# Ab Initio Construction of
Poincaré and AdS Particles

TaeHwan OH

###### Abstract

We study the construction of a manifestly covariant worldline action from a
coadjoint orbit. A coadjoint orbit is a submanifold in the dual vector space
of the Lie algebra and generated by coadjoint actions. Since a coadjoint orbit
is a symplectic space, we derive the worldline particle action from the
symplectic two-form. One subtlety in formulating worldline particle actions
from coadjoint orbits is choosing the coordinate system that sufficiently
illustrates physical properties of the particles. We introduce Hamiltonian
constraints by the defining conditions of the isometry. This allows us to
write a manifestly covariant worldline action. We demonstrate our method to
the both massive and massless particles in Minkowski and AdS spacetime.

##  1 Introduction

There have been many attempts to formulate spinning particles since the 1920s,
when a classical description of an electron was first tried [1]. Modern
methods using the supersymmetry were developed in the late 1970s [2, 3, 4, 5,
6] (see also [7, Section 1] for additional references). Over the past century,
wordline particle actions have played important roles in various areas of
physics, e.g., [8, 9, 10, 11, 12]. Despite their importance, it is not well
developed to construct worldline actions for various particle species or
particles obeying various isometries in a systematic way.

In our previous paper [7], we introduced a method of construction for
worldline theory starting from the coadjoint orbit, which we will employ in
this paper. Coadjoint orbits have been studied since 60-70s [13, 14, 15, 16,
17, 18] to obtain unitary irreducible representations through the quantization
of coadjoint orbits, which are symplectic. These efforts provided inspiration
for the development of worldline actions for particles [19, 11, 20, 7].

A significant challenge to construct worldline actions from the orbit method
is the choice of an appropriate coordinate system of the orbit. We choose the
covariance of the coordinate system as the appropriateness. To formulate the
worldline action in a manifestly covariant way, Hamiltonian constraints should
be introduced in the action. For instance, as we will see later on, a Poincaré
scalar particle action obtained from a scalar coadjoint orbit is written as

$S=\int\langle m{\cal P}^{0}\,,\,g^{-1}{\rm d}g\rangle\,,$ (1.1)

where $g$ is an $ISO(1,d-1)$ element and ${\cal P}^{0}$ is the dual of Lie
algebra generator $P_{0}$. A manifestly covariant worldline action for a
scalar particle is written as

$S[x\,,p\,,e]=\int\,p_{\mu}{\rm
d}x^{\mu}+\frac{e}{2}\left(p^{2}+m^{2}\right)\,.$ (1.2)

This action contains a Hamiltonian constraint $p^{2}+m^{2}\approx 0$. We will
see that the constraint can be interpreted as a defining condition of the
$SO(1,d-1)$ group.

In our previous paper [7], we explained the generalities about how worldline
actions can be driven from the coadjoint orbit. The main objective of this
paper is demonstrating the method explicitly, which we explained in the
previous paper, on both massive and massless particles in Minkowski and AdS
spacetime. In Section 2, we will explain how to construct worldline particle
actions of $ISO(1,d-1)$ and $SO(2,d-1)$ isometries from the orbit method. For
this, we will give a pedagogical review on the orbit method in the beginning
of the section. In Section 3, we will apply the orbit method to analyse these
particles. Finally, in Section 4, we will show how worldline actions of these
particles by applying the method introduced in earlier sections. In Appendix
A, we will give a proof for a mathematical property used in the Section 2.2.

##  2 Covariant Action from the Coadjoint Orbit

In this section, we will briefly review how to construct a covariant worldline
action from a coadjoint orbit. To find more detailed discussions about orbit
method and constructing worldline actions from coadjoint orbits, see e.g. [18,
14, 21, 22, 23, 7].

###  2.1 Review on the Coadjoint Orbit

For a Lie group $G$, there is a left-invariant vector space called Lie algebra
${\mathfrak{g}}$, and there exists a Lie algebra dual space
${\mathfrak{g}}^{*}$ which form a pairing $\langle\cdot\,,\cdot\rangle$ with
${\mathfrak{g}}$ such that

$\langle{\cal T}^{a}\,,T_{b}\rangle=\delta^{a}_{b}\,,$ (2.1)

where $T_{a}$ and ${\cal T}^{a}$ are generators of ${\mathfrak{g}}$ and
${\mathfrak{g}}^{*}$, respectively. Note that when $G$ is a simple group, then
${\mathfrak{g}}^{*}\simeq{\mathfrak{g}}$ since there exists a non-degenerated
bilinear form. A group action on ${\mathfrak{g}}^{*}$, denoted by ${\rm
Ad}^{*}_{g}$, is called a coadjoint action, and given as a dual of the adjoint
action:

$\langle{\rm Ad}_{g}^{*}\phi\,,X\rangle=\langle\phi\,,{\rm
Ad}_{g^{-}1}X\rangle\,.$ (2.2)

where $\phi\in{\mathfrak{g}}^{*}\,,\ X\in{\mathfrak{g}}\,,$ and $g\in G$. A
coadjoint orbit ${\cal O}^{\phi}$ is the submanifold generated by coadjoint
actions on $\phi$, such that

${\cal O}^{\phi}=\\{\varphi\in{\mathfrak{g}}^{*}\,|\,{\rm
Ad}^{*}_{g}\phi=\varphi\,,\ g\in G\\}\,,$ (2.3)

and the stabiliser $G^{\phi}$ is the isotropic subgroup for the coadjoint
action:

$G^{\phi}=\\{g\in G\,|\,{\rm Ad}^{*}_{g}\phi=\phi\\}\,.$ (2.4)

Since different $\varphi$’s within ${\cal O}^{\phi}$ belong to the same
equivalence class with equivalent relation $\phi\sim\varphi={\rm
Ad}_{g}^{*}\phi$, a specific vector $\phi$ can serve as a representative
vector of ${\cal O}^{\phi}$.

One can immediately find that the coadjoint action ${\rm ad}^{*}_{X}$ by the
Lie algebra vector $X\in{\mathfrak{g}}$ is given as

$\langle{\rm ad}_{X}^{*}\phi\,,Y\rangle=-\langle\phi\,,{\rm
ad}_{X}Y\rangle=-\langle\phi\,,[X\,,Y]\rangle\,.$ (2.5)

The tangent space of the coadjoint orbit is given as

$T_{\varphi}{\cal O}^{\phi}=\\{X\in{\mathfrak{g}}\ |\,{\rm
ad}^{*}_{X}\phi=\varphi\,,\ \varphi\in{\cal O}^{\phi}\\}\,,$ (2.6)

and the stabiliser subalgebra is given as

${\mathfrak{g}}^{\phi}=\\{X\in G\,|\,{\rm ad}^{*}_{X}\phi=0\\}\,.$ (2.7)

From (2.5), a stabiliser subalgebra element $X\in{\mathfrak{g}}^{\phi}$ should
satisfy

$\langle\phi\,,[X,Y]\rangle=0\,,$ (2.8)

for all $Y\in{\mathfrak{g}}$.

