# Some remarks on projectile motion with a linear resistance force

C. A. Morales1,2, J. H. Muñoz1,† , C. E. Vera1

###### Abstract

In this article we revisit the projectile motion assuming a retarding force
proportional to the velocity, $\vec{F_{r}}=-mk\vec{V}$. We obtain an
analytical expression for the set of maxima of the trajectories, in Cartesian
coordinates, without using the Lambert $W$ function. Also, we investigate the
effect of the parameter $k$ on the radial distance of the projectile showing
that the radial distance oscillates from a certain critical launch angle and
find an approximate expression for it. In our analysis, we consider the impact
of the parameter $k$ in the kinetic energy, the potential energy, the total
energy and the rate of energy loss, and in the phase space. Our results can be
included in an intermediate-level classical mechanics course.

Keywords: parabolic motion, linear resistance force, radial distance, maximum
height

1Departamento de Física, Universidad del Tolima, Código Postal 730006299,
Ibagué, Colombia 2 Institución Educativa Francisco de Miranda, Rovira-Tolima,
Colombia † Corresponding author

##  1 Introduction

The projectile motion in a constant gravitational field is an important topic
that is studied in introductory physics courses at university level. It is
considered in absence of medium resistance in almost all fundamental physics
textbooks (see for example [1, 2, 3, 4]). To consider a more realistic
situation, it is necessary to include retarding forces ($F_{r}$). A good
approximation, in this case, is to assume that they are proportional to some
power of the speed ($F_{r}\propto v^{n}$) [5, 6, 7, 8, 9, 10, 11].

In this paper, we revisit the motion of a projectile in a fluid considering
the effect of a retarding force proportional to the velocity ($n=1$):
$\vec{F_{r}}=-mk\vec{V}=-mk(\dot{x}\,\widehat{i}+\dot{y}\,\widehat{j})$, where
$\dot{x}=dx/dt$, $\dot{y}=dy/dt$, $m$ is the mass of the projectile and $k$ is
a positive constant that specifies the strength of the resisting force. The
unit of $k$ is s${}^{-1}.$ The linear drag model, $\vec{F_{r}}\sim\vec{V}$, is
a good approximation when the dimensionless Reynolds number is small,
indicating that the inertia forces are negligible with respect to the viscous
forces and the fluid has a laminar flow. Under these conditions, the Stokes’
law is valid for a sphere of radio $R$ moving in a fluid and $k=6\pi R\eta$,
where $\eta$ is the viscosity of the medium [12, 13, 14].

The projectile motion with linear resistance force has been extensively
discussed in the literature a long time ago using different approaches.
According to our bibliographical review, this literature can be classified in
two groups. In one of the them, the studies were performed by means of
approximate methods, whereas in the other group the researches were performed
by introducing the Lambert $W$ function in order to obtain analytical
expressions.

In References [6, 7, 8, 9, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27,
28, 29, 30, 31, 32, 33, 34] it is obtained, by means of approximate procedures
or through computational tools, the trajectory of the particle, the time of
flight, the maximum height, the range, the curve of safety or the path length.
On the other hand, in the last two decades the interest in the projectile
motion with a retarding force proportional to the velocity has increased
because it is a good scenario to apply the Lambert W function since it is
necessary to solve transcendental equations in this problem. In this
direction, several authors have obtained analytical expressions for the shape
of the trajectory, the range, the angle that gives the maximum range (the
optimal launch angle), the time of flight, the time of ascent (or descent),
the time of fall, the maximum height, the locus of the apexes (in Cartesian
and polar coordinates) in terms of the Lambert $W$ function [35, 36, 37, 38,
39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51].

Stewart [40] and Hernandez-Saldaña [43] obtained, using the Lambert $W$
function, the locus of the set of apexes corresponding to the maximum heights
in Cartesian and polar coordinates, respectively, for the projectile motion
with a retarding force proportional to the velocity. Motivated by these two
works, we have extended them obtaining an analytical expression for this locus
in Cartesian coordinates without introducing the Lambert $W$ function, through
a simple and didactical procedure, with the help of Mathematica (as in Ref.
[20]). According to our knowledge, this result has not been previously
reported.

In addition, we have scrutinized the effect of the parameter $k$ on the radial
distance of the projectile motion with a retarding force proportional to the
velocity, motivated by the previous works of Walker [52] (without friction)
and, Ribeiro and Sousa [33]. Our results complement these two works. Moreover,
we have performed a pedagogical and didactical overview on typical observables
associated to the projectile motion with a friction force proportional to the
velocity, including in our analysis observables as kinetic energy, potential
energy, total energy an the rate of energy loss that have not been considered
in the literature up to now.

The paper is organized as follows. In section 2 we present an overview on
several observables of the projectile motion with a retarding force
proportional to the velocity, including the kinetic energy, the potential
energy, the total energy, the rate of energy loss and the trajectory in the
phase space. In section 3, we analyze the impact of the parameter $k$ on the
radial distance and the critical angle for obtaining a radial oscillation. In
section 4, the most important, we study the evolution of the apexes of the
trajectories in function of the launch angle, in Cartesian coordinates without
using the Lambert $W$ function. Finally, in section 5 we summarize our
principal results.

##  2 General results: An overview

Let us assume that in $t=0$ the projectile is launched from the origin of the
coordinate system with the initial velocity $\vec{V_{0}}$ and the angle of
elevation $\theta_{0}$. The equations of motion in the horizontal and vertical
directions, respectively, are

$\displaystyle m\ddot{x}$ $\displaystyle=$ $\displaystyle-mk\dot{x},$ (1)
$\displaystyle m\ddot{y}$ $\displaystyle=$ $\displaystyle-mk\dot{y}-mg,$ (2)

where $m$ is the mass of the projectile, $g$ is the acceleration of gravity,
$\dot{x}$ ($\dot{y}$) is the horizontal (vertical) velocity and $\ddot{x}$
($\ddot{y}$) is the horizontal (vertical) acceleration of the projectile.
These differential equations are solving making $\ddot{x}=d\dot{x}/dt$ and
$\ddot{y}=d\dot{y}/dt$ and integrating to obtain $\dot{x}$ and $\dot{y}$ in
function of time. The solutions for these equations are well known [6, 8, 9,
16, 17, 18, 19, 20, 21, 23, 24, 25, 29]:

$x(t)=\dfrac{U}{k}\left(1-e^{-kt}\right),$ (3)

$y(t)=-\dfrac{gt}{k}+\dfrac{kV+g}{k^{2}}\left(1-e^{-kt}\right),$ (4)

where $U=V_{0}\cos\theta_{0}$ and $V=V_{0}\sin\theta_{0}$.

The horizontal and vertical components of the velocity are

$\dot{x}=Ue^{-kt},$ (5)

$\dot{y}=-\dfrac{g}{k}+\dfrac{kV+g}{k}e^{-kt},$ (6)

and the components of the acceleration are given by

$\displaystyle\ddot{x}$ $\displaystyle=$ $\displaystyle-kUe^{-kt},$ (7)
$\displaystyle\ddot{y}$ $\displaystyle=$ $\displaystyle-(kV+g)e^{-kt}.$ (8)

We will now investigate the phase space of projectile motion under the
influence of a linear resistance force, which has not been studied so far. Our
aim is to enhance our understanding of the temporal evolution of some
variables and find recurring patterns.

The trajectory in the phase space $\dot{x}$ vs $x$ is obtained from
expressions (3) and (5) for $x(t)$ and $\dot{x}(t)$, respectively. It is:

$\dot{x}=U-kx.$ (9)

This trajectory corresponds to a straight line (Figure 1 (left)). Its slope is
the resistance coefficient $k$. Experimentally, this expression could be
useful for obtaining the parameter $k$ of a laminar fluid by plotting
$\dot{x}$ versus $x$.

In a similar way, the trajectory in the phase space $\dot{y}$ vs $y$ is
obtained from equations (4) and (6) for $y(t)$ and $\dot{y}(t)$, respectively,
giving

$\frac{g}{k}\text{ln}\left[\frac{k}{kV+g}\left(\dot{y}+\frac{g}{k}\right)\right]-\dot{y}=ky-V.$
(10)

The Figure 1 (middle) shows this phase plot. When there is no friction, it is
a parabola. We can see that the graph is not symmetric with respect to the
line $\dot{y}=0$ because during the motion, some of the projectile’s
mechanical energy dissipates, converting into thermal energy as a result of
collisions between the molecules composing the fluid and the projectile.
Therefore, for a given height, there are two values of $\dot{y}$ that
correspond to when the projectile goes up and goes down, such that
$|\dot{y}(going\ up)|>|\dot{y}(going\ down)|.$

Additionally, in the Figure 1 (right) it is displayed the phase plot $v$ vs
$r$, where $v=\sqrt{\dot{x}^{2}+\dot{y}^{2}}$ is the magnitude of the velocity
and $r=\sqrt{x^{2}+y^{2}}$ is the radial distance. The minimum value of $v$
does not correspond to the point of maximum height. From the radial distance
at which $v$ is minimum, the projectile falls, increasing its vertical
velocity and decreasing its horizontal velocity until it reaches a constant
value equal to $g/k$. As a result, the curve asymptotically tends towards the
line $v=g/k$. For $k=2$, this value is approximately $(9.8\
\text{m/s}^{2})/(2\text{s}^{-1})=4.9\ \text{m/s}$ (This value can be verified
in Figure 1 (right)).

Figure 1: Phase plots $\dot{x}$ vs $x$ (left), $\dot{y}$ vs $y$ (middle) and
$v$ vs $r$ (right) with $k=2$ s-1, $V_{0}=10$ m/s and $\theta_{0}=60^{\circ}$.

Now we are going to get the time required for the projectile to arrive at the
maximum height ($=t_{peak}$). Taking $\dot{y}=0$ in equation (6) it is
obtained that the projectile reaches the peak at the time

$t_{peak}=\dfrac{1}{k}\text{ln}\left[\dfrac{kV+g}{g}\right].$ (11)

The maximum height $(Y_{m})$ and the horizontal position corresponding to this
height $(X_{m})$ are obtained substituting equation (11) in $y(t)$ and $x(t)$,
respectively:

$Y_{m}=-\dfrac{g}{k^{2}}\text{ln}\left[1+\dfrac{kV}{g}\right]+\dfrac{V}{k}$
(12)

$X_{m}=\dfrac{UV}{g}\left(1+\dfrac{kV}{g}\right)^{-1}.$ (13)

The point $(X_{m},Y_{m})$ gives the apex of the trajectory. We will discuss in
section 4 the evolution of these points as a function of the launch angle.

