the model yields Dirac and Klein-Gordon equations for the particle's wave function that is a homogeneous function of degree zero.
In this note we continue to study the above model and examine the possibility of including interactions with background gauge fields. As the starting point we take Hamiltonian first-class constraints of the free spinning particle model. Then we seek for the generalizations of the odd constraint, that is the world-line supersymmetry generator, by the terms depending on the background gauge fields and calculate its DB relations with itself that define bosonic con-straint generating world-line reparametrizations. Then linear combination of these constraints and the generator of the space-time dilatations with the Lagrange multipliers is used to write down the Lagrangian of the interacting spinning particle model in terms of the phase-space variables. These Lagrange multipliers play the role of the gauge fields for local world-line supersymmetry, reparametrizations and space-time dilatations. Integrating out space-time momentum and some of the Lagrange multipliers we derive various representations of the spinning particle Lagrangian. After that we discuss Dirac quantization of the proposed models. We find Hermitian operators associated with the classical first-class constraints from the requirement that they satisfy quantum world-line supersymmetry algebra. Then the substitution of the realization of the Hermitian momentum operator as a differential operator in configuration space produces Dirac-and Klein-Gordon-type equations for the wave function of the spinning particle in homogeneous coordinates. We also write these equations in the inhomogeneous and intrinsic coordinates on Ad S d .
Section 2 is devoted to the spinning particle's interaction with the background electromagnetic field. 2 In Sect. 3 we discuss gauge-invariant interaction with the rank r − 1 antisymmetric gauge field. Like in the case of electromagnetic interaction closed algebra of the constraints is obtained and various forms of Dirac-and Klein-Gordon-type equations for the particle's wave function are found. Let us remark that the spinning particle model with minimal world-line supersymmetry interacting with odd-rank antisymmetric tensor gauge fields in (2d + 1)-dimensional Minkowski space was studied in [12] from the perspective of the Kaluza-Klein dimensional reduction. In 2d dimensions it results in the particle's interactions with both rank 2r and 2r + 1 antisymmetric gauge fields as well as with the electromagnetic field. Curiously antisymmetric gauge fields appear in quantization of the spinning particle model with extended world-line supersymmetry [53].

