DIP 2406-01

Theory of Complex Particle without Extra Dimensions

Takayuki Hori ***email<EMAIL_ADDRESS>
Doyo-kai Institute of Physics, 3-46-4, Kitanodai,

Hachiouji-shi, Tokyo 192-0913, Japan

###### Abstract

Complex particle is a kind of bilocal particle having unexpected symmetry,
which was proposed by the authour. In the present paper, we show that critical
dimension of the complex particle in Minkowski spacetime is $D=4$, while
$D=2,4$ or $6$ are permitted in Euclid spacetime. The origin of the
restriction to the dimension is the existence of tertiary constraint in the
canonical theory, quantization of which leads to an eigenvalue equation having
single-valued and bounded solutions only in particular dimension of spacetime.
The derivation is based on a detailed analysis of Laplace-Beltrami operator on
$S^{1,D-2}$ or $S^{D-1}$.

##  1 Introduction

In the past two decades the investigations of string theorists may be devoted
to the ploblem of the vacuum structure of the world, called landscape [1],[2],
which is consistent to the low energy phenomenology. The motivation for that
work is based on the premise that the extra dimensions of spacetime exist,
which is far from the present experimental evidence. On the other hand, no
realistic model of particle theory has been proposed, whose critical dimension
equals to four.

In the present paper we show that there is a model without extra dimensions,
though it is a toy model at present. The key element leading to the
restriction to the dimension is the tertiary constraint which exists, in
general, in such a model that the number of gauge degrees of freedom is less
than that of the first class constraints in the canonical theory, i.e., the
breakdown of Dirac’s conjecture [3]-[13].

The model is complex particle proposed by the author, and a preliminary study
has been reported [14],[15]. The dynamical variables of the model are complex
coordinates of an object in $D$-dimensional flat spacetime, the real and the
imaginary parts of which may be representing two relativistic particles.
However, the two coordinates are not independent on each other due to a
bilinear interaction term in the lagrangian. In that sense the complex
particle is resembling to the bilocal particle [16]-[18] which has unexpected
global symmetry. The lagrangians of these models have gauge symmetry
generalising the reparametrization invariance of the relativistic particle,
and have the corresponding BRST symmetry. One unsolved problem in the bilocal
particle model is that the ghost numbers, associated with the BRST symmetry,
of the states in the physical space are not zero [18], and the ordinary
quantization scheme may not be applicable. Fortunately the complex particle
has not the problem of the quantization appeared in the bilocal particle.

In the quantum theory the tertiary constraint in the complex particle is
replaced by differential equations. In solving them we get, among other
equations, an eigenvalue equation of d’Alembert operator, and solving it
results in the eigenvalue problem of the Laplace-Beltrami(LB) operator on
$S^{D-1}$ and $S^{1,D-2}$, corresponding to the Euclid and the Minkowski
metric, respectively.

The LB operator in $D=3$ is the square of the angular momentum operator, and
the eigenvalues are $\ell(\ell+1),~{}(\ell\in\mathbb{Z}_{\geq 0}$) as is well
known in the elementary course of quantum mechanics. The eigenvalue problem of
LB in $D$-dimensional Euclid space has been well investigated in the last
century, published in a mathematical literature in the context of the
representation theory of the orthogonal groups [23]-[26]. However, the
eigenvalue problem in the Minkowski spacetime has not been well studied
because of the less application to physics. We find some papers in this era
[27],[28].

In the present paper we find the complete solution to the eigenvalue problem
of LB in the $D$-dimensional Minkowski or Euclid spacetime, with some physical
conditions such as single-valuedness or boudedness. On the basis of this
result we conclude that the dimension of spacetime being allowed in the
complex particle in the Minkowski spacetime is 4, and $D=2,4$ or $6$ in the
Euclid spacetime.

In Section 2 and 3 we give an outline of the classical theory of the complex
particle. In Section 4 we give the quantization of the model and the equations
representing the physical state condition. In Section 5 the mathematical
aspect of LB operator is illustrated. Section 6 is devoted to derivation of
the critical dimension of spacetime. Finally in Section 7 we give some
speculations for future study of the present model.

##  2 Lagrangian of complex particle

Let us start with a system with complex coordinates,
$\mbox{\boldmath$z$}=(z^{\mu}),~{}(0\leq\mu\leq D-1)$, which describes an
object moving in $D$-dimensional complex manifold. The trajectory of it is
parametrized by time variable, $\tau$. Introducing the complex einbein, $g$,
we can write the action as

$\displaystyle I=\int d\tau~{}L,\qquad
L=\frac{\mbox{\boldmath$w$}^{2}}{2g}+i\kappa\mbox{\boldmath$w$}\cdot\bar{\mbox{\boldmath$z$}}+{\rm
c.c.},$ (2.1)

where c.c. is the complex conjugate and $\kappa$ is a constant with dimension
of mass square 222Here and hereafter we use the notation for $D$-dimensional
vectors $\mbox{\boldmath$a$},~{}\mbox{\boldmath$b$}$, as
$\mbox{\boldmath$a$}\cdot\mbox{\boldmath$b$}=a_{\mu}b^{\mu}$. The flat metric
of spacetime is $\eta^{\mu\nu}={\rm diag}(\pm 1,1,..,1)$, where the $\pm$ sign
is $+1$ and $-1$ for Euclid and Minkowski spacetime, respectively.. The real
and the imaginary parts of $z$ describe real coordinates of two relativistic
particles traveling in our $D$-dimensional spacetime manifold. However, we
call the system a single complex particle instead of two particles, because
the Poisson bracket between the two momenta does not vanish, as is shown
later, and they have not simultaneous eigenvalues in quantum theory. Actually
one of the two particles has wrong sign in the kinetic term, i.e., behaves as
ghost.

The Euler-Lagrange(EL) equation obtained by minimizing the action under the
variation of $z$ is

$\displaystyle{\bf[EL]}$ $\displaystyle\stackrel{{\scriptstyle\rm def}}{{=}}$
$\displaystyle\frac{d}{d\tau}\left(\frac{\mbox{\boldmath$w$}}{g}+2i\kappa\bar{\mbox{\boldmath$z$}}\right)=0.\qquad({\rm
and}~{}{\rm c.c.})$ (2.2)

Another equation is obtained by varying the einbein $g$:

$\displaystyle\ell$ $\displaystyle\stackrel{{\scriptstyle\rm def}}{{=}}$
$\displaystyle-g^{2}{\bf[EL]}_{g}=\mbox{\boldmath$w$}\cdot\mbox{\boldmath$w$}=0.\qquad({\rm
and}~{}{\rm c.c.})$ (2.3)

The dynamical variables are the coordinate variables,
$(\mbox{\boldmath$z$},g)$ and their velocities,
$(\mbox{\boldmath$w$},w=\dot{g})$, (and their complex conjugates). The Euler-
Lagrange equations which are first rank differential equations with respect to
$\tau$, determine the time development of the dynamical variables under some
initial value conditions. In that sense Eq.(2.3) is not an equation of motion
determining time-development but a constraint equation for the dynamical
variables. Such a constraint is called lagrangian constraints [21]. The
lagrangian constraint should be preserved not only at initial time but along
the trajectory,. Since the time derivative of $\ell$ is

$\displaystyle\dot{\ell}=\frac{2w}{g}\ell-4i\kappa
g\mbox{\boldmath$w$}\cdot\bar{\mbox{\boldmath$w$}},$ (2.4)

we have the condition

$\displaystyle\ell_{0}\stackrel{{\scriptstyle\rm
def}}{{=}}\mbox{\boldmath$w$}\cdot\bar{\mbox{\boldmath$w$}}=0,$ (2.5)

provided $\ell=0$. The time preservation condition of a lagrangian constraint
is called 2nd order lagrangian constraint, which should be also preserved,
yielding 3rd order lagrangian constraint, and repeated on. In the present case
the lagrangian constraint is over at the 2nd order. It is worth while to note
that the 2nd order lagrangian constraint (2.5) exists only if $\kappa\neq 0$.

The 1st integral of (2.2) is

$\displaystyle\mbox{\boldmath$w$}=-2ig\kappa(\bar{\mbox{\boldmath$z$}}-\bar{\mbox{\boldmath$C$}})$
(2.6)

where $C$ is an arbitrary constant vector. Substituting it back into (2.2) we
get the second rank differential equation

$\displaystyle\ddot{y}-\rho\dot{y}-\eta y=0,$ (2.7)

where we put $\mbox{\boldmath$z$}-\mbox{\boldmath$C$}\rightarrow
y,~{}\rho=q/g,~{}\eta=4\kappa^{2}g\bar{g}$. The solution for $z$ is

$\displaystyle\mbox{\boldmath$z$}(\tau)=\mbox{\boldmath$C$}+\mbox{\boldmath$z$}_{(0)}f(\tau)$
(2.8)

where $y=f(\tau)$ is a solution to (2.7) and $\mbox{\boldmath$z$}_{(0)}$ is
arbitrary constant vector. It is not necessary to solve (2.7) generally in
order to get expression of the trajectory in the target space as is shown
below.

From the 0-th component of (2.8) we have

$\displaystyle f(\tau)=\frac{1}{z_{(0)}^{0}}(z^{0}(\tau)-C^{0}),$ (2.9)

and substituting it to the spacial component of (2.8) we get

$\displaystyle z^{i}(\tau)$ $\displaystyle=$ $\displaystyle
c^{i}+v^{i}z^{0}(\tau),$ (2.10)

where

$\displaystyle c^{i}=\frac{C^{[i}z_{(0)}^{0]}}{z_{(0)}^{0}},\qquad
v^{i}=\frac{z_{(0)}^{i}}{z_{(0)}^{0}},\qquad(i=1,..,D-1).$ (2.11)

Now let us consider the physical content of the above solution in the
Minkowski spacetime ($\eta_{00}=-1$). With respect to the time component of
$z$, one must set a relation between the time, $z^{0}$, of the target space
and the internal time, $\tau$. We set

$\displaystyle\Re z^{0}(\tau)=t,\qquad\Im z^{0}(\tau)=0.$ (2.12)

Then the real and imaginary parts of (2.10), $z^{i}=c^{i}+v^{i}t$, represent a
particle moving in the target space with constant velocities, $\Re v^{i}$ and
$\Im v^{i}$, respectively, with respect to the time $t$.

The condition (2.12) should be used after solving the lagrangian constraints.
Since (2.8) holds on the trajectory satisfying EL equation, we have
$\mbox{\boldmath$w$}=\mbox{\boldmath$z$}_{(0)}\dot{f}(\tau),$ on the
trajectory. Then the 1st order lagrangian constraint becomes
$(\mbox{\boldmath$z$}_{(0)})^{2}=0$. Hence from the second equation of (2.11)
we have

$\displaystyle\sum_{i=1}^{D-1}v^{i}v_{i}=1,$ (2.13)

here we use $\eta_{00}=-1$. Putting $v_{1}^{i}=\Re v^{i},~{}v_{2}^{i}=\Im
v^{i}$, this condition becomes

$\displaystyle\sum_{i=1}^{D-1}(v_{1}^{i}v_{1i}-v_{2}^{i}v_{2i})=1,\qquad\sum_{i=1}^{D-1}v_{1}^{i}v_{2i}=0.$
(2.14)

This means that the real part of the complex coordinates represent a particle
moving faster than light, i.e. a tachyon, while the imaginary part represent
one moving perpendicular to the former.

The 2nd order lagrangian constraint (2.5), which exists if $\kappa\neq 0$,
changes the above picture drastically. The similar manipulation as that of the
1st order lagrangian constraint gives

$\displaystyle\sum_{i=1}^{D-1}(v_{1}^{i}v_{1i}+v_{2}^{i}v_{2i})=1,$ (2.15)

and, in conjunction with (2.14), we conclude

$\displaystyle\sum_{i=1}^{D-1}v_{1}^{i}v_{1i}=1,\qquad v_{2}^{i}=0.$ (2.16)

That is, the real part of $z$ represent a moving particle in a constant
direction with the velocity of light, and the imaginary part of $z$ represent
a particle staying at rest. In another words the latter may be interpreted as
a kind of conserved quantity. The coordinates of the complex particle in the
Minkowski spacetime may be viewed as composite of the ordinary coordinates of
particle moving with velocity of light and a conserved quantity, the
conservation of the latter is guaranteed by the EL equations. In the above
discussion we see the vanishing limit of $\kappa$ does not coincide with the
system which has $\kappa=0$ from outset. This situation is seen in many
respects of the complex particle.