Note that $G$ can be regarded as a principle $G^{\phi}$–bundle over ${\cal
O}^{\phi}$ with projection $\pi_{\phi}$:

$\displaystyle\pi_{\phi}\,:\ G$ $\displaystyle\rightarrow{\cal O}^{\phi}\,,$
(2.9) $\displaystyle g$ $\displaystyle\mapsto\pi_{\phi}(g)={\rm
Ad}^{*}_{g}\phi\,,$

then ${\cal O}^{\phi}$ is a homogeneous space, given as

${\cal O}^{\phi}\simeq G/G^{\phi}\,.$ (2.10)

One important feature of coadjoint orbits are their nature as a symplectic
space, that is, ${\cal O}^{\phi}$ is always even-dimensional manifold, and a
symplectic form $\omega$, known as a Kirillov-Kostant-Souriau(KKS) two-form,
exist. KKS two-form on $\varphi\in{\cal O}^{\phi}$, denoted by
$\omega_{\varphi}$, is expressed as

$\omega_{\varphi}\left({\rm ad}^{*}_{X}\varphi\,,{\rm
ad}^{*}_{Y}\varphi\right)=\langle\varphi\,,[X,Y]\rangle\,.$ (2.11)

Using the projection operation in (2.9), the pull-back of the KKS two-form
$\omega$ onto group $G$ is written as

$\Omega\equiv\pi^{*}_{\phi}(\omega)=\langle\phi\,,[\Theta(X),\Theta(Y)]\rangle=-{\rm
d}\langle\phi\,,\Theta\rangle\,,$ (2.12)

where $\Theta$ is a ${\mathfrak{g}}$-valued left-invariant one-form on $G$,
called Maurer-Cartan one-form, which is expressed as $\Theta=g^{-1}{\rm d}g$.
We used the Maurer-Cartan equation in the last equality in (2.12).

Since the two-form $\Omega$ is an exact form on $G$, we can find a symplectic
potential $\theta$ as

$\theta=\langle\phi\,,\Theta\rangle\,,$ (2.13)

so a worldline action is written in terms of the symplectic potential by

$S=\int_{\gamma}\theta=\int_{\gamma}\langle\phi\,,\Theta\rangle=\int_{\gamma}\langle\phi\,,g^{-1}{\rm
d}g\rangle\,,$ (2.14)

where $\gamma\in G$ is a path lying in the group.

###  2.2 Manifestly Covariant Worldline Action

So far, we discussed how to derive a worldline action from a coadjoint orbit.
However, a subtlety arises when finding an appropriate coordinate system of
the orbit. We have chosen covariance as our criterion for the appropriateness
of the coordinate system. Manifestly covariant worldline actions are
formulated by incorporating Hamiltonian constraints into the action. As we
will show later, we will use Hamiltonian constraints as the defining
conditions of the group. To see more detailed discussions about the choice of
the coordinates, see Section 2.5 of [7].

Manifestly covariant worldline actions for various groups, classical Lie
groups and semi-direct groups, are formulated systematically in our previous
paper (see [7, Section 4]). In the rest of the section, we will focus on
$SO(2,d-1)$ and $ISO(1,d-1)$ cases which are isometry groups of the
$d$-dimensional AdS and flat spacetime, respectively.

####  2.2.1 Covariant Formulation of $SO(2,d-1)$ action

For a matrix group, (2.14) can be expressed in terms of the matrices. So the
worldline action for $SO(2,d-1)$ will be rephrased as

$S[X]=\int\tr(\phi\,X^{-1}{\rm
d}X)=\int\phi^{a}{}_{b}\,(X^{-1})^{b}{}_{c}\,{\rm d}X^{c}{}_{a}\,,$ (2.15)

for $X\in SO(2,d-1)$. To make a manifestly covariant worldline action, we
introduce the defining condition of $SO(2,d-1)$, that is, $X^{T}\eta X=\eta$
where $\eta$ is the metric $\eta=(-\,,-\,,+\,,+\,,\ldots)$, then the action
(2.15) becomes

$S[X,A]=\int\left[\phi^{a}{}_{b}\,(X^{-1})^{b}{}_{c}\,{\rm
d}X^{c}{}_{a}+A^{ab}\big{(}X^{c}{}_{b}\,\eta_{cd}\,X^{d}{}_{a}-\eta_{ba}\big{)}\right]\,,$
(2.16)

where $a\,,b\,,\cdots=0^{\prime}\,,0\,,1\,,\cdots\,,d-1$ are
$(d+1)$-dimensional indices, and $A^{ab}$ is a Lagrange multiplier. Then $X$
is regarded as a $GL(d+1,{\mathbb{R}})$ element. Using the constraint,
$X^{-1}$ can be substituted with $\eta\,X^{T}\eta$ so

$\displaystyle S[X,A]$
$\displaystyle=\int\left[\phi^{a}{}_{b}\,\eta^{bc}\,X_{dc}\eta^{de}\,{\rm
d}X_{ea}+A^{ab}\big{(}X^{c}{}_{b}\,\eta_{cd}\,X^{d}{}_{a}-\eta_{ba}\big{)}\right]$
(2.17) $\displaystyle=\int\left[\phi^{ab}\,X^{c}{}_{b}\,{\rm
d}X_{ca}+A^{ab}\big{(}X^{c}{}_{b}\,\eta_{cd}\,X^{d}{}_{a}-\eta_{ba}\big{)}\right]\,.$

Since $\phi^{ab}$ is an ${\mathfrak{so}}(2,d-1)^{*}$ element, it is an anti-
symmetric matrix and $\phi^{ab}$ can be expressed as a symplectic matrix
through a congruent transformation: let $\rank\phi=M\leq d+1$ 111Since $\phi$
is an anti-symmetric and non-degenerated matrix, $M$ must be even. then $\phi$
can be expressed as

$\phi^{ab}=T^{a}{}_{\alpha}\Omega^{\alpha\beta}T^{b}{}_{\beta}\,,$ (2.18)

where $\alpha\,,\beta=1\,,\cdots\,,M$ are sympectic indices, $\Omega$ is an
$M\times M$ symplectic matrix, and $T$ is an $M\times(d+1)$ matrix. Note that
we can always find the matrix $T$ which transforms the symplectic matrix to
any anti-symmetric matrix (See Appendix A to find a proof of this statement.).

Once we define the variables as

$Y^{c}{}_{\alpha}=X^{c}{}_{a}T^{a}{}_{\alpha}\,,\quad
A^{\alpha\beta}=A^{ab}\,T^{b}{}_{\beta}\,T^{a}{}_{\alpha}\,,\quad\varphi_{\beta\alpha}=T^{a}{}_{\alpha}\,T_{a\beta}\,,$
(2.19)

then the worldline action can be rewritten as

$S[Y,A]=\int\left[\Omega^{\alpha\beta}Y^{c}{}_{\beta}\,{\rm
d}Y_{c\alpha}+A^{\alpha\beta}\big{(}Y^{c}{}_{\beta}Y_{c\alpha}-\varphi_{\beta\alpha}\big{)}\right]\,,$
(2.20)

after integrating out the spurious variables $X^{a}{}_{i}$ such that
$X^{a}{}_{i}\,T^{i}{}_{\alpha}=0$ for all $\alpha$ since they are non-
dynamical.

Remark that the form of the action (2.20) is universal and it is sufficient to
identify $\varphi$ from the representative $\phi$ of the orbit to obtain a
manifestly covariant worldline particle action associated with the orbit
${\cal O}^{\phi}$

####  2.2.2 Covariant Formulation of $ISO(1,d-1)$ action

For Poincaré group $ISO(1,d-1)$, a representative vector $\phi$ is composed of
an inhomogeneous part $\phi_{I}\in{\mathbb{R}}^{1,d-1}{}^{*}$ and a
homogeneous part $\phi_{H}\in{\mathfrak{so}}(1,d-1)^{*}$ (i.e.
$\phi=\phi_{I}+\phi_{H}$). Let $\Lambda$ be an $SO(1,d-1)$ element and $x$ be
a ${\mathbb{R}}^{1,d-1}$ element, then the worldline action is written as
follows

$S[x\,,\Lambda]=\int\tr[\phi_{I}\Lambda^{-1}{\rm d}x+\phi_{H}\Lambda^{-1}{\rm
d}\Lambda\big{]}\,.$ (2.21)