Charged spinning particle in background electromagnetic field
Consider odd constraint as the classical analogue of the Dirac equation that includes interaction with external electromagnetic field. We take it as the generator of the minimal world-line supersymmetry.
In the absence of the interaction it coincides with odd constraint introduced in [46]. Observe that the minimality principle fixes the homogeneity degree of . Transverse strength of the electromagnetic field by definition is and the last equality follows by taking into account transversality (x · A(x)) = 0 and homogeneity properties of the electromagnetic potential. After introduction of the PB (DB) relations it is easy to see that odd constraint (2) has zero PB with the constraint that generates dilatations of the embedding-space coordinates. Also the DB relations of the supersymmetry generator with itself define bosonic constraint that is the generator of the world-line reparametrizations. Eq. (6) appears to be the only non-trivial relation of the world-line supersymmetry algebra extended by the spacetime dilatations.
Having introduced the classical first-class constraints we can write down the spinning particle's Hamiltonian as their linear combination with evenẽ, a and odd χ Lagrange multipliers. Then the action is defined as the integral of the Lagrangian expressed in terms of the phase-space variables where Integrating out the momentum p m yields configuration-space form of the particle's Lagrangian The Lagrange multiplier a plays the role of the gauge field for the scale transformations of x and p. Integrating it out allows to bring the Lagrangian to the form that manifests the realization of Ad S d as the projective space R P d parametrized by the homogeneous coordinates with the degenerate metric θ mn = η mn + 1 |x| 2 x m x n . In quantum theory classical observables are replaced by the Hermitian operators and their PB (DB) relations-by the (anti)commutators. The operators associated with the phasespace variables satisfy the (anti)commutation relations 3 From the anticommutation relations of ξ m it follows that they are proportional to γ −matrices in (d + 1) dimensions: ξ m = 2 −1/2 γ m and their Hermiticity is understood in the same sense as that of γ m , i.e. (γ m ) † = (−) t Aγ m A −1 , where A = γ 0 1 γ 0 2 · · · γ 0 t and t is the number of time-like dimensions (t = 2 for the realization of Ad S d as the hyperboloid in the ambient space-time). Classical constraints become Hermitian operators that select physical subspace in the space of quantum states of the spinning particle. We choose Hermitian operator associated with the classical supersymmetry generator in the form where the second summand arises as a result of moving the momentum operator to the right in the manifestly Hermitian representation for the first summand. The square of (e)H 2 defines Hermitian operator associated with the classical con- where is the Hermitian operator for the generator of the space-time dilatations. Note the relation that makes obvious the contact with the classical constraint (7). For the case of flat configuration-space Hermitian momentum operator can be realized as the coordinate partial derivative acting on the wave function (x). Whenever configuration space is a curved manifold, Hermitian momentum operator is given by where g is the determinant of the configuration-space metric tensor. In the realization of anti-de Sitter space-time as the projective manifold, the scale-invariant measure is proportional to |x| −d−1 ε m 1 m 2 ···m d+1 x m 1 dx m 2 ∧ · · · ∧ dx m d+1 , so as the definition of the Hermitian momentum operator we take Then the constraint (14) translates into the Dirac-type equation for the particle's wave function (x) that is the 2 [ d+1 2 ]component spinor field. It has the homogeneity degree zero D H (x) = (x · ∂) = 0 and also satisfies the second-order equation 4 Discussion of the ambiguities in the definition of Hermitian operators in locally supersymmetric models can be found, e.g., in [54][55][56].
To conclude this section let us discuss how the conventional form of spin 1/2 particle's equations in Ad S d in terms of intrinsic coordinates can be derived from the equations given above. As an intermediate step let us present Eqs. (21) and (22) in the inhomogeneous coordinates y m = |x| −1 x m . For Eq. (21) we obtain where ∇ m = θ m n (y)∂/∂ y n , θ m n (y) = δ n m + y m y n , and Eq. (22) becomes The electromagnetic potential and field strength in the homogeneous and inhomogeneous coordinates are related as Transverse field strength in the inhomogeneous coordinates is defined by and the last equality follows by using the transversality property of the potential y · A(y) = 0. Above equations for the particle's wave function in the inhomogeneous coordinates can be transformed to intrinsic coordinates using the transition formulae [24,25,45]. In particular, we use the relation between the derivatives of the coordinate functions where g mn (z) = ∂ m y m ∂ n y m and g mn (z) = ∇ m z m ∇ m z n are the Ad S d metric and its inverse in the intrinsic coordinates. Also the spinning particle's wave functions in the inhomogeneous and intrinsic coordinates are connected by In Eqs. (29) (31) and σ ab = 1 4 (ρ a ρ b −ρ b ρ a ) that span the so(1, d −1) algebra. Useful consequences of Eqs. (29) and (30) are To where is the spinor covariant derivative extended by the external electromagnetic potential. The commutator of the covariant derivatives appears in the transformation of the second-order equation (24) to the intrinsic coordinates. To find the final form we substituted explicit expression for the Riemann tensor R klmn = g kn g lm − g km g ln , (R = R mn mn = −d(d − 1)) that provides solution of the Einstein equations in the form widely used in the literature on the Ad S/C FT correspondence: with the cosmological constant Note that [(γ · y) (e)H ] 2 differs from 2 (e)H by the linear combination of the constraints (e)H and D H .

Spinning particle interactions with antisymmetric gauge fields
In this section we discuss gauge-invariant coupling of the spinning particle to external (r − 1)-form gauge field A m[r −1] (x) 5

that we assume to be transverse x n A nm[r −2] (x)
= 0 and homogeneous of degree −(r − 1). The definition of the transverse field strength generalizes that for the electromagnetic field (3). Since the form of the coupling is sensitive to the value of r we start with the case of odd r and then turn to even r .

r odd
In this case the fermionic constraint naturally generalizes that for the free spinning particle. q stands for the particle's charge, ξ m[r ] = ξ m 1 . . . ξ m r and the factor |x| r makes the last term homogeneous of degree zero like the first is, while the factor i n , n = r −1 2 − 2[ r −1 4 ] makes it real under the complex conjugation. DB relations of this constraint with itself generate classical world-line supersymmetry algebra with being the world-line reparametrization generator in the presence of the interaction. Similarly to the previously considered case of the interaction with the background electromagnetic field, one can write down the spinning particle's Hamiltonian and the action functional where the Lagrangian expressed in terms of the phase-space variables has the form Integrating consecutively momentum p m and dilatation gauge field a yields two representations of the configurationspace Lagrangian: and In quantum theory the Hermitian operator associated with the odd constraint (40) is where the antisymmetrized product of r γ -matrices is defined as Squaring the constraint (48) allows to obtain quantum version of the world-line supersymmetry algebra (41) 2 (q, r odd)H = T (q, r odd)H (50) and define the Hermitian operator corresponding to the generator of the world-line reparametrizations Let us note in passing that the square of γ m[r ] F m [r ] can be expanded over the basis of the antisymmetrized products of γ -matrices using the relations given, e.g. in [57] Clearly which of the (r + 1)/2 terms actually contribute to the sum depends on the values of r and the space-time dimension d.
Substitute now the realization (20) of the momentum as the differential operator to impose (48) and (51) on the configuration space wave function (x). So we come to the first-order Dirac-type equation and the second-order Klein-Gordon-type equation For transformation of the above equations to the intrinsic coordinates let us first rewrite them in the inhomogeneous coordinates. Eq. (53) takes the form where the r -form field strength in the homogeneous and inhomogeneous coordinates is related in the following way generalizing (25) and (26) for the electromagnetic field. The Klein-Gordon-type equation (54) in the inhomogeneous coordinates reads Then using the relations (27)-(32) we find wave equations describing gauge-invariant interaction of the spin 1/2 field on the Ad S d background with the odd-rank antisymmetric gauge field in the intrinsic coordinates and The definition of the spinor covariant derivative coincides with (34) in the absence of electromagnetic field and the r −form field strength in the ambient-space and intrinsic coordinates is related as

