# A classical mechanical model of two interacting massless particles in de
Sitter space and its quantization

Naohiro Kanda<EMAIL_ADDRESS>Satoshi Okano
<EMAIL_ADDRESS>Research Institute of Science and Technology,
College of Science and Technology, Nihon University, Chiyoda-ku, Tokyo
101-8308, Japan

###### Abstract

A conformally invariant model of two interacting massless particles in
Minkowski space was proposed by Casalbuoni and Gomis [1]. We generalize this
model to the case of de Sitter space from the perspective of geodesic
distance, in such a way that the resulting, generalized action reduces to the
original action in a limit that de Sitter radius goes to infinity. We analyze
the Hamiltonian formulation in accordance with Dirac’s prescription for
constrained Hamiltonian systems and carry out its subsequent canonical
quantization in coordinate representation following DeWitt. As the result, we
derive a fourth-order differential wave equation for bilocal fields that, in
the infinite radius limit, reproduces one obtained in the original model for
Minkowski space case.

††journal: Physics Letters B

##  1 Introduction

The early universe during cosmological inflation can be approximated as de
Sitter space [2, 3, 4]. On the other hand, the universe after inflation
approaches Minkowski space, which can be regarded as a limit that de Sitter
radius $R$ goes to infinity. During inflation and for a while after its end,
the energy scale of the universe remains much higher than the electroweak
scale. In the period, except for the Higgs particle, all the elementary
particles included in the standard model remain massless. In particular, after
inflation, their dynamics should have conformal invariance in Minkowski space.
However, until the end of inflation, during which the arena of physics is de
Sitter space, it is not clear whether conformal invariance is realized. This
observation motivates us to consider a massless system that does not have
conformal invariance at the beginning but restores it in the limit
$R\rightarrow\infty$. In this letter, we construct such a model at classical
and quantum mechanical levels.

A conformally invariant model of interacting massless particles in Minkowski
space was given by Casalbuoni and Gomis [1]. An action for two free massless
particles in Minkowski space is

$\displaystyle S_{0}=\int d\tau L_{0},\;\quad
L_{0}=-\sum\limits^{2}_{i=1}\frac{\eta_{\mu\nu}\dot{x}^{\mu}_{i}\dot{x}^{\nu}_{i}}{2e_{i}}\>,$
(1)

where $\mu,\nu=0,1,2,3$, $\eta_{\mu\nu}=\eta^{\mu\nu}=\mathrm{diag}(+,-,-,-)$,
$x^{\mu}_{i}(\tau),i=1,2$ are the spacetime coordinates for the two particles,
$e_{i}(\tau)$ are einbeins, $\tau$ is the worldline parameter common to the
two particles and $\dot{x}^{\mu}_{i}=dx_{i}^{\mu}/d\tau$. The free action
$S_{0}$ is invariant under both reparameterization of $\tau$ and conformal
transformations in Minkowski space. An interaction term $S_{\mathrm{int}}$
that has the same invariance as $S_{0}$ and does not include the velocities
$\dot{x}_{i}$ is uniquely determined [1] as

$\displaystyle S_{\mathrm{int}}=\int d\tau V(r)\>,\quad
V(r)=-\frac{\alpha^{2}}{4}\frac{\sqrt{e_{1}e_{2}}}{r^{2}}\>,$ (2)

where $\alpha$ is the coupling constant, $r^{\mu}:=x_{1}^{\mu}-x_{2}^{\mu}$
are the relative coordinates and $r^{2}=\eta_{\mu\nu}r^{\mu}r^{\nu}$. Thus,
the conformally invariant action for two interacting massless particles is
given as

$\displaystyle S:=S_{0}+S_{\mathrm{int}}=\int
d\tau\left(-\sum^{2}_{i=1}\frac{1}{2}\frac{\eta_{\mu\nu}\dot{x}^{\mu}_{i}\dot{x}^{\nu}_{i}}{e_{i}}-\frac{\alpha^{2}}{4}\frac{\sqrt{e_{1}e_{2}}}{r^{2}}\right)\>.$
(3)

In the Hamiltonian formulation based on this action $S$, first-class
constraints associated with the reparameterization invariance occur. After
canonical quantization, they are replaced with the physical state conditions
and read as a fourth-order differential equation:

$\displaystyle\left(\square_{1}\square_{2}-\frac{\alpha^{4}}{16r^{4}}\right)\varphi(x_{1},x_{2})=0\>,$
(4)

where $\square_{i}=\partial^{2}/\partial x_{i\mu}\partial x^{\mu}_{i}$ and
$\varphi(x_{1},x_{2})$ is a wave function of the system, which we call a
bilocal field in the sense that the wave function depends on two spacetime
points. Our purpose is to generalize the action $S$ and its consequent wave
equation to the case of de Sitter space.

In generalizing the action (3), we rely on the fact that the squared relative
coordinates $r^{2}$ are considered as a special case of a squared geodesic
distance. A geodesic distance $\sigma(x_{1},x_{2})$ between two points
$x_{1}^{\mu}(\tau)$ and $x_{2}^{\mu}(\tau)$ in a general spacetime endowed
with a metric $g_{\mu\nu}(x)$ is given by

$\displaystyle\sigma(x_{1},x_{2})=\int_{s_{1}}^{s_{2}}ds\sqrt{\pm
g_{\mu\nu}(x){x}^{\prime\mu}{x}^{\prime\nu}}\>,$ (5)

with $x_{1}^{\mu}(\tau)=x^{\mu}(s_{1}),\;x_{2}^{\mu}(\tau)=x^{\mu}(s_{2})$,
where $x^{\mu}(s)$ is the geodesic curve joining the two points, $s$ is a
parameter of the geodesic connecting the two points,
$x^{\prime\mu}:=dx^{\mu}/ds$, and the positive (negative) sign is chosen if
the geodesic is timelike (spacelike). In particular,
$\sigma^{2}(x_{1},x_{2})=\pm r^{2}$ holds for Minkowski space. Hence, the
system described by the action (3) can be interpreted as a system of two
particles interacting with each other along the geodesic connecting them. (For
example, such a 2-particle system in a curved spacetime has been considered
for the shockwave background in Ref. [5].)

From this point of view, we generalize the action (3) to a general curved
background as

$\displaystyle S_{g}$ $\displaystyle=\int
d\tau\left(-\sum^{2}_{i=1}\frac{1}{2}\frac{g_{\mu\nu}(x_{i})\dot{x}^{\mu}_{i}\dot{x}^{\nu}_{i}}{e_{i}}\mp\frac{\alpha^{2}}{4}\frac{\sqrt{e_{1}e_{2}}}{\sigma^{2}(x_{1},x_{2})}\right)\>,$
(6)

where the negative (positive) sign is chosen if the two particles are timelike
(spacelike) sepalated. For a spacetime such that $\sigma^{2}\rightarrow\pm
r^{2}$ as $g_{\mu\nu}\rightarrow\eta_{\mu\nu}$, this action smoothly reduces
to the conformally invariant action (3) in Minkowski space. However, it is
noted that the generalized action (6) itself does not necessarily have
conformal symmetry before taking such a limit.

This letter is organized as follows: In the next section, we write the action
$S_{g}$ for the case of 4-dimensional de Sitter space by using embedding
coordinates, commonly used to describe (anti-)de Sitter space. (See for
example [6, 7].) This description enables us to express the geodesic distance
in a simple form without taking a particular coordinate system of de Sitter
space. In Sec. 3, we consider the Hamiltonian formulation in accordance with
Dirac’s procedure [8] for constrained systems. In Sec. 4, new canonical
variables constituting canonical pairs in the sense of Dirac brackets are
introduced. In Sec. 5, we apply the canonical quantization procedure in
coordinate representation for curved configuration spaces following DeWitt
[9]. Sec. 6 is devoted to conclude this letter.