Further insight is obtained by dividing lagrangian in terms of the real and
imaginary parts of $w$ and $w$ (hereafter we consider both of the Euclid and
Minkowski signature). The kinetic terms are expressed as

$\displaystyle L_{\rm
kin}=\frac{2}{g_{1}^{2}+g_{2}^{2}}[g_{1}(\mbox{\boldmath$u$}^{2}-\mbox{\boldmath$f$}^{2})+2g_{2}\mbox{\boldmath$u$}\cdot\mbox{\boldmath$f$}],$
(2.17)

where $\mbox{\boldmath$w$}=\mbox{\boldmath$u$}+i\mbox{\boldmath$f$}$ and
$g=g_{1}+ig_{2}$. If we put conventionally $g_{1}>0$, the signs of the kinetic
terms are plus for the real part of $z$ and minus for the imaginary part.
Hence the imaginary part of the complex coordinates are the ghost variables.

Thus the classical behaviour of the complex particle is restricted in a
somehow curious manner. This is a consequence of existence of unexpected
symmetry of the action. Generalizing the reparametrization invariance of the
action of two relativistic particles, we consider the following transformation
of $z$ and $g$:

$\displaystyle\delta\mbox{\boldmath$z$}=\varepsilon\mbox{\boldmath$w$}+\frac{\varepsilon_{0}}{\bar{g}}\bar{\mbox{\boldmath$w$}},\qquad\delta
g=\xi,$ (2.18)

and c.c. of them, where $\varepsilon,~{}\varepsilon_{0}$ and $\xi$ are
$\tau$-dependent infinitesimal parameters, and $\varepsilon_{0}$ is a real
parameter. After straightforward but rather lengthy calculation (see Appendix
A.1), we see the variation of the lagrangian under the transformation (2.18)
is

$\displaystyle\delta L$ $\displaystyle=$
$\displaystyle\frac{\ell_{0}}{2g\bar{g}}[\dot{\varepsilon}_{0}-2i\kappa
g\bar{g}(\varepsilon-\bar{\varepsilon})]+A\ell+\frac{dE}{d\tau}+{\rm c.c.},$
(2.19)

where

$\displaystyle A\stackrel{{\scriptstyle\rm
def}}{{=}}\frac{1}{2g^{2}}\left(\frac{d}{d\tau}(\varepsilon g)+4i\kappa
g\varepsilon_{0}-\xi\right),\qquad E\stackrel{{\scriptstyle\rm
def}}{{=}}\left(\frac{\mbox{\boldmath$w$}}{2g}+i\kappa\bar{\mbox{\boldmath$z$}}\right)\cdot\delta\mbox{\boldmath$z$}.$
(2.20)

The transformation under which the action is invariant apart from the
lagrangian constraints is called semi-gauge transformation [20]. Hence (2.18)
is a semi-gauge transformation for arbitrary infinitesimal parameters,
$\varepsilon,\varepsilon_{0}$ and $\xi$.

In particular if we choose the parameters so that

$\displaystyle\dot{\varepsilon}_{0}-2i\kappa
g\bar{g}(\varepsilon-\bar{\varepsilon})=0,$ (2.21)
$\displaystyle\xi=\frac{d}{d\tau}(\varepsilon g)+4i\kappa g\varepsilon_{0},$
(2.22)

then the transformation is a gauge transformation, i.e., the action is
invariant under it. Hereafter we set $\xi$ as (2.22) so that $A=0$. It is
important to note that (2.21) determines $\varepsilon_{0}$, except its
constant mode, completely by $\varepsilon,~{}\bar{\varepsilon}$. In that sense
the number of the gauge degrees of freedom of the complex particle is
essentially two. The global invariance under the transformation generated by
the constant mode of $\varepsilon_{0}$ is a hidden one, which is in fact a
generalization of what exists in the system of free relativistic two
particles.

What is the conservation law derived from the above global symmetry? We see
from (2.19) that the lagrangian varies under the global transformation as

$\displaystyle\delta
L=\frac{d}{d\tau}\left[\frac{\varepsilon_{0}\bar{\mbox{\boldmath$w$}}}{\bar{g}}\cdot\left(\frac{\mbox{\boldmath$w$}}{2g}+i\kappa\bar{\mbox{\boldmath$z$}}\right)\right]+{\rm
c.c.}$ (2.23)

On the other hand we have, up to EL-equations,

$\displaystyle\delta
L=\frac{d}{d\tau}\left(\frac{\varepsilon_{0}\bar{\mbox{\boldmath$w$}}}{\bar{g}}\cdot\frac{\partial
L}{\partial\mbox{\boldmath$w$}}\right)+{\rm c.c.}.$ (2.24)

Equating them we see that the quantity

$\displaystyle
Q=\frac{\bar{\mbox{\boldmath$w$}}}{\bar{g}}\cdot\left(\frac{\mbox{\boldmath$w$}}{2g}+i\kappa\bar{\mbox{\boldmath$z$}}-\frac{\partial
L}{\partial\mbox{\boldmath$w$}}\right)=\frac{\bar{\mbox{\boldmath$w$}}\cdot\mbox{\boldmath$w$}}{2\bar{g}g}$
(2.25)

is conserved along the trajectory satisfying the EL-equations. However, $Q$
vanishes by the 2nd order lagrangian constraint. Thus we get no nontrivial
conserved quantity from the global invariance. Another conserved quantity is
the $D$-dimensional momentum

$\displaystyle\mbox{\boldmath$p$}\stackrel{{\scriptstyle\rm
def}}{{=}}\frac{\mbox{\boldmath$w$}}{g}+2i\kappa\bar{\mbox{\boldmath$z$}}.$
(2.26)

derived from the translation invariance of the action.

##  3 Canonical theory

Now, let us proceed to the canonical theory. Denote the canonical momenta of
$z$ and the complex einbein $g$ as $\pi$ and $\pi$, respectively. The
characteristic equations $\mbox{\boldmath$\pi$}=\partial
L/\partial\mbox{\boldmath$w$}$ and $p=\partial L/\partial w$ are written as

$\displaystyle\mbox{\boldmath$\pi$}=\frac{\mbox{\boldmath$w$}}{g}+i\kappa\bar{z},\qquad\pi=0.$
(3.1)

The general solution, $\mbox{\boldmath$w$}=\mbox{\boldmath$U$},~{}w=U$, to the
above equations is

$\displaystyle\mbox{\boldmath$U$}=(\mbox{\boldmath$\pi$}-i\kappa\bar{z})g,\qquad
U=c(\mbox{\boldmath$z$},\mbox{\boldmath$\pi$},g,\pi),$ (3.2)

where $c$ is an arbitrary function of the canonical variables. The primary
constraint is

$\displaystyle\varphi\stackrel{{\scriptstyle\rm def}}{{=}}\pi=0,$ (3.3)

and its c.c., which correspond to the lagrangian relation, $\partial
L/\partial w=\partial L/\partial\bar{w}=0$.

The hamiltonian is

$\displaystyle H$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$\pi$}\cdot\mbox{\boldmath$U$}+\pi
U-\left(\frac{\mbox{\boldmath$U$}^{2}}{2g}+i\kappa\mbox{\boldmath$U$}\cdot\bar{\mbox{\boldmath$z$}}\right)+{\rm
c.c.}$ (3.4) $\displaystyle=$ $\displaystyle g\chi+c\varphi+{\rm c.c.},$

where

$\displaystyle\chi\stackrel{{\scriptstyle\rm
def}}{{=}}\frac{1}{2}(\mbox{\boldmath$\pi$}-i\kappa\bar{\mbox{\boldmath$z$}})^{2}=\frac{\mbox{\boldmath$U$}^{2}}{2g}.$
(3.5)

The equation of motion for a canonical variable, $q$, is the canonical
equation expressed as

$\displaystyle\dot{q}=\\{q,H\\},$ (3.6)

where the Poisson bracket is defined by

$\displaystyle\\{A,B\\}=\frac{\partial
A}{\partial\mbox{\boldmath$z$}}\cdot\frac{\partial
B}{\partial\mbox{\boldmath$\pi$}}-\frac{\partial
A}{\partial\mbox{\boldmath$\pi$}}\cdot\frac{\partial
B}{\partial\mbox{\boldmath$z$}}+\frac{\partial A}{\partial g}\frac{\partial
B}{\partial p}-\frac{\partial A}{\partial p}\frac{\partial B}{\partial g}.$
(3.7)

We denote

$\displaystyle A^{\sim}\stackrel{{\scriptstyle\rm def}}{{=}}\\{A,H\\},$ (3.8)

which is frequently used in what follows. For example the canonical equation
is written as $\dot{q}=q^{\sim}$.

In order to compare the above relations to those of the lagrangian theory, we
define the pullback of canonical variables by

$\displaystyle q_{\rm pb}\stackrel{{\scriptstyle\rm
def}}{{=}}q|_{\mbox{\boldmath$\pi$}=\mbox{\boldmath$W$},~{}\pi=W},\qquad\mbox{\boldmath$W$}\stackrel{{\scriptstyle\rm
def}}{{=}}\frac{\partial L}{\partial\mbox{\boldmath$w$}},\qquad
W\stackrel{{\scriptstyle\rm def}}{{=}}\frac{\partial L}{\partial w}=0.$ (3.9)

The pullback of the canonical equations for $\pi$ and $z$ are

$\displaystyle 0$ $\displaystyle=$
$\displaystyle(\dot{\mbox{\boldmath$\pi$}}-\mbox{\boldmath$\pi$}^{\sim})_{\rm
pb}=[{\bf EL}]_{u=U_{\rm pb}},$ (3.10) $\displaystyle 0$ $\displaystyle=$
$\displaystyle(\dot{\mbox{\boldmath$z$}}-\mbox{\boldmath$z$}^{\sim})_{\rm
pb}=\dot{\mbox{\boldmath$z$}}-\mbox{\boldmath$U$}_{\rm pb},$ (3.11)

which are equivalent to the EL-equations, since $\mbox{\boldmath$U$}_{\rm pb}$
is dummy variable in this expression.

The pullback of the primary constraint, $\varphi=0$, is identity, and it
should be preserved in the orbit, where EL-equations are satisfied.. The
preservation of the primary constraint requires the 1st order secondary
constraint which is expressed in the present case as

$\displaystyle\chi=0,$ (3.12)

and its c.c.. Furthermore there is the 2nd order secondary constraint222 Here
we call all constraints apart form the primary one as secondary constraints.
In a literature the secondary constraints of higher order may be called, e.g.,
tertiary constraints. :

$\displaystyle\chi_{0}\stackrel{{\scriptstyle\rm
def}}{{=}}(\mbox{\boldmath$\pi$}-i\kappa\bar{\mbox{\boldmath$z$}})\cdot(\bar{\mbox{\boldmath$\pi$}}+i\kappa\mbox{\boldmath$z$})=0,$
(3.13)

which preserves $\chi=0$. There is no other constraints.

It has been proven, in general gauge theories, that the pullback of the
secondary constraints to the lagrangian theory are all lagrangian constraints
[19]. In the present model we have the lagrangian constraints, (2.3) and
(2.5), which correspond to the secondary constraints, (3.12) and (3.13),
respectively. The 2nd order secondary constraint, (3.13), plays an important
role in deriving the critical dimension of the complex particle as is shown
later.

The primary and the secondary constraints in the present model are 1st-class,
i.e., their Poisson brackets are closed in themselves:

$\displaystyle\\{\varphi,\chi\\}=\\{\varphi,\bar{\chi}\\}=0,$ (3.14)
$\displaystyle\\{\chi,\bar{\chi}\\}=-2i\kappa\chi_{0},\qquad\\{\chi,\chi_{0}\\}=-4i\kappa\chi.$
(3.15)

If we use the following basis

$\displaystyle L_{1}=-\frac{1}{4\kappa}\chi,\qquad
L_{-1}=-\frac{1}{4\kappa}\bar{\chi},\qquad L_{0}=-\frac{1}{16\kappa}\chi_{0},$
(3.16)

(3.15) are written in a compact form:

$\displaystyle\\{L_{n},L_{m}\\}=i(n-m)L_{n+m},\qquad(n,m=0,\pm 1)$ (3.17)

which makes apparent the sl$(2,\mathbb{R})$ algebra of the constraint
functions.