As we did in Section 2.2.1, we introduce a Hamiltonian constraint as the
defining conditions of $SO(1,d-1)$ then $\Lambda$ becomes a
$GL(d,{\mathbb{R}})$ element. Using the Hamiltonian constraint, $\Lambda^{-1}$
can be replaced to $\eta\,\Lambda^{T}\eta$, then we find

$\displaystyle S[x\,,\Lambda\,,A]$
$\displaystyle=\int\left[(\phi_{I})_{a}\,(\Lambda^{-1})^{a}{}_{b}\,{\rm
d}x^{b}+(\phi_{H})^{a}{}_{b}(\Lambda^{-1})^{b}{}_{c}\,{\rm
d}\Lambda^{c}{}_{a}+A^{ab}\big{(}\Lambda_{cb}\,\Lambda^{c}{}_{a}-\eta_{ba}\big{)}\right]\,,$
(2.22) $\displaystyle=\int\left[(\phi_{I})^{a}\,\Lambda_{ba}\,{\rm
d}x^{b}+(\phi_{H})^{ab}\,\Lambda^{c}{}_{b}{\rm
d}\Lambda_{ca}+A^{ab}\big{(}\Lambda_{cb}\Lambda^{c}{}_{a}-\eta_{ba}\big{)}\right]\,,$

where $a,b=0,1\,,\cdots\,,d-1$. As we discussed in Section 2.2.1 and Appendix
A, $\phi_{H}$ can be given from the symplectic matrix by congruence
transformation
$\phi_{H}{}^{ab}=T^{a}{}_{\alpha}\,\Omega^{\alpha\beta}\,T^{b}{}_{\beta}$
where $\alpha,\beta=1\,,\cdots\,,M$ are symplectic indices for
$M=\rank\phi_{H}\leq d$. Before redefining the variables along the
transformation, the Lagrangian multiplier $A^{ab}$ can be decomposed into
$A^{ab}=\phi_{I}{}^{a}\phi_{I}{}^{b}A+\phi_{I}{}^{(a}A^{b)}+A^{ab}$, then the
action (2.22) is re-expressed to

$\displaystyle S[x\,,\Lambda\,,A]=\int\Big{[}p_{a}\,{\rm d}x^{a}$
$\displaystyle+\Omega^{\alpha\beta}\,\Sigma_{c\beta}\,{\rm
d}\Sigma^{c}{}_{\alpha}+A\big{(}p^{a}p_{a}-\varphi_{M}\big{)}$ (2.23)
$\displaystyle+A^{\alpha}\big{(}p_{a}\Sigma^{a}{}_{\alpha}-\varphi_{I}{}_{\alpha}\big{)}+A^{\alpha\beta}(\Sigma_{c\beta}\Sigma^{c}{}_{\alpha}-\varphi_{H}\,{}_{\beta\alpha})\Big{]}\,,$

after integrating out spurious variables like $\Lambda^{a}{}_{i}$ such that
$\Lambda^{a}{}_{i}\,T^{i}{}_{\alpha}=0$ for all $\alpha$ and
$(\phi_{I})^{i}=0$ since they are non-dynamical. We redefine the variables in
(2.23) as

$\begin{gathered}p_{a}=(\phi_{I})^{b}\Lambda_{ab}\,,\quad\Sigma^{c}{}_{\alpha}=\Lambda^{c}{}_{\alpha}\,,\\\
\varphi_{M}=\phi_{I}{}_{a}\phi_{I}{}^{a}\,,\quad\varphi_{I}{}_{\alpha}=\phi_{I}{}_{a}T^{a}{}_{\alpha}\,,\quad\varphi_{H}{}_{\alpha\beta}=T^{a}{}_{\alpha}\,T^{b}{}_{\beta}\,\eta_{ab}\,.\end{gathered}$
(2.24)

Similarly to the previous case of $SO(2,d-1)$ action, the above action (2.23)
is universal and it is sufficient to extract $\varphi_{M}\,,\varphi_{I}\,,$
and $\varphi_{H}$ from the original representative vector $\phi$ of the
coadjoint orbit ${\cal O}^{\phi}$, in order to obtain a manifestly covariant
worldline action of the particle corresponding to ${\cal O}^{\phi}$.

##  3 Stabiliser Analysis of Coadjoint Orbits

In this section, we will study the coadjoint orbits of Poincaré and AdS
coadjoint orbits from the stabiliser algebras. These stabiliser algebras can
be determined by solving a system of linear equations.

For the sake of simplicity, we will focus on the three particular cases:
Poincaré massive particles, represented as $\phi=m\,{\cal P}^{0}+s\,{\cal
J}^{12}$, Poincaré massless particles, represented as $\phi=E\,{\cal
P}^{+}+s\,{\cal J}^{12}$, and AdS particles represented as $\phi=m\,{\cal
J}^{0^{\prime}0}+s\,{\cal J}^{12}$.

###  3.1 Stabiliser Algebra of Poincaré Particles

Since representative vectors of ${\mathfrak{iso}}(1,d-1)$ are composed of the
inhomogeneous parts $\phi_{I}$ and the homogeneous parts $\phi_{H}$, they can
be classified by each part. $\phi_{I}$ characterises the mass type of
particle. For instance, $\phi_{I}=m\,{\cal P}^{0}$ describes a massive
particle, $\phi_{I}=E\,{\cal P}^{+}$ describes a massless particle, and
finally $\phi_{I}=\mu\,{\cal P}^{1}$ describes a tachyonic particle. On the
other hand, $\phi_{H}$ determines the spin of the particles from the Young
diagram, except the continuous spinning particle case, $\phi=E\,{\cal
P}^{+}+\sigma\,{\cal J}^{-1}$. For instance, when the homogeneous part is
given as

$\phi_{H}=s_{1}\,{\cal J}^{12}+s_{2}\,{\cal J}^{34}+\cdots\,,$ (3.1)

then the spin of the particle is given by the Young diagram 222The components
of $\phi_{H}$ can take only integer values due to the consistency of
quantization condition, such that

${\mathbb{Y}}=(s_{1}\,,s_{2}\,,\ldots)\,.$ (3.2)

Here, we will focus on the massive and the massless particles. Remark that
this classification of $\phi$ is accordance with the Wigner’s classification
[24, 25].

Recall that stabiliser algebras can be obtained from (2.8): let us parametrise
$\phi\,,\ X\,,$ and $Y$ as

$\phi=\alpha_{a}{\cal P}^{a}+\beta_{ab}{\cal J}^{ab}\,,\quad
X=A^{a}\,P_{a}+B^{ab}\,J_{ab}\,,\quad Y=C^{a}\,P_{a}+D^{ab}\,J_{ab}\,.$ (3.3)

Imposing (2.7)333We used commutator relations between
${\mathfrak{iso}}(1,d-1)$ generators are given by
$\begin{gathered}{}[J_{ab}\,,J_{cd}]=\eta_{ac}J_{bd}-\eta_{ad}J_{bc}-\eta_{bc}J_{ad}+\eta_{bd}J_{ac}\,,\\\
[P_{a}\,,J_{cd}]=\eta_{ac}P_{d}-\eta_{ad}P_{c}\,,\quad[P_{a}\,,P_{b}]=0\,.\end{gathered}$
(3.4) to these variables, we find

$\displaystyle\langle\phi\,,$ $\displaystyle[X\,,Y]\rangle$ (3.5)
$\displaystyle=C^{c}(2B^{ab}\,\eta_{ac}\,\alpha_{b})+D^{cd}\big{[}2B^{ab}\,(\eta_{ac}\,\beta_{bd}-\eta_{ad}\,\beta_{bc})-A^{a}\,(\eta_{ac}\,\alpha_{d}-\eta_{ad}\,\alpha_{c})\big{]}\,.$

If $X$ belong to ${\mathfrak{g}}^{\phi}$ then (3.5) must be vanished for any
$C^{c}$ and $D^{cd}$. It defines a system of linear equations

$B^{ab}\,\eta_{ac}\,\alpha_{b}=0\,,$ (3.6a)
$2B^{ab}\,(\eta_{ac}\,\beta_{bd}-\eta_{ad}\,\beta_{bc})-A^{a}\,(\eta_{ac}\,\alpha_{d}-\eta_{ad}\,\alpha_{c})=0\,.$
(3.6b)

In conclusion, once the data $\\{\alpha_{a}\,,\beta_{\alpha\beta}\\}$ are
determined from the representative, the stabiliser subalgebra can be found out
from the equation (3.6).