##  2 Lagrangian formulation

In this section, we specify the form of the action (6) for the case of a
4-dimensional de Sitter space in terms of embedding coordinates and then
consider its limit $R\rightarrow\infty$.

A $4$-dimensional de Sitter space $\mathrm{dS}_{4}$ of radius $R$ can be
embedded into a $(1+4)$-dimensional flat space $\mathbb{R}^{1,4}$ as a
hyperboloid

$\displaystyle\mathrm{dS}_{4}=\\{X^{A}\in\mathbb{R}^{1,4}\;|\;\eta_{AB}X^{A}X^{B}=-R^{2}\>\\}\>,$
(7)

where $X^{A}\>(A=0,1,2,3,5)$ are embedding coordinates and
$\eta_{AB}=\mathrm{diag}(+1,-1,-1,-1,-1)$. This hypersurface has a
parameterization

$\displaystyle X^{\mu}=x^{\mu}\>,\quad
X^{5}=\pm\sqrt{\eta_{\mu\nu}x^{\mu}x^{\nu}+R^{2}}$ (8)

with $\mu,\nu=0,1,2,3$. Then the induced metric on the de Sitter space for
this parameterization has the form

$\displaystyle
g_{\mu\nu}=\eta_{\mu\nu}-\frac{x_{\mu}x_{\nu}}{\eta_{\alpha\beta}x^{\alpha}x^{\beta}+R^{2}}\>,$
(9)

which approaches the 4-dimensional Minkowski metric $\eta_{\mu\nu}$ in the
limit that $R\rightarrow\infty$.

In terms of embedding coordinates, a Lagrangian for two free massless
particles in de Sitter space is written as

$\displaystyle
L_{0(\mathrm{dS})}=-\sum^{2}_{i=1}\frac{1}{2}\frac{\dot{X}^{2}_{i}}{e_{i}}-\sum^{2}_{i=1}\lambda_{i}(X^{2}_{i}+R^{2}),\rule[0.0pt]{0.0pt}{14.22636pt}$
(10)

where $X_{i}^{A}=X^{A}_{i}(\tau)\>(i=1,2)$ are the positions of two particles
and $\lambda_{i}\>(i=1,2)$ are the multipliers.

In this letter, we restrict ourselves to the case that the particles are
spacelike separated, in which the geodesic distance $\sigma$ is written in
terms of the embedding coordinates as (cf. Ref. [10])

$\displaystyle\sigma(X_{1},X_{2})$
$\displaystyle=R\cos^{-1}\left(-\frac{X_{1}\cdot X_{2}}{R^{2}}\right).$ (11)

Thus the interaction term is given as

$\displaystyle
V_{\mathrm{dS}}=\frac{\alpha^{2}}{4}\frac{\sqrt{e_{1}e_{2}}}{\sigma^{2}(X_{1},X_{2})}\>.$
(12)

From Eqs. (10) and (12), the generalized action (6) for the case of de Sitter
space becomes

$\displaystyle S_{\mathrm{dS}}:=\int d\tau\,L_{\mathrm{dS}}\>,$ (13)

with the Lagrangian is expressed as

$\displaystyle L_{\text{dS}}$
$\displaystyle:=L_{0(\mathrm{dS})}+V_{\mathrm{dS}}$
$\displaystyle=-\sum^{2}_{i=1}\frac{1}{2}\frac{\dot{X}^{2}_{i}}{e_{i}}+\frac{\alpha^{2}}{4}\frac{\sqrt{e_{1}e_{2}}}{\sigma^{2}}-\sum^{2}_{i=1}\lambda_{i}(X^{2}_{i}+R^{2})\>.$
(14)

The action $S_{\text{dS}}$ is invariant under the reparameterization
$\tau\rightarrow\tau^{\prime}(\tau)$ with the transformation rules of
$X_{i}^{A},e_{i}$ and $\lambda_{i}$:

$\displaystyle X^{A}_{i}(\tau)$ $\displaystyle\>\rightarrow\>X^{\prime
A}_{i}(\tau^{\prime})=X^{A}_{i}(\tau)\>,$ (15a) $\displaystyle e_{i}(\tau)$
$\displaystyle\>\rightarrow\>e^{\prime}_{i}(\tau^{\prime})=\frac{d\tau}{d\tau^{\prime}}e_{i}(\tau)\>,$
(15b) $\displaystyle\lambda_{i}(\tau)$
$\displaystyle\>\rightarrow\>\lambda^{\prime}_{i}(\tau^{\prime})=\frac{d\tau}{d\tau^{\prime}}\lambda_{i}(\tau)\>.$
(15c)

The action $S_{\mathrm{dS}}$ also leaves invariant under the isometry of de
Sitter space

$\displaystyle
X_{i}^{A}\>\rightarrow\>X^{\prime}{}_{i}^{A}=L^{A}{}_{B}X_{i}^{B}\>,$ (16)

where $e_{i}$ and $\lambda_{i}$ are left unchanged and $(L^{A}{}_{B})\in
SO(4,1)$ is a constant matrix. Although the action $S$, given in Eq. (3), is
invariant under conformal transformations in Minkowski space, its generalized
action $S_{\mathrm{dS}}$ for de Sitter space cannot be invariant under an
infinitesimal conformal isometry [11]

$\displaystyle\delta X_{i}^{A}$
$\displaystyle=(b_{B}X_{i}^{B})X_{i}^{A}+R^{2}b^{A}\>,$ (17)

with no sum over $i$, where $b^{A}$ are infinitesimal constant parameters. 111
With the transformation law $\delta e_{i}=2b_{A}X_{i}^{A}e_{i}$ concerning the
infinitesimal conformal isometry (17), we see that the following action is
conformally invariant:  $\displaystyle\tilde{S}=$ $\displaystyle\int
d\tau\left\\{L_{0(dS)}-\frac{\alpha^{2}}{4}\frac{\sqrt{e_{1}e_{2}}}{(X_{1}-X_{2})^{2}}\right\\}\>.$
However, $(X_{1}-X_{2})^{2}$ does not represent the squared geodesic distance
between the two points $X_{1}^{A}$ and $X_{2}^{A}$ in de Sitter space.

We show that the action $S_{\mathrm{dS}}$ reproduces the action $S$ for
Minkowski space in a limit as $R\rightarrow\infty$. By using the
parameterization (8) and the induced metric (9), the free Lagrangian (10) is
written as

$\displaystyle
L_{0(\mathrm{dS})}=-\sum_{i=1}^{2}\left(\eta_{\mu\nu}-\frac{x_{i\mu}x_{i\nu}}{\eta_{\alpha\beta}x_{i}^{\alpha}x_{i}^{\beta}+R^{2}}\right)\frac{\dot{x}_{i}^{\mu}\dot{x}^{\nu}_{i}}{2e_{i}}\>,$
(18)

while the geodesic distance (11) can be expanded around the point
$\sqrt{-r^{2}}/\sqrt{2}R=0$ as

$\displaystyle\sigma\big{(}X_{1}(x_{1}),X_{2}(x_{2})\big{)}=\sqrt{-r^{2}}+\mathcal{O}\left(R^{-1}\right)\>\>$
(19)

with $r^{\mu}=x_{1}^{\mu}-x_{2}^{\mu}$ and
$r^{2}=\eta_{\mu\nu}r^{\mu}r^{\nu}<0$. From Eqs. (18) and (19) we see that
$L_{0(\mathrm{dS})}\rightarrow\>L_{0}$ and
$V_{\mathrm{dS}}\>\rightarrow\>V(r)$ as $R\rightarrow\infty$. Therefore the
action $S_{\mathrm{dS}}$ reproduces the action $S$ for Minkowski space case in
the limit of the infinite de Sitter radius.

