Journal of Modern Physics, 2013, 4, 950-962
http://dx.doi.org/10.4236/jmp.2013.47128 Published Online July 2013 (http://www.scirp.org/journal/jmp)
Noneuclidean Tessellations and Their Relation to Regge
Trajectories
B. H. Lavenda
Università degli Studi, Camerino, Italy
Received January 14, 2013; revised February 17, 2013; accepted March 14, 2013
Copyright © 2013 B. H. Lavenda. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
The coefficients in the confluent hypergeometric equation specify the Regge trajectories and the degeneracy of the an-
gular momentum states. Bound states are associated with real angular momenta while resonances are characterized by
complex angular momenta. With a centrifugal potential, the half-plane is tessellated by crescents. The addition of an
electrostatic potential converts it into a hydrogen atom, and the crescents into triangles which may have complex con-
jugate angles; the angle through which a rotation takes place is accompanied by a stretching. Rather than studying the
properties of the wave functions themselves, we study their symmetry groups. A complex angle indicates that the group
contains loxodromic elements. Since the domain of such groups is not the disc, hyperbolic plane geometry cannot be
used. Rather, the theory of the isometric circle is adapted since it treats all groups symmetrically. The pairing of circles
and their inverses is likened to pairing particles with their antiparticles which then go on to produce nested circles, or a
proliferation of particles. A corollary to Laguerre’s theorem, which states that the euclidean angle is represented by a
pure imaginary projective invariant, represents the imaginary angle in the form of a real projective invariant.
Keywords: Tessellations; Reggie Trajectories
1. Introduction
Poincaré discovered his conformal models of hyperbolic
geometry in an attempt to understand whether there were
solutions to the hypergeometric equation of higher peri-
odicities than the then known circular and elliptic func-
tions (of genus 0 and 1, respectively). The names, Fuch-
sian and Kleinian functions, which he coined, belong to a
certain class of automorphic functions that live on tiles
that tessellate the half-plane or disc, depending on which
model is chosen.
The indicial equation is a solution to a differential equ-
ation in the neighborhood of a singular point. An equa-
tion of second-order can have at most three branch points,
which are conveniently taken to be 0, 1, and ∞. The use
of a matrix to describe how an algebraic function is
branched had been introduced by Hermite, but it was
Riemann who first considered products of such matrices.
Frobenius showed that the hypergeometric equation is
completely determined by its exponents at its singular
point. If the only effect of analytically continuing two
solutions around a singular point is to multipy them by a
constant, then the differences in the exponents must all
be integers, without the solution containing a logarithmic
term. In other words, the matrices are rotations about
each of the singular points where performing a complete
circuit multiplies the solution by a constant factor. We
will generalize these rotation matrices through real an-
gles to complex ones, and in so doing elliptic transforma-
tions will become loxodromic ones.
The monodromy matrices of Riemann are generators
of a group, and are either homothetic (magnification) or
rotation. These are related to the hyperbolic and elliptic
geometries, respectively, both of which preserve the unit
circle. According to the Riemann mapping theorem, any
arbitrary region bounded by a closed curve can be map-
ped in a one-to-one fashion onto the interior of a unit disc
by an analytic function. Riemann arrived at his theorem
as an intuitive conjecture in 1852, and it is hardly com-
prehensible that it took almost fifty years to prove it.
According to Schwarz, the only biunique analytic map-
pings of the interior of the unit disc onto itself is a linear,
fractional (Möbius) transform of the hyperbolic or ellip-
tic type; that is, magnification or rotation.
The relevant equation for quantum theory is the con-
fluent hypergeometric equation, where two the branch
points of the hypergeometric equation merge at ∞ to be-
C
opyright © 2013 SciRes. JMP

B. H. LAVENDA 951
come an essential singularity. The origin is the regular
singularity. The two parameters of the equation deter-
mine the Regge trajectories, and the degeneracy of the
angular momentum states. The Regge trajectories express
the angular momentum in terms of the energy, and if the
potential is real, or the energy is greater than the potential
energy, the angular momentum will be real and discrete.
The parameter that determines the trajectories discrimi-
nates between quantized motion of bound states and un-
stable resonance states. In the former case it is negative,
and identified as the radial quantum