Let us consider first massive scalar particles. A massive scalar particle is
described by the following representative vector

$\phi_{I}=m\,{\cal P}^{0}\,,$ (3.7)

so $\alpha_{0}=m$, $\alpha_{i}=0$, and $\beta_{ab}=0$ for
$i=1\,,\ldots\,,d-1$. Imposing these data to (3.6a), we find

  * •

$B^{a0}\,\eta_{ac}\,\alpha_{0}=0\,\Rightarrow\,B^{0i}=0\,.$

Also, we find another result

  * •

$A^{a}\,\eta_{ai}\,\alpha_{0}=0\,\Rightarrow\,A^{i}=0\,,$

from (3.6b). From two results, the stabiliser algebra of a massive scalar
particle ${\mathfrak{g}}^{\phi_{I}}$ is given as

${\mathfrak{g}}^{\phi_{I}}={\rm Span}\left\\{P_{0}\,,\ J_{ij}\right\\}\,,$
(3.8)

for $i,j=1\,,\ldots\,,d-1$. Since ${\mathfrak{g}}^{\phi_{I}}$ is isomorphic to
${\mathbb{R}}\oplus{\mathfrak{so}}(d-1)$, it is a
$\frac{1}{2}(d^{2}-3d+4)$-dimensional vector space. This implies that the
coadjoint orbit of a massive scalar particle ${\cal O}^{\phi_{I}}$ is a
$2(d-1)$-dimensional phase space and it is consistent with the dimension of
the phase space of a scalar particle.

A massless scalar particle corresponds to the following representative vector

$\phi_{I^{\prime}}=E\,{\cal P}^{+}\,,$ (3.9)

where $X_{\pm}=X_{0}\pm X_{d-1}$, then $\alpha_{+}=E$, $\alpha_{i}=0$, and
$\beta_{ab}=0$ for $i=-\,,1\,,\ldots\,,d-2$. Using these data, we obtain

  * •

$B^{a+}\,\eta_{ac}\,\alpha_{+}=0\,\Rightarrow\,B^{+i}=0\,,$

from (3.6a) and

  * •

$A^{a}\,\eta_{ai}\,\alpha_{+}=0\,\Rightarrow\,A^{a}=0\,,$ unless $a=-\,.$

from (3.6b). Therefore, the stabilser subalgebra of a massless particle is
given as

${\mathfrak{g}}^{\phi_{I^{\prime}}}={\rm
Span}\left\\{P_{-}\,,J_{-\alpha}\,,J_{\alpha\beta}\right\\}\,,$ (3.10)

for $\alpha\,,\beta=1\,,\cdots\,,d-2$. The stabiliser subalgebra
${\mathfrak{g}}^{\phi_{I^{\prime}}}$ is isomorphic to
${\mathbb{R}}\oplus{\mathfrak{iso}}(d-2)$ so the coadjoint orbit of a massless
scalar particle ${\cal O}^{\phi_{I^{\prime}}}$ is also a $2(d-1)$-dimensional
phase space.

Note that the two coadjoint orbits ${\mathfrak{g}}^{\phi_{I}}$ and
${\mathfrak{g}}^{\phi_{I^{\prime}}}$ possess different characteristics. The
stabiliser of the massive scalar (3.8) includes its dual generator $P_{0}$. In
contrast, the stabiliser of the massless scalar (3.10) does not include its
dual generator $P_{+}$. Mathematically speaking, ${\cal O}^{\phi}$ is a
nilpotent orbit when the condition $\langle\phi\,,X\rangle=0$ holds for all
$X\in{\mathfrak{g}}^{\phi}$, and if ${\cal O}^{\phi}$ is not a nilpotent
orbit, then it is called a semisimple orbit. According to this classification,
the coadjoint orbit of the massive scalar particle is classified as a
semisimple orbit, whereas that of the massless scalar particle is considered a
nilpotent orbit.

We continue our discussion to the spinning cases. To describe a massive
spinning particle, we should append additional dual vector $\phi_{H}$ to (3.7)
and they should be chosen in the dual of little algebra elements. Thus, we
choose an additional dual vector as $\phi_{H}={\cal J}^{12}$ besides to the
massive $\phi_{I}$, so the representative vector for the massive spinning
particle will be given as

$\phi=m\,{\cal P}^{0}+s\,{\cal J}^{12}\,.$ (3.11)

Applying (3.6) with the data $\alpha_{0}=m$, $\beta_{12}=s$, and setting all
other $\alpha$ and $\beta$ to zero, we find the solutions from (3.6a)

  * •

$B^{a0}\,\eta_{ac}\,\alpha_{0}=0\,\Rightarrow\,B^{0a}=0\,,$

and from (3.6b) for each $(c\,,d)$ pairs

  * •

$(0\,,1)\quad 2s\,B^{02}-m\,A^{1}=0\quad\Rightarrow A^{1}=0\,;$

  * •

$(0\,,2)\quad-2s\,B^{01}-m\,A^{2}=0\quad\Rightarrow A^{2}=0\,;$

  * •

$(0\,,\mu)\quad m\,A^{\mu}=0\quad\Rightarrow A^{\mu}=0\,;$

  * •

$(1\,,\mu)\quad s\,B^{2\mu}=0\quad\Rightarrow B^{2\mu}=0\,;$

  * •

$(2\,,\mu)\quad s\,B^{1\mu}=0\quad\Rightarrow B^{1\mu}=0\,,$

where $\mu=3\,,\ldots\,,d-1$. As a result, the stabiliser subalgebra is given
as

${\mathfrak{g}}^{\phi}={\rm
Span}\left\\{P_{0}\,,J_{12}\,,J_{\mu\nu}\right\\}\,,$ (3.12)

where $\mu\,,\nu=3\,,\ldots\,,d-1$, and ${\mathfrak{g}}^{\phi}$ is identical
to ${\mathbb{R}}\oplus{\mathfrak{u}}(1)\oplus{\mathfrak{so}}(d-3)$. So it is a
$\frac{1}{2}(d^{2}-7d+16)$-dimensional algebra. This means that the coadjoint
orbit ${\cal O}^{\phi}$ is a $2(2d-4)$-dimensional phase space.