##  3 Hamiltonian formulation

In this section, we consider the constrained Hamilton system governed by the
Lagrangian (2) in accordance with Dirac’s procedure.

We treat $(X_{1}^{A},X_{2}^{A},e_{1},e_{2},\lambda_{1},\lambda_{2})$ as the
canonical coordinates and define the canonical momenta
$(P_{1}{}_{A},P_{2}{}_{A},P_{e_{1}},P_{e_{2}},P_{\lambda_{1}},P_{\lambda_{2}})$
conjugate to them as

$\displaystyle P_{1}{}_{A}$ $\displaystyle:=\frac{\partial
L_{\mathrm{dS}}}{\partial\dot{X}_{1}^{A}}=-\frac{\dot{X}_{1}{}_{A}}{e_{1}}\>,$
$\displaystyle P_{2}{}_{A}$ $\displaystyle:=\frac{\partial
L_{\mathrm{dS}}}{\partial\dot{X}_{2}^{A}}=-\frac{\dot{X}_{2}{}_{A}}{e_{2}}\>,$
(20a) $\displaystyle P_{e_{1}}$ $\displaystyle:=\frac{\partial
L_{\mathrm{dS}}}{\partial\dot{e}_{1}}=0\>,$ $\displaystyle P_{e_{2}}$
$\displaystyle:=\frac{\partial L_{\mathrm{dS}}}{\partial\dot{e}_{2}}=0\>,$
(20b) $\displaystyle P_{\lambda_{1}}$ $\displaystyle:=\frac{\partial
L_{\mathrm{dS}}}{\partial\dot{\lambda}_{1}}=0\>,$ $\displaystyle
P_{\lambda_{2}}$ $\displaystyle:=\frac{\partial
L_{\mathrm{dS}}}{\partial\dot{\lambda}_{2}}=0\>.$ (20c)

The nonvanishing Poisson brackets between all the canonical variables are

$\displaystyle\\{X_{1}^{A}\;,\>P_{1B}\\}$ $\displaystyle=\delta^{A}_{B}\>,$
$\displaystyle\\{X_{2}^{A}\;,\>P_{2B}\\}$ $\displaystyle=\delta^{A}_{B}\>,$
(21a) $\displaystyle\\{e_{1}\;,\>P_{e_{1}}\\}$ $\displaystyle=1\>,$
$\displaystyle\\{e_{2}\;,\>P_{e_{2}}\\}$ $\displaystyle=1\>,$ (21b)
$\displaystyle\\{\lambda_{1}\;,\>P_{\lambda_{1}}\\}$ $\displaystyle=1\>,$
$\displaystyle\\{\lambda_{2}\;,\>P_{\lambda_{2}}\\}$ $\displaystyle=1\>.$
(21c)

From Eq. (20), the canonical Hamiltonian $H_{\mathrm{C}}$ is obtained by the
Legendre transform of $L_{\mathrm{dS}}$ as

$\displaystyle H_{\mathrm{C}}:=$
$\displaystyle\>P_{1A}\dot{X}_{1}^{A}+P_{2A}\dot{X}_{2}^{A}+P_{e_{1}}\dot{e}_{1}+P_{e_{2}}\dot{e}_{2}$
$\displaystyle+P_{\lambda_{1}}\dot{\lambda}_{1}+P_{\lambda_{2}}\dot{\lambda}_{2}-L_{\mathrm{dS}}$
$\displaystyle=$
$\displaystyle\;-\frac{e_{1}}{2}P_{1}^{2}-\frac{e_{2}}{2}P_{2}^{2}-\frac{\alpha^{2}}{4}\frac{\sqrt{e_{1}e_{2}}}{\,\sigma^{2}\,}$
$\displaystyle\>+\lambda_{1}\Big{(}X_{1}^{2}+R^{2}\Big{)}+\lambda_{2}\Big{(}X_{2}^{2}+R^{2}\Big{)}\>.$
(22)

Eqs. (20b)-(20c) are treated as primary constraints

$\displaystyle\phi_{e_{1}}:=\>$ $\displaystyle P_{e_{1}}\approx 0\>,$
$\displaystyle\phi_{e_{2}}:=\>$ $\displaystyle P_{e_{2}}\approx 0\>,$ (23)
$\displaystyle\phi_{\lambda_{1}}:=\>$ $\displaystyle P_{\lambda_{1}}\approx
0\>,$ $\displaystyle\phi_{\lambda_{2}}:=\>$ $\displaystyle
P_{\lambda_{2}}\approx 0\>,$ (24)

where the symbol “$\approx$” denotes weak equality and $\phi$’s are referred
to as constraint quantities for primary constraints. Introducing Lagrange
multipliers $u_{e_{1}},u_{e_{2}},u_{\lambda_{1}},u_{\lambda_{2}}$ for the
primary constraints, we define the total Hamiltonian as

$\displaystyle H_{\mathrm{T}}:=\>$ $\displaystyle
H_{\mathrm{C}}+u_{e_{1}}\phi_{e_{1}}+u_{e_{2}}\phi_{e_{2}}+u_{\lambda_{1}}\phi_{\lambda_{1}}+u_{\lambda_{2}}\phi_{\lambda_{2}}\>.$
(25)

The $\tau$ evolution of a function $f$ of the canonical variables is governed
by

$\displaystyle\dot{f}=\big{\\{}f\,,\,H_{\mathrm{T}}\big{\\}}\>.$ (26)

Using this equation and Eq. (21), the $\tau$ evolution of the constraint
quantities for primary constraints are evaluated as

$\displaystyle\dot{\phi}_{e_{1}}$
$\displaystyle=\frac{P_{1}^{2}}{2}+\frac{\alpha^{2}}{8}\sqrt{\frac{e_{2}}{e_{1}}}\frac{1}{\sigma^{2}}\>,$
(27a) $\displaystyle\dot{\phi}_{e_{2}}$
$\displaystyle=\frac{P_{2}^{2}}{2}+\frac{\alpha^{2}}{8}\sqrt{\frac{e_{1}}{e_{2}}}\frac{1}{\sigma^{2}}\>,$
(27b) $\displaystyle\dot{\phi}_{\lambda_{1}}$
$\displaystyle=-\Big{(}X_{1}^{2}+R^{2}\Big{)}\>,$ (27c)
$\displaystyle\dot{\phi}_{\lambda_{2}}$
$\displaystyle=-\Big{(}X_{2}^{2}+R^{2}\Big{)}\>.$ (27d)

The consistency conditions
$(\dot{\phi}_{e_{1}},\dot{\phi}_{e_{2}},\dot{\phi}_{\lambda_{1}},\dot{\phi}_{\lambda_{2}})\approx
0$ require secondary constraints:

$\displaystyle\chi_{e_{1}}$
$\displaystyle:=\>P_{1}^{2}+\frac{\alpha^{2}}{4}\sqrt{\frac{e_{2}}{e_{1}}}\frac{1}{\sigma^{2}}\approx
0\>,$ (28a) $\displaystyle\chi_{e_{2}}$
$\displaystyle:=\>P_{2}^{2}+\frac{\alpha^{2}}{4}\sqrt{\frac{e_{1}}{e_{2}}}\frac{1}{\sigma^{2}}\approx
0\>,$ (28b) $\displaystyle\chi_{\lambda_{1}}$
$\displaystyle:=\>X_{1}^{2}+R^{2}\approx 0\>,$ (28c)
$\displaystyle\chi_{\lambda_{2}}$ $\displaystyle:=\>X_{2}^{2}+R^{2}\approx
0\>,$ (28d)

where $\chi$’s are constraint quantities for secondary constraints. The $\tau$
evolution of $\chi$’s are obtained as