Now let us get the trajectory of the projectile. To do this, we obtain the
time $t$ from the equation $x=x(t)$:

$t=\dfrac{1}{k}\text{ln}\left[\left(1-\dfrac{k}{U}x\right)^{-1}\right],$ (14)

and substitute it in $y=y(t)$:

$y=\dfrac{g}{k^{2}}\text{ln}\left(1-\dfrac{k}{U}x\right)+\dfrac{kV+g}{kU}x.$
(15)

The range $R$ of the projectile is obtained by setting $y=0$ in equation (15).
By doing this, it is obtained

$\frac{g}{k^{2}}\text{ln}\left(1-\frac{k}{U}R\right)+\frac{V+\frac{g}{k}}{U}R=0.$
(16)

Obtaining an analytical solution for the range $R$ from the last equation is
not easy because in this expression a linear function for $R$ is equated to a
logarithmic function for $R$. However, it is possible to obtain an approximate
expression for it using the expansion $\text{ln}(1+z)=z-z^{2}/2+z^{3}/3-...$.
By doing this, it is obtained:

$R\approx\dfrac{2UV}{g}\left(1-\dfrac{kV}{4g}\right).$ (17)

For $k=0$, we obtain $R\approx 2UV/g=V_{0}^{2}\sin 2\theta_{0}/g$, which is
the well-known expression for the case where there is no friction.

Next, let us consider the time of flight $(T_{f})$, which is the required time
for the projectile returns to the ground. It is obtained taking $y=0$ in
$y(t)$ (see equation (4). Doing this, it is obtained

$gT_{f}-\dfrac{kV+g}{k}\left(1-e^{-kT_{f}}\right)=0.$ (18)

Using the expansion $e^{z}=1+x+x^{2}/2+\ldots$ and considering only terms
through $k^{2}$ in the last equation, the time of flight is, approximately,

$T_{f}\approx\frac{2V}{kV+g}=\frac{2V}{g(1+kV/g)}=\frac{2V}{g}(1+kV/g)^{-1},$

$T_{f}\approx\frac{2V}{g}(1-kV/g).$ (19)

This result is a good approximation only for small values of $k$. If $k=0$, it
is obtained the well-known expression for the time flight in the ideal case.

Another way to analyze the equation (18) is by rewriting it as

$gT_{f}=\dfrac{kV+g}{k}\left(1-e^{-kT_{f}}\right)\ \ \ \ \equiv\ \ \ \
h(T_{f})=f(T_{f}),$ (20)

where the units of $h$ and $f$ are m/s. The left-hand side of the last
equation ($h(T_{f})$) is a linear function whereas the right-hand side
($f(T_{f})$) is a transcendental expression. Therefore, it is not possible to
get an analytic expression for $T_{f}$ by means of an easy and simple
mathematical procedure. One approach to obtain an approximate solution to the
last equation is using numerical methods. On the other hand, as we mentioned
in the introduction, it is possible to find analytical solutions for the time
of flight in terms of the Lambert $W$ function which is available in some
computational tools as Geogebra, Maple and Mathematica.

Considering that the procedure to obtain an analytical expression for $T_{f}$
in terms of the Lambert $W$ function is difficult, abstract and cumbersome for
undergraduate students from different fields other than physics or
mathematics, or those who have not taken a course on special functions, we use
the Plot command in Mathematica, without employing the Lambert $W$ function,
to obtain the flight time easily. We get numerical values for $T_{f}$ taking
the coordinates of the intersection point, namely, where $h(T_{f})=f(T_{f})$,
for several values of the retarding force parameter $k$ assuming that the
launch angle is fixed. Figure 2 shows the graphs of $h(T_{f})$ and $f(T_{f})$
(see equation (20)) with $k=2$ s-1 and $\theta_{0}=60^{\circ}$. In this case,
we obtain that the solution is given by $T_{f}=1.274$ s.

Figure 2: The linear $h(t)$ (blue) and the transcendental $f(t)$ (orange)
expressions as a function of $t$, given by equation (20), with $k=2$ s-1,
$V_{0}=10$ m/s and $\theta_{0}=60^{\circ}$.

Now, we are going to analyze the effect of the parameter $k$ on kinetic,
potential and total energies. Figure 3 shows the kinetic energy
$T=\frac{1}{2}m(\dot{x}^{2}+\dot{y}^{2})$ in function of time taking $m=1$ kg.
Similar to the case without friction, kinetic energy has a minimum value, but
it is not obtained at the point of maximum height. To obtain the time at which
the kinetic energy is minimum, we set $dT/dt=0$ and obtain

$t_{min}=\dfrac{1}{k}\text{ln}\left[\dfrac{kV+g}{g}\left(1+\left(\dfrac{kU}{kV+g}\right)^{2}\right)\right]=t_{peak}+\dfrac{1}{k}\text{
ln}\left[1+\left(\dfrac{kU}{kV+g}\right)^{2}\right].$ (21)

Thus, it is demonstrated that the time at which the kinetic energy is minimum
$(t_{min})$ is different from the time required to reach maximum height
$(t_{peak})$. As $t_{min}\geq t_{peak}$, the kinetic energy is minimum when
the projectile is descending. When the time is bigger than the time for
reaching the maximum height $(t\gg t_{peak})$, the kinetic energy becomes
constant because $\dot{x}\rightarrow 0$ (see equation (5)) and
$\dot{y}\rightarrow-g/k$ (see equation (6)), where $-g/k$ is the terminal
velocity.

The potential energy $U=mgy$ as a function of time has the same behavior as
$y=y(t)$. The only difference between the $U$ vs $t$ and $y$ vs $t$ graphs is
the vertical scale. Figure 4 shows the total energy $E=T+U$ of the particle in
function of time. At $t=0$, the total energy is $E(t=0)=mV_{0}^{2}/2$, so all
the curves start from the same point for different values of $k$. In the ideal
case, a horizontal line is obtained at this value because during the ascent
(descent), potential energy $U$ increases (decreases ) at the same rate that
kinetic energy $T$ decreases (increases). When there is a resistance medium,
the loss of kinetic energy $T$ during ascent is not compensated by an increase
in potential energy $U$, resulting in energy dissipation. This dissipation is
greater during ascent because it is proportional to the projectile’s velocity,
causing the curve to have a steeper slope at each point. During descent, the
projectile falls more slowly, resulting in less energy loss and a less steep
slope of the curve. If the projectile continues its motion below $y=0$, the
curve asymptotically tends towards a line with a slope of $-mg^{2}/k$
(proportional to the terminal velocity).

The colored curves in figure 3 do not intersect each other for different
values of k. In the case of figure 4, the colored curves can cross each other
for different values of $k$ only when $E<0$, that is, for times greater than
the time needed to reach the range $R$.

For completeness, we consider the rate of energy-loss $(dE/dt)$ that it is the
slope of Figure 4. It is given by

$\frac{dE}{dt}=\frac{dT}{dt}+\frac{dU}{dt}=m(\dot{x}\ddot{x}+\dot{y}\ddot{y})+mg\dot{y}.$

For large times $\dot{x}\rightarrow 0$, $\dot{y}\rightarrow-g/k$,
$\ddot{x}\rightarrow 0$ and $\ddot{y}\rightarrow 0$ (see equations (5), (6),
(7) and (8)). Thus, $dE/dt$ is dominated by $dU/dt$ and tends to the constant
value of $-mg^{2}/k$.

Figure 3: Kinetic energy of the projectile with $V_{0}=10$ m/s,
$\theta_{0}=60^{\circ}$ and different values of the parameter $k$ in s-1.
Figure 4: Total energy of the projectile with $V_{0}=10$ m/s,
$\theta_{0}=60^{\circ}$ and several values of the parameter $k$ in s-1.

Finally, we are going to obtain, in an approximate way, the expressions for
$t_{peak}$, $Y_{m}$, $X_{m}$ and the trajectory. Using the expansions
$\text{ln}(1+z)=z-z^{2}/2+z^{3}/3-...$ in equations (11), (12) and (15), and
$(1+z)^{-1}=1-z+z^{2}-...$ in equation (13), it is obtained

$\displaystyle t_{peak}$ $\displaystyle=$
$\displaystyle\dfrac{V}{g}\left[1-\frac{1}{2}\left(\dfrac{kV}{g}\right)+\frac{1}{3}\left(\dfrac{kV}{g}\right)^{2}-\cdots\right]$
(22) $\displaystyle Y_{m}$ $\displaystyle=$
$\displaystyle\dfrac{V^{2}}{2g}\left[1-\dfrac{2}{3}\left(\dfrac{kV}{g}\right)+\cdots\right]$
(23) $\displaystyle X_{m}$ $\displaystyle=$
$\displaystyle\dfrac{2UV}{2g}\left[1-\dfrac{kV}{g}+\left(\dfrac{kV}{g}\right)^{2}-\cdots\right]$
(24) $\displaystyle y$ $\displaystyle=$
$\displaystyle(tan\theta_{0})x-\dfrac{g}{2U^{2}}x^{2}-\dfrac{gk}{3U^{3}}x^{3}+\cdots,$
(25)

where we have assumed that the dimensionless perturbative parameter
$\frac{kV}{g}$ is small. Taking $k=0$ in the above equations, it is obtained
the well-known expressions for the projectile motion in absence of medium
resistance.

We provide [53] a didactic simulation in GeoGebra to obtain the graphs of all
the equations obtained in this section. With this computational tool, the
reader can easily verify the presented results in this section by manipulating
the values of the parameter $k$, the initial velocity $V_{0}$, and the launch
angle $\theta_{0}$.

##  3 Behavior of the radial distance $(r)$ and the critical angle
$(\theta_{c})$

The radial distance from the origin of the coordinate system to the projectile
can be expressed as a function of time ($r=\sqrt{x^{2}(t)+y^{2}(t)}$) or the
horizontal distance ($r=\sqrt{x^{2}+y^{2}(x)}$). Some years ago, Walker [52]
reported an interesting phenomenon in the projectile motion in absence of a
friction force. He found that projectiles are "coming and going" for launch
angles greater than $70.5288^{\circ}$. From this critical angle
($\theta_{c}$), the radial distance exhibits an oscillation: it increases,
decreases and increase again. Recently, Ribeiro and Sousa [33] extended the
Walker’s work and demonstrated that the "coming and going" phenomenon is also
present in the projectile motion with a linear resistance force. Motivated by
the previous works of Walker [52] and Ribeiro-Sousa [33] we have scrutinized
this fascinating result about the oscillation of the radial distance.