Finally, to describe a massless spinning particle, we add additional dual
vector $\phi_{H^{\prime}}$ to (3.9). There are two possibilities,
$\phi_{H^{\prime}}={\cal J}^{12}$ and $\phi_{H^{\prime}}={\cal J}^{-1}$, where
the first case corresponds to the massless spinning particle and the second
case corresponds to the continuous spinning particle according to the Wigner’s
classification. The massless spinning particle is then described by the
representative vector

$\phi^{\prime}=E\,{\cal P}^{+}+s\,{\cal J}^{12}\,,$ (3.13)

that is, $\alpha_{+}=E\,,$ $\beta_{12}=s\,,$
$\alpha_{-}=\alpha_{1}=\alpha_{2}=\alpha_{i}=0$, and $\beta_{ij}=\beta_{-i}=0$
for $i\,,j=3\,,\cdots\,,d-2$. From these data, (3.6a) gives

  * •

$B^{a+}\,\eta_{ac}\,\alpha_{+}=0\,\Rightarrow\,B^{+-}=B^{+1}=B^{+2}=B^{+i}=0\,,$

and (3.6b) gives several equations for each $(c\,,d)$ pairs, as follows

  * •

$(+\,,-)\quad A^{+}\,\eta_{-+}\,\alpha_{+}=0\quad\Rightarrow A^{+}=0\,;$

  * •

$(+\,,1)\quad
2B^{-2}\,\eta_{-+}\,B_{21}+A^{1}\,\eta_{11}\,\alpha_{+}=0\quad\Rightarrow
s\,B^{-2}+E\,A^{1}=0\,;$

  * •

$(+\,,2)\quad
2B^{-1}\,\eta_{-+}\,B_{12}+A^{2}\,\eta_{22}\,\alpha_{+}=0\quad\Rightarrow-s\,B^{-1}+E\,A^{2}=0\,;$

  * •

$(+\,,i)\quad A^{i}\,\eta_{ii}\,\alpha_{+}=0\quad\Rightarrow A^{i}=0\,;$

  * •

$(-\,,1)\quad 2B^{+2}\,\eta_{+-}\,B_{21}=0\quad\Rightarrow B^{+2}=0\,;$

  * •

$(-\,,2)\quad 2B^{+1}\,\eta_{+-}\,B_{12}=0\quad\Rightarrow B^{+1}=0\,;$

  * •

$(1\,,i)\quad 2B^{i2}\,\eta_{ii}\,B_{21}=0\quad\Rightarrow B^{2i}=0\,;$

  * •

$(2\,,i)\quad 2B^{i1}\,\eta_{ii}\,B_{12}=0\quad\Rightarrow B^{1i}=0\,.$

Then the stabiliser subalgebra ${\mathfrak{g}}^{\phi^{\prime}}$ is given as

${\mathfrak{g}}^{\phi^{\prime}}={\rm
Span}\left\\{-E\,J_{-2}+s\,P_{1}\,,E\,J_{-1}+s\,P_{2}\,,J_{12}\,,P_{-}\,,J_{-i}\,,J_{ij}\right\\}\,,$
(3.14)

which is isomorphic to
$({\mathfrak{heis}}_{2}\niplus{\mathfrak{u}}(1))\oplus{\mathfrak{iso}}(d-4)$.

The stabiliser algebras contains information for the geometry of the coadjoint
orbit. In the group level, the stabiliser of the massive scalar particle is
given as $G^{\phi_{I}}={\mathbb{R}}\times SO(d-1)$, so its coadjoint orbit is

${\cal O}^{\phi_{I}}\simeq\frac{ISO(1,d-1)}{{\mathbb{R}}\times
SO(d-1)}\simeq{\mathbb{R}}^{2(d-1)}\,.$ (3.15)

On the other hand, for the massive spinning particle, the coadjoint orbit can
be understood as a fibre bundle over the base manifold ${\cal O}^{\phi_{I}}$
with spin fibre

${\cal O}^{S}\simeq\frac{SO(d-1)}{SO(2)\times SO(d-3)}\simeq{\rm
Gr}_{\mathbb{R}}(d-1,2)\,,$ (3.16)

where we denoted the phase space of the spin degrees of freedom as ${\cal
O}^{S}$ and a Grassmannian manifold as ${\rm Gr}_{\mathbb{R}}$. For $d=4$, the
phase space of the spinning particle will be given by
${\mathbb{R}}^{6}\times{\rm Gr}_{\mathbb{R}}(3,2)\simeq{\mathbb{R}}^{6}\times
S^{2}$, and the spin phase spaces $S^{2}$ can be decomposed into various modes
with discrete radii. Note that Poincaré coadjoint orbits are also classified
and analysed in several studies [26, 27, 28, 29, 30, 31, 32, 20, 33, 7].

We will close this section with one comment on a particle with a complicated
spin structure. We append $\phi_{H}=s\,{\cal J}^{12}$ to the massive scalar
$\phi_{I}$ to describe the massive spinning particle. This procedure can be
repeated to add more vectors to $\phi_{H}$, resulting in, $\phi=m\,{\cal
P}^{0}+s_{1}\,{\cal J}^{12}+s_{2}\,{\cal J}^{34}+\cdots$. This particle is
interpreted as a particle version of a mixed symmetric field where spin is
given by the Young diagram ${\mathbb{Y}}=(s_{1}\,,s_{2}\,,\ldots)$. Stabiliser
subalgebra for this representative vector is given by

${\mathfrak{g}}^{\phi}={\mathbb{R}}\oplus{\mathfrak{so}}(d-1-2n)\oplus\bigoplus_{i=1}^{R}{\mathfrak{u}}(r_{i})\,,$
(3.17)

when the Young diagram of this particle is given as

${\mathbb{Y}}=(\underbrace{s_{1}\,,\ldots\,,s_{1}}_{r_{1}}\,,\underbrace{s_{2}\,,\ldots\,,s_{2}}_{r_{2}}\,,\ldots)\,.$
(3.18)

###  3.2 Stabiliser Algebra of AdS Particles

There are various particle species enjoying AdS isometry (see e.g. [7, Section
6]), but in this paper we will focus on the AdS particle with time-like
momentum, and space-like spin,

$\phi=m\,{\cal J}^{0^{\prime}0}+s\,{\cal J}^{12}\,.$ (3.19)

These particles can be regarded as a counterpart of Poincaré massive particles
with transvection ${\cal J}^{0^{\prime}a}\sim{\cal P}^{a}$, but in fact, this
naive correspondence is subtle for spinning case as we will see later. For
convenience, we will call (3.19) as AdS spinning particles in the rest of
paper.

Similarly to the Poincaré case, we can find the stabiliser algebra of (3.19)
solving a system of linear equations

$B^{ab}\,\eta_{ac}\,\beta_{bd}-B^{ab}\,\eta_{ad}\,\beta_{bc}=0\,,$ (3.20)

which can be obtained by ignoring the inhomogeneous parts of (3.6).

Let us first consider the scalar case with $s=0$,

$\phi_{0}=m\,{\cal J}^{0^{\prime}0}\,.$ (3.21)

Putting $\beta_{0^{\prime}0}=0$ and other $\beta=0$ to (3.20), we find two
equations for $C=0\,,1$ and $D=i$ cases

  * •

$(0^{\prime}\,,i)\quad m\,B^{0i}=0\quad\Rightarrow B^{0i}=0\,;$

  * •

$(0\,,i)\quad m\,B^{0^{\prime}i}=0\quad\Rightarrow B^{0^{\prime}i}=0\,,$

where $i=1\,,\ldots\,,d-1$, then the stabiliser algebra is given as

${\mathfrak{g}}^{\phi_{0}}={\rm
Span}\left\\{J_{0^{\prime}0}\,,J_{ij}\right\\}\,,$ (3.22)

for $i,j=1\,,\ldots\,,d-1$. The stabiliser algebra is isomorphic to
${\mathfrak{u}}(1)\oplus{\mathfrak{so}}(d-1)$. As a consequence, the dimension
of the corresponding coadjoint orbit ${\cal O}^{\phi_{0}}$ is $2(d-1)$ which
is the same degrees of freedom in Poincaré scalar cases.