Any function, $Q(q)$, of the canonical variables generates transformation
through the Poisson bracket as $\delta q=\\{q,Q\\}$, which in fact is one of
the canonical transformations defined by Goldstein [22]. In particular if it
satisfies

$\displaystyle Q^{\sim}\equiv 0\qquad{\rm mod}~{}\varphi,$ (3.18)

it is called the canonical gauge transformation.

Consider the generating function, $Q$, which is a linear combination of the
constraint functions:

$\displaystyle
Q=g\epsilon\chi+\bar{g}\bar{\epsilon}\bar{\chi}+\epsilon_{0}\chi_{0}+\xi\varphi+\bar{\xi}\bar{\varphi},$
(3.19)

where $\epsilon,\epsilon_{0}$ and $\xi$ are infinitesimal parameters, then the
transformation generated by $Q$ is

$\displaystyle\delta_{Q}\mbox{\boldmath$z$}$ $\displaystyle=$ $\displaystyle
g\epsilon(\mbox{\boldmath$\pi$}-i\kappa\bar{\mbox{\boldmath$z$}})+\epsilon_{0}(\bar{\mbox{\boldmath$\pi$}}+i\kappa\beta
bz),$ (3.20) $\displaystyle\delta_{Q}g$ $\displaystyle=$ $\displaystyle\xi,$
(3.21)

here we omit terms which vanish by the constraints. The pullback of the above
transformation to the lagrangian theory is

$\displaystyle\delta_{\rm L}\mbox{\boldmath$z$}$ $\displaystyle=$
$\displaystyle\epsilon\mbox{\boldmath$U$}_{\rm
pb}+\frac{\epsilon_{0}}{\bar{g}}\bar{\mbox{\boldmath$U$}}_{\rm pb},$ (3.22)
$\displaystyle\delta_{\rm L}g$ $\displaystyle=$ $\displaystyle\xi,$ (3.23)

which is equivalent to the semi-gauge transformation, (2.18). We see by a
simple calculation that the condition for the canonical geuge transformation,
(3.18), becomes

$\displaystyle\epsilon^{\sim}_{0}-2i\kappa
g\bar{g}(\epsilon-\bar{\epsilon})=0,$ (3.24)
$\displaystyle~{}~{}\xi=(g\epsilon)^{\sim}+4i\kappa\epsilon_{0}g,$ (3.25)

the pullback of which precisely coincide with (2.21) and (2.22), the condition
for the gauge transformation.

From Eqs.(2.26) and (3.1), we see the conserved momentum of the complex
particle is expressed in terms of the canonical variables by

$\displaystyle\mbox{\boldmath$p$}=\mbox{\boldmath$\pi$}+i\kappa\bar{\mbox{\boldmath$z$}}.$
(3.26)

In closing this section we point out that the Poisson bracket between the
conserved momenta and its c.c. is

$\displaystyle\\{(\mbox{\boldmath$p$})^{\mu},(\bar{\mbox{\boldmath$p$}})^{\nu}\\}=2i\kappa\eta^{\mu\nu},$
(3.27)

which does not vanish for $\kappa\neq 0$. This means that $z$ and
$\bar{\mbox{\boldmath$z$}}$ do not represent two independent quantities.

##  4 Quantization

In order to quantize the complex particle we represent the canonical variables
as linear operator on some Hilbert space of state vectors which are expressed
by functions of $z$ and $\bar{\mbox{\boldmath$z$}}$. We assume the functions
are single valued and bounded everywhere. The inner product of two states
$\phi_{1}$ and $\phi_{2}$ is defined by

$\displaystyle\langle\phi_{1}|\phi_{2}\rangle=\int
d^{D}zd^{D}\bar{z}~{}\phi_{1}^{*}(z,\bar{z})\phi_{2}(z,\bar{z}).$ (4.1)

The operators corresponding to canonical variables,
$\mbox{\boldmath$\pi$},\bar{\mbox{\boldmath$\pi$}}$, are

$\displaystyle\hat{\mbox{\boldmath$\pi$}}=-i\mbox{\boldmath$\partial$}\stackrel{{\scriptstyle\rm
def}}{{=}}-i\frac{\partial{}}{\partial{\mbox{\boldmath$z$}}},\qquad\hat{\bar{\mbox{\boldmath$\pi$}}}=-i\bar{\mbox{\boldmath$\partial$}}\stackrel{{\scriptstyle\rm
def}}{{=}}-i\frac{\partial{}}{\partial{\bar{\mbox{\boldmath$z$}}}}.$ (4.2)

The primary constraint function, $\varphi=\pi$, may be replaced by
$-i\partial/\partial g$, but we set $\varphi=0$ simply by assuming the state
function is independent of $g$.

The constraint functions, $L_{n},~{}(n=0,\pm 1)$, are replaced by the
operators

$\displaystyle\hat{L}_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(-i\mbox{\boldmath$\partial$}-i\kappa\bar{\mbox{\boldmath$z$}})^{2},$
(4.3) $\displaystyle\hat{L}_{-1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(-i\bar{\mbox{\boldmath$\partial$}}+i\kappa\mbox{\boldmath$z$})^{2},$
(4.4) $\displaystyle\hat{L}_{0}$ $\displaystyle=$
$\displaystyle(-i\mbox{\boldmath$\partial$}-i\kappa\bar{\mbox{\boldmath$z$}})\cdot(-i\bar{\mbox{\boldmath$\partial$}}+i\kappa\mbox{\boldmath$z$})+\alpha,$
(4.5)

where $\alpha$ is a constant representing the ordering ambiguity of operators.
The commutation relation among $\hat{L}_{n}$ is

$\displaystyle[\hat{L}_{n},~{}\hat{L}_{m}]=(n-m)\left(\hat{L}_{n+m}-\left(\alpha-\frac{D}{4}\right)\delta_{n+m}\right).\qquad(n,m=0,\pm
1)$ (4.6)

According to the usual quantization scheme we introduce the ghost variables,
$c_{n},~{}(n=0,\pm 1)$, and the BRST operator expressed as

$\displaystyle Q_{\rm BRST}=\sum_{n=0,\pm
1}c_{n}\hat{L}_{n}-\frac{1}{2}\sum_{n,m=0,\pm
1}(n-m)c_{n}c_{m}\frac{\partial{}}{\partial{c_{n+m}}}.$ (4.7)

Requiring the nilpotency of $Q_{\rm BRST}$ we fix the ordering ambiguity as

$\displaystyle\alpha=\frac{D}{4}.$ (4.8)

Then $\hat{L}_{n}$ becomes a basis of sl$(2,\mathbb{R})$ without central term.

In the classical theory the constraints assure the equivalence between the
lagrangian and the canonical theories, and are employed to define the physical
subspace of whole phase space. In quantum theory they cannot be put as
operator equations. Then this condition is so relaxed that the matrix elements
of any products of $\hat{L}_{n}$ between the physical states, $|{\rm
phys}\rangle=|\phi\rangle,|\varphi\rangle$, vanish:

$\displaystyle\langle\phi|\hat{L}_{n_{1}}\cdots\hat{L}_{n_{k}}|\varphi\rangle=0.$
(4.9)

This condition is realized by the requirement

$\displaystyle\hat{L}_{1}|{\rm phys}\rangle=\hat{L}_{0}|{\rm phys}\rangle=0,$
(4.10)

since $\hat{L}_{n},~{}(n=0,\pm 1)$ close under the commutation relations, and
the fact, $\hat{L}^{\dagger}_{1}=\hat{L}_{-1}$, giving $\langle{\rm
phys}|\hat{L}_{-1}=0$.

In the quantum theory the classical conserved momentum
$\mbox{\boldmath$p$}=\mbox{\boldmath$\pi$}+i\kappa\bar{\mbox{\boldmath$z$}}$,
is replaced by

$\displaystyle\hat{\mbox{\boldmath$p$}}$ $\displaystyle=$
$\displaystyle-i\mbox{\boldmath$\partial$}+i\kappa\bar{\mbox{\boldmath$z$}},$
(4.11)

and its hermitian conjugate. In order that the model is physically meaningful
there must be eigenvector of the conserved momentum with real eigenvalues in
the physical sector of states. Such a operator is
$\hat{\mbox{\boldmath$P$}}=\beta\hat{\mbox{\boldmath$p$}}+\beta\hat{\mbox{\boldmath$p$}}$,
with some complex value $\beta$. Since we can set $\beta=1$ by a global
rotation, we choose

$\displaystyle\hat{\mbox{\boldmath$P$}}=\hat{\mbox{\boldmath$p$}}+\hat{\mbox{\boldmath$p$}}^{\dagger},$
(4.12)

as the conserved momentum without loss of generality.

Now let us obtain the function representing the physical state, which
satisfies the constraint equations and is the eigenvector of
$\hat{\mbox{\boldmath$P$}}$. For this purpose we must solve the following
equations:

$\displaystyle\hat{L}_{1}|k\rangle=\hat{L}_{0}|k\rangle=0,$ (4.13)
$\displaystyle\hat{\mbox{\boldmath$P$}}|k\rangle=\mbox{\boldmath$k$}|k\rangle.$
(4.14)

Let us put

$\displaystyle|k\rangle=e^{-\kappa\mbox{\boldmath$z$}\cdot\bar{\mbox{\boldmath$z$}}}f(z,\bar{z}).$
(4.15)

Then the condition $\hat{L}_{1}|k\rangle=0$ becomes
$\mbox{\boldmath$\partial$}^{2}f(z,\bar{z})=0$, i.e., $f(z,\bar{z})$ is an
harmonic function of $z$. The eigenvalue equation (4.14) is written in terms
of $f(z,\bar{z})$ as

$\displaystyle(\mbox{\boldmath$\partial$}+\bar{\mbox{\boldmath$\partial$}}-2\kappa\bar{\mbox{\boldmath$z$}}-i\mbox{\boldmath$k$})f(z,\bar{z})=0.$
(4.16)

This is the form of separating variables with respect to $z$ and
$\bar{\mbox{\boldmath$z$}}$. The solution is written as

$\displaystyle
f(z,\bar{z})=e^{i\mbox{\boldmath$k$}_{1}\cdot\mbox{\boldmath$z$}+i\left(\mbox{\boldmath$k$}-\mbox{\boldmath$k$}_{1}\right)\cdot\bar{\mbox{\boldmath$z$}}+\kappa\bar{\mbox{\boldmath$z$}}^{2}}$
(4.17)

where $\mbox{\boldmath$k$}_{1}$ is an arbitrary separation constant. The
general solution is obtained by multiplying arbitrary function, $a(k_{1})$, to
$f(z,\bar{z})$ and integrating over $k_{1}$:

$\displaystyle
f(z,\bar{z})=e^{i\mbox{\boldmath$k$}\cdot\bar{\mbox{\boldmath$z$}}+\kappa\bar{\mbox{\boldmath$z$}}^{2}}g(y),\qquad\mbox{\boldmath$y$}=i(\mbox{\boldmath$z$}-\bar{\mbox{\boldmath$z$}}),$
(4.18)

where $g(y)$ is the function of real variable $y$, whose Fourier coefficient
is $a(k_{1})$. Since $f(z,\bar{z})$ is harmonic with respect to $z$, $g(y)$
should be a real analytic function.