Figure 5 shows $r$ as a function of $t$ (left) and of $x$ (right),
respectively, for several values of the launch angle with $k=0.5$ s-1,
$V_{0}=10$ m/s and $g=9.81$ m/s2. In both graphs, a radial oscillation is
observed starting from a launch angle close to $65^{\circ}$, approximately.
The radial distance increases, decreases and increases again from this launch
angle.

It is interesting to highlight that the behavior of the radial distance $r$
with respect to the time $t$ and the horizontal distance $x$ is different (see
figure 5). This behavior depends on the condition $dr/dt$ = 0 and $dr/dx=0$.
They are related by
$\frac{dr}{dt}=\frac{dr}{dx}\frac{dx}{dt}=\dot{x}\frac{dr}{dx}.$ When friction
is present, $\dot{x}$ depends on time; therefore, the behavior of $dr/dt$ and
$dr/dx$ is different. In the opposite case, when there is no friction,
$\dot{x}$ is constant. Hence, the behavior of $dr/dt$ and $dr/dx$ is the same.

Figure 5: Radial distance as a function of time (left) and radial distance as
a function of horizontal position (right) with $k=0.5$ s-1, $V_{0}=10$ m/s and
$g=9.81$ m/s2.

Now we are going to obtain, in an approximate way, an expression for the
launch critical angle, $\theta_{c}$, from which the radial oscillation is
manifested. We begin by requiring that the radial velocity be zero:

$\frac{dr}{dt}=\frac{x\dot{x}+y\dot{y}}{\sqrt{x^{2}+y^{2}}}=0.$ (26)

Assuming that $kt\leq 1$, we use the expansion $e^{-kt}\approx
1-kt+\dfrac{(kt)^{2}}{2}-\dfrac{(kt)^{3}}{6}$ and substitute it in the
expression $x\dot{x}+y\dot{y}=0$. Thus, we obtain the condition

$(7k^{2}V_{0}^{2}+10kVg+3g^{2})t^{2}-9(kV_{0}^{2}+Vg)t+6V_{0}^{2}=0.$ (27)

Solving for $t$ yields

$t=\frac{9(kV_{0}^{2}+Vg)\pm\sqrt{81(kV_{0}^{2}+Vg)^{2}-24V_{0}^{2}(7k^{2}V_{0}^{2}+10kVg+3g^{2})}}{2(7k^{2}V_{0}^{2}+10kVg+3g^{2})}.$
(28)

As the phenomenon of radial oscillation begins when the last equation has a
single solution, it is required that the quantity inside the square root be
zero. This gives:

$27g^{2}\sin^{2}{\theta_{c}}-26kgV_{0}\sin{\theta_{c}}-(29k^{2}V_{0}^{2}+24g^{2})=0,$
(29)

where we have used $V=V_{0}\sin\theta_{c}$. Solving the last equation for
$\sin\theta_{c}$, it is obtained

$\sin\theta_{c}=\frac{13}{27}\Gamma\pm\frac{2}{9}\sqrt{\frac{238}{9}\Gamma^{2}+18},$
(30)

where the dimensionless parameter $\Gamma$ is defined by
$\Gamma\equiv\frac{kV_{0}}{g}$. The solution with sign minus gives negative
angles. So, we take the solution with sign plus and obtain the expression for
the critical angle in function of the parameter $\Gamma$:

$\theta_{c}(\Gamma)=\sin^{-1}\left(\frac{13}{27}\Gamma+\frac{2}{9}\sqrt{\frac{238}{9}\Gamma^{2}+18}\right).$
(31)

This expression is only valid for small values of $k$. Figure 6 displays the
critical angle for small values of the parameter $\Gamma$. Let us mention that
for $k=0$ it is obtained $\theta_{c}\approx 70.53$. So, this result agrees
with the value obtained in Ref. [52] without the linear resistance force.
However, our result is different from the one reported in the equation (9) of
[33]. According to this reference, it is reported that
$\theta_{c}(\Gamma)=\sin^{-1}\left(-\frac{5}{27}\Gamma+\frac{2}{9}\sqrt{\frac{40}{9}\Gamma^{2}+18}\right).$
We consider that there is a mistake in this expression which likely arises
from the algebraic manipulation of the Taylor series used to expand
$x\dot{x}+y\dot{y}=0$.

Finally, let us mention that if we start by demanding that $dr/dx=0$, where
$r=\sqrt{x^{2}+y^{2}(x)}$, we obtain, through a similar algebraic procedure,
the following expression for the critical angle assuming that
$\Gamma\equiv\frac{kV_{0}}{g}\ll 1$:

$\alpha_{c}(\Gamma)=\sin^{-1}\left(-\frac{32}{27}\Gamma+\frac{2}{9}\sqrt{\frac{256}{9}\Gamma^{2}+18}\right).$
(32)

Figure 6: Behavior of the critical angle in function of $\Gamma$.

##  4 Evolution of the apexes

When the fluid resistance is not considered $(k=0)$, the locus conformed by
the set of points corresponding to the maximum height,
$(Y_{m}(\theta_{0}),X_{m}(\theta_{0}))$, where

$Y_{m}(\theta_{0})=(V_{0}\sin{\theta_{0}})^{2}/2g$ and
$X_{m}(\theta_{0})=(V_{0}^{2}\sin{2\theta_{0}})/g$,

describes an ellipse in function of the launch angle (see, for example, Refs.
[24, 52, 54, 55, 56, 57]). If the retarding force is considered, i.e., $k\neq
0$, it is natural to extend the same analysis to this type of motion and pose
the following question: what is the locus that describes the maxima of all
projectile trajectories as a function of the launch angle?

Stewart [40] and Hernandez [43] demonstrated by means of the Lambert $W$
function in Cartesian form and polar coordinates, respectively, that the locus
of the set of maxima of the projectile motion in a linear resisting medium is
not an ellipse. The procedure to obtain an explicit analytical expression
$y=y(x)$ for the locus of the apexes is not easy or trivial in terms of the
Lambert $W$ function. It is rather something tedious and cumbersome. For that
reason, we have revisited it and obtained this locus in Cartesian coordinates,
in an easier way, without using the Lambert $W$ function.

In order to derive an expression for the locus of the set of maxima in
Cartesian coordinates without using the Lambert function, it is necessary to
obtain $Y_{m}$ as a function of $X_{m}$ from equations (12) and (13). To
accomplish this, we substitute $V=V_{0}\sin\theta_{0}$ into equation (13) and
then clear $\sin\theta_{0}$ in function of $X_{m}$ obtaining, after some
algebraic manipulations, the following expression

$\sin^{4}{\theta_{0}}+\left(\frac{X_{m}^{2}k^{2}}{V_{0}^{2}}-1\right)\sin^{2}{\theta_{0}}+\frac{2gkX_{m}^{2}}{V_{0}^{3}}\sin{\theta_{0}}+\frac{X_{m}^{2}g^{2}}{V_{0}^{4}}=0.$
(33)

This is a equation of fourth degree in $\sin{\theta_{0}}$. With the help of
Mathematica and some manipulation by hand, we find that the two physical
solutions of this equation, in function of $X_{m}$, are given by

$\sin{\theta_{1}}=\frac{1}{2}\left(\sqrt{E}-\sqrt{-6A-E-\frac{4gkX_{m}^{2}}{V_{0}^{3}\sqrt{E}}}\right)$
(34)

$\sin{\theta_{2}}=\frac{1}{2}\left(\sqrt{E}+\sqrt{-6A-E-\frac{4gkX_{m}^{2}}{V_{0}^{3}\sqrt{E}}}\right),$
(35)

where

$A=\frac{k^{2}X_{m}^{2}-V_{0}^{2}}{3V_{0}^{2}},$ (36)

$E=-2A+\frac{B}{3\left(C+\sqrt{D+C^{2}}\right)^{1/3}}+\frac{1}{3(2^{1/3})V_{0}^{4}}\left(C+\sqrt{D+C^{2}}\right)^{1/3},$
(37)

with

$B=2^{1/3}\left(V_{0}^{4}-2k^{2}X_{m}^{2}V_{0}^{2}+12g^{2}X_{m}^{2}+k^{4}X_{m}^{4}\right),$
(38)

$C=108g^{2}k^{2}X_{m}^{4}V_{0}^{6}-72g^{2}X_{m}^{2}V_{0}^{6}\left(k^{2}X_{m}^{2}-V_{0}^{2}\right)+2V_{0}^{6}\left(k^{2}X_{m}^{2}-V_{0}^{2}\right)^{3},$
(39)

$D=-4\left(12g^{2}X_{m}^{2}V_{0}^{4}+V_{0}^{4}(k^{2}X_{m}^{2}-V_{0}^{2})^{2}\right)^{3}.$
(40)

After, we replace the expressions for $\sin{\theta_{1}}$ and
${\sin\theta_{2}}$ in equation (12). Thus, we obtain in Cartesian coordinates
the trajectory of the set of maxima of the projectile motion, $Y_{m}(X_{m})$:

$Y_{m}(X_{m})=-\dfrac{g}{k^{2}}\text{ln}\left[1+\dfrac{kV_{0}\sin\theta_{1(2)}}{g}\right]+\dfrac{V_{0}\sin\theta_{1(2)}}{k}.$
(41)

Figure 7 displays this trajectory for different values of the parameter $k$
with $V_{0}=10$ m/s. It shows that the trajectory of the peak is not an
ellipse when the retarding force $\vec{F_{r}}=-mk\vec{V}$ is included. The
upper part of each graph is obtained replacing $\sin{\theta_{1}}$ in $Y_{m}$,
while the lower part of each graph is obtained using $\sin{\theta_{2}}$ in
$Y_{m}$. We highlight that these trajectories are obtained without the Lambert
$W$ function.

Figure 7: Trajectory of the highest point of the projectile for $k=0.0001$ s-1
(blue), $k=0.5$ s-1 (red), $k=1$ s-1 (green), $k=2$ s-1 (purple), $k=3$ s-1
(orange), and $k=4$ s-1 (black).

##  5 Summary

In this article we presented a didactical overview on the projectile motion
considering a retarding force proportional to the velocity,
$\vec{F_{r}}=-mk\vec{V}$, where the parameter $k$ gives the strength of the
force. We obtained an analytical expression for the set of maxima of the
trajectories, in Cartesian coordinates, without using the Lambert $W$
function, showing that the trajectory of the locus of the apexes in function
of the launch angle seems a deformed ellipse. This calculation complements
previous results obtained in Refs. [40, 43], in Cartesian and polar
coordinates, but employing the Lambert $W$ function. According to our
literature review, this result has not yet been published.