$\displaystyle\dot{\chi}_{e_{1}}$
$\displaystyle=-\frac{\alpha^{2}}{8\sigma^{2}}\sqrt{\frac{e_{2}}{e_{1}}}\left(\frac{u_{e_{1}}}{e_{1}}-\frac{u_{e_{2}}}{e_{2}}+\frac{4R}{\sigma
K}\varepsilon_{-}\right)$ $\displaystyle\quad\,-4\lambda_{1}P_{1}\cdot
X_{1}\>,$ (29a) $\displaystyle\dot{\chi}_{e_{2}}$
$\displaystyle=\frac{\alpha^{2}}{8\sigma^{2}}\sqrt{\frac{e_{1}}{e_{2}}}\left(\frac{u_{e_{1}}}{e_{1}}-\frac{u_{e_{2}}}{e_{2}}+\frac{4R}{\sigma
K}\varepsilon_{-}\right)$ $\displaystyle\quad\,-4\lambda_{2}P_{2}\cdot
X_{2}\>,$ (29b) $\displaystyle\dot{\chi}_{\lambda_{1}}$
$\displaystyle=-2e_{1}P_{1}\cdot X_{1}\>,\qquad$ (29c)
$\displaystyle\dot{\chi}_{\lambda_{2}}$ $\displaystyle=-2e_{2}P_{2}\cdot
X_{2}\>,$ (29d)

where $K:=\sqrt{R^{4}-(X_{1}\cdot X_{2})^{2}}$ and
$\varepsilon_{-}:=e_{1}P_{1}\cdot X_{2}-e_{2}P_{2}\cdot X_{1}.$

From the consistency conditions for Eqs. (29c) and (29d), two more constraints
are required:

$\displaystyle\chi_{5}:=\>P_{1}\cdot X_{1}\approx
0\>\>,\qquad\chi_{6}:=\>P_{2}\cdot X_{2}\approx 0\>\>.$ (30)

Together with these, the consistency conditions $\dot{\chi}_{e_{1}}\approx 0$
and $\dot{\chi}_{e_{2}}\approx 0$ become identical and give the single
condition

$\displaystyle
u_{-}:=\frac{u_{e_{1}}}{e_{1}}-\frac{u_{e_{2}}}{e_{2}}=-\frac{4R}{\sigma
K}\varepsilon_{-}\>\>.$ (31)

The consistency conditions $\dot{\chi}_{5}\approx 0$ and
$\dot{\chi}_{6}\approx 0$ give further secondary constraints

$\displaystyle\chi_{7}:=$
$\displaystyle\>e_{1}P_{1}^{2}+\frac{\alpha^{2}}{2}\sqrt{e_{1}e_{2}}\frac{RX_{1}\\!\cdot
X_{2}}{\sigma^{3}K}+2\lambda_{1}X_{1}^{2}\approx 0\>\>,$ (32)
$\displaystyle\chi_{8}:=$
$\displaystyle\>e_{2}P_{2}^{2}+\frac{\alpha^{2}}{2}\sqrt{e_{1}e_{2}}\frac{RX_{1}\\!\cdot
X_{2}}{\sigma^{3}K}+2\lambda_{2}X_{2}^{2}\approx 0\>\>.$ (33)

From Eq. (28) and these equations, it can be shown that

$\displaystyle X_{1}^{2}\,\approx
X_{2}^{2}\approx-R^{2}\>,\qquad\lambda_{1}\,\approx\lambda_{2}\>.$ (34)

Using these weak equalities, we find that the consistency conditions
$\dot{\chi}_{7}\approx 0$ and $\dot{\chi}_{8}\approx 0$ determine
$u_{\lambda_{1}}$ and $u_{\lambda_{2}}$ as

$\displaystyle
u_{\lambda_{1}}=u_{\lambda_{2}}=-\frac{\alpha^{2}\sqrt{e_{1}e_{2}}}{4R\sigma^{3}K}\Big{(}F+1\Big{)}\varepsilon_{+}+\frac{1}{4}\lambda_{+}u_{+}\>,$
(35)

where

$\displaystyle\varepsilon_{+}:=$ $\displaystyle\,\>e_{1}P_{1}\cdot
X_{2}+e_{2}P_{2}\cdot X_{1}\>,$ (36) $\displaystyle F:=$
$\displaystyle\,\>1+\frac{(X_{1}\cdot X_{2})^{2}}{K^{2}}+\frac{3RX_{1}\cdot
X_{2}}{\sigma K}\>,$ (37) $\displaystyle\lambda_{+}:=$
$\displaystyle\,\>\lambda_{1}+\lambda_{2}\>,$ (38) $\displaystyle u_{+}:=$
$\displaystyle\,\>\frac{u_{e_{1}}}{e_{1}}+\frac{u_{e_{2}}}{e_{2}}\>.$ (39)

Thus, no new constraints are derived at this point. While the multipliers
$u_{-},\>u_{\lambda_{1}}$, and $u_{\lambda_{2}}$ are determined, $u_{+}$
remains undetermined due to the $\tau$ reparameterization invariance of the
system.

To classify all the constraints into first- and second-class, we consider the
following set of linear combinations of the constraint functions
$\psi_{a}\;(a=1,2,\ldots,12)$:

$\displaystyle\psi_{1}$
$\displaystyle=e_{1}\phi_{e_{1}}-e_{2}\phi_{e_{2}}+\frac{\mathcal{P}}{2R^{2}\lambda_{+}}(\lambda_{1}\phi_{\lambda_{1}}-\lambda_{2}\phi_{\lambda_{2}})\>,$
(40a) $\displaystyle\psi_{2}$
$\displaystyle=e_{1}\chi_{e_{1}}-e_{2}\chi_{e_{2}}-\frac{\mathcal{P}}{R^{2}\lambda_{+}}(\lambda_{1}\chi_{\lambda_{1}}-\lambda_{2}\chi_{\lambda_{2}})$
$\displaystyle\qquad+\frac{\alpha^{2}\sqrt{e_{1}e_{2}}\varepsilon_{-}F}{R\lambda_{+}\sigma^{3}K}(\lambda_{1}\phi_{\lambda_{1}}+\lambda_{2}\phi_{\lambda_{2}})\>,$
(40b) $\displaystyle\psi_{3}$
$\displaystyle=\lambda_{1}\phi_{\lambda_{1}}+\lambda_{2}\phi_{\lambda_{2}}\>,$
(40c) $\displaystyle\psi_{4}$
$\displaystyle=\chi_{7}+\chi_{8}+\frac{\alpha^{2}\sqrt{e_{1}e_{2}}\varepsilon_{-}F}{R\lambda_{+}\sigma^{3}K}(\lambda_{1}\phi_{\lambda_{1}}-\lambda_{2}\phi_{\lambda_{2}})\>,$
(40d) $\displaystyle\psi_{5}$
$\displaystyle=\lambda_{1}\phi_{\lambda_{1}}-\lambda_{2}\phi_{\lambda_{2}}\>,$
(40e) $\displaystyle\psi_{6}$ $\displaystyle=\chi_{7}-\chi_{8}\>,$ (40f)
$\displaystyle\psi_{7}$
$\displaystyle=\lambda_{1}\chi_{\lambda_{1}}+\lambda_{2}\chi_{\lambda_{2}}\>,$
(40g) $\displaystyle\psi_{8}$
$\displaystyle=\left\\{\frac{\alpha^{2}\sqrt{e_{1}e_{2}}(F+1)X_{1}\cdot
X_{2}}{R\lambda_{+}\sigma^{3}K}-4\right\\}(\lambda_{1}\phi_{\lambda_{1}}+\lambda_{2}\phi_{\lambda_{2}})$
$\displaystyle\qquad+\chi_{5}+\chi_{6}\>,$ (40h) $\displaystyle\psi_{9}$
$\displaystyle=\lambda_{1}\chi_{\lambda_{1}}-\lambda_{2}\chi_{\lambda_{2}}\>,$
(40i) $\displaystyle\psi_{10}$
$\displaystyle=\chi_{5}-\chi_{6}-\left(\frac{\mathcal{P}}{R^{2}\lambda_{+}}+2\right)(\lambda_{1}\phi_{\lambda_{1}}-\lambda_{2}\phi_{\lambda_{2}})\>,$
(40j) $\displaystyle\psi_{11}$
$\displaystyle=e_{1}\phi_{e_{1}}+e_{2}\phi_{e_{2}}+\lambda_{1}\phi_{\lambda_{1}}+\lambda_{2}\phi_{\lambda_{2}}\>,$
(40k) $\displaystyle\psi_{12}$
$\displaystyle=e_{1}\chi_{e_{1}}+e_{2}\chi_{e_{2}}-2(\lambda_{1}\chi_{\lambda_{1}}+\lambda_{2}\chi_{\lambda_{2}})$
$\displaystyle\quad+\frac{4R\varepsilon_{-}}{\sigma
K}(e_{1}\phi_{e_{1}}-e_{2}\phi_{e_{2}})$
$\displaystyle\quad+\frac{\alpha^{2}\sqrt{e_{1}e_{2}}\varepsilon_{+}(1+F)}{R\lambda_{+}\sigma^{3}K}(\lambda_{1}\phi_{\lambda_{1}}+\lambda_{2}\phi_{\lambda_{2}})\>,$
(40l) with $\mathcal{P}:=e_{1}P_{1}^{2}+e_{2}P_{2}^{2}\>.$