At last the condition $\hat{L}_{0}|k\rangle=0$ is expressed as

$\displaystyle(\mbox{\boldmath$\partial$}\cdot\bar{\mbox{\boldmath$\partial$}}-2\kappa\mbox{\boldmath$z$}\cdot\mbox{\boldmath$\partial$}-4k\alpha)f(z,\bar{z})=0.$
(4.19)

Substituting (4.18) into this relation, we get

$\displaystyle\left[\left(y^{\mu}-\frac{k^{\mu}}{2\kappa}\right)\frac{\partial{}}{\partial{y^{\mu}}}\right]g(y)=0.$
(4.20)

Defining function, $h(u)$, by

$\displaystyle
g(y)=\left[\left(\mbox{\boldmath$y$}-\frac{\mbox{\boldmath$k$}}{2\kappa}\right)^{2}\right]^{-\alpha}h\left(y-\frac{k}{2\kappa}\right),$
(4.21)

we see that Eq.(4.20) is written as

$\displaystyle u^{\mu}\frac{\partial{}}{\partial{u^{\mu}}}h(u)$
$\displaystyle=$ $\displaystyle 0.$ (4.22)

Using this equation the real analyticity of $g(y)$ is expressed as

$\displaystyle\left(\rule[-2.84526pt]{0.28453pt}{11.38109pt}\rule[8.53581pt]{11.38109pt}{0.28453pt}\rule[-2.84526pt]{11.38109pt}{0.28453pt}\rule[-2.84526pt]{0.28453pt}{11.38109pt}\hskip
2.84526pt_{u}-\frac{K}{\mbox{\boldmath$u$}^{2}}\right)h(u)$ $\displaystyle=$
$\displaystyle 0,$ (4.23) $\displaystyle K$ $\displaystyle=$ $\displaystyle
2\alpha(D-2(\alpha+1))=\frac{1}{4}D(D-4),$ (4.24)

where
$\rule[-2.84526pt]{0.28453pt}{11.38109pt}\rule[8.53581pt]{11.38109pt}{0.28453pt}\rule[-2.84526pt]{11.38109pt}{0.28453pt}\rule[-2.84526pt]{0.28453pt}{11.38109pt}\hskip
2.84526pt_{u}=\partial^{2}/\partial u^{\mu}\partial u_{\mu}$ is the d’Alembert
operator with respect to $D$-dimensional real coordinates, $u^{\mu}$.

Our task is to solve Eqs. (4.22) and (4.23) with some physical assumptions. In
order to see the physical meaning of these equations, let us remember the
corresponding equations in the case of single relativistic particle. In that
case there is one secondary constraint, $\mbox{\boldmath$\pi$}^{2}=0$, and the
corresponding quantum equation is
$\rule[-2.84526pt]{0.28453pt}{11.38109pt}\rule[8.53581pt]{11.38109pt}{0.28453pt}\rule[-2.84526pt]{11.38109pt}{0.28453pt}\rule[-2.84526pt]{0.28453pt}{11.38109pt}\hskip
2.84526pt|k\rangle=0$. The solution is the plane wave,
$|k\rangle=e^{i\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}}$, with
$\mbox{\boldmath$k$}^{2}=0$. In the case of the complex particle the
corresponding state function is

$\displaystyle|k\rangle$ $\displaystyle=$ $\displaystyle
e^{i(\mbox{\boldmath$k$}+\kappa\mbox{\boldmath$y$})\cdot\bar{\mbox{\boldmath$z$}}}\left[\left(\mbox{\boldmath$y$}-\frac{\mbox{\boldmath$k$}}{2\kappa}\right)^{2}\right]^{-D/4}h\left(\mbox{\boldmath$y$}-\frac{\mbox{\boldmath$k$}}{2\kappa}\right),\qquad
y=i(\mbox{\boldmath$z$}-\bar{\mbox{\boldmath$z$}}),$ (4.25)

as is seen by the definition of $h$. A natural assumption for the function
$h(u)$ is that it is single-valued and bounded everywhere.

The solution for $h(u)$ is obtained by using the eigenvalue equation of the
Laplace-Beltrami(LB) operator on $S^{1,D-2}$ and $S^{D-1}$ in the case of
Minkowski and Euclid metric, respectively. LB is connected with the
(pseudo-)rotation in the real spacetime, which is generated by

$\displaystyle\ell^{\mu\nu}=-i\left(u^{\mu}\frac{\partial{}}{\partial{u_{\nu}}}-u^{\nu}\frac{\partial{}}{\partial{u_{\mu}}}\right),$
(4.26)

acting on the real coordinates $u^{\mu}$ as

$\displaystyle\left(\frac{i}{2}\epsilon_{\mu\nu}\ell^{\mu\nu}\right)u_{\lambda}=\epsilon_{\mu\lambda}u^{\mu},\qquad\epsilon_{\nu\mu}=-\epsilon_{\mu\nu},$
(4.27)

where $\epsilon_{\nu\mu}$ are infinitesimal anti-symmetric parameters.
$\ell^{\mu\nu}$ satisfy the commutation relation

$\displaystyle[\ell^{\mu\nu},\ell^{\mu^{\prime}\nu^{\prime}}]=i(\eta^{\nu^{\prime}[\mu}\ell^{\nu]\mu^{\prime}}-\eta^{\mu^{\prime}[\mu}\ell^{\nu]\nu^{\prime}}).$
(4.28)

The definition of LB is222 In the case of $D=3$ Euclidean space
${\varDelta}_{\rm LB}$ is minus square of the angular momentum operators. Here
the minus sign is put in accordance with mathematical literatures.

$\displaystyle{\varDelta}_{\rm LB}\stackrel{{\scriptstyle\rm
def}}{{=}}-\frac{1}{2}\ell_{\mu\nu}\ell^{\mu\nu},$ (4.29)

and it commutes with all $\ell^{\mu\nu}$, i.e., one of Casimir operators of
the (pseudo-)rotation group.

LB has properties which are easily treated in the (pseudo-)spherical
coordinates, $(r,\theta_{1},..,\theta_{D-1})$, as described in the next
section. An important relation between the d’Alembert operator and LB in the
(pseudo-)spherical coordinates is

$\displaystyle
s~{}\rule[-2.84526pt]{0.28453pt}{11.38109pt}\rule[8.53581pt]{11.38109pt}{0.28453pt}\rule[-2.84526pt]{11.38109pt}{0.28453pt}\rule[-2.84526pt]{0.28453pt}{11.38109pt}\hskip
2.84526pt_{u}=\frac{1}{r^{2}}{\varDelta}_{\rm
LB}+\frac{1}{r^{D-1}}\frac{\partial{}}{\partial{r}}r^{D-1}\frac{\partial{}}{\partial{r}},$
(4.30)

where

$\displaystyle s\stackrel{{\scriptstyle\rm def}}{{=}}{\rm
sgn}(u_{\mu}u^{\mu}).$ (4.31)

The proof of (4.30) is simple, but we give it in Appendix A.2, for
completeness. The condition (4.22) becomes $r$ independence of $h(u)$ in the
(pseudo-)spherical coordinates (see (5.28) in the next section). Hence the
second term in r.h.s. of (4.30) vanishes in acting to $h(u)$. In Section 6 we
solve the eigenvalue equation of ${\varDelta}_{\rm LB}$,

$\displaystyle{\varDelta}_{\rm LB}h=\lambda h,$ (4.32)

where the eigenvalue $\lambda$ must have restricted values. Therefore, from
Eq. (4.23) we get the relation between the dimension of spacetime and the
eigenvalue $\lambda$,

$\displaystyle\frac{1}{4}D(D-4)=\lambda,$ (4.33)

here we use $\mbox{\boldmath$u$}^{2}=sr^{2}$.

##  5 Laplace-Beltrami operator

In the theory of spherical harmonics the LB operator plays an important role,
which maps $S^{D-1}$ into itself in the case of the Euclid metric. For the
purpose of the present paper it is convenient to express LB in terms of the
(pseudo-)spherical coordinates.

The relation between the $D$-dimensional orthogonal coordinates, $u^{\mu}$, in
the Euclid spacetime, and the spherical coordinates,
$(r,\theta_{1},..,\theta_{D-1})$, is defined by

$\displaystyle u^{m}$ $\displaystyle=$ $\displaystyle
rC_{m+1}P_{m},\qquad(0\leq m\leq D-2),$ (5.1) $\displaystyle u^{D-1}$
$\displaystyle=$ $\displaystyle rP_{D-1},$ (5.2)

here we use the abbreviated notation

$\displaystyle C_{m}\\!\\!$ $\displaystyle\stackrel{{\scriptstyle\rm
def}}{{=}}$ $\displaystyle\\!\\!\cos{\theta_{m}},\qquad
S_{m}\stackrel{{\scriptstyle\rm def}}{{=}}\sin{\theta_{m}},$ (5.3)
$\displaystyle P_{m}\\!\\!$ $\displaystyle\stackrel{{\scriptstyle\rm
def}}{{=}}$ $\displaystyle\\!\\!\prod_{k=1}^{m}S_{k},\quad(1\leq m\leq
D-1),\qquad P_{0}\stackrel{{\scriptstyle\rm def}}{{=}}1.$ (5.4)

It is convenient to set $C_{D}=1,~{}S_{D}=0$, since then only (5.1) is
necessary in that notation. The ranges of the variables in the spherical
coordinates are

$\displaystyle 0\leq r<\infty,\qquad 0\leq\theta_{m}\leq\pi,\quad(1\leq m\leq
D-2),\qquad 0\leq\theta_{D-1}\leq 2\pi,$ (5.5)

here we must set $0\leq\theta_{1}\leq 2\pi$ in the case of $D=2$.

The pseudo-spherical coordinates in the Minkowski spacetime can be defined in
a similar manner. First we divide the whole spacetime into three connected
subspaces:

$\displaystyle M_{+}$ $\displaystyle=$
$\displaystyle\\{u|~{}u_{\mu}u^{\mu}\leq 0,~{}u^{0}\geq 0\\},$ (5.6)
$\displaystyle M_{-}$ $\displaystyle=$
$\displaystyle\\{u|~{}u_{\mu}u^{\mu}\leq 0,~{}u^{0}\leq 0\\},$ (5.7)
$\displaystyle M_{0}$ $\displaystyle=$
$\displaystyle\\{u|~{}u_{\mu}u^{\mu}\geq 0\\}.$ (5.8)

The set of points whose radius vector from the origin is time-like belong to
$M_{\pm}$, and those of space-like ones belong to $M_{0}$. The boundaries are
the upper light-cone for $M_{+}$, the lower light-cone for $M_{-}$ and both of
them for $M_{0}$. For each connected subspace the pseudo-spherical coordinates
are defined by

$\displaystyle u^{m}=rC_{m+1}P_{m},\quad(0\leq m\leq D-1),$ (5.9)

where

$\displaystyle(C_{1},S_{1})=\left\\{\begin{array}[]{c}(\pm\cosh{\tau},~{}\pm\sinh{\tau})\qquad{\rm
for}~{}u\in M_{\pm}\\\ (\sinh{\tau},~{}-\cosh{\tau})\qquad\quad{\rm
for}~{}u\in M_{0}\end{array}\right.,\qquad\tau=\theta_{1},$ (5.12)
$\displaystyle C_{k}\stackrel{{\scriptstyle\rm
def}}{{=}}\cos{\theta_{k}},\qquad S_{k}\stackrel{{\scriptstyle\rm
def}}{{=}}\sin{\theta_{k}},\qquad(2\leq k\leq D-1)$ (5.13) $\displaystyle
P_{m}\stackrel{{\scriptstyle\rm def}}{{=}}\prod_{k=1}^{m}S_{k},\qquad(1\leq
m\leq D-1),$ (5.14) $\displaystyle\qquad C_{D}=1,\qquad S_{D}=0,\qquad
P_{0}=1.$ (5.15)

Here we use $\tau$ instead of $\theta_{1}$, since $\tau$ is the argument of
hyperbolic functions. The ranges of $(r,\tau.\theta_{k}),~{}(2\leq k\leq D-1)$
are

$\displaystyle 0<r<\infty,\quad-\infty<\tau<\infty,\quad
0\leq\theta_{k}\leq\pi,\quad(2\leq k\leq D-2),\quad 0\leq\theta_{D-1}\leq
2\pi.$ (5.16)

Let us list up some useful formulas with simple proofs of them.

$\displaystyle\bullet$
$\displaystyle\quad\eta_{00}C_{1}^{2}+S_{1}^{2}=s,\qquad\partial_{1}\left(\begin{array}[]{c}C_{1}\\\
S_{1}\end{array}\right)=s\left(\begin{array}[]{c}-S_{1}\\\
\eta_{00}C_{1}\end{array}\right),\qquad\partial_{m}\stackrel{{\scriptstyle\rm
def}}{{=}}\frac{\partial{}}{\partial{\theta_{m}}},$ (5.21)
$\displaystyle\bullet$
$\displaystyle\quad\sum_{k=m}^{D-1}C_{k+1}^{2}P_{k}^{2}=P_{m}^{2},\qquad(1\leq
m\leq D-1)$ (5.22) $\displaystyle\bullet$ $\displaystyle\quad
r=\tilde{r}~{}\stackrel{{\scriptstyle\rm
def}}{{=}}~{}\sqrt{s~{}u_{\mu}u^{\mu}}~{},\qquad s\stackrel{{\scriptstyle\rm
def}}{{=}}{\rm sgn}(u_{\mu}u^{\mu}),$ (5.23) $\displaystyle\bullet$
$\displaystyle\quad C_{1}=\frac{u^{0}}{\sqrt{s~{}u_{\mu}u^{\mu}}}~{},\qquad$
(5.24) $\displaystyle\bullet$ $\displaystyle\quad
C_{n}=\frac{s_{n-1}u^{n-1}}{\sqrt{\sum_{k=n-1}^{D-1}u_{k}^{2}}}~{},\qquad(2\leq
n\leq D-1),\qquad s_{n}\stackrel{{\scriptstyle\rm def}}{{=}}{\rm sgn}(P_{n}).$
(5.25)

———————————————————————————–

pr.)