Also, we extended the analysis performed in Ref. [52] for the ideal case
$(k=0)$ and complemented the results of Ref. [33] related with the effect of
the parameter $k$ on the radial distance, showing that there is a radial
oscillation from certain critical angle which depends of $k$ and obtained, in
an approximate way, the expression for this angle.

In our analysis, we have included the impact of the parameter $k$ in the
kinetic energy, the potential energy, the total energy and the rate of energy
loss, and in the $\dot{x}-x$, $\dot{y}-y$ and $v-r$ phase plots. Although this
is straightforward with the help of the Mathematica package, up to now the
analysis of these important observables have not been included in the
literature. We provide a simulation [53], very easy to manipulate, in which
the reader can verify all the results and graphs obtained in this work by
changing the numerical values of the parameter $k$, the initial velocity
$V_{0}$, and the launch angle $\theta$.

In general, our results extend the knowledge of the projectile motion assuming
a retarding force proportional to the velocity and can be included in an
intermediate-level classical mechanics course.

Acknowledgment.
The authors would like to express their gratitude to the referees for their
invaluable comments, which significantly contributed to improving the
manuscript.

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# Derivation of breakup probabilities from experimental elastic backscattering
data

V.V.Sargsyan1, G.G.Adamian1, N.V.Antonenko1, W. Scheid2, and H.Q.Zhang3 1Joint
Institute for Nuclear Research, 141980 Dubna, Russia
2Institut für Theoretische Physik der Justus–Liebig–Universität, D–35392
Giessen, Germany
3China Institute of Atomic Energy, Post Office Box 275, Beijing 102413, China

###### Abstract

We suggest simple and useful method to extract breakup probabilities from the
experimental elastic backscattering probabilities in the reactions with
toughly and weakly bound nuclei.

###### pacs:

25.70.Jj, 24.10.-i, 24.60.-k
Key words: sub-barrier capture, neutron transfer, quantum diffusion approach

The fusion (capture) dynamics induced by loosely bound radioactive ion beams
is being extensively studied Gomes ; EPJSub4 . However, the long-standing
question whether fusion (capture) is enhanced or suppressed with these beams
has not yet been answered unambiguously. The study of the fusion reactions
involving nuclei close to the drip-lines has led to contradictory results. The
lack of a clear systematic behavior EPJSub4 ; PRSGomes5  of the breakup
probability as a function of the target charge requires additional
experimental and theoretical studies. The quasi-elastic backscattering has
been suggested EPJSub4  as an alternative to investigate the breakup
probability. Since the quasi-elastic experiments are usually not as complex as
the fusion (capture) and breakup measurements, they are well suited to survey
the breakup probability.

In the present report we will show that by employing the experimental elastic
backscattering data, one can extract the breakup probabilities of weakly bound
nuclei. So, new method for the study of the breakup probability will be
suggested.

There is a direct relationship between the capture, quasi-elastic scattering
and breakup processes, since any losses from the elastic scattering and
breakup channels contribute directly to other channels (the conservation of
the total reaction flux at given bombarding energy $E_{\rm c.m.}$ and angular
momentum $J$):

$\displaystyle P_{qe}(E_{\rm c.m.},J)+P_{cap}(E_{\rm c.m.},J)+P_{BU}(E_{\rm
c.m.},J)=$ $\displaystyle=P_{el}(E_{\rm c.m.},J)+P_{rest}(E_{\rm
c.m.},J)+P_{BU}(E_{\rm c.m.},J)=1,$ (1)

where $P_{rest}=P_{cap}+P_{in}+P_{tr}$, $P_{qe}=P_{el}+P_{in}+P_{tr}$ is the
quasi-elastic scattering probability, $P_{BU}$ is the breakup probability, and
$P_{cap}$ is the capture probability. The quasi-elastic scattering ($P_{qe}$)
is the sum of all direct reactions, which include elastic ($P_{el}$),
inelastic ($P_{in}$), and a few nucleon transfer ($P_{tr}$) processes. In Eq.
(1), we neglect the deep-inelastic collision process, since we are concerned
with low energies.

Equation (1) can be rewritten as

$\displaystyle\frac{P_{el}(E_{\rm c.m.},J)}{1-P_{BU}(E_{\rm
c.m.},J)}+\frac{P_{rest}(E_{\rm c.m.},J)}{1-P_{BU}(E_{\rm
c.m.},J)}=P_{el}^{noBU}(E_{\rm c.m.},J)+P_{rest}^{noBU}(E_{\rm c.m.},J)=1,$
(2)

where

$P_{el}^{noBU}(E_{\rm c.m.},J)=\frac{P_{el}(E_{\rm c.m.},J)}{1-P_{BU}(E_{\rm
c.m.},J)}$

and

$P_{rest}^{noBU}(E_{\rm c.m.},J)=\frac{P_{rest}(E_{\rm
c.m.},J)}{1-P_{BU}(E_{\rm c.m.},J)}$

are the elastic scattering and other channels probabilities, respectively, in
the absence of the breakup process. From these expressions we obtain the
useful formulas

$\displaystyle\frac{P_{el}(E_{\rm c.m.},J)}{P_{rest}(E_{\rm
c.m.},J)}=\frac{P_{el}^{noBU}(E_{\rm c.m.},J)}{P_{rest}^{noBU}(E_{\rm
c.m.},J)}=\frac{P_{el}^{noBU}(E_{\rm c.m.},J)}{1-P_{el}^{noBU}(E_{\rm
c.m.},J)}.$ (3)

Using Eqs. (1) and (3), one can find the relationship between the breakup and
elastic scattering processes:

$\displaystyle P_{BU}(E_{\rm c.m.},J)=1-\frac{P_{el}(E_{\rm
c.m.},J)}{P_{el}^{noBU}(E_{\rm c.m.},J)}.$ (4)

The last equation is the main result of the present report. Note that similar
formula

$\displaystyle P_{BU}(E_{\rm c.m.},J)=1-\frac{P_{qe}(E_{\rm
c.m.},J)}{P_{qe}^{noBU}(E_{\rm c.m.},J)}$ (5)

was derived in Ref. EPJSub4  to relate the breakup and quasi-elastic
scattering processes.

The reflection elastic or quasi-elastic backscattering probability
$P_{el,qe}(E_{\rm c.m.},J=0)=d\sigma_{el,qe}/d\sigma_{Ru}$ for bombarding
energy $E_{\rm c.m.}$ and angular momentum $J=0$ is given by the ratio of the
elastic or quasi-elastic scattering differential cross section
$\sigma_{el,qe}$ and Rutherford differential cross section $\sigma_{Ru}$ at
180 degrees Timmers . Employing Eq. (4) or (5) and the experimental elastic or
quasi-elastic backscattering data in the reactions with toughly and weakly
bound isotopes-projectiles and the same or almost the same compound nucleus,
one can extract the breakup probability of the exotic nucleus. For example,
using Eq. (4) or (5) at backward angle, the experimental
$P_{el,qe}^{noBU}$[4He+AX] of the 4He+AX reaction with toughly bound nuclei
(without breakup), and the experimental $P_{el,qe}$[6He+A-2X] of the 6He+A-2X
reaction with weakly bound projectile (with breakup), and assuming approximate
equality $V_{b}$(4He+AX)$\approx V_{b}$(6He+A-2X) for the Coulomb barriers of
very asymmetric systems, one can extract the breakup probability of the 6He:

$\displaystyle P_{BU}(E_{\rm c.m.},J=0)=1-\frac{P_{el,qe}(E_{\rm
c.m.},J=0)[^{6}{\rm He}+^{A-2}{\rm X}]}{P_{el,qe}^{noBU}(E_{\rm
c.m.},J=0)[^{4}{\rm He}+^{A}{\rm X}]}$ (6)

or

$\displaystyle P_{BU}(E_{\rm c.m.},J=0)=1-\frac{P_{el,qe}(E_{\rm
c.m.},J=0)[^{6}{\rm He}+^{A-2}{\rm X}]}{P_{el,qe}^{noBU}(E_{\rm
c.m.},J=0)[^{4}{\rm He}+^{A-2}{\rm X}]}.$ (7)

Comparing the experimental elastic or quasi-elastic backscattering
probabilities in the presence and absence of breakup data in the reaction
pairs 6He+68Zn and 4He+68,70Zn, 6He+122Sn and 4He+122,124Sn, 6He+236U and
4He+236,238U, 8He+204Pb and 4He+204,208Pb, 8Li+207Pb and 7Li+207,208Pb,
7Be+207Pb and 10Be+204,207Pb, 9Be+208Pb and 10Be+207,208Pb, 11Be+206Pb and
10Be+206,207Pb, 8B+208Pb and 10B+206,208Pb, 8B+207Pb and 11B+204,207Pb,
9B+208Pb and 11B+206,208Pb, 15C+204Pb and 12C+204,207Pb, 15C+206Pb and
13C+206,208Pb, 15C+207Pb and 14C+207,208Pb, 17F+208Pb and 19F+206,208Pb,
leading to the same or almost the same corresponding compound nuclei, one can
analyse the role of the breakup channels in the reactions with the light
weakly bound projectiles 6,8He, 8Li, 7,9,11Be, 8,9B, 15C, and 17F at energies
near and above the Coulomb barrier.

One concludes that the elastic or quasi-elastic backscattering technique could
be a very useful tool in the study of breakup. The breakup probabilities can
be extracted from the elastic or quasi-elastic backscattering probabilities of
systems mentioned above.

We thank P.R.S. Gomes and A. Lépina-Szily for fruitful discussions and
suggestions. This work was supported by NSFC, RFBR, and JINR grants. The
IN2P3(France)-JINR(Dubna) and Polish - JINR(Dubna) Cooperation Programmes are
gratefully acknowledged.

## References

  * (1) L.F. Canto, P.R.S. Gomes, R. Donangelo, and M.S. Hussein, Phys. Rep. 424, (2006) 1.
  * (2) V.V. Sargsyan, G.G. Adamian, N.V. Antonenko, W. Scheid, and H.Q. Zhang, Phys. Rev. C 86, 054610 (2012).
  * (3) P.R.S. Gomes et al., Phys. Rev. C 84, 014615 (2011); P.R.S. Gomes, J. Lubian, and L.F. Canto, Phys. Rev. C 79, 027606 (2009).
  * (4) H. Timmers et al., Nucl. Phys. A584, 190 (1995); H.Q. Zhang, F. Yang, C. Lin, Z. Liu, and Y. Hu, Phys. Rev. C 57, R1047 (1998); A.A. Sonzogni et al., Phys. Rev. C 57, 722 (1998); O.A. Capurro et al., Phys. Rev. C 61, 037603 (2000); S. Santra et al., Phys. Rev. C 64, 024602 (2001); R.F. Simões et al., Phys. Lett. B 527, 187 (2002); S. Sinha et al., Phys. Rev. C 64, 024607 (2001); E. Piasecki et al., Phys. Rev. C 65, 054611 (2002).