The Poisson brackets between the new constraint functions form the $12\times
12$ matrix

$\displaystyle\\{\psi_{a},\psi_{b}\\}=\begin{pmatrix}C&0\\\
0&0\end{pmatrix}\>,$ (41)

where $C=(C_{ab})\>(a,b=1,2\ldots,10)$ is given by

$\displaystyle C=\begin{pmatrix}C_{1}&&&&\\\ &C_{2}&&&\\\ &&C_{2}&&\\\
&&&C_{2}&\\\ &&&&C_{2}\end{pmatrix}$

with

$\displaystyle C_{1}=\begin{pmatrix}0&2V_{\mathrm{dS}}\\\
-2V_{\mathrm{dS}}&0\end{pmatrix}\>,\quad
C_{2}=\begin{pmatrix}0&-2R^{2}\lambda_{+}\\\
2R^{2}\lambda_{+}&0\end{pmatrix}\>,$

where $V_{\mathrm{dS}}$ has been defined in Eq. (12). Therefore, the
constraints $\psi_{11}\approx 0$ and $\psi_{12}\approx 0$ are classified as
first class and the constraints $\psi_{a}\approx 0\;(a=1,2,\ldots,10)$ are
classified as second class. The inverse matrix $C^{-1}=[(C^{-1})^{ab}]$ of $C$
reads as

$\displaystyle C^{-1}=\begin{pmatrix}C_{1}^{-1}&&&&\\\ &C_{2}^{-1}&&&\\\
&&C_{2}^{-1}&&\\\ &&&C_{2}^{-1}&\\\ &&&&C_{2}^{-1}\end{pmatrix}\>.$

Using this inverse matrix, we can define the Dirac bracket
$\\{F,\,G\\}_{\mathrm{D}}$ between any functions of canonical variables as

$\displaystyle\Big{\\{}F\,,\,G\Big{\\}}_{\mathrm{D}}$
$\displaystyle:=\Big{\\{}F,G\Big{\\}}-\Big{\\{}F,\psi_{a}\Big{\\}}(C^{-1})^{ab}\Big{\\{}\psi_{b},G\Big{\\}}$
$\displaystyle\>=\Big{\\{}F\,,\,G\Big{\\}}+\frac{1}{2V_{\mathrm{dS}}}\bigg{(}\Big{\\{}F,\psi_{1}\Big{\\}}\Big{\\{}\psi_{2},G\Big{\\}}-\Big{\\{}F,\psi_{2}\Big{\\}}\Big{\\{}\psi_{1},G\Big{\\}}\bigg{)}$
$\displaystyle\qquad-\frac{1}{2R^{2}\lambda_{+}}\bigg{(}\Big{\\{}F\,,\,\psi_{3}\Big{\\}}\Big{\\{}\psi_{4}\,,\,G\Big{\\}}-\Big{\\{}F\,,\,\psi_{4}\Big{\\}}\Big{\\{}\psi_{3}\,,\,G\Big{\\}}$
$\displaystyle\quad\quad+\Big{\\{}F\,,\,\psi_{5}\Big{\\}}\Big{\\{}\psi_{6}\,,\,G\Big{\\}}-\Big{\\{}F\,,\,\psi_{6}\Big{\\}}\Big{\\{}\psi_{5}\,,\,G\Big{\\}}$
$\displaystyle\quad\quad+\Big{\\{}F\,,\,\psi_{7}\Big{\\}}\Big{\\{}\psi_{8}\,,\,G\Big{\\}}-\Big{\\{}F\,,\,\psi_{8}\Big{\\}}\Big{\\{}\psi_{7}\,,\,G\Big{\\}}$
$\displaystyle\quad\quad+\Big{\\{}F\,,\,\psi_{9}\Big{\\}}\Big{\\{}\psi_{10}\,,\,G\Big{\\}}-\Big{\\{}F\,,\,\psi_{10}\Big{\\}}\Big{\\{}\psi_{9}\,,\,G\Big{\\}}\bigg{)}\>.$
(42)

The nonvanishing Dirac brackets between the canonical variables are

$\displaystyle\Big{\\{}X_{i}^{A},\;P_{i}^{B}\Big{\\}}_{\mathrm{D}}$
$\displaystyle=\eta^{AB}-\frac{1}{X_{i}^{2}}X_{i}^{A}X_{i}^{B}\>\>,$ (43a)
$\displaystyle\Big{\\{}X_{i}^{A},\;e_{j}\Big{\\}}_{\mathrm{D}}$
$\displaystyle=\frac{1}{V_{\mathrm{dS}}}\left(\delta_{ij}-\varepsilon_{ij}\right)P_{i}^{A}e_{i}e_{j}\>\>,$
(43b) $\displaystyle\Big{\\{}P_{i}^{A},\;e_{j}\Big{\\}}_{\mathrm{D}}$
$\displaystyle=\frac{\mathcal{P}}{2V_{\mathrm{dS}}R^{2}}\left(\delta_{ij}-\varepsilon_{ij}\right)X_{i}^{A}e_{j}\>\>,$
(43c) $\displaystyle\Big{\\{}e_{i},\;P_{e_{j}}\Big{\\}}_{\mathrm{D}}$
$\displaystyle=\delta_{ij}+\frac{1}{2V_{\mathrm{dS}}}\left(\delta_{ij}-\sigma{}_{ij}\right)e_{i}P_{j}^{2}\>\>,$
(43d) $\displaystyle\Big{\\{}e_{i},\;\lambda_{j}\Big{\\}}_{\mathrm{D}}$
$\displaystyle=(-1)^{i}\frac{F\varepsilon_{-}}{KR\sigma}\delta_{ij}e_{i}$
(43e)

with no summation over $i,j$, where
$\sigma_{11}=\sigma_{22}=0,\>\sigma_{12}=\sigma_{21}=1$ and
$\varepsilon_{ij}=-\varepsilon_{ji}$ with $\varepsilon_{12}=1$.