Eqs.(5.21) are apparent from the definitions. Eq.(5.22) is derived by summing
the square of $C_{k+1}P_{k},~{}(k\geq 1)$ from $k=D-1$ to $k=m,~{}(m\geq 1)$.
Using this relation with $m=1$ we have

$\displaystyle
s~{}u_{\mu}u^{\mu}=s~{}\left(u_{0}u^{0}+\sum_{k=1}^{D-1}(u_{k})^{2}\right)=sr^{2}(\eta_{00}C_{1}^{2}+S_{1}^{2})=r^{2},$
(5.26)

hence (5.23) follows. From (5.21) and $u^{0}=rC_{1}$ we have (5.24). Finally,
for $n\geq 2$ we have

$\displaystyle
C_{n}=\frac{u^{n-1}}{rP_{n-1}}=\frac{s_{n-1}u^{n-1}}{r|P_{n-1}|},$ (5.27)

and this coincides with (5.25) by (5.22). $\blacksquare$
———————————————————————————–

The derivative operators of $u^{\mu}$ are expressed by those of
$(r,\theta_{1},..,\theta_{D-1})$ as

$\displaystyle\bullet$
$\displaystyle\quad\frac{u^{\mu}}{\tilde{r}}\frac{\partial{}}{\partial{u^{\mu}}}=\frac{\partial{}}{\partial{{r},}}$
(5.28) $\displaystyle\bullet$
$\displaystyle\quad\frac{\partial{}}{\partial{u_{0}}}\\!=\\!sC_{1}\frac{\partial{}}{\partial{r}}-\eta_{00}\frac{S_{1}}{r}\partial_{1},\qquad\partial_{m}\stackrel{{\scriptstyle\rm
def}}{{=}}\frac{\partial{}}{\partial{\theta_{m}}},$ (5.29)
$\displaystyle\bullet$
$\displaystyle\quad\frac{\partial{}}{\partial{u_{m}}}\\!=\\!C_{m+1}P_{m}\left(s\frac{\partial{}}{\partial{r}}+\frac{1}{r}\sum_{p=1}^{m}\frac{C_{p}S_{p}}{P_{p}^{2}}\partial_{p}\right)-\frac{S_{m+1}}{rP_{m}}\partial_{m+1},\quad(1\leq
m\leq D-1)$ (5.30)

where

$\displaystyle C_{D}\\!=\\!1,\qquad S_{D}=0,\qquad
P_{m}\stackrel{{\scriptstyle\rm def}}{{=}}\prod_{k=1}^{m}S_{k},\qquad
P_{0}=1.$ (5.31)

———————————————————————————–

pr.)

Eq.(5.28) is apparent from the definition. As is seen from (5.25) $C_{p}$ does
not depend on $u^{0},..,u^{p-2}$. Hence we have for $1\leq m\leq D-2$

$\displaystyle\frac{\partial{}}{\partial{u_{m}}}$ $\displaystyle=$
$\displaystyle\frac{\partial\tilde{r}}{\partial
u_{m}}\frac{\partial{}}{\partial{r}}+\left(\frac{\partial{}}{\partial{u_{m}}}\frac{u^{0}}{\sqrt{su_{\mu}u^{\mu}}}\right)\left(\frac{dC_{1}}{d\theta_{1}}\right)^{-1}\partial_{1}+\sum_{p=2}^{m+1}\left(\frac{\partial{}}{\partial{u_{m}}}\frac{s_{p-1}u^{p-1}}{\sqrt{\sum_{k=p-1}^{D-1}u_{k}u^{k}}}\right)\left(\frac{dC_{p}}{d\theta_{p}}\right)^{-1}\partial_{p}.$
(5.32)

The second term in r.h.s. of this equation is

$\displaystyle({\rm 2nd}~{}{\rm term}~{}{\rm in}~{}{\rm r.h.s.})$
$\displaystyle=$ $\displaystyle\frac{u^{0}u^{m}}{r^{3}S_{1}}\partial_{1}.$
(5.33)

By a straightforward but somehow lengthy calculation, using the above
equation, we have (5.30) in the case of $(1\leq m\leq D-2)$.

In the case of $m=0$ we have

$\displaystyle\frac{\partial{}}{\partial{u_{0}}}=\frac{\partial\tilde{r}}{\partial
u_{0}}\frac{\partial{}}{\partial{r}}+\left(\frac{\partial{}}{\partial{u_{0}}}\frac{u^{0}}{\sqrt{su_{\mu}u^{\mu}}}\right)\left(\frac{dC_{1}}{d\theta_{1}}\right)^{-1}\partial_{1},$
(5.34)

here the 2nd term of r.h.s. is

$\displaystyle({\rm 2nd}~{}{\rm term}~{}{\rm in}~{}{\rm r.h.s.})$
$\displaystyle=$
$\displaystyle\frac{\eta_{00}s}{\tilde{r}^{3}}(u_{\mu}u^{\mu}-u_{0}u^{0})\left(\frac{dC_{1}}{d\theta_{1}}\right)^{-1}\partial_{1}$
(5.35) $\displaystyle=$
$\displaystyle-\frac{\eta_{00}}{\tilde{r}}\left(\sum_{m=1}^{D-1}C_{m+1}^{2}P_{m}^{2}\right)\left(S_{1}\right)^{-1}\partial_{1}.$

Using (5.22) we see

$\displaystyle\quad=-\eta_{00}\frac{S_{1}}{r}\partial_{1},$ (5.36)

hence (5.29) follows.

The case of $m=D-1$ is the same as that of $m\leq D-2$ apart from that the
upper limit of the sum in (5.32) is $m$. Hence we have

$\displaystyle\frac{\partial{}}{\partial{u_{D-1}}}$ $\displaystyle=$
$\displaystyle
P_{D-1}\left(s\frac{\partial{}}{\partial{r}}+\frac{1}{r}\sum_{p=1}^{D-1}\frac{C_{p}S_{p}}{P_{p}^{2}}\partial_{p}\right).$
(5.37)

After all (5.30) holds for $(0\leq m\leq D-1)$. $\blacksquare$
———————————————————————————–

Since, in the pseudo-spherical coordinates, $C_{1},~{}S_{1}$ are the
hyperbolic function, they may appear in various relations in different way
from $C_{p},~{}S_{p},~{}(p\neq 1)$. For example Eq.(5.30) with $m=0$ does not
coincide with (5.29).

The main purpose of this section is to represent the LB operator in the
(pseudo-)spherical coordinates. In the case of Euclid spacetime the expression
of the LB operator, and the proof of it in another context, have been given in
[23]-[26]. Here we extend them to the case of Minkowski spacetime. The result
is

$\displaystyle{\varDelta}_{\rm
LB}=\sum_{n=1}^{D-1}\frac{\sigma_{n}S_{n}^{n-D+3}}{P_{n}^{2}}\partial_{n}S_{n}^{D-n-1}\partial_{n},\qquad\sigma_{n}\stackrel{{\scriptstyle\rm
def}}{{=}}\left\\{\begin{array}[]{c}\eta_{00}\qquad{\rm for}~{}n=1\\\
~{}s\qquad~{}{\rm for}~{}n\geq 2\end{array}\right..$ (5.40)

If we put $\sigma_{n}=1$ in this equation we get ${\varDelta}_{\rm LB}$ in the
Euclid metric. Using (5.30) and the definition of $\ell^{\mu\nu}$, it might be
possible to obtain (5.40). Since the direct substitution of $\ell^{\mu\nu}$
into the definition of ${\varDelta}_{\rm LB}$ gives very tedious expression to
handle, it may not be a smart method to prove (5.40) (if not impossible). We
give here a proof by the induction with respect to the dimension $D$.

In the case of $D=2$ we see

$\displaystyle u^{0}=rC_{1},\quad
u^{1}=rS_{1},\quad\frac{\partial{}}{\partial{u_{0}}}=C_{1}\frac{\partial{}}{\partial{r}}-\frac{\eta_{00}S_{1}}{r}\partial_{1},\quad\frac{\partial{}}{\partial{u_{1}}}=S_{1}\frac{\partial{}}{\partial{r}}+\frac{C_{1}}{r}\partial_{1},$
(5.41)
$\displaystyle\ell^{01}=-ir\left[C_{1}\left(S_{1}\frac{\partial{}}{\partial{r}}+\frac{C_{1}}{r}\partial_{1}\right)-S_{1}\left(C_{1}\frac{\partial{}}{\partial{r}}-\frac{\eta_{00}S_{1}}{r}\partial_{1}\right)\right]$
$\displaystyle\quad~{}\\!=-i(C_{1}^{2}+\eta_{00}S_{1}^{2})\partial_{1}=-is\eta_{00}\partial_{1}.$
(5.42)

Hence we have ${\varDelta}_{\rm
LB}=-\ell^{01}\ell_{01}=\eta_{00}\partial_{1}^{2}$, and Eq.(5.40) holds. Let
us prove Eq.(5.40) in $D$-dimension by assuming it holds in $(D-1)$-dimension.

For that purpose it is convenient to define the (pseudo-)spherical coordinates
through the following two steps:

$\displaystyle u^{m}=\rho C_{m+1}P_{m},\qquad(1\leq m\leq D-1),$ (5.43)

and

$\displaystyle u^{0}$ $\displaystyle=$ $\displaystyle
rC_{1},\qquad\qquad\qquad\qquad\qquad\qquad\quad$ (5.44) $\displaystyle\rho$
$\displaystyle=$ $\displaystyle rS_{1},$ (5.45)

where $C_{1},~{}S_{1}$ are defined by (5.12). The two Eqs. (5.44) and (5.45)
are the same as the relation between two-dimensional orthogonal coordinates
$(u^{0},\rho)$ and the (pseudo-)spherical coordinates $(r,\theta_{1})$. Since
in the case of the Minkowski metric, $C_{1},~{}S_{1}$ are hyperbolic
functions, and being different from the spacial part of (5.43), this two step
definition makes the calculations easy.