Group theoretical derivation of consistent particle theories

Giuseppe Nisticò1,

1 Università della Calabria, Italy

2 INFN, gr. collegato di Cosenza, Italy

*<EMAIL_ADDRESS>
34th International Colloquium on Group Theoretical Methods in Physics
Strasbourg, 18-22 July 2022 10.21468/SciPostPhysProc.?

## Abstract

Current quantum theories of an elementary free particle assume unitary space
inversion and anti-unitary time reversal operators. In so doing robust classes
of possible theories are discarded. The present work shows that consistent
theories can be derived through a strictly deductive development from the
principle of relativistic invariance and position covariance, also with anti-
unitary space inversion and unitary time reversal operators. In doing so the
class of possible consistent theories is extended for positive but also zero
mass particles. In particular, consistent theories for a Klein-Gordon particle
are derived and the non-localizability theorem for a non zero helicity
massless particle is extended.

###### Contents

  1. 1 Introduction
  2. 2 General implications of Poincaré invariance
    1. 2.1 Prerequisites and notation
    2. 2.2 Quantum theoretical implications for an elementary free particle
  3. 3 Quantum theories of positive mass elementary free particle
    1. 3.1 Positive mass irreducible triplets
    2. 3.2 Theories of elementary free particle with positive mass
    3. 3.3 Conclusions for the positive mass case
  4. 4 Quantum theories of zero mass elementary free particle
    1. 4.1 Zero mass irreducible triplets
    2. 4.2 The theories of elementary free particle with zero mass
    3. 4.3 Conclusions for the zero mass case

##  1 Introduction

Relativistic quantum theories of single free particle can be deductively
derived from the principles of relativistic invariance and covariance [1]
-[4]; the first principle implies that the Hilbert space of the quantum theory
of a free particle must admit a transformer triplet
$(U,{{}_{\triangleleft}\kern-2.0pt\texttt{S}},{{}^{\triangleleft}\kern-1.5pt\texttt{T}})$
formed by a unitary representation $U$ of the universal covering group
$\tilde{\mathcal{P}}_{+}^{\uparrow}$ of the proper orthochronus Poincaré group
${\mathcal{P}}_{+}^{\uparrow}$ and by the operators
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ and
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$, which realize the quantum
transformations implied by the transformations of
${\mathcal{P}}_{+}^{\uparrow}$, by space inversion
${{}_{\triangleleft}\kern-2.2pt\texttt{s}}$ and by time reversal
${{}^{\triangleleft}\kern-3.2pt\texttt{t}}$, respectively. Yet the literature,
except some works [5][6] with specific aims different from the present one,
excludes transformer triplets with ${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$
anti-unitary or with ${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ unitary, from
the pionering works of Wigner, Bargmann [1] -[3], to subsequent investigations
[4] -[9] . In so doing robust classes of triplets, and hence of possible
theories, are lost. For instance, there is no such a triplet for a consistent
theory of Klein-Gordon particles111Klein-Gordon theory, indeed, was obtained
through canonical quantization [10],[11], but it predicts inconsistencies,
such as negative probabilities[12]..

The motivation for the exclusion of
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ unitary or
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ anti-unitary was their implication
of negative spectral values for the hamiltonian operator $P_{0}$, values
deemed inconsistent because $P_{0}$ was identified with the positive
relativistic kinetic energy operator $E_{kin}=\mu(1-{\dot{\bf
Q}}^{2})^{-1/2}$, where $\dot{\bf Q}$ is the “velocity” operator. But remark
3.1 shall show that the hamiltonian operator $P_{0}$ does not always coincide
with $E_{kin}$, so that a unitary ${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$
or an anti-unitary ${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ can be
consistent.

In the present article we show how a strictly deductive development of
consistent quantum theories of elementary free particle can be successfully
carried out without apriori preclusions about the unitary or anti-unitary
character of ${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ or
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$. As results, classes of consistent
possible theories for a positive mass particle are expicitly identified, which
meaningfully extend the class of the current theories; in particular,
consistent theories of Klein-Gordon particle are derived. Also in the case of
a massless particle the approach extends the class of possible theories.
Furthermore, the non-localizability theorem for non zero helicity massless
particles is extended to the new theories with
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ unitary or
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ anti-unitary. Section 2 shows how
the relativistic invariance principle implies that every theory of elementary
free particle admits a transformer triplet. In section 3 the class of possible
consistent theories for a positive mass particle is identified; this class
contains consistent theories with ${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$
anti-unitary, e.g. consistent theories of Klein-Gordon particle. Section 4
identifies the class of consistent theories for a zero mass elementary free
particle; once again, besides the current theories, it contains theories with
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ anti-unitary or
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ unitary. A more accurate and more
general argument is presented, which denies localizability of non zero
helicity mass zero particles

##  2 General implications of Poincaré invariance

###  2.1 Prerequisites and notation

First of all, it is worth to fix the notaion for any quantum theory based on a
Hilbert space $\mathcal{H}$:

\- $\Omega({\mathcal{H}})$ denotes the set of all self-adjoint operators
representing observables;

\- ${\mathcal{S}}({\mathcal{H}})$ denotes the set of all density operators
$\rho$ identified with quantum states;

\- ${\mathcal{U}}({\mathcal{H}})$ denotes the group of all unitary unitary
operators;

\- ${\mathcal{V}}({\mathcal{H}})$ is the larger group of all unitary or anti-
unitary operators.

The Poincaré group $\mathcal{P}$ is a very important mathematical structure
for the present work, because it is the group of symmetry transformations for
a free particle. $\mathcal{P}$ is the group generated by
${\mathcal{P}}_{+}^{\uparrow}\cup\\{{{}^{\triangleleft}\kern-3.2pt\texttt{t}},{{}_{\triangleleft}\kern-2.2pt\texttt{s}}\\}$,
where ${\mathcal{P}}_{+}^{\uparrow}$ is the proper orthochronus Poincaré
group, ${{}^{\triangleleft}\kern-3.2pt\texttt{t}}$ and
${{}_{\triangleleft}\kern-2.2pt\texttt{s}}$ are the time reversal and space
inversion transformations. The proper orthochronus group
${\mathcal{P}}_{+}^{\uparrow}$ is a connected group generated by 10 one-
parameter subgroups, namely the subgroup ${\mathcal{T}}_{0}$ of time
translations, the three subgroups ${\mathcal{T}}_{j}$ ($j=1,2,3$) of spatial
translations, the three subgroups ${\mathcal{R}}_{j}$ of spatial rotations,
the three subgroups ${\mathcal{B}}_{j}$ of Lorentz boosts, relative to the
three spatial axes $x_{j}$. Time reversal
${{}^{\triangleleft}\kern-3.2pt\texttt{t}}$ and space inversion
${{}_{\triangleleft}\kern-2.2pt\texttt{s}}$ are not connected with the
identity transformation $e\in\mathcal{P}$. Given any vector
$\underline{x}=(x_{0},{\bf x})\in\mathop{{\rm I}\kern-1.99997pt{\rm
R}}\nolimits^{4}$, where $x_{0}$ is called the time component of
$\underline{x}$ and ${\bf x}=(x_{1},x_{2},x_{3})$ is called the spatial
component of $\underline{x}$, time reversal
${{}^{\triangleleft}\kern-3.2pt\texttt{t}}$ transforms
$\underline{x}=(x_{0},{\bf x})$ into $(-x_{0},{\bf x})$ and space inversion
${{}_{\triangleleft}\kern-2.2pt\texttt{s}}$ transforms
$\underline{x}=(x_{0},{\bf x})$ into $(x_{0},-{\bf x})$.

The universal covering group of ${\mathcal{P}}_{+}^{\uparrow}$ is the
semidirect product $\tilde{\mathcal{P}}_{+}^{\uparrow}=\mathop{{\rm
I}\kern-1.99997pt{\rm R}}\nolimits^{4}\hbox{$\bigcirc$\kern-6.00006pt\hbox{s}
}SL(2,\hbox{{{I}}\kern-4.70001pt\hbox{C}})$ of the time-space translation
group $\mathop{{\rm I}\kern-1.99997pt{\rm R}}\nolimits^{4}$ and the group
$SL(2,\hbox{{{I}}\kern-4.70001pt\hbox{C}})=\\{\underline{\Lambda}\in
GL(2,\hbox{{{I}}\kern-4.70001pt\hbox{C}})\mid\det\underline{\Lambda}=1\\}$.
Accordingly, $\tilde{\mathcal{P}}_{+}^{\uparrow}$ is simply connected and
there is a canonical homomorphism
${\textsf{h}}:\tilde{\mathcal{P}}_{+}^{\uparrow}\to{\mathcal{P}}_{+}^{\uparrow}$,
$\tilde{g}\to{\textsf{h}}(\tilde{g})\in{\mathcal{P}}_{+}^{\uparrow}$, which
restricts to an isomorphism within a small enough neighborhood of the identity
$(0,\hbox{\sl
1\kern-2.5pt\hbox{I}}_{{{}_{\hbox{{{I}}\kern-2.35pt\hbox{C}}^{2}}}})$ of
$\tilde{\mathcal{P}}_{+}^{\uparrow}$. By $\tilde{\mathcal{T}}_{0}$,
$\tilde{\mathcal{T}}_{j},\tilde{\mathcal{R}}_{j}$, $\tilde{\mathcal{B}}_{j}$,
$\tilde{\mathcal{L}}_{+}^{\uparrow}$ we denote the subgroups of
$\tilde{\mathcal{P}}_{+}^{\uparrow}$ which correspond to the subgoroups
${\mathcal{T}}_{0}$, ${\mathcal{T}}_{j},{\mathcal{R}}_{j}$,
${\mathcal{B}}_{j}$, ${\mathcal{L}}_{+}^{\uparrow}$ of
${\mathcal{P}}_{+}^{\uparrow}$, through the homomorphism h.

###  2.2 Quantum theoretical implications for an elementary free particle

Since a free particle is a particular kind of isolated system, we begin by
showing the derivation of the general structure of the quantum theory of an
isolated system. By $\mathcal{F}$ we denote the class of the (inertial)
reference frames that move uniformly with respect to each other. A physical
system is an isolated system if the following invariance principle holds.