In order to find stabiliser subalgebra of (3.19), one tries to solve (3.20)
for $\beta_{0^{\prime}0}=m\,,$ $\beta_{12}=s\,,$ and other $\beta=0$ and finds
a system of linear equations, as follows

  * •

$(0^{\prime}\,,1)\quad s\,B^{0^{\prime}2}-m\,B^{01}=0\,;$

  * •

$(0\phantom{{}^{\prime}}\,,1)\quad s\,B^{02}+m\,B^{0^{\prime}1}=0\,;$

  * •

$(0^{\prime}\,,2)\quad s\,B^{0^{\prime}1}+m\,B^{02}=0\,;$

  * •

$(0\phantom{{}^{\prime}}\,,2)\quad s\,B^{01}-m\,B^{0^{\prime}2}=0\,;$

  * •

$(0^{\prime}\,,i)\quad B^{0i}=0\,;$

  * •

$(0\phantom{{}^{\prime}}\,,i)\quad B^{0^{\prime}i}=0\,;$

  * •

$(1\phantom{{}^{\prime}}\,,i)\quad B^{2i}=0\,;$

  * •

$(2\phantom{{}^{\prime}}\,,i)\quad B^{1i}=0\,,$

where $i,j=3\,,\ldots\,,d-1$. This system of linear equations has a
singularity at $m=s$ which requires a separate analysis. With $m\neq s$, the
first four equations provide the solutions
$B^{0^{\prime}1}=B^{0^{\prime}2}=B^{01}=B^{02}=0$. Consequently, the
stabiliser subalgebra is obtained as

${\mathfrak{g}}^{\phi}={\rm Span}\left\\{J_{0^{\prime}0}\,,\ J_{12}\,,\
J_{ij}\right\\}\,,$ (3.23)

where $i,j=3\,,\ldots\,,d-1$, and it is isomorphic to
${\mathfrak{u}}(1)\oplus{\mathfrak{u}}(1)\oplus{\mathfrak{so}}(d-3)$. Since
${\mathfrak{g}}^{\phi}$ is a $\frac{1}{2}(d^{2}-7d+16)$-dimensional algebra,
${\cal O}^{\phi}$ is a $2(2d-4)$-dimensional phase space which has the same
degrees of freedom with the Poincaré massive spinning particle.

On the other hand, for $m=s$ case, the representative is given as

$\phi^{\prime}=s({\cal J}^{0^{\prime}0}+{\cal J}^{12})\,,$ (3.24)

and the solutions from the first four equation are $B^{0^{\prime}1}=-B^{02}$
and $B^{0^{\prime}2}=B^{01}$. Therefore, the corresponding stabiliser algebra
is given as

${\mathfrak{g}}^{\phi^{\prime}}={\rm Span}\left\\{J_{0^{\prime}1}-J_{02}\,,\
J_{0^{\prime}2}+J_{01}\,,\ J_{0^{\prime}0}+J_{12}\,,\
J_{0^{\prime}0}-J_{12}\,,\ J_{0i}\,,\ \ J_{ij}\right\\}\,,$ (3.25)

and it is isomorphic to ${\mathfrak{u}}(1,1)\oplus{\mathfrak{so}}(d-3)$.

Note that $\dim{\mathfrak{g}}^{\phi^{\prime}}$ is bigger than
$\dim{\mathfrak{g}}^{\phi}$ by two, so ${\cal O}^{\phi^{\prime}}$ is a smaller
phase space with $2(2d-5)$ dimension. This indicates that $\phi^{\prime}$ is
the counterpart of the Poincaré massless particle.

In conclusion, $\phi=m\,{\cal J}^{0^{\prime}0}+s\,{\cal J}^{12}$ is
interpreted as a massive spinning particle in AdS for $m\neq s$ otherwise
$\phi^{\prime}=s({\cal J}^{0^{\prime}0}+{\cal J}^{12})$ can be regarded as a
massless particle in AdS spacetime and this implies that $m=s$ is a classical
version of the AdS massless condition. 444Actually, there is a subtlety from
the magnitude of $m$ and $s$. In comparison to the quantum theory, $m=s$ is an
unitary bound and $m<s$ can be regarded as a non-unitary region. However,
according to the orbit method, each coadjoint orbit has the relation to the
unitary representation. This apparent inconsistency can be resolved through
the existence of novel unitary representations in the $m<s$ region. This issue
had been discussed in [7] and we will skip it here. See [34] for more
discussions on the contraction of AdS coadjoint orbits to Poincaré ones.
Therefore, we will call the representative $\phi$ as a AdS massive spinning
particle and $\phi^{\prime}$ to a AdS massless spinning particle in the rest
of the paper.

##  4 Worldline Actions

In Section 2, we have discussed about how to formulate manifestly covariant
worldline actions obeying either $ISO(1,d-1)$ or $SO(2,d-1)$ isometry. In this
section, we will construct the worldline action of Poincaré and AdS spinning
particles from the representative vectors discussed in the previous section.

###  4.1 Poincaré Spinning Particle

Let us consider the massive spinning particle of Poincaré group. From the
representative (3.11), the homogeneous part will be given by the rank two
matrix

$\phi_{H}=\frac{1}{2}\left(\begin{array}[]{cccc}0&s&0&\cdots\\\
-s&0&0&\cdots\\\ 0&0&0&\cdots\\\
{}\hfil&\vdots&{}\hfil&\ddots\end{array}\right)\,,$ (4.1)

where the first column corresponds to the $(\phi_{H})^{a1}$ and the second
column corresponds to the $(\phi_{H})^{a2}$. Note that the factor $1/2$ in
(4.1) is coming from the normalisation condition (2.1). From (4.1), one can
find $2\times(d+1)$ matrix $T$,

$\displaystyle
T=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}\sqrt{s}&0&\cdots\\\
0&\sqrt{s}&\cdots\end{array}\right)\,,$ (4.2)

such that (2.18) holds. From this $T$ matrix, $\varphi$ are given as

$\varphi_{M}=m^{2}\,,\quad(\varphi_{I})_{\alpha}=0\,,\quad(\varphi_{H})_{\alpha\beta}=\frac{1}{2}\left(\begin{array}[]{cc}s&0\\\
0&s\end{array}\right)\,.$ (4.3)

Following the substitutions (2.24), we set the variables for our convenience
as

$p_{a}=m\Lambda_{a0}\,,\ \pi^{a}=\frac{1}{\sqrt{s}}\Lambda^{a}{}_{1}\,,\
\chi^{a}=\sqrt{s}\,\Lambda^{a}{}_{2}\,,$ (4.4)

and put them into (2.23) then we can find the worldline action of the massive
spinning particle as

$\displaystyle S[x\,,p\,,\pi\,,\chi\,,A_{I}]=\int\,\Big{[}$ $\displaystyle
p_{a}{\rm d}x^{a}+\pi_{a}{\rm d}\chi^{a}$ (4.5)
$\displaystyle\qquad+A_{pp}\left(p^{2}+m^{2}\right)+A_{\pi\pi}\left(\pi^{2}-s^{2}\right)+A_{\chi\chi}\left(\chi^{2}-1\right)$
$\displaystyle\qquad+A_{p\pi}\left(p\cdot\pi\right)+A_{p\chi}\left(p\cdot\chi\right)+A_{\pi\chi}\left(\pi\cdot\chi\right)\Big{]}\,.$

Note that the worldline action of the massive spinning particle was also
discussed in many literatures, such that [35, 36, 11, 20, 33].

The six constraints have the following non-trivial Poisson brackets:

$\\{\pi^{2}\,,\pi\cdot\chi\\}=2\pi^{2}\approx 2s^{2}\,,\
\\{\chi^{2}\,,\pi\cdot\chi\\}=-2\chi^{2}\approx-2\,,\
\\{p\cdot\pi\,,p\cdot\chi\\}=p^{2}\approx-m^{2}\,.$ (4.6)

Among the six constraints, the first-class constraints are

$p^{2}\approx-m^{2}\,,\quad\pi^{2}+s^{2}\,\chi^{2}\approx 2s^{2}\,,$ (4.7)

satisfying Poisson bracket

$\\{p^{2}\,,\pi^{2}+s^{2}\,\chi^{2}\\}=0\,.$ (4.8)

This indicates that the first-class constraints form
${\mathbb{R}}\oplus{\mathfrak{u}}(1)$ Lie algebra. The embedding phase space
has $4d$ degrees of freedom but constraints kill $8$ degrees of freedom, so
the reduced phase space is $2(2d-4)$-dimensional. This number is identical to
the dimension of the coadjoint orbit of the representative (3.11).