Let us write again the relation (4.30),

$\displaystyle
s~{}\rule[-2.84526pt]{0.28453pt}{11.38109pt}\rule[8.53581pt]{11.38109pt}{0.28453pt}\rule[-2.84526pt]{11.38109pt}{0.28453pt}\rule[-2.84526pt]{0.28453pt}{11.38109pt}\hskip
2.84526pt^{(D)}=\frac{{\varDelta}_{\rm
LB}^{(D)}}{r^{2}}+\frac{1}{r^{D-1}}\frac{\partial{}}{\partial{r}}r^{D-1}\frac{\partial{}}{\partial{r}},$
(5.46)

which holds in arbitrary dimension, $D$. For the spacial part of the
(pseudo-)spherical coordinates in the definition of the above two step
definition, which is $(D-1)$-dimensional Euclid space ($s=+1$), Eq.(5.46) is
written as

$\displaystyle\rule[-2.84526pt]{0.28453pt}{11.38109pt}\rule[8.53581pt]{11.38109pt}{0.28453pt}\rule[-2.84526pt]{11.38109pt}{0.28453pt}\rule[-2.84526pt]{0.28453pt}{11.38109pt}\hskip
2.84526pt^{(D-1)}=\frac{{\varDelta}_{\rm
LB}^{(D-1)}}{\rho^{2}}+\frac{1}{\rho^{D-2}}\frac{\partial{}}{\partial{\rho}}\rho^{D-2}\frac{\partial{}}{\partial{\rho}}.$
(5.47)

For the two dimensional part of the (pseudo-)spherical coordinates, it is
written as

$\displaystyle s\left(\eta_{00}\frac{\partial^{2}}{\partial
u_{0}^{2}}+\frac{\partial^{2}}{\partial\rho^{2}}\right)=\frac{\eta_{00}}{r^{2}}\partial_{1}^{2}+\frac{1}{r}\frac{\partial{}}{\partial{r}}r\frac{\partial{}}{\partial{r}}.$
(5.48)

From Eqs. (5.46) $-$ (5.48), and
$\rule[-2.84526pt]{0.28453pt}{11.38109pt}\rule[8.53581pt]{11.38109pt}{0.28453pt}\rule[-2.84526pt]{11.38109pt}{0.28453pt}\rule[-2.84526pt]{0.28453pt}{11.38109pt}\hskip
2.84526pt^{D}=\rule[-2.84526pt]{0.28453pt}{11.38109pt}\rule[8.53581pt]{11.38109pt}{0.28453pt}\rule[-2.84526pt]{11.38109pt}{0.28453pt}\rule[-2.84526pt]{0.28453pt}{11.38109pt}\hskip
2.84526pt^{(D-1)}+\partial^{2}/\partial\rho^{2}$, we have

$\displaystyle{\varDelta}_{\rm LB}^{(D)}$ $\displaystyle=$ $\displaystyle
sr^{2}\left(\frac{{\varDelta}_{\rm
LB}^{(D-1)}}{\rho^{2}}-\frac{s}{r^{D-1}}\frac{\partial{}}{\partial{r}}r^{D-1}\frac{\partial{}}{\partial{r}}+\frac{1}{\rho^{D-2}}\frac{\partial{}}{\partial{\rho}}\rho^{D-2}\frac{\partial{}}{\partial{\rho}}+\frac{s}{r}\frac{\partial{}}{\partial{r}}r\frac{\partial{}}{\partial{r}}+\frac{s\eta_{00}}{r^{2}}\partial_{1}^{2}-\frac{\partial^{2}}{\partial\rho^{2}}\right)$
(5.49) $\displaystyle=$ $\displaystyle sr^{2}\left(\frac{{\varDelta}_{\rm
LB}^{(D-1)}}{\rho^{2}}-\frac{s(D-1)}{r}\frac{\partial{}}{\partial{r}}+\frac{D-2}{\rho}\frac{\partial{}}{\partial{\rho}}+\frac{s\eta_{00}}{r^{2}}\partial_{1}^{2}\right).$

From the definition of the two dimensional part of the (pseudo-)spherical
coordinates, we see $\rho=rS_{1}$ and the differential operator with respect
to $\rho$ is obtained by putting $\rho=u^{1},~{}C_{2}=1,~{}S_{2}=0$ in
Eq.(5.30),

$\displaystyle\frac{\partial{}}{\partial{\rho}}=sS_{1}\frac{\partial{}}{\partial{r}}+\frac{C_{1}}{r}\partial_{1}.$
(5.50)

Substituting it to (5.49), we see the dependence on $r$ and the terms of
$\partial/\partial r$ cancels out, giving

$\displaystyle{\varDelta}_{\rm LB}^{(D)}$ $\displaystyle=$
$\displaystyle\frac{s}{S_{1}^{2}}{\varDelta}_{\rm
LB}^{(D-1)}+\frac{s(D-2)C_{1}}{S_{1}}\partial_{1}+\eta_{00}\partial_{1}^{2}$
(5.51) $\displaystyle=$ $\displaystyle\frac{s}{S_{1}^{2}}{\varDelta}_{\rm
LB}^{(D-1)}+\frac{\eta_{00}}{S_{1}^{D-2}}\partial_{1}S_{1}^{D-2}\partial_{1}.$

By the assumption of the deduction Eq.(5.40) holds in $(D-1)$-dimensional
Euclid spacetime, hence we have

$\displaystyle{\varDelta}_{\rm
LB}^{(D-1)}=\sum_{n=1}^{D-2}\frac{S_{n}^{n-D+4}}{P_{n}^{2}}\partial_{n}S_{n}^{D-n-2}\partial_{n}=S_{1}^{2}\sum_{n=2}^{D-1}\frac{S_{n}^{n-D+3}}{P_{n}^{2}}\partial_{n}S_{n}^{D-n-1}\partial_{n}.$
(5.52)

Substituting it to Eq.(5.51) we get

$\displaystyle{\varDelta}_{\rm LB}^{(D)}$ $\displaystyle=$ $\displaystyle
s\sum_{n=2}^{D-1}\frac{S_{n}^{n-D+3}}{P_{n}^{2}}\partial_{n}S_{n}^{D-n-1}\partial_{n}+\frac{\eta_{00}}{S_{1}^{D-2}}\partial_{1}S_{1}^{D-2}\partial_{1}$
(5.53) $\displaystyle=$
$\displaystyle\sum_{n=1}^{D-1}\frac{\sigma_{n}S_{n}^{n-D+3}}{P_{n}^{2}}\partial_{n}S_{n}^{D-n-1}\partial_{n}.$

Hence (5.40) holds in $D-$dimensional Minkowski as well as Euclid spacetime.
$\blacksquare$
———————————————————————————–

The expression (5.40) for the Minkowski spacetime has not been published in a
mathematical literature as far as we know. In the next section we find all
possible eigenvalues of ${\varDelta}_{\rm LB}$, and deduce the critical
dimension of spacetime.

##  6 Critical dimension

In a mathematical literature the eigenvalue problem of LB is often discussed
through d’Alembert equation,
$\rule[-2.84526pt]{0.28453pt}{11.38109pt}\rule[8.53581pt]{11.38109pt}{0.28453pt}\rule[-2.84526pt]{11.38109pt}{0.28453pt}\rule[-2.84526pt]{0.28453pt}{11.38109pt}\hskip
2.84526pt\Psi=0$. In this section we solve the eigenvalue equation of
${\varDelta}_{\rm LB}$ directly in terms of the angle variables in the
(pseudo-)spherical coordinates, since ${\varDelta}_{\rm LB}$ is expressed only
by those variables irrelevant to the radius, $r$.

Let us solve the eigenvalue equation of ${\varDelta}_{\rm LB}$:

$\displaystyle{\varDelta}_{\rm LB}\Psi=\lambda\Psi.$ (6.1)

We assume the eigen-function $\Psi$ is single-valued and bounded everywhere.
The eigenvalue $\lambda$ is related to the dimension of spacetime $D$ as
Eq.(4.33).

In the case of $D=2$ the eigenvalue equation is

$\displaystyle\eta_{00}\partial_{1}^{2}\Psi=\lambda\Psi,$ (6.2)

and the solution is222 Here and hereafter we omit the normalization constant
of $\Psi$.

$\displaystyle\Psi=e^{m\theta_{1}},\qquad\lambda=\eta_{00}m^{2}.$ (6.3)

For the Euclid metric ($\eta_{00}=1$) the range of the spherical coordinate,
$\theta_{1}$, is $0\leq\theta_{1}\leq 2\pi$. Hence, from the single-valuedness
of $\Psi$, we see

$\displaystyle m=iL,\qquad L\in\mathbb{Z},\qquad\lambda=-L^{2}\leq
0,\qquad({\rm for}~{}{\rm Euclid}~{}{\rm metric}).$ (6.4)

On the other hand for the Minkowski metric ($\eta_{00}=-1$) the range of the
pseudo-spherical coordinate, $\tau=\theta_{1}$, is $-\infty<\tau<\infty$.
Hence, from the boundedness of $\Psi$ at infinity ($\tau=\pm\infty$), we see
$\Re m=0$, i.e.,

$\displaystyle m=iL,\qquad L\in\mathbb{R},\qquad\lambda=L^{2}\geq
0.\qquad({\rm for}~{}{\rm Minkowski}~{}{\rm metric}).$ (6.5)

Thus the sign of the eigenvalue depends on whether the invariant space of
(pseudo-)rotation is compact ($S^{D-1}$) or non-compact ($S^{1,D-2}$).

Next let us consider the case of $D\geq 3$. The eigenvalue equation is

$\displaystyle\left(\sum_{n=1}^{D-1}\frac{\sigma_{n}S_{n}^{n-D+3}}{P_{n}^{2}}\partial_{n}S_{n}^{D-n-1}\partial_{n}\right)\Psi=\lambda\Psi,$
(6.6)

which is of the form of separated variables with respect to
$(\theta_{1},..,\theta_{D-2})$ and $\theta_{D-1}$. Putting
$\Psi=G_{1}(\theta_{1},..,\theta_{D-2})M_{1}(\theta_{D-1})$, Eq.(6.6) reduces
to

$\displaystyle\left(\sum_{n=1}^{D-2}\frac{\sigma_{n}S_{n}^{n-D+3}}{P_{n}^{2}}\partial_{n}S_{n}^{D-1-n}\partial_{n}-\lambda+\frac{c_{1}}{P_{D-2}^{2}}\right)G_{1}$
$\displaystyle=$ $\displaystyle 0,$ (6.7)
$\displaystyle\left(s\partial_{D-1}^{2}-c_{1}\right)M_{1}$ $\displaystyle=$
$\displaystyle 0,$ (6.8)

where $c_{1}$ is the separation constant. If $D=3$ the solving step is
completed here. But for $D\geq 4$ Eq.(6.7) should be solved, which is again of
the separated type with respect to $(\theta_{1},..,\theta_{D-3})$ and
$\theta_{D-2}$. Putting
$G_{1}=G_{2}(\theta_{1},..,\theta_{D-3})M_{2}(\theta_{D-2})$, we have

$\displaystyle\left(\sum_{n=1}^{D-3}\frac{\sigma_{n}S_{n}^{n-D+3}}{P_{n}^{2}}\partial_{n}S_{n}^{D-1-n}\partial_{n}-\lambda+\frac{c_{2}}{P_{D-3}^{2}}\right)G_{2}$
$\displaystyle=$ $\displaystyle 0,$ (6.9)
$\displaystyle\left(\frac{s}{S_{D-2}}\partial_{D-2}S_{D-2}\partial_{D-2}+\frac{c_{1}}{S_{D-2}^{2}}-c_{2}\right)M_{2}$
$\displaystyle=$ $\displaystyle 0,$ (6.10)

where $c_{2}$ is the separation constant.

Going on similarly until $m=D-2$ we get the following set of ordinary
differential equations:

$\displaystyle\left(\eta_{00}S_{1}^{2-D}\partial_{1}S_{1}^{D-2}\partial_{1}-\lambda+\frac{c_{D-2}}{P_{1}^{2}}\right)G_{D-2}$
$\displaystyle=$ $\displaystyle 0,$ (6.11)
$\displaystyle\left(\frac{s}{S_{D-m}^{m-1}}\partial_{D-m}S_{D-m}^{m-1}\partial_{D-m}+\frac{c_{m-1}}{S_{D-m}^{2}}-c_{m}\right)M_{m}$
$\displaystyle=$ $\displaystyle 0,\qquad(m=3,..,D-2),$ (6.12)

where
$G_{m}=G_{m+1}(\theta_{1},..,\theta_{D-m-2})M_{m+1}(\theta_{D-m-1}),~{}(m=3,4,..,D-3)$,
and $c_{m},~{}(m=3,..,D-2)$ are the separation constants. Since (6.8) and
(6.10) coincide with (6.12) if we put $c_{0}=0,~{}m=1,2$ there, the equations
to be solved are (6.11) and (6.12) with $(m=1,2,..,D-2)$, the total number of
which is $D-1$. Using the above solutions, the eigen-function of
${\varDelta}_{\rm LB}$ is written as

$\displaystyle\Psi=G_{D-2}\prod_{m=1}^{D-2}M_{m}.$ (6.13)

Now let us solve (6.11) and (6.12) with $(m=1,2,..,D-2)$. As a solution to
Eq.(6.12) for $D\geq 3$ put

$\displaystyle
M_{m}(\theta_{D-m})=\left\\{\begin{array}[]{c}e^{in_{1}\theta_{D-1}},\qquad\qquad(m=1){}{}\\\
S_{D-m}^{n_{m}}(\theta_{D-m}),\qquad(2\leq m\leq D-2)\end{array}\right.,$
(6.16)

where $n_{m},(m=1,..,D-2)$ are complex numbers. Then the equations reduce to

$\displaystyle(sn_{1}^{2}+c_{1})e^{n_{1}\theta_{D-1}}=0,$ (6.17)
$\displaystyle(sn_{m}(n_{m}+m-2)+c_{m-1})S_{D-m}^{n_{m}-2}-(sn_{m}(n_{m}+m-1)+c_{m})S_{D-m}^{n_{m}}=0.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad(2\leq
m\leq D-2)$ (6.18)

Hence if $n_{m},~{}(1\leq m\leq D-2)$ satisfy

$\displaystyle n_{1}^{2}$ $\displaystyle=$ $\displaystyle-sc_{1},$ (6.19)
$\displaystyle c_{m}$ $\displaystyle=$ $\displaystyle-
sn_{m+1}(n_{m+1}+m-1),\quad(1\leq m\leq D-3)$ (6.20) $\displaystyle c_{m}$
$\displaystyle=$ $\displaystyle-sn_{m}(n_{m}+m-1),\qquad\quad(2\leq m\leq
D-2),$ (6.21)

for given $c_{m},(1\leq m\leq D-2)$, then $M_{m}$ are solutions of (6.12) with
$(m=1,2,..,D-2)$. Note that Eqs. (6.19) $-$ (6.21) do not change by replacing
$n_{m}$ to $-(n_{m}+m-1)$ in them. Thus other solutions are obtained by the
above replacement. Since Eq.(6.12) are homogeneous ordinary differential
equations of second rank, the dimension of the solution space as a vector
space is two. The above two special solutions are linearly independent, hence
the general solutions are linear combinations of them.