$\mathcal{IP}$

The theory of an isolated system is invariant with respect to changes of
frames within $\mathcal{F}$.

If $\Sigma$ belongs to $\mathcal{F}$, then $\Sigma_{g}$ denotes the frame
related to $\Sigma$ by such $g$, for every $g\in\mathcal{P}$. Given an
observable $\mathcal{A}$ represented by the operator
$A\in\Omega({\mathcal{H}})$, let ${\mathcal{M}}_{A}$ be a procedure to measure
$\mathcal{A}$; then the invariance principle implies that another measuring
procedure ${\mathcal{M}}_{A}^{\prime}$ must exist, which is with respect to
$\Sigma_{g}$ identical to what is ${\mathcal{M}}_{A}$ with respect to
$\Sigma$, otherwise the principle $\mathcal{IP}$ would be violated. Hence,
$\mathcal{IP}$ implies the existence [12] of the so called quantum
transformation associated to $g$, i.e., of a mapping

$S_{g}:\Omega({\mathcal{H}})\to\Omega({\mathcal{H}})\,,\quad A\to S_{g}[A]\,,$

where ${\it S_{g}}[A]$ is the self-adjoint operator that represents the
observable measured by ${\mathcal{M}}_{A}^{\prime}$.

To every element $\tilde{g}$ of the covering group
$\tilde{\mathcal{P}}_{+}^{\uparrow}$ we can associate the quantum
transformation $S_{\textsf{h}(\tilde{g})}\equiv S_{\tilde{g}}$ through the
canonical homomorphism h. In [12] it is proved that the properties of quantum
transformations, under a continuity condition for $\tilde{g}\to
S_{\tilde{g}}$, imply that

Imp.1. a continuous unitary representation $U$ of
$\tilde{\mathcal{P}}_{+}^{\uparrow}$ exists such that
$S_{\tilde{g}}[A]=U_{\tilde{g}}AU_{\tilde{g}}^{-1}$, and

Imp.2. two operators
${{}_{\triangleleft}\kern-2.0pt\texttt{S}},{{}^{\triangleleft}\kern-1.5pt\texttt{T}}\in{\mathcal{V}}({\mathcal{H}})$
exist such that
$S_{{}_{\triangleleft}\kern-2.2pt\texttt{s}}[A]={{}_{\triangleleft}\kern-2.0pt\texttt{S}}A{{}_{\triangleleft}\kern-2.0pt\texttt{S}}^{-1}$
and
$S_{{}^{\triangleleft}\kern-3.2pt\texttt{t}}[A]={{}^{\triangleleft}\kern-1.5pt\texttt{T}}A{{}^{\triangleleft}\kern-1.5pt\texttt{T}}^{-1}$.

Thus, the principle $\mathcal{IP}$ has the following fundamental implication.

(FI)

The quantum theory of an isolated system admits a transformer triplet
$(U,{{}_{\triangleleft}\kern-2.0pt\texttt{S}},{{}^{\triangleleft}\kern-1.5pt\texttt{T}})$
such that implications Imp.1 and Imp.2 hold.

Given a transformer triplet
$(U,{{}_{\triangleleft}\kern-2.0pt\texttt{S}},{{}^{\triangleleft}\kern-1.5pt\texttt{T}})$,
let $P_{0},P_{j},J_{j},K_{j}\in\Omega({\mathcal{H})}$ be the selfadjoint
generators of $U$; so [12], if $\tilde{g}\in\tilde{\mathcal{T}}_{0}$ (resp.,
$\tilde{\mathcal{T}}_{j}$, $\tilde{\mathcal{R}}_{j}$,
$\tilde{\mathcal{B}}_{j}$) is identified by the parameter $t$ (resp., $a$,
$\theta$, $u$), then

$U_{\tilde{g}}=e^{iP_{0}t},\quad(\hbox{resp.,
}U_{\tilde{g}}=e^{iP_{j}a},\;U_{\tilde{g}}=e^{J_{j}\theta},\;U_{\tilde{g}}=e^{iK_{j}\frac{1}{2}\ln\frac{1+u}{1-u}}).$
$None$

The generator $P_{0}$ relative to time translations is the hamiltonian
operator, so that

$(i)\quad\frac{d}{dt}A_{t}\equiv\dot{A}_{t}=i[P_{0},A_{t}],\qquad(ii)\quad\frac{d}{dt}\rho_{t}\equiv\dot{\rho}_{t}=-i[P_{0},\rho_{t}].$
$None$

By “elementary” free particle we mean an isolated system whose quantum theory
has a unique three-operator ${\bf Q}\equiv(Q_{1},Q_{2},Q_{3})$ with
$Q_{j}\in\Omega({\mathcal{H}})$, called position operator, such that
$(U(\tilde{\mathcal{P}}_{+}^{\uparrow}),{{}_{\triangleleft}\kern-2.0pt\texttt{S}},{{}^{\triangleleft}\kern-1.5pt\texttt{T}};{\bf
Q})$ is an irreducible system of operators, and satisfying the following
conditions.

  * (Q.1)

$\;[Q_{j},Q_{k}]=\hbox{{{I}}\kern-4.70001pt\hbox{O}}$, for all $j,k=1,2,3$;
this condition establishes that a measurement of position yields all three
values of the coordinates of the same specimen of the system.

  * (Q.2)

For every $g\in\mathcal{P}$, the position operator ${\bf Q}$ and the
transformed position operator $S_{g}[{\bf Q}]$ satisfy the transformation
properties of position with respect to $g$.

As proved in [12], the transformer triplet
$(U,{{}_{\triangleleft}\kern-2.0pt\texttt{S}},{{}^{\triangleleft}\kern-1.5pt\texttt{T}})$
of the quantum theory of an elementary free particle must be irreducible.
Thus, the identification of all possible theories of an elementary free
particle can be carried out in two steps: first by identifying all irreducible
transformer triplets
$(U,{{}_{\triangleleft}\kern-2.0pt\texttt{S}},{{}^{\triangleleft}\kern-1.5pt\texttt{T}})$,
and then selecting those triplets for which a unique position operator $\bf Q$
exists. The mathematical group structural properties of $\mathcal{P}$ imply
[12],[16] that each irreducible triplet
$(U,{{}_{\triangleleft}\kern-2.0pt\texttt{S}},{{}^{\triangleleft}\kern-1.5pt\texttt{T}})$
is characterized by a number $\mu\in\hbox{{{I}}\kern-4.70001pt\hbox{C}}$,
called mass, with $\mu^{2}\in\mathop{{\rm I}\kern-1.99997pt{\rm R}}\nolimits$,
such that $P_{0}^{2}-{\bf P}^{2}=\mu^{2}\hbox{\sl 1\kern-2.5pt\hbox{I}}$.

##  3 Quantum theories of positive mass elementary free particle

To identify the positive mass possible theories, we shall identify the
irreducible triplets with $\mu>0$; then, the triplets admitting a three-
operator $\bf Q$ satisfying (Q.1), (Q.2) are singled out.

###  3.1 Positive mass irreducible triplets

Following [12], for any pair $(\mu,s)$, where $\mu>0$ and $s$ is an integral
or half-integral number $s\in\frac{1}{2}\hbox{I\kern-1.99997pt\hbox{N}}$
called spin, there is at least one irreducible triplet. Conversely, every
irreducible triplet is characterized by one such a pair. The following theorem
yields a first classification. Theorem 3.1. If
$(U,{{}_{\triangleleft}\kern-2.0pt\texttt{S}},{{}^{\triangleleft}\kern-1.5pt\texttt{T}})$
is an irreducible triplet with non-negative mass $\mu\geq 0$, then

i) $\sigma(P_{0})=(-\infty,-\mu]$ or $\sigma(P_{0})=[\mu,\infty)$ or
$\sigma(P_{0})=(-\infty,-\mu]\cup[\mu,\infty)$, where $\sigma(P_{0})$ is the
spectrum of $P_{0}$.

Moreover, $\sigma(P_{0})=(-\infty,-\mu]\cup[\mu,\infty)$ if and only is
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ is unitary or
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ is anti-unitary.

ii) Each class ${\mathcal{I}}(\mu,s)$ of all irreducible triplets with
positive mass $\mu>0$ decomposes as

${\mathcal{I}}(\mu,s)={\mathcal{I}}^{-}(\mu,s)\cup{\mathcal{I}}^{+}(\mu,s)\cup{\mathcal{I}}^{-+}(\mu,s).$
$None$

where ${\mathcal{I}}^{-}(\mu,s)$, ${\mathcal{I}}^{+}(\mu,s)$ and
${\mathcal{I}}^{-+}(\mu,s)$ are respectively the classes of irreducible
triplets with $\sigma(P_{0})=(-\infty,-\mu]$, $\sigma(P_{0})=[\mu,\infty)$ and
$\sigma(P_{0})=(-\infty,-\mu]\cup[\mu,\infty)$. The representation $U$ of a
triplet in ${\mathcal{I}}^{+}(\mu,s)$ or ${\mathcal{I}}^{-}(\mu,s)$ can be
irreducible or not. We refer to [12] for a complete identification of the
irreducible triplets of ${\mathcal{I}}^{\pm}(\mu,s)$ with $U$ irreducible.
Therein also instances of triplet in ${\mathcal{I}}^{+}(\mu,s)$ and
${\mathcal{I}}^{+}(\mu,s)$ with $U$ reducible are explicitly shown. The
representation $U$ of a triplet in ${\mathcal{I}}^{-+}(\mu,s)$ is always
reducible [12], namely $U=U^{+}\oplus U^{-}$ where $U^{\pm}$ belongs to a
triplet in ${\mathcal{I}}^{\pm}(\mu,s)$. Moreover, $U^{+}$ is reducible if and
only if $U^{-}$ is reducible.

The class of all irreducible triplets of ${\mathcal{I}}^{-+}(\mu,s)$ with
$U^{+}$ irreducible can be found in [12], where also triplets of
${\mathcal{I}}^{-+}(\mu,s)$ with $U^{+}$ reducible are concretely shown.