As long as the Dirac bracket is used, the second-class constraints can be set
to zero strongly. Thus, the first-class constraint $\psi_{12}\approx 0$
becomes

$\displaystyle\psi_{12}$ $\displaystyle=e_{1}\chi_{e_{1}}+e_{2}\chi_{e_{2}}$
$\displaystyle=e_{1}P_{1}^{2}+e_{2}P_{2}^{2}+\frac{\alpha^{2}}{2}\frac{\sqrt{e_{1}e_{2}}}{\sigma^{2}}\approx
0\>.$ (44)

The second-class constraints (40) yields

$\displaystyle e_{1}^{2}P_{1}^{4}+e_{2}^{2}P_{2}^{4}$
$\displaystyle=2e_{1}e_{2}P_{1}^{2}P_{2}^{2}\>.$ (45)

Using this equation, the first-class constraint (3) yields

$\displaystyle\varLambda:=P_{1}^{2}P_{2}^{2}-\frac{\alpha^{4}}{16}\frac{1}{\sigma^{4}}$
$\displaystyle\approx 0\>.$ (46)

##  4 New canonical variables

As seen from Eq. (43), the pairs $(X^{A}_{i},P_{iA})$, $(e_{i},P_{e_{i}})$ and
$(\lambda_{i},P_{\lambda_{i}})$, which consist of the 28 canonical variables,
are no longer canonically conjugate pairs under the Dirac bracket. This is due
to the 10 second-class constraints and we have only 18 independent degrees of
freedom that may constitute canonical pairs in the sense of Dirac brackets. To
identify them, we define new variables as follows:

$\displaystyle{x}^{\mu}_{i}$ $\displaystyle:=X^{\mu}_{i},$
$\displaystyle{p}_{i\mu}$ $\displaystyle:=g_{\mu\nu}(x_{i})P^{\nu}_{i},$ (47a)
$\displaystyle{e}$ $\displaystyle:=\frac{e_{1}e_{2}}{2},$ $\displaystyle{\pi}$
$\displaystyle:=\frac{e_{1}P_{e_{1}}+e_{2}P_{e_{2}}}{e_{1}e_{2}},$ (47b)

with

$\displaystyle
g_{\mu\nu}(x_{i}):=\eta_{\mu\nu}-\frac{x_{i\mu}x_{i\nu}}{\eta_{\alpha\beta}x^{\alpha}_{i}x^{\beta}_{i}+R^{2}}\;,$
(48)

where $\mu,\nu=0,1,2,3$ and $x_{i\mu}:=\eta_{\mu\nu}x^{\nu}_{i}$. By using Eq.
(43), it is easily checked that the nonvanishing Dirac brackets between the
new variables become

$\displaystyle\left\\{{x}^{\mu}_{i},{p}_{j\nu}\right\\}_{\mathrm{D}}$
$\displaystyle=\delta_{ij}\delta^{\mu}_{\nu}\;,$
$\displaystyle\left\\{e,\;\pi\right\\}_{\mathrm{D}}$ $\displaystyle=1\>\>.$
(49)

Therefore each pair of $({x}^{\mu}_{i},{p}_{i\mu})$ and $(e,\pi)$ forms a pair
of canonical variables in the sense of Dirac brackets.

It is noted that taking the pairs $(x_{i}^{\mu},p_{i\mu})$ as canonical pairs
corresponds to choosing a particular coordinate system $(x^{\mu})$ for the de
Sitter space. To see this, we solve Eqs. (28c) and (28d) with respect to
$X^{5}_{1}$ and $X^{5}_{2}$, respectively, and write them in terms of
$x^{\mu}_{i}$ as

$\displaystyle X_{i}^{5}(x_{i})$
$\displaystyle=\pm\sqrt{\eta_{\mu\nu}X^{\mu}_{i}X^{\nu}_{i}+R^{2}}=\pm\sqrt{\eta_{\mu\nu}x^{\mu}_{i}x^{\nu}_{i}+R^{2}}\>.$
(50)

This is the same as the parameterization of the de Sitter space (8) and Eq.
(48) is the induced metric (9). By solving Eq. (30) with respect to
$P^{5}_{i}$, we get

$\displaystyle P_{i}^{5}({x}_{i},p_{i})$
$\displaystyle=\frac{\eta_{\mu\nu}X^{\mu}_{i}P^{\nu}_{i}}{X_{i}^{5}}=\pm\frac{g^{\mu\nu}(x_{i})x_{i\mu}p_{i\nu}}{\sqrt{\eta_{\mu\nu}x^{\mu}_{i}x^{\nu}_{i}+R^{2}}}\>,$
(51)

where $g^{\mu\nu}(x_{i})=\eta^{\mu\nu}+x^{\mu}_{i}x^{\nu}_{i}/R^{2}$ is the
inverse of $g_{\mu\nu}(x)$ at $x=x_{i}$.

Using new canonical variables (47), the first-class constraint
$\psi_{11}\approx 0$ is written as $e\pi\approx 0$, which is equivalent to
$\pi\approx 0$. Together with Eq. (46), the first-class constraints are now
listed as

$\displaystyle\pi$ $\displaystyle\approx 0\>,$ (52a)
$\displaystyle\varLambda(x_{1},x_{2},p_{1},p_{2})=P_{1}^{2}P_{2}^{2}-\frac{\alpha^{4}}{16}\frac{1}{\sigma^{4}(x_{1},x_{2})}$
$\displaystyle\approx 0\>\>,$ (52b)

where $P_{i}^{2}$ and $\sigma(x_{1},x_{2})$ are written as

$\displaystyle P_{i}^{2}$ $\displaystyle=g^{\mu\nu}(x_{i})p_{i\mu}p_{i\nu}\>,$
(53) $\displaystyle\sigma(x_{1},x_{2})$
$\displaystyle=R\mathrm{cos}^{-1}\left(-\frac{\eta_{\mu\nu}x^{\mu}_{1}x^{\nu}_{2}+X^{5}_{1}(x_{1})X^{5}_{2}(x_{2})}{R^{2}}\right)\>.$
(54)

In the next sectoin, we carry out the canonical quantization of our system
based on the Dirac brackets (49) for the new variables and the first-class
constraints (52).

##  5 Canonical quantization

We set the commutation relations between $\hat{f}$ and $\hat{g}$ corresponding
to functions $f$ and $g$ of canonical variables, in such manner

$\displaystyle[\hat{f}\,,\>\hat{g}]=i\widehat{\\{f,\>g\\}}_{\mathrm{D}}$ (55)

in units $\hbar=1$. The symbol $\widehat{\\{f,\>g\\}}_{\mathrm{D}}$ denotes
the operator corresponding to the Dirac brackets $\\{f,g\\}_{\mathrm{D}}$.
From Eq. (49), we have the commutation relations

$\displaystyle\big{[}\hat{x}_{i}^{\mu},\;\hat{p}_{j\nu}\big{]}$
$\displaystyle=i\delta_{ij}\delta^{\mu}_{\nu}\>,$
$\displaystyle\big{[}\hat{e},\;\hat{\pi}\big{]}$ $\displaystyle=i\>.$ (56)

We consider the quantum mechanics characterized by the above commutation
relations in the coordinate representation, where the configuration space is
the de Sitter space endowed with the metric (9). The coordinate representation
for curved configuration spaces was considered by DeWitt [9]. Following
DeWitt, we take the simultaneous eigenstates of $\hat{x}_{i},\;\hat{e}$