For the massless case (3.13), (4.1) and (4.2) are given in the same as the
massive case, so $\varphi$ are written in almost same way as the massive case,
except $\varphi_{M}$, such that

$\varphi_{M}=0\,,\quad(\varphi_{I})_{\alpha}=0\,,\quad(\varphi_{H})_{\alpha\beta}=\frac{1}{2}\left(\begin{array}[]{cc}s&0\\\
0&s\end{array}\right)\,.$ (4.9)

When we take same variables as (4.4), we find the worldline action for the
massless spinning particle as

$\displaystyle S[x\,,p\,,\pi\,,\chi\,,A_{I}]=\int\,\Big{[}$ $\displaystyle
p_{a}{\rm d}x^{a}+\pi_{a}{\rm d}\chi^{a}$ (4.10)
$\displaystyle\qquad+A_{pp}\left(p^{2}\right)+A_{\pi\pi}\left(\pi^{2}-s^{2}\right)+A_{\chi\chi}\left(\chi^{2}-1\right)$
$\displaystyle\qquad+A_{p\pi}\left(p\cdot\pi\right)+A_{p\chi}\left(p\cdot\chi\right)+A_{\pi\chi}\left(\pi\cdot\chi\right)\Big{]}\,.$

The difference between (4.5) and (4.10) is that in the latter, $m^{2}$ is set
to zero. Moreover, while the parameter $m$ from the representative vector
(3.11) appears in the action (4.5), the parameter $E$ from the representative
vector (3.13) does not appears in the action (4.10), whereas the parameter $m$
of the massive particle (3.11) is appeared in the action . This implies that
$E$ does not parametrise the coadjoint orbit thus it is a dummy variable which
can be tuned by coadjoint action with $J_{+-}$, acting as a scaling change of
$E$.

Among the six constraints of massless particle action (4.10), the non-trivial
Poisson brackets are

$\\{\pi^{2}\,,\pi\cdot\chi\\}=2\pi^{2}\approx 2s^{2}\,,\
\\{\chi^{2}\,,\pi\cdot\chi\\}=-2\chi^{2}\approx-2\,,\
\\{p\cdot\pi\,,p\cdot\chi\\}=p^{2}\approx 0\,,$ (4.11)

and the first-class constraints are

$p^{2}\approx 0\,,\quad\pi^{2}+s^{2}\,\chi^{2}\approx 2s^{2}\,,\quad
p\cdot\pi\approx 0\,,\quad p\cdot\chi\approx 0\,,$ (4.12)

satisfying Poisson brackets

$\begin{gathered}\\{p\cdot\pi\,,p\cdot\chi\\}=p^{2}\,,\quad\\{p\cdot\pi\,,\pi^{2}+s^{2}\,\chi^{2}\\}=2s^{2}\,p\cdot\chi\,,\quad\\{p\cdot\chi\,,\pi^{2}+s^{2}\,\chi^{2}\\}=2s^{2}\,p\cdot\pi\,,\\\
\\{p^{2}\,,\pi^{2}+s^{2}\,\chi^{2}\\}=\\{p^{2}\,,p\cdot\pi\\}=\\{p^{2}\,,p\cdot\chi\\}=0\,.\end{gathered}$
(4.13)

This indicates that the first-class constraints form
${\mathfrak{u}}(1)\niplus{\mathfrak{heis}}_{2}$ algebra. Since constraints
kills 10 degrees of freedom, the physical degrees of freedom is $2(2d-5)$.

Remark that, in both massive and massless actions, the Lie algebra, which
corresponds to the first-class constraints, are same with stabiliser algebra
when the $d$-dependent part is removed. It tells us about strong relation
between the coadjoint orbit and the constraints, known as the dual pair
correspondence [7, 37].

###  4.2 AdS Spinning Particle

In this section, we concentrate on the AdS spinning particle with the
representative vector (3.19). The representative vector (3.19) can be
expressed as the $(d+1)\times(d+1)$ matrix

$\phi=\frac{1}{2}\left(\begin{array}[]{cccccc}0&m&0&0&0&\cdots\\\
-m&0&0&0&0&\cdots\\\ 0&0&0&s&0&\cdots\\\ 0&0&-s&0&0&\cdots\\\
{}\hfil&{}\hfil&\vdots&{}\hfil&{}\hfil&\ddots\end{array}\right)\,.$ (4.14)

Since the rank of $\phi$ is 4, $\phi$ can be transformed into the $4\times 4$
symplectic matrix $\Omega$ by a $4\times(d+1)$ transformation matrix,

$T=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccccc}\sqrt{m}&0&0&0&\cdots\\\
0&\sqrt{m}&0&0&\cdots\\\ 0&0&\sqrt{s}&0&\cdots\\\
0&0&0&\sqrt{s}&\cdots\end{array}\right)\,.$ (4.15)

From the above, we find $\varphi$

$\varphi=\frac{1}{2}\left(\begin{array}[]{cccc}-m&0&0&0\\\ 0&-m&0&0\\\
0&0&s&0\\\ 0&0&0&s\end{array}\right)\,,$ (4.16)

which appear next to the constraints. From (2.19), we substitute $X\in
SO(2,d-1)$, as follows

$X^{A}=\sqrt{m}\,X^{A}{}_{0^{\prime}}\,,\quad
P^{A}=\frac{1}{\sqrt{m}}X^{A}{}_{0}\,,\quad\chi^{A}={\sqrt{s}}\,X^{A}{}_{1}\,,\quad\pi^{A}=\frac{1}{\sqrt{s}}X^{A}{}_{2}\,,$
(4.17)

where $A,B=0^{\prime}\,,0\,,1\,,\ldots\,,d-1$ are indices for ambient space.
Implementing (4.16) and (4.17) into (2.20), we can find the manifestly
covariant worldline action for the AdS spinning particle as

$\displaystyle S[X\,,P\,,\pi\,,\chi\,,A_{I}]=\int\,\Big{[}$ $\displaystyle
P_{A}{\rm d}X^{A}+\pi_{A}{\rm
d}\chi^{A}+A_{PP}\left(P^{2}+m^{2}\right)+A_{XX}\left(X^{2}+1\right)$ (4.18)
$\displaystyle\qquad+A_{\pi\pi}\left(\pi^{2}-s^{2}\right)+A_{\chi\chi}\left(\chi^{2}-1\right)$
$\displaystyle\qquad+A_{PX}\left(P\cdot
X\right)+A_{P\pi}\left(P\cdot\pi\right)+A_{p\chi}\left(P\cdot\chi\right)$
$\displaystyle\qquad+A_{X\pi}\left(X\cdot\pi\right)+A_{X\chi}\left(X\cdot\chi\right)+A_{\pi\chi}\left(\pi\cdot\chi\right)\Big{]}\,.$

Note that the constraint $P^{2}\approx m^{2}$ can be understood as a mass-
shell constraint, $P\cdot X\approx 0$ can be understood as a homogeneity
condition, and $X^{2}=-1$ can be interpreted that $X$ is an ambient space
coordinate of AdS spacetime. The action of AdS particles are also discussed in
[38, 39, 40]

There are eight non-trivial Poisson brackets between the constraints:

$\begin{gathered}\\{P^{2}\,,P\cdot X\\}=2P^{2}\approx-2m^{2}\,,\
\\{X^{2}\,,P\cdot X\\}=-2X^{2}\approx 2\,,\\\
\\{\pi^{2}\,,\pi\cdot\chi\\}=2\pi^{2}\approx 2s^{2}\,,\
\\{\chi^{2}\,,\pi\cdot\chi\\}=-2\chi^{2}\approx-2\,,\\\
\\{P\cdot\pi\,,P\cdot\chi\\}=P^{2}\approx-m^{2}\,,\
\\{P\cdot\pi\,,X\cdot\pi\\}=\pi^{2}\approx s^{2}\,,\\\
\\{X\cdot\pi\,,X\cdot\chi\\}=X^{2}\approx-1\,,\
\\{P\cdot\chi\,,X\cdot\chi\\}=\chi^{2}\approx 1\,.\end{gathered}$ (4.19)

Among the constraints, there are two first-class constraints given as

$P^{2}+X^{2}\,,\quad\pi^{2}+\chi^{2}\,,$ (4.20)

for any $m$ and $s$, and two more first-class constraints are appeared

$P\cdot\chi+X\cdot\pi\,,\quad X\cdot\chi-P\cdot\pi\,,$ (4.21)

only if $s=m$. As we discussed in Section 3.2, $m=s$ is the massless condition
of AdS particle. Since the embedding phase space has $4(d+1)$ degrees of
freedom, genuine degrees of freedom for the massive case is $2(2d-4)$. On the
other hand, the number of degrees of freedom for the massless case is
$2(2d-5)$ due to the two additional first-class constraints (4.21).

For the massive case, the first-class constraints (4.20) satisfy the Poisson
bracket,

$\\{P^{2}+X^{2}\,,\pi^{2}+\chi^{2}\\}=0\,,$ (4.22)

so they form ${\mathfrak{u}}(1)\oplus{\mathfrak{u}}(1)$ algebra. In the case
of the massless particle, (4.20) and (4.21) satisfy

$\begin{gathered}\\{P^{2}+X^{2}\,,P\cdot\chi+X\cdot\pi\\}=2(P\cdot\pi-X\cdot\chi)\,,\
\\{P^{2}+X^{2}\,,X\cdot\chi-P\cdot\pi\\}=2(P\cdot\chi+X\cdot\pi)\,,\\\
\\{\pi^{2}+\chi^{2}\,,P\cdot\chi+X\cdot\pi\\}=2(P\cdot\pi-X\cdot\chi)\,,\
\\{\pi^{2}+\chi^{2}\,,X\cdot\chi-P\cdot\pi\\}=2(X\cdot\pi+P\cdot\chi)\,,\\\
\\{P^{2}+X^{2}\,,\pi^{2}+\chi^{2}\\}=0\,,\
\\{P\cdot\chi+X\cdot\pi\,,X\cdot\chi-P\cdot\pi\\}=P^{2}+X^{2}+\chi^{2}+\pi^{2}\,.\end{gathered}$
(4.23)

As consequences, these constraints form ${\mathfrak{u}}(1,1)$ algebra.
Similarly to the Poincaré case, the Lie algebras formed from the first-class
constraints are identical to the stabiliser algebras without $d$-dependent
pieces. These facts also implies the dual pair correspondence [7, 37].

##  5 Conclusion

We have discussed the construction of manifestly covariant worldline actions
using orbit method by introducing Hamiltonian constraints that make possible
to formulate ab initio worldlline action. To describe Poincaré particles, we
choose the inhomogeneous representative vector for the mass type of the
particle then we find the homogeneous representative vectors from the scalar
particle stabiliser subalgebra. From the representative vector, one can easily
find the stabiliser algebra and identify the geometry of the coadjoint orbit.

For AdS particles, we chose the representatives which can be regarded as
analogues of the Poincaré massive particles. Remarkably, when the mass
parameter becomes equal to the spin one, we find a smaller coadjoint orbit
which can be interpreted as the phase space of the AdS massless particle.

Starting with these representatives, we can formulate worldline actions. One
systematic and convenient approach to construct worldline actions with more
appropriate coordinate is by imposing the defining condition of the isometry
as the Hamiltonian constraint. In this manner, we can construct a manifestly
covariant worldline action. When we compare the stabiliser subalgebra with the
first-class constraint, not only the coadjoint orbits and the physical phase
spaces of particle have the same degrees of the freedom, but also the
stabiliser algebras are equivalent to the first-class constraints if we ignore
$d$-dependent parts of the stabiliser algebras like ${\mathfrak{so}}(d-n)$ or
${\mathfrak{iso}}(d-n)$. This equivalence between two objects gives a hint of
the dual pair correspondence, which implies that there exist one-to-one
correspondence between coadjoint orbit of the isometry and the constraint.

The method discussed in this paper can be adapted to different contexts, such
as twistorial particles or particles obeying different isometries like dS or
Galilean symmetry. Furthermore, we hope that this work shed new light on the
various physics topics making use of worldline action.

##  6 Acknowledgement

I am very glad to have the kind discussion with Euihun Joung and also thanks
to Sejin Kim and Minkyeu Cho for discussions about the draft. The authors
declare no competing financial interest. The author was supported by the
National Research Foundation of Korea (NRF) grant funded by the Korea
government (MSIT) (No. 2022R1F1A1074977).

##  Appendix A Transformation of an Anti-symmetic Matrix into a Symplectic
Matrix

Let suppose $\phi$ to be an $N\times N$ anti-symmetric matrix with real
entities. Then $\phi^{T}\,\phi$ is a symmetric matrix, so it should have
positive or null eigenvalues:

$\displaystyle\phi^{T}\phi\,v_{i}=\lambda_{i}^{2}v_{i}\,,$ (A.1)
$\displaystyle\phi^{T}\phi\,\kappa_{a}=0\,,$

where $i=1\,,\cdots\,,M=\rank\phi$, We assume that $\lambda_{i}$ are positive,
without loss of generality. Let us define new $M$ vectors $w_{i}$ as

$w_{i}=\frac{1}{\lambda_{i}}\phi\,v_{i}\,.$ (A.2)

Since $\phi$ is an anti-symmetric matrix, $w_{i}$ is also an eigenvector of
$\phi^{T}\phi$ with the eigenvalue $\lambda_{i}^{2}$, as follows

$\phi^{T}\phi\,w_{i}=\frac{1}{\lambda_{i}}\phi^{T}\phi\phi\,v_{i}=-\lambda_{i}\phi^{T}\,v_{i}=\lambda_{i}^{2}w_{i}\,.$
(A.3)

But one can check that $v_{i}$ and $w_{i}$ are degenerated eigenvectors

$\displaystyle v_{i}\cdot w_{i}$
$\displaystyle=\left(-\frac{1}{\lambda_{i}}\phi\,w_{i}\right)\cdot\left(\frac{1}{\lambda_{i}}\phi\,v_{i}\right)$
(A.4)
$\displaystyle=-\frac{1}{\lambda_{i}^{2}}\,w_{i}^{T}\,\phi^{T}\phi\,v_{i}$
$\displaystyle=-w_{i}\cdot v_{i}\,.$

Thus, if we choose the matrix $Q$ as

$Q=\left(\frac{1}{\sqrt{\lambda_{1}}}\vec{v}_{1}\,,\
\frac{1}{\sqrt{\lambda_{1}}}\vec{w}_{1}\,,\
\cdots\frac{1}{\sqrt{\lambda_{M}}}\vec{v}_{M}\,,\
\frac{1}{\sqrt{\lambda_{M}}}\vec{w}_{M}\,,\
\vec{\kappa}_{1}\,,\cdots\vec{\kappa}_{N-2M}\right)$ (A.5)

then it satisfies

$Q^{T}\phi\,Q=\Omega\,.$ (A.6)

By construction, $Q$ is a non-degenerated matrix, so it can be inverted
easily. Then, we can find $T$ matrix which satisfies

$T^{T}\,\Omega\,T=\phi\,.$ (A.7)

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