r even
In the case of r even, the odd constraint takes the form In this subsection n = r Similarly to the previously considered models, DB relations of this constraint with itself generate the world-line supersymmetry algebra where the world-line reparametrization generator equals The constraints (61), (63) and D are the first-class constraints of the model and are used to define the spinning particle's Lagrangian and action functional S (q, r even) = dτ L (q, r even) ph : (64) Substituting explicit expressions for these constraints and integrating out the momentum allows to transfer from the phase-space to the configuration-space form of the Lagrangian Further integrating out the dilatation gauge field, one finds the Lagrangian that corresponds to the realization of the Ad S d as a projective manifold with the degenerate metric L (q, r even) R P d = 1 2ẽ|x| 2 (ẋθẋ) Now we come to the discussion of the Dirac quantization of the model. Let us define the Hermitian operator associated with the odd constraint (61) as As in the previous sections expression for the Hermitian operator that corresponds to the world-line reparametrization generator is obtained by requiring the closure of the world-line supersymmetry algebra 2 (q, r even)H = T (q, r even)H , where T (q, r even)H = |x|(γ · p) Realizing momentum operator as the differential operator in configuration space allows to obtain Dirac-type and Klein-Gordon-type equations for the particle's wave function in the homogeneous coordinates and Note that (x) is homogeneous of degree zero since D H (x) = 0. In terms of the inhomogeneous coordinates y m = |x| −1 x m these equations acquire the form Connection between the r -form field strength in the homogeneous and inhomogeneous coordinates is given in (56).
Using the transition relations (27) The relation between the r -form field strength in the inhomogeneous and intrinsic coordinates is given in (60). Analogously the second-order equation (74) transforms into the generalization of the Klein-Gordon equation

Conclusion
In this note we have studied interactions with background electromagnetic or rank (r − 1) antisymmetric gauge fields of the minimally-supersymmetric massless spinning particle in anti-de Sitter space-time. d−dimensional anti-de Sitter space-time has been realized as a real projective manifold parametrized by the homogeneous coordinates. For all of the considered interactions we have found the set of three first-class constraints, one odd and two even, that generate extended world-line supersymmetry algebra. The constraints are the classical generators of 1d supersymmetry, reparametrizations and rescalings of the space-time homogeneous coordinates. Various forms of the spinning particle's Lagrangian both in terms of the phase-space and configuration-space variables have been derived. Then the quantum realization of the classical constraint algebra by the Hermitian operators has been found. The form of the Hermitian operator associated with the classical generator of the world-line reparametrizations is unambiguously fixed by the closure of the quantum algebra of the constraints. The realization of the Hermitian momentum operator as the differential operator in configuration space yields first-and secondorder equations for the particle's wave function in the presence of background electromagnetic field or antisymmetric gauge fields. These equations have been presented both in the homogeneous and inhomogeneous coordinates of the ambient space. Finally using known transition relations between the ambient and intrinsic coordinates they have been written in the conventional form of extended Dirac and Klein-Gordon equations in Ad S d .
Let us note that although we treated independently interactions of the spinning particle with electromagnetic and (r − 1)-form gauge fields, along the same lines it is possible to consider simultaneous coupling to a number of gauge fields and electromagnetic field. One can also consider interactions with mixed symmetry fields that carry even number of indices in each set of the antisymmetrized indices.
The part of our discussion that concerned transition of the equations for particle's wave function from the ambient-space to intrinsic coordinates assumed implicitly that Spin(1, d −1) and Spin(2, d −1) spinor representations have equal dimension that is the case for d even. So one of possible generalizations is to consider the case of d odd that presumably requires introduction of additional (odd) variables and constraints to impose chirality projection on the spinor wave function in d + 1 dimensions.
As a further development of the results reported here it is possible to consider interaction of the spinning particle with the Yang-Mills field, to look for the superfield formulation and to describe particles with other values of spin.

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