$\displaystyle\hat{x}^{\mu}_{i}|x_{1},x_{2},e\rangle$
$\displaystyle=x_{i}^{\mu}|x_{1},x_{2},e\rangle\>,$ (57)
$\displaystyle\hat{e}|x_{1},x_{2},e\rangle$
$\displaystyle=e|x_{1},x_{2},e\rangle\>$ (58)

as a basis of the Hilbert space. Then we expand an arbitrary state
$|\varphi\rangle$ in terms of the basis vectors in the form

$\displaystyle|\varphi\rangle=\int\Big{(}\prod_{i=1,2}\sqrt{-g(x_{i})}d^{4}x_{i}\Big{)}de\,\varphi(x_{1},x_{2},e)|x_{1},x_{2},e\,\rangle\>,$
(59)

with $g(x)=\det g_{\mu\nu}(x)$ and wave functions
$\varphi(x_{1},x_{2},e)\equiv\langle x_{1},x_{2},e|\varphi\rangle$. In
addition, we assume that the eigenvectors $|x_{1},x_{2},e\,\rangle$ satisfy
the normalization condition

$\displaystyle\langle
x^{\prime\prime}_{1},x^{\prime\prime}_{2},e^{\prime\prime}|x^{\prime}_{1},x^{\prime}_{2},e^{\prime}\rangle=\delta(e^{\prime\prime}-e^{\prime})\prod_{i=1,2}\frac{\delta^{4}(x_{i}^{\prime\prime}-x_{i}^{\prime})}{\sqrt{-g(x_{i}^{\prime})}}\>,$
(60)

which is consistent with the completeness relation

$\displaystyle
1=\int\prod_{i=1,2}\sqrt{-g(x_{i})}d^{4}x_{i}de\,|x_{1},x_{2},e\rangle\langle
x_{1},x_{2},e|\>\>.$ (61)

On the basis vectors satisfying Eq. (60), the momentum operators
$\hat{p}_{i}{}_{\mu}$ and $\hat{\pi}$ can be represented [9] as

$\displaystyle\langle x_{1},x_{2},e|\hat{p}_{i\mu}$
$\displaystyle=\left(-i\frac{\partial}{\partial
x^{\mu}_{i}}-\frac{i}{2}\Gamma_{\mu}(x_{i})\right)\langle x_{1},x_{2},e|\>,$
(62) $\displaystyle\langle x_{1},x_{2},e|\hat{\pi}$
$\displaystyle=-i\frac{\partial}{\partial e}\langle x_{1},x_{2},e|\>,$ (63)

where $\Gamma_{\mu}(x):=\Gamma^{\nu}_{\nu\mu}$ and the Christoffel symbol
$\Gamma^{\sigma}_{\mu\nu}$ is defined as

$\Gamma^{\sigma}_{\mu\nu}=\frac{1}{2}g^{\sigma\tau}\left(\frac{\partial}{\partial
x^{\nu}}g_{\tau\mu}+\frac{\partial}{\partial
x^{\mu}}g_{\tau\nu}-\frac{\partial}{\partial x^{\tau}}g_{\mu\nu}\right)\>.$

For confirming the hermiticity of $\hat{p}_{i\mu}$, namely
$\langle\psi|\hat{p}_{i\mu}\varphi\rangle=\langle\hat{p}_{i\mu}\psi|\varphi\rangle$,
it is useful to use an alternative expression

$\displaystyle\langle
x_{1},x_{2},e|\hat{p}_{i\mu}=(-g(x_{i}))^{-\frac{1}{4}}\left(-i\frac{\partial}{\partial
x_{i}^{\mu}}\right)(-g(x_{i}))^{\frac{1}{4}}\langle x_{1},x_{2},e|\>.$ (64)

This is also convenient to verity that the Laplace-Beltrami operator
$\square^{(dS)}_{i}$ at a point $x_{i}^{\mu}$ on ${\rm dS}_{4}$ can be written
as

$\displaystyle\square^{(dS)}_{i}$
$\displaystyle:=\frac{1}{\sqrt{-g(x_{i})}}\frac{\partial}{\partial
x_{i}^{\mu}}\sqrt{-g(x_{i})}g^{\mu\nu}(x_{i})\frac{\partial}{\partial
x_{i}^{\nu}}$
$\displaystyle=-(-g(x_{i}))^{-\frac{1}{4}}\hat{p}_{i\mu}\sqrt{-g(x_{i})}g^{\mu\nu}(x_{i})\hat{p}_{i\nu}(-g(x_{i}))^{-\frac{1}{4}}\>\>$
(65)

in the $x$-representation.

Now, let us derive a wave equation associated with the first class
constraints. We replace Eqs. (52a) and (52b) with the conditions that single
out the physical states, respectively:

$\displaystyle\hat{\pi}\,|\varphi\rangle$ $\displaystyle=0\>,$ (66a)
$\displaystyle\hat{\varLambda}(\hat{x}_{1},\hat{x}_{2},\hat{p}_{1},\hat{p}_{2})\,|\varphi\rangle$
$\displaystyle=0\>\>,$ (66b)

where

$\displaystyle\hat{\varLambda}(\hat{x}_{1},\hat{x}_{2},\hat{p}_{1},\hat{p}_{2}):=\widehat{P_{1}^{2}}\widehat{P_{2}^{2}}-\frac{\alpha^{4}}{16\sigma^{4}(\hat{x}_{1},\hat{x}_{2})}\>\>,$
(67)

and $\widehat{P_{i}^{2}}$ is the quantum mechanical operator corresponding to
the classical quantity $P_{i}^{2}=g^{\mu\nu}(x_{i})p_{i\mu}p_{i\nu}$ in Eq.
(53). Obviously, there is ordering ambiguity in determining an explicit
expression of $\widehat{P_{i}^{2}}$. Since $P_{i}^{2}$ is a scalar under
general coordinate transformations, we simply require $\widehat{P_{i}^{2}}$ to
be also a scalar for a particular ordering of $\widehat{P_{i}^{2}}$. 222The
Weyl ordering rule, commonly taken in canonical quantization procedure, does
not satisfy our requirement because the Weyl ordered operator,
$\widehat{P_{i}^{2}}_{(W)}$, for $P_{i}^{2}=g_{\mu\nu}p^{\mu}_{i}p^{\nu}_{i}$
is explicitly evaluated as  $\displaystyle\widehat{P_{i}^{2}}_{(W)}$
$\displaystyle=\hat{p}_{i}^{\mu}g_{\mu\nu}(x_{i})\hat{p}^{\nu}_{i}-\frac{1}{4}\frac{\partial
g_{\mu\nu}(x_{i})}{\partial x_{i}^{\mu}\partial x_{i}^{\nu}}$
$\displaystyle=-\square_{i}-\frac{1}{4}\mathcal{R}(x_{i})-\frac{1}{4}g^{\mu\nu}(x_{i})\Gamma^{\alpha}_{\beta\mu}\Gamma^{\beta}_{\alpha\nu}(x_{i})\>$
with no sum over $i$, where $\mathcal{R}(x)$ is the Ricci scalar. The last
term in the last line is not invariant under general coordinate
transformations.