###  3.2 Theories of elementary free particle with positive mass

To determine the possible theories of positive mass elemetary free particle,
we have to select irreducible triplets of ${\mathcal{I}}(\mu,s)$ identified in
[12] for which a position $\bf Q$ satisfying (Q.1) and (Q.2) exists. Condition
(Q.2) can be only partially imposed. In fact, while the covariance properties
with respect to translations, rotations, time reversal and space inversion are
known and explicitly expressed by the following relations[12]

$(i)\;\;[Q_{j},P_{k}]=i\delta_{jk},\quad(ii)\;\;[J_{j},Q_{k}]=i\epsilon_{jkl}Q_{l},\quad(iii)\;\;{{}^{\triangleleft}\kern-1.5pt\texttt{T}}{\bf
Q}={\bf
Q}{{}^{\triangleleft}\kern-1.5pt\texttt{T}},\quad(iv)\;\;{{}_{\triangleleft}\kern-2.0pt\texttt{S}}{\bf
Q}=-{\bf Q}{{}^{\triangleleft}\kern-1.5pt\texttt{T}},$ $None$

the explicit relations that establish the transformation properties of
position with respect to boosts are not available, yet [12]. However,
conditions (4) are sufficient to uniquely identify $\bf Q$ for some subclasses
of irreducible triplets, according to the following theorem [12]. Theorem 3.2.
Given a triplet in ${\mathcal{I}}^{+}(\mu,0)$ with $U$ irreducible there is a
unique three-operator $\;\bf Q$, satisfying (Q.1) and (4). Modulo unitary
isomorphism, the resulting theory has Hilbert space
${\mathcal{H}}=L_{2}(\mathop{{\rm I}\kern-1.99997pt{\rm
R}}\nolimits^{3},\hbox{{{I}}\kern-4.70001pt\hbox{C}}^{2s+1},d\nu)$, where
$d\nu({\bf p})=\frac{dp_{1}dp_{2}dp_{3}}{p_{0}}$ with
$p_{0}=\sqrt{\mu^{2}+{\bf p}^{2}}$,

–

generators defined by $(P_{j}\psi)({\bf p})=p_{j}\psi({\bf p})$,
$(P_{0}\psi)(\underline{p})=p_{0}\psi(\underline{p})$,
$J_{k}={\textsf{J}}_{k}^{(0)}$, $K_{j}={\textsf{K}}_{j}^{(0)}$, $(5)$

where ${\textsf{J}}_{k}^{(0)}=-i\left(p_{l}\frac{\partial}{\partial
p_{j}}-p_{j}\frac{\partial}{\partial p_{l}}\right)$,
${\textsf{K}}_{j}^{(0)}=ip_{0}\frac{\partial}{\partial p_{j}}$;

–

${{}_{\triangleleft}\kern-2.0pt\texttt{S}}=\Upsilon$,
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}={\mathcal{K}}\Upsilon$, where
${\mathcal{K}}$ and $\Upsilon$ are defined by ${\mathcal{K}}\psi({\bf
p})=\overline{\psi({\bf p})}$, $(\Upsilon\psi)({\bf p})=\psi(-{\bf p})$.

–

The position operator is ${\bf Q}={\bf F}$, where $\bf F$ is the Newton-Wigner
[13] operator defined by

$F_{j}=i\frac{\partial}{\partial p_{j}}-\frac{i}{2p_{0}^{2}}p_{j}.$ $None$

Analogously, there is only one theory based on a triplet in
$\;{\mathcal{I}}^{-}(\mu,0)$ with $U$ irreducible. It differs from that in
$\;{\mathcal{I}}^{+}(\mu,0)$ by $P_{0}=-p_{0}$ and
$K_{j}=-{\textsf{K}}_{j}^{(0)}$. There are only two theories based on triplets
of ${\mathcal{I}}^{-+}(\mu,0)$ with $U^{+}$ irreducible. They share the
Hilbert space and generators222 If $\psi\in L_{2}(\mathop{{\rm
I}\kern-1.99997pt{\rm
R}}\nolimits^{3},\hbox{{{I}}\kern-4.70001pt\hbox{C}}^{2s+1},d\nu)\oplus
L_{2}(\mathop{{\rm I}\kern-1.99997pt{\rm
R}}\nolimits^{3},\hbox{{{I}}\kern-4.70001pt\hbox{C}}^{2s+1},d\nu)$, we write
$\psi\equiv\psi_{1}\oplus\psi_{2}\equiv\left[\begin{array}[]{c}\psi_{1}\cr\psi_{2}\end{array}\right]$,
$\psi_{1},\psi_{2}\in L_{2}(\mathop{{\rm I}\kern-1.99997pt{\rm
R}}\nolimits^{3},\hbox{{{I}}\kern-4.70001pt\hbox{C}}^{2s+1},d\nu))$.:
${\mathcal{H}}=L_{2}(\mathop{{\rm I}\kern-1.99997pt{\rm
R}}\nolimits^{3},\hbox{{{I}}\kern-4.70001pt\hbox{C}}^{2s+1},d\nu)\oplus
L_{2}(\mathop{{\rm I}\kern-1.99997pt{\rm
R}}\nolimits^{3},\hbox{{{I}}\kern-4.70001pt\hbox{C}}^{2s+1},d\nu)$
$P_{j}=\left[\begin{array}[]{cc}p_{j}&0\\\ 0&p_{j}\end{array}\right],\quad
P_{0}=\left[\begin{array}[]{cc}p_{0}&0\\\ 0&-p_{0}\end{array}\right],\quad
J_{k}=\left[\begin{array}[]{cc}{\textsf{J}}^{(0)}_{k}&0\\\
0&{\textsf{J}}^{(0)}_{k}\end{array}\right],\quad
K_{j}=\left[\begin{array}[]{cc}{\textsf{K}}^{(0)}_{j}&0\\\
0&-{\textsf{K}}^{(0)}_{j}\end{array}\right]$. (7)

The two theories differ for the different pairs
$({{}_{\triangleleft}\kern-2.0pt\texttt{S}}_{1},{{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{1})$,
$({{}_{\triangleleft}\kern-2.0pt\texttt{S}}_{2},{{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{2})$
of space inversion and time reversal operators; indeed
$\;{{}_{\triangleleft}\kern-2.0pt\texttt{S}}_{1}={{}_{\triangleleft}\kern-2.0pt\texttt{S}}_{2}=\left[\begin{array}[]{cc}0&1\\\
1&0\end{array}\right]{\mathcal{K}}$ while
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{1}={\mathcal{K}}\Upsilon\left[\begin{array}[]{cc}1&0\\\
0&1\end{array}\right]$ and
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{2}=\left[\begin{array}[]{cc}0&1\\\
1&0\end{array}\right]$.

For both theories the position operator is $\;{\bf
Q}=\left[\begin{array}[]{cc}{\bf F}&0\cr 0&{\bf F}\end{array}\right]$.  For
all triplets with $s>0$ (Q.1) and (4) are not sufficient [12] to completely
identify $\bf Q$. Remark 3.1. In both theories based on
${\mathcal{I}}^{-+}(\mu,0)$ the hamiltonian operator $P_{0}$ has also negative
spectral values. But since the “velocity” is $\dot{\bf Q}=\frac{d}{dt}{\bf
Q}=i[P_{0},{\bf Q}]=\left[\begin{array}[]{cc}\frac{\bf p}{p_{0}}&0\\\
0&-\frac{\bf p}{p_{0}}\end{array}\right]$, we compute that
$E_{kin}=\mu(1-{\dot{\bf
Q}}^{2})^{-1/2}=p_{0}>\hbox{{{I}}\kern-4.70001pt\hbox{O}}$, i.e. the theories
are consistent.

###  3.3 Conclusions for the positive mass case

According to section 3.2, four classes of possible consistent theories are
completely determined by following the present approach, with $U$ or $U^{+}$
irreducible. However, the class of theories based on
${\mathcal{I}}^{\pm}(\mu,0)$ with $U$ reducible and the class of theories
based on ${\mathcal{I}}^{-+}(\mu,0)$ with $U^{+}$ reducible are not empty;
concrete examples are given in [12]. They are new species theories, i.e. they
correspond to none of the known theories. Hence, our approach extends the
class of consistent theories of positive mass elementary spin 0 free particle.

Moreover, it provides consistent theories for Klein-Gordon particles. Indeed,
by means of a unitary transformation, operated by the operator $Z=Z_{1}Z_{2}$,
where $Z_{2}=\frac{1}{\sqrt{p_{0}}}\hbox{\sl 1\kern-2.5pt\hbox{I}}$ and
$Z_{1}$ is the inverse of the Fourier-Plancherel operator, the theories based
on ${\mathcal{I}}^{-+}(\mu,0)$ turn out to be equivalent to two theories with
Hilbert space $\hat{\mathcal{H}}=Z\left(L_{2}(\mathop{{\rm
I}\kern-1.99997pt{\rm R}}\nolimits^{3},d\nu)\oplus L_{2}(\mathop{{\rm
I}\kern-1.99997pt{\rm R}}\nolimits^{3},d\nu)\right)\equiv L_{2}(\mathop{{\rm
I}\kern-1.99997pt{\rm R}}\nolimits^{3})\oplus L_{2}(\mathop{{\rm
I}\kern-1.99997pt{\rm R}}\nolimits^{3})$, with the self-adjoint generators
${\hat{P}}_{j}=\left[\begin{array}[]{cc}-i\frac{\partial}{\partial x_{j}}&0\cr
0&-i\frac{\partial}{\partial x_{j}}\end{array}\right]$,
${\hat{P}}_{0}=\sqrt{\mu^{2}-\nabla^{2}}\left[\begin{array}[]{cc}1&0\cr
0&-1\end{array}\right]$,

${\hat{J}}_{j}=-i\left(x_{k}\frac{\partial}{\partial
x_{l}}-x_{l}\frac{\partial}{\partial
x_{k}}\right)\left[\begin{array}[]{cc}1&0\cr 0&1\end{array}\right]$,
${\hat{K}}_{j}=\frac{1}{2}\left(x_{j}\sqrt{\mu^{2}-\nabla^{2}}+\sqrt{\mu^{2}-\nabla^{2}}x_{j}\right)\left[\begin{array}[]{cc}1&0\cr
0&-1\end{array}\right]$,

while
$\hat{{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{1}={\mathcal{K}}\left[\begin{array}[]{cc}1&0\cr
0&1\end{array}\right]$ ,
$\hat{{}_{\triangleleft}\kern-2.0pt\texttt{S}}_{1}={\mathcal{K}}\Upsilon\left[\begin{array}[]{cc}0&1\cr
1&0\end{array}\right]$, and
$\hat{{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{2}=\left[\begin{array}[]{cc}0&1\cr
1&0\end{array}\right]$ and
$\hat{{}_{\triangleleft}\kern-2.0pt\texttt{S}}_{2}={\mathcal{K}}\Upsilon\left[\begin{array}[]{cc}0&1\cr
1&0\end{array}\right]$. The position operator is
${\hat{Q}}_{j}=\left[\begin{array}[]{cc}x_{j}&0\cr 0&x_{j}\end{array}\right]$.