Since $S_{m}(\theta_{m})=\sin{\theta_{m}}$ for $m\geq 2$, the boundedness at
origin requires $\Re n_{m}\geq 0,~{}(2\leq m\leq D-2)$. Also the single-
valuedness of the solution requires $n_{1}\in\mathbb{Z}$. After all, $n_{m}$
which satisfy the boundedness and the single-valuedness are determined by

$\displaystyle n_{1}^{2}=n_{2}^{2},\qquad n_{1}\in\mathbb{Z}$ (6.22)
$\displaystyle n_{m+1}(n_{m+1}+m-1)=n_{m}(n_{m}+m-1),\qquad(2\leq m\leq D-2),$
(6.23) $\displaystyle\qquad\Re n_{m}\geq 0,\qquad(2\leq m\leq D-2).$ (6.24)

Firstly from (6.22) and $\Re n_{2}\geq 0$ we have
$n_{2}=|n_{1}|\in\mathbb{Z}_{\geq 0}$. Since (6.23) is written as

$\displaystyle(n_{m+1}-n_{m})(n_{m+1}+n_{m}+m-1)=0,$ (6.25)

it has two solutions $n_{m+1}=n_{m}$ and $n_{m+1}=-n_{m}-m+1$ as a 2nd order
algebraic equation for $n_{m+1}$. For $m=2$ the latter solution is excluded
because $\Re n_{3}<0$. Similarly we see the possible solutions are

$\displaystyle\ell\stackrel{{\scriptstyle\rm
def}}{{=}}|n_{1}|=n_{2}=\cdots=n_{D-2}\in\mathbb{Z}_{\geq 0}.$ (6.26)

In order to solve the remaining Eq.(6.11) we put
$G_{D-2}(\theta_{1})=S_{1}^{p}(\theta_{1})$ with a complex number $p$.
Substituting it to Eq.(6.11), we have

$\displaystyle[s~{}p(D-3+p)+c_{D-2}]S_{1}^{p-2}-[p(D-2+p)+\lambda]S_{1}^{p}=0.$
(6.27)

Hence we see that $G_{D-2}(\theta_{1})=S_{1}^{p}(\theta_{1})$ is a solution to
Eq.(6.11) if the following relations hold:

$\displaystyle c_{D-2}$ $\displaystyle=$ $\displaystyle-s~{}p(p+D-3),$ (6.28)
$\displaystyle\lambda$ $\displaystyle=$ $\displaystyle-p(p+D-2).$ (6.29)

Furthermore it is easy to show that there is another solution to Eq.(6.11)
with the same $\lambda$ and $c_{D-2}$, expressed as

$\displaystyle G^{\prime}_{D-2}=S_{1}^{p}\int S_{1}^{-2p-D+2}d\theta_{1},$
(6.30)

where $p$ satisfies (6.28) and (6.29). Since $G_{D-2}$ and $G^{\prime}_{D-2}$
are linearly independent, the general solutions to Eq.(6.11) are linear
combinations of the above two special solutions.

From Eqs. (6.28), (6.29), and (6.26) we see

$\displaystyle c_{D-2}=-s\ell(\ell+D-3),\qquad\ell\in\mathbb{Z}_{\geq 0},$
(6.31) $\displaystyle\ell(\ell+D-3)=p(p+D-3),$ (6.32)

i.e.

$\displaystyle(p-\ell)(p+\ell+D-3)=0.$ (6.33)

Corresponding to the two solutions, $p=\ell$ and $p=-\ell-D+3$, to the above
2nd order algebraic Eq. for $p$, the eigenvalues of ${\varDelta}_{\rm LB}$ are

$\displaystyle\lambda=\left\\{\begin{array}[]{c}-\ell(\ell+D-2),\qquad\qquad
p=\ell\\\ -(\ell-1)(\ell+D-3),\qquad
p=-\ell-D+3\end{array}\right.,\qquad\ell\in\mathbb{Z}_{\geq 0}.$ (6.36)

There is no other eigenvalue.

Let us show that $\lambda$ is further constrained by the requirement of the
boundedness of $G_{D-2}$ and $G^{\prime}_{D-2}$. We divide the argument for
the Euclid and the Minkowski metric.

(1) Euclid metric:

Since $S_{1}=\sin{\theta_{1}}$ in this case, the behavior of
$G^{\prime}_{D-2}$ around origin is $G^{\prime}_{D-2}\sim\theta_{1}^{-p-D+3}$.
Corresponding to the solutions, $p=\ell$ and $p=-\ell-D+3$, we have
$G^{\prime}_{D-2}\sim\theta_{1}^{-\ell-D+3}$ and
$G^{\prime}_{D-2}\sim\theta_{1}^{\ell}$, respectively. From the boundedness at
origin the former cases with $D\geq 4$ are excluded for all
$\ell\in\mathbb{Z}_{\geq 0}$, and the latter cases are all permitted and have
the eigenvalues $\lambda=-(\ell-1)(\ell+D-3)$, as is seen by Eq.(6.36).

Since the behavior of the function $G_{D-2}=S_{1}^{p}$ at origin is
$G_{D-2}\sim\theta_{1}^{p}$, $p=\ell$ is permitted and have the eigenvalue
$\lambda=-\ell(\ell+D-2)$. The solutions with $p=-\ell-D+3$ are permitted only
in the case $D\leq 3-\ell$, i.e., the cases of $D=2,3$ ††† Here we omit the
trivial case , $D=1$.. After all the permitted eigenvalues are

$\displaystyle\lambda=-(\ell-1)(\ell+D-3),\qquad\ell\in\mathbb{Z}_{\geq
0},\qquad({\rm for}~{}{\rm Euclid}~{}{\rm metric}).$ (6.37)

The eigenvalues $\lambda=-\ell(\ell+D-2)$ are contained in (6.37). In the case
of $D=2$ the eigenvalues expressed in Eq.(6.4) are permitted, which are also
contained in (6.37). Corresponding to these eigenvalues there are eigen-
functions which are single-valued and bounded at origin.

(2) Minkowski metric:

In this case there are three connected domains defined by (5.6) \- (5.8). In
$M_{\pm}$ we have $S_{1}=\pm\sinh{\tau}$, and in $M_{0}$ we have
$S_{1}=-\cosh{\tau}$. Contrary to the case of Euclid metric we should require
the boundedness condition at infinity ($\tau\rightarrow\pm\infty$) not only at
origin.

The eigen-functions should have the same eigenvalues in all domains, hence
those functions blowing up in some domains are excluded. The behavior of the
functions under consideration is the same for the three domains, and the most
singular domain is $M_{\pm}$, hence it is sufficient to consider there.

Firstly for the eigen-function $G^{\prime}_{D-2}$ the behavior of it at origin
and infinity is $G^{\prime}_{D-2}\sim e^{\tau(-\ell-D+2)}$ or
$G^{\prime}_{D-2}\sim e^{\tau(\ell-1)}$ for the solutions $p=\ell$ or
$p=-\ell-D+3$, respectively. Hence from the condition of the boudedness for
the solution of $p=\ell$ we have $\ell=2-D$ and $\lambda=0$. For the solution
of $p=-\ell-D+3$ we have $\ell=1$ and also $\lambda=0$. Concerning to
$G_{D-2}=S_{1}^{p}$, from the boundedness for the solution of $p=\ell$, we
have $\ell=0$ and $\lambda=0$. The solution of $p=-\ell-D+3$ gives $\ell=3-D$
and also $\lambda=0$. In the case of $D=2$ the permitted eigenvalue is given
by Eq.(6.5). After all the eigenvalues in the Minkowski metric is expressed as

$\displaystyle\lambda=\left\\{\begin{array}[]{c}~{}~{}0,\qquad\qquad\quad~{}({\rm
for}~{}D\neq 2,~{}{\rm Minkowski}~{}{\rm metric})\\\ L^{2},\quad
L\in\mathbb{R},\quad~{}({\rm for}~{}D=2,~{}{\rm Minkowski}~{}{\rm
metric})\end{array}\right..$ (6.40)

Corresponding to these eigenvalues there are eigen-functions which are bounded
at origin and infinity.

We have thus prepared to get the critical dimension of spacetime, through
Eq.(4.33), i.e., $D(D-4)/4=\lambda$. In the case of Euclid metric this gives

$\displaystyle\frac{1}{4}D(D-4)=-(\ell-1)(\ell+D-3),\qquad\ell\in\mathbb{Z}_{\geq
0}$ (6.41)

i.e.,

$\displaystyle(D+2\ell-6)(D+2\ell-2)=0.$ (6.42)

Hence we get the solution $D=2,4,6,$ for the Euclid metric.

In the case of Minkowski metric we have two cases, i.e., $D=2$ with
$\lambda\geq 0$, and the case of other $D$’s with $\lambda=0$. The former case
is excluded since $D(D-4)/4=-1$ which does not coincide to $\lambda\geq 0$.
Thus we see the permitted case is only $D=4$ with $\lambda=0$, the main result
of the present paper. This result is a consequence of the 2nd-order secondary
constraint, since in other dimension there exists no solution of the quantized
equation of the constraint. In that sense we can say that the critical
dimension of the complex particle in the Minkowski spacetime is four.

##  7 Conclusion and speculations

In the present paper we establish the fact that the first quantized theory of
the complex particle in the Minkowski spacetime is consistent if and only if
the dimension is four. The result is different in the case of the Euclid
spacetime, where $D=2,4$ and $6$ are permitted. The classical theory of the
model has some pathological features such as emergence of the higher order
secondary constraint, which is the key condition of deducing the critical
dimension and is absent in the physically established models. Such constraint
may be related to the breakdown of Dirac’s conjecture [3],[4]. Concerning to
the conjecture some authors have been studied in various contexts [5]-[12],
and there seems not firm agreement on it (see [19],[20] about the opinion of
the author).

The next step is to build a field theory of the model, in order to promote the
complex particle to a physically viable model. A plausible candidate is the
Chern-Simons type action proposed by Witten [29] in the string model. It is
expressed, for example, as

$\displaystyle{\cal I}=\int d\mu~{}(\psi\star Q\psi+g\psi\star\psi\star\psi),$
(7.1)

where $\star$ represents the so-called star-product, satisfying some
properties, $d\mu$ is some measure and $Q$ is the BRST operator. The
nilpotency of $Q$ makes it possible to interpret $Q$ as something similar to
the exterior derivative $d$ in the theory of the differential form. If $\star$
and $d\mu$ are properly defined the action may be invariant under
$\delta\psi=Q\Lambda+g\psi\star\Lambda,$ where $\Lambda$ is infinitesimal
parameter depending on the spacetime coordinates. Required properties are the
Leibniz rule, $Q(\psi_{1}\star\psi_{2})=(Q\psi_{1})\star\psi_{2}+\psi_{1}\star
Q\psi_{2}$, and vanishing of integral of exact form, $\int d\mu~{}Q\psi=0$. In
the case of the complex particle we proposed a plausible candidate of $\star$
and $d\mu$ in ref.[15]. The Leibniz rule is easily satisfied but the vanishing
of $\int d\mu~{}Q\psi$ restricts the spacetime dimension. Surprisingly, the
above requirement leads to $D=4$. However, the field theory of the complex
particle is under construction at present because of some problems to be
overcome.