$\displaystyle\widehat{P_{i}^{2}}:=(-g(\hat{x}_{i}))^{-\frac{1}{4}}\hat{p}_{i\mu}\sqrt{-g(\hat{x}_{i})}g^{\mu\nu}(\hat{x}_{i})\hat{p}_{i\nu}(-g(\hat{x}_{i}))^{-\frac{1}{4}}\>.$
(68)

On a bra vector, this is represented as

$\displaystyle\langle
x_{1},x_{2},e|\widehat{P_{i}^{2}}=-\square^{(dS)}_{i}\langle
x_{1},x_{2},e|\>\>.$ (69)

Using Eq. (67) and Eq. (69), we have

$\displaystyle\langle
x_{1},x_{2},e|\hat{\varLambda}(\hat{x}_{1},\hat{x}_{2},\hat{p}_{1},\hat{p}_{2})$
$\displaystyle=\left(\square^{(dS)}_{1}\square^{(dS)}_{2}-\frac{\alpha^{4}}{16\sigma^{4}(x_{1},x_{2})}\right)\langle
x_{1},x_{2},e|\>.$ (70)

Multiplying Eqs. (66a) and (66b) by $\langle x_{1},x_{2},e|$ and using Eqs.
(63) and (5), respectively, we have simultaneous partial differential
equations for wave functions $\varphi(x_{1},x_{2},e)$:

$\displaystyle\frac{\partial}{\partial e}\varphi(x_{1},x_{2},e)$
$\displaystyle=0\>,$ (71a)
$\displaystyle\left(\square^{(dS)}_{1}\square^{(dS)}_{2}-\frac{\alpha^{4}}{16\sigma^{4}(x_{1},x_{2})}\right)\varphi(x_{1},x_{2},e)$
$\displaystyle=0\>.$ (71b)

Eq. (71a) tells us that the wave functions do not depend on the variable $e$.
Therefore we can write $\varphi(x_{1},x_{2},e)=\varphi(x_{1},x_{2})$ and Eq.
(71b) becomes the wave equation for bilocal fields $\varphi(x_{1},x_{2})$:

$\displaystyle\left(\square^{(dS)}_{1}\square^{(dS)}_{2}-\frac{\alpha^{4}}{16\sigma^{4}(x_{1},x_{2})}\right)\varphi(x_{1},x_{2})=0\>.$
(72)

This is precisely a de Sitter version of the wave equation (4) originally
derived for the case of Minkowski space in Ref. [1]. The squared geodesic
distance has the property $\sigma^{2}\rightarrow-r^{2}$ as
$R\rightarrow\infty$ as shown in Sec. 2, so that Eq. (72) also reduces to Eq.
(4).

Although it is important to consider the well-posedness of an initial value
problem for the wave equation (72), further investigations for it do not enter
into the scope of this letter.

##  6 Conclusion

In this letter, we have generalized the conformally invariant action $S$, Eq.
(3), for two interacting massless particles in Minkowski space to the case of
de Sitter space from the perspective of geodesic distance. To begin with, we
have defined the action $S_{g}$, Eq. (6), as a generalization of $S$ to
general spacetimes that become close to Minkowski space in a certain limit.
After that, we have specified the action $S_{\mathrm{dS}}$, Eq. (13), as
$S_{g}$ for the case of de Sitter space in terms of the embedding coordinates
$X_{i}^{A}$ with the explicit expression of the geodesic distance (11) and
also confirmed that $S_{\mathrm{dS}}$ really reproduces $S$ in the limit
$R\rightarrow\infty$ with the coordinate chart (8). In the Hamilton analysis,
according to Dirac’s prescription [8] for the constrained Hamiltonian systems,
we have carried out the classification of the constraints and have defined the
Dirac bracket from the second-class constraints. Then we have found that the
new variables (47) constitute the canonically conjugate pairs in the sense of
Dirac brackets. Following DeWitt [9], we have performed the subsequent
canonical quantization of the system in coordinate representation where the
configuration space is $\mathrm{dS}_{4}$ endowed with the metric (9). By
choosing the operator ordering (68) for $\widehat{P}_{i}^{2}$, the physical
state conditions (66) associated with the first-class constraints (52) has
been read as the fourth-order differential equation (72) for bilocal fields
$\varphi(x_{1},x_{2})$, which can be recognized as a de Sitter version of the
wave equation (4) derived based on the action (3) in Ref. [1].

It seems worthwhile to quantize the model governed by the action (6) in path
integral formulation in curved spaces, which was studied for a nonrelativistic
particle in Ref. [12] and for a relativistic particle in a manifestly
covariant way in Ref. [13].

It is also interesting to consider the dynamics of bilocal fields satisfying
the wave equation (72), including to confirm the well-posedness of (72), which
may possibly be relevant to higher derivative theories of scalar fields in
curved spacetime [14] and/or higher spin theories [15].

## Acknowledgements

The authors thank the members of the theoretical group in Nihon University for
providing the research environment. N.K. is also grateful to Kengo Maeda and
Shigefumi Naka for their useful comments in the early stage of this work and
appreciate kind support from Haruki Toyoda. S.O. would like to thank Shinichi
Deguchi for his insightful comments.

## References

  * Casalbuoni and Gomis [2014] R. Casalbuoni, J. Gomis,  Conformal symmetry for relativistic point particles,  Phys. Rev. D 91 (2014) 1–10.
  * Kolb and Turner [1990] E. W. Kolb, M. S. Turner,  The early universe,  Front. Phys. 69 (1990) 1–547.
  * Imponente and Montani [2004] G. Imponente, G. Montani,  General Aspects of the de Sitter phase (2004).
  * Brandenberger [2011] R. H. Brandenberger,  Introduction to Early Universe Cosmology,  in: Proc. 4th Int. Conf. Fundam. Interact. — PoS(ICFI 2010), Sissa Medialab, Trieste, Italy, 2011, p. 001\. URL: http://arxiv.org/abs/1103.2271https://pos.sissa.it/124/001. doi:10.22323/1.124.0001. arXiv:1103.2271.
  * Kanda and Naka [2015] N. Kanda, S. Naka,  Bi-local field in gravitational shockwave background,  Prog. Theor. Exp. Phys. 2015 (2015) 33B10--0.
  * Kuzenko and Yarevskaya [1995] S. Kuzenko, J. Yarevskaya,  Conformal invariance, N-extended supersymmetry and massless spinning particles in Anti-de Sitter space,  arXiv Prepr. hep-th/9512115 2 (1995).
  * Dorn and Jorjadze [2006] H. Dorn, G. Jorjadze,  Massless scalar particle on AdS spacetime: Hamiltonian reduction and quantization,  J. High Energy Phys. 2006 (2006).
  * Dirac [1964] P. A. M. Dirac, Lectures on quantum mechanics, volume 2 of Belfer Graduate School of Science Monographs Series, Belfer Graduate School of Science, New York, 1964.
  * Dewitt [1952] B. S. Dewitt,  Point transformations in quantum mechanics,  Phys. Rev. 85 (1952) 653--661.
  * Cunningham et al. [2017] W. J. Cunningham, D. Rideout, J. Halverson, D. Krioukov,  Exact geodesic distances in FLRW spacetimes,  Phys. Rev. D 96 (2017) 103538.
  * Chernikov and Tagirov [1968] N. A. Chernikov, E. A. Tagirov,  Quantum theory of scalar field in de Sitter space-time,  Ann. l’I.H.P. Phys. théorique 9 (1968) 109--141.
  * Cheng [1972] K. Cheng,  Quantization of a General Dynamical System by Feynman’s Path Integration Formulation,  J. Math. Phys. 13 (1972) 1723--1726.
  * Bastianelli [1992] F. Bastianelli,  The path integral for a particle in curved spaces and Weyl anomalies,  Nucl. Phys. B 376 (1992) 113--126.
  * Gibbons et al. [2019] G. W. Gibbons, C. N. Pope, S. Solodukhin,  Higher derivative scalar quantum field theory in curved spacetime,  Phys. Rev. D 100 (2019) 105008.
  * Sorokin [2005] D. Sorokin,  Introduction to the classical theory of higher spins,  AIP Conf. Proc. 767 (2005) 172--202.