These ${\hat{P}}_{j},\;{\hat{J}}_{j},\;\;{\hat{K}}_{j},\;\hat{\mathcal{H}}$
are generators and Hilbert space of Klein-Gordon theory of spin-0 particle
[10][11]. However, since the position operator is the multiplication operator,
the position probability density must be $\rho(t,{\bf
x})=|\hat{\psi}_{1}(t,{\bf x})|^{2}+|\hat{\psi}_{2}(t,{\bf x})|^{2}$, hence
non-negative. Thus, our extended class includes consistent theories for Klein-
Gordon particle free from the inconsistent negative probabilities of the early
theory. It turns out [12] that in all triplets with non zero spin, the
position operator $\bf Q$ is not uniquely determined by (Q.1) and (4). On the
other hand, the transformation properties of position with respect to boosts,
expressed for instance by a relation for $[K_{j},Q_{k}]$, are not available in
order to better identify $\bf Q$ by imposing them.

To each solution $\bf Q$ of (Q.1) and (4) there correspond a different
$[K_{j},Q_{k}]$, in general. For instance, Dirac theory for spin 1/2 particle
[14] is completely characterized by the relation
$[K_{j},Q_{k}]=-\frac{i}{2}(Q_{j}\dot{Q}_{k}+\dot{Q}_{k}Q_{j})$ satisfied by
the posistion operator of Dirac theory; however, other solutions $\bf Q$
yielding other relations for $[K_{j},Q_{k}]$ are theoretically consistent too.

##  4 Quantum theories of zero mass elementary free particle

Analogously to the positive mass case, the possible quantum theories of zero
mass particle are determined first by identifying the class
${\mathcal{I}}_{0}$ of the irreducible transformer triplets with $\mu=0$, and
then by selecting those triplets that admit a unique position operator.
According to theorem 3.1.i the class ${\mathcal{I}}_{0}$ decomposes as
${\mathcal{I}}^{+}_{0}={\mathcal{I}}^{+}_{0}\cup{\mathcal{I}}^{-}_{0}\cup{\mathcal{I}}^{-+}_{0}$,
where ${\mathcal{I}}^{-}_{0}$ (resp., ${\mathcal{I}}^{+}_{0}$,
${\mathcal{I}}^{-+}_{0}$) denotes the class of irreducible triplets with
$\sigma(P_{0})=(-\infty,0]$ (resp., $\sigma(P_{0})=[0,\infty)$,
$\;\sigma(P_{0})=\mathop{{\rm I}\kern-1.99997pt{\rm R}}\nolimits$).

###  4.1 Zero mass irreducible triplets

In [16] the irreducible triplets of ${\mathcal{I}}^{+}_{0}$ and
${\mathcal{I}}^{-}_{0}$ with $U$ irreducible, and of ${\mathcal{I}}^{-+}_{0}$
with $U^{+}$ irreducible are completely identified. The results are collected
by the following statement. Theorem 4.1. Modulo unitary isomorphisms, there is
only one triplet
$(U,{{}_{\triangleleft}\kern-2.0pt\texttt{S}},{{}^{\triangleleft}\kern-1.5pt\texttt{T}})$
in ${\mathcal{I}}^{+}_{0}$ and in ${\mathcal{I}}^{-}_{0}$ with $U$
irreducible, whose Hilbert space is ${\mathcal{H}}=L_{2}(\mathop{{\rm
I}\kern-1.99997pt{\rm R}}\nolimits^{3},d\nu)$, and

$(P_{j}\psi)({\bf p})=p_{j}\psi({\bf p})$, $P_{0}\psi({\bf p})=\pm
p_{0}\psi({\bf p})$, $J_{j}={\textsf{J}}_{j}^{(0)}$,
$K_{j}=\pm{\textsf{K}}_{j}^{(0)}$,
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}=\Upsilon$,
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}={\mathcal{K}}\Upsilon$. If
$(U,{{}_{\triangleleft}\kern-2.0pt\texttt{S}},{{}^{\triangleleft}\kern-1.5pt\texttt{T}})$
is an irreducible triplet of ${\mathcal{I}}^{-+}_{0}$ with $U^{+}$
irreducible, then $m\in{{\rm Z}\kern-2.79999pt{\rm Z}}$ exists such that
${\mathcal{H}}=L_{2}(\mathop{{\rm I}\kern-1.99997pt{\rm
R}}\nolimits^{3},d\nu)\oplus L_{2}(\mathop{{\rm I}\kern-1.99997pt{\rm
R}}\nolimits^{3},d\nu)$ and

$P_{0}=\left[\begin{array}[]{cc}p_{0}&0\cr 0&-p_{0}\end{array}\right]$,
$P_{j}=\left[\begin{array}[]{cc}p_{j}&0\cr 0&p_{j}\end{array}\right]$,

$J_{j}=\left[\begin{array}[]{cc}{\textsf{J}}_{j}^{(0)}+\textsf{j}_{j}&0\cr
0&{\textsf{J}}_{j}^{(0)}-\textsf{j}_{j}\end{array}\right]$,
$K_{j}=\left[\begin{array}[]{cc}{\textsf{K}}_{j}^{(0)}+\textsf{k}_{j}&0\cr
0&-{\textsf{K}}_{j}^{(0)}+\textsf{k}_{j}\end{array}\right]$,

where $\textsf{j}_{1}=\frac{m}{2}\frac{p_{1}p_{0}}{p_{1}^{2}+p_{2}^{2}}$,
$\textsf{j}_{2}=\frac{m}{2}\frac{p_{2}p_{0}}{p_{1}^{2}+p_{2}^{2}}$,
$\textsf{j}_{3}=0$,
$\textsf{k}_{1}=-\frac{m}{2}\frac{p_{2}p_{3}}{p_{1}^{2}+p_{2}^{2}}$,
$\textsf{k}_{2}=\frac{m}{2}\frac{p_{3}p_{1}}{p_{1}^{2}+p_{2}^{2}}$,
$\textsf{k}_{3}=0$. With $m=0$ there are six triplets, each characterized by a
different pair
$({{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{n},{{}_{\triangleleft}\kern-2.0pt\texttt{S}}_{n})$,
$n=1,2,...,6$. For every $m\neq 0$ in ${\mathcal{I}}^{-+}_{0}$ there are two
triplets with different pairs
$({{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{a},{{}_{\triangleleft}\kern-2.0pt\texttt{S}}_{a})$
and
$({{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{b},{{}_{\triangleleft}\kern-2.0pt\texttt{S}}_{b})$.
Remark 4.1. For the zero mass case the helicity operator
$\hat{\lambda}=\frac{{\bf J}\cdot{\bf P}}{p_{0}}$ plays an important role.
Theorem 4.1 and (5) imply [16] that $\hat{\lambda}=0$ for the triplets in
${\mathcal{I}}_{0}^{\pm}$.

Using theorem 4.1 we see that $\hat{\lambda}=-\frac{m}{2}$ for every triplet
of ${\mathcal{I}}_{0}^{-+}$ with $U^{+}$ irreducible.

###  4.2 The theories of elementary free particle with zero mass

The possible theories of elementary free particle with zero mass can now be
identified by selecting the triplets that admit a unique position operator.
One conclusion shared by the past approaches states that no position operator
exists for massless particles with non-zero helicity. Yet, the theoretical
structures where such non-existence is proven [7][9] are triplets where
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ is unitary and
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ is anti-unitary. The present
approach highlights that this is a serious shortcoming, because according to
theorem 3.1 these structures must be triplets in ${\mathcal{I}}_{0}^{+}$ or
${\mathcal{I}}_{0}^{-}$. But according to section 4.1 irreducible triplets
with non-zero helicity can exist only in ${\mathcal{I}}_{0}^{-+}$. Therefore,
these proofs do not apply.

In fact our approach proves the following theorems [16]. Theorem 4.2. If
$\hat{\lambda}\neq 0$, then in every triplet of ${\mathcal{I}}_{0}$ there is
no three-operator satisfying $({\it Q}.1)$ and (4.i), (4.ii). Theorem 4.3. For
the triplet of $\;{\mathcal{I}}_{0}^{+}$ or of $\;{\mathcal{I}}_{0}^{-}$, with
$U$ irreducible, there is only one three-operator satisfying $({\it Q}.1)$ and
$(4)$, namely Newton-Wigner operator ${\bf Q}={\bf F}$. Since the search for a
position operator must be restricted to triplets with $\hat{\lambda}=0$, in
${\mathcal{I}}_{0}^{-+}$ only triplets with $m=0$ have to be checked. Theorem
4.4. The triplets of ${\mathcal{I}}_{0}^{-+}$ with a three-operator satisfying
$({\it Q}.1)$ and $(4)$ are three of the six triplets with $m=0$ of theorem
4.1, characterized by
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{1}=\left[\begin{array}[]{cc}0&1\cr
1&0\end{array}\right]$,
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}_{1}={\mathcal{K}}\left[\begin{array}[]{cc}0&1\cr
1&0\end{array}\right]$, by
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{2}={{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{1}$,${{}_{\triangleleft}\kern-2.0pt\texttt{S}}_{2}=\Upsilon\left[\begin{array}[]{cc}1&0\cr
0&-1\end{array}\right]$ and by
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}_{3}={\mathcal{K}\Upsilon}\left[\begin{array}[]{cc}0&1\\\
1&0\end{array}\right]$;
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}_{3}=\left[\begin{array}[]{cc}0&1\\\
-1&0\end{array}\right]{\mathcal{K}}$.

In all the three theories ${\bf Q}=\left[\begin{array}[]{cc}{\bf F}&0\cr
0&{\bf F}\end{array}\right]$.

###  4.3 Conclusions for the zero mass case

The current literature in fact restricts the search for theories of massless
elementary free particle to triplets with
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ anti-unitary and
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ unitary, i.e. to triplets of
${\mathcal{I}}_{0}^{+}$ and ${\mathcal{I}}_{0}^{-}$. Our approach proves that
consistent theories can be developed also if
${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ is unitary or
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ is anti-unitary. As a consequence,
the class of possible theories extends to include a subclass of
${\mathcal{I}}_{0}^{-+}$.

Furthermore, the non-existence proofs of a position operator for non zero
helicity massless particles extends to the larger class of possible theories,
because the operators ${{}^{\triangleleft}\kern-1.5pt\texttt{T}}$ and
${{}_{\triangleleft}\kern-2.0pt\texttt{S}}$ play no role in the new theorem
4.2.

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