The string theories in 10 or 26 dimensions inevitably have extra dimensions
which brings on the model with Calabi-Yau manifold [30] or the theory of
landscape [2]. They may incorporate a quantum theory of gravity, but there
exists absolutely no experimental evidence for the phenomenology of them.
Although the present stage of the complex particle is a toy model, there may
be possibility that it provides an alternative starting line to a realistic
model without extra dimensions.

##  A ppendix

###### A.1

Under the transformation of $z$ and $g$,

$\displaystyle\delta\mbox{\boldmath$z$}=\varepsilon\mbox{\boldmath$w$}+\frac{\varepsilon_{0}}{\bar{g}}\bar{\mbox{\boldmath$w$}},\qquad\delta
g=\xi,$ (A.1)

the lagrangian of the complex particle varies as

$\displaystyle\delta L$ $\displaystyle=$
$\displaystyle\frac{\ell_{0}}{2g\bar{g}}[\dot{\varepsilon}_{0}-2i\kappa
g\bar{g}(\varepsilon-\bar{\varepsilon})]+A\ell+\frac{dE}{d\tau}+{\rm c.c.},$
(A.2)

where $A$ and $E$ are defined by Eqs.(2.20).
———————————————————————————–

pr.)

Starting with the expression

$\displaystyle\delta L$ $\displaystyle=$
$\displaystyle\frac{1}{g}\mbox{\boldmath$w$}\cdot\left(\varepsilon\mbox{\boldmath$w$}+\frac{\varepsilon_{0}}{\bar{g}}\bar{\mbox{\boldmath$w$}}\right)^{\bullet}-\frac{\mbox{\boldmath$w$}^{2}}{2g^{2}}\xi+i\kappa\left[\left(\varepsilon\mbox{\boldmath$w$}+\frac{\varepsilon_{0}}{\bar{g}}\bar{\mbox{\boldmath$w$}}\right)^{\bullet}\cdot\bar{\mbox{\boldmath$z$}}+\mbox{\boldmath$w$}\cdot\left(\bar{\varepsilon}\bar{\mbox{\boldmath$w$}}+\frac{\varepsilon_{0}}{g}\mbox{\boldmath$w$}\right)\right]+{\rm
c.c.}$ (A.3) $\displaystyle=$
$\displaystyle\varepsilon\left\\{\underline{\frac{(\mbox{\boldmath$w$}^{2})^{\bullet}}{2g}}+i\kappa(\dot{\mbox{\boldmath$w$}}\cdot\bar{\mbox{\boldmath$z$}}-\mbox{\boldmath$w$}\cdot\bar{\mbox{\boldmath$w$}})\right\\}+\dot{\varepsilon}\left(\underline{\frac{\mbox{\boldmath$w$}^{2}}{g}}+i\kappa\mbox{\boldmath$w$}\cdot\bar{\mbox{\boldmath$z$}}\right)-\frac{\mbox{\boldmath$w$}^{2}}{2g^{2}}\xi$
$\displaystyle\qquad+\varepsilon_{0}\left\\{\left(\uwave{\frac{\mbox{\boldmath$w$}}{g}}+i\kappa\bar{\mbox{\boldmath$z$}}\right)\cdot\left(\frac{\bar{\mbox{\boldmath$w$}}}{\bar{g}}\right)^{\bullet}+i\kappa\frac{1}{g}\mbox{\boldmath$w$}^{2}\right\\}+\dot{\varepsilon}_{0}\left\\{\frac{\mbox{\boldmath$w$}\cdot\bar{\mbox{\boldmath$w$}}}{g\bar{g}}+i\kappa\frac{\bar{\mbox{\boldmath$w$}}\cdot\bar{\mbox{\boldmath$z$}}}{\bar{g}}\right\\}+{\rm
c.c.},$

the terms of total derivatives are picked up by using the following relations:

$\displaystyle\varepsilon\frac{(\mbox{\boldmath$w$}^{2})^{\bullet}}{2g}+\dot{\varepsilon}\frac{\mbox{\boldmath$w$}^{2}}{g}$
$\displaystyle=$
$\displaystyle\frac{\mbox{\boldmath$w$}^{2}}{2g^{2}}(\varepsilon
g)^{\bullet}+\frac{d}{d\tau}\left(\varepsilon\frac{\mbox{\boldmath$w$}^{2}}{2g}\right),$
(A.4)
$\displaystyle\varepsilon(\dot{\mbox{\boldmath$w$}}\cdot\bar{\mbox{\boldmath$z$}}-\mbox{\boldmath$w$}\cdot\bar{\mbox{\boldmath$w$}})+\dot{\varepsilon}\mbox{\boldmath$w$}\cdot{\bar{\mbox{\boldmath$z$}}}$
$\displaystyle=$
$\displaystyle-2\varepsilon\mbox{\boldmath$w$}\cdot\bar{\mbox{\boldmath$w$}}+\frac{d}{d\tau}(\varepsilon\mbox{\boldmath$w$}\cdot\bar{\mbox{\boldmath$z$}}),$
(A.5)
$\displaystyle\varepsilon_{0}\frac{\mbox{\boldmath$w$}}{g}\cdot\left(\frac{\bar{\mbox{\boldmath$w$}}}{\bar{g}}\right)^{\bullet}+{\rm
c.c.}$ $\displaystyle=$
$\displaystyle-\frac{\dot{\varepsilon}_{0}}{2}\frac{\mbox{\boldmath$w$}\cdot\bar{\mbox{\boldmath$w$}}}{g\bar{g}}+\frac{d}{d\tau}\left(\varepsilon_{0}\frac{\mbox{\boldmath$w$}\cdot\bar{\mbox{\boldmath$w$}}}{2g\bar{g}}\right)+{\rm
c.c.},$ (A.6)
$\displaystyle\varepsilon_{0}\bar{\mbox{\boldmath$z$}}\cdot\left(\frac{\bar{\mbox{\boldmath$w$}}}{\bar{g}}\right)^{\bullet}+\dot{\varepsilon}_{0}\frac{\bar{\mbox{\boldmath$z$}}\cdot\bar{\mbox{\boldmath$w$}}}{\bar{g}}$
$\displaystyle=$
$\displaystyle-\varepsilon_{0}\frac{\bar{\mbox{\boldmath$w$}}^{2}}{\bar{g}}+\frac{d}{d\tau}\left(\varepsilon_{0}\frac{\bar{\mbox{\boldmath$z$}}\cdot\bar{\mbox{\boldmath$w$}}}{\bar{g}}\right).$
(A.7)

Then

$\displaystyle\delta L$ $\displaystyle=$
$\displaystyle\frac{\mbox{\boldmath$w$}\cdot\bar{\mbox{\boldmath$w$}}}{2g\bar{g}}(\dot{\varepsilon}_{0}-4i\varepsilon\kappa
g\bar{g})+((\varepsilon g)^{\bullet}+4i\kappa
g\varepsilon_{0}-\xi)\frac{\mbox{\boldmath$w$}^{2}}{2g^{2}}$ (A.8)
$\displaystyle\qquad+\frac{d}{d\tau}\left\\{\varepsilon\left(\frac{\mbox{\boldmath$w$}^{2}}{2g}+i\kappa\mbox{\boldmath$w$}\cdot\bar{\mbox{\boldmath$z$}}\right)+\varepsilon_{0}\left(i\kappa\frac{\bar{\mbox{\boldmath$w$}}\cdot\bar{\mbox{\boldmath$z$}}}{\bar{g}}+\frac{\mbox{\boldmath$w$}\cdot\bar{\mbox{\boldmath$w$}}}{2g\bar{g}}\right)\right\\}+{\rm
c.c.},$

where we used $i(\bar{\mbox{\boldmath$w$}})^{2}/\bar{g}+{\rm
c.c.}=-i\mbox{\boldmath$w$}^{2}/g+{\rm c.c.}$. Writing the term of c.c.
explicitly, we obtain Eq.(A.2). $\blacksquare$
———————————————————————————–

###### A.2

The d’Aembert and the Laplace-Beltrami operator are related by

$\displaystyle
s~{}\rule[-2.84526pt]{0.28453pt}{11.38109pt}\rule[8.53581pt]{11.38109pt}{0.28453pt}\rule[-2.84526pt]{11.38109pt}{0.28453pt}\rule[-2.84526pt]{0.28453pt}{11.38109pt}\hskip
2.84526pt_{u}=\frac{1}{r^{2}}{\varDelta}_{\rm
LB}+\frac{1}{r^{D-1}}\frac{\partial{}}{\partial{r}}r^{D-1}\frac{\partial{}}{\partial{r}},\qquad
s\stackrel{{\scriptstyle\rm def}}{{=}}{\rm sgn}(u_{\mu}u^{\mu}).$ (A.9)

———————————————————————————–

pr.)

From the definition of ${\varDelta}_{\rm LB}$ we have

$\displaystyle{\varDelta}_{\rm LB}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(u^{\mu}\frac{\partial{}}{\partial{u_{\nu}}}-u^{\nu}\frac{\partial{}}{\partial{u_{\mu}}}\right)\left(u_{\mu}\frac{\partial{}}{\partial{u^{\nu}}}-u_{\nu}\frac{\partial{}}{\partial{u^{\mu}}}\right)$
(A.10) $\displaystyle=$ $\displaystyle
u_{\mu}u^{\mu}\rule[-2.84526pt]{0.28453pt}{11.38109pt}\rule[8.53581pt]{11.38109pt}{0.28453pt}\rule[-2.84526pt]{11.38109pt}{0.28453pt}\rule[-2.84526pt]{0.28453pt}{11.38109pt}\hskip
2.84526pt_{u}-(D-1)\tilde{r}D_{r}-u^{\mu}u^{\nu}\frac{\partial^{2}}{\partial
u^{\mu}\partial u^{\nu}},$

where

$\displaystyle D_{r}\stackrel{{\scriptstyle\rm
def}}{{=}}\frac{u^{\mu}}{\tilde{r}}\frac{\partial{}}{\partial{u^{\mu}}},\qquad\tilde{r}\stackrel{{\scriptstyle\rm
def}}{{=}}\sqrt{s~{}u_{\mu}u^{\mu}}.$ (A.11)

Writting $D_{r}^{2}$ as

$\displaystyle D_{r}^{2}$ $\displaystyle=$
$\displaystyle\frac{u^{\mu}}{\tilde{r}}\left(\frac{\partial{}}{\partial{u^{\mu}}}\frac{u^{\nu}}{\tilde{r}}\right)\frac{\partial{}}{\partial{u^{\nu}}}+\frac{u^{\mu}u^{\nu}}{\tilde{r}^{2}}\frac{\partial^{2}{}}{\partial{u^{\nu}}\partial{u^{\mu}}},$
(A.12)

we see the 1st term in r.h.s. of the above eqation is

$\displaystyle({\rm 1st}~{}{\rm term}~{}{\rm in}~{}{\rm
r.h.s.})=\frac{u^{\mu}}{\tilde{r}}\left(\frac{\delta^{\nu}_{\mu}}{\tilde{r}}-\frac{s~{}u^{\nu}u_{\mu}}{\tilde{r}^{3}}\right)\frac{\partial{}}{\partial{u^{\nu}}}=0.$
(A.13)

Hence

$\displaystyle
D_{r}^{2}=\frac{u^{\mu}u^{\nu}}{\tilde{r}^{2}}\frac{\partial^{2}{}}{\partial{u^{\nu}}\partial{u^{\mu}}},$
(A.14)

and from Eq.(A.10) we get

$\displaystyle{\varDelta}_{\rm
LB}=s\tilde{r}^{2}\rule[-2.84526pt]{0.28453pt}{11.38109pt}\rule[8.53581pt]{11.38109pt}{0.28453pt}\rule[-2.84526pt]{11.38109pt}{0.28453pt}\rule[-2.84526pt]{0.28453pt}{11.38109pt}\hskip
2.84526pt_{u}-(D-1)\tilde{r}D_{r}-\tilde{r}^{2}D_{r}^{2}.$ (A.15)

Using Eq.(5.28) we get Eq.(A.9). $\blacksquare$

———————————————————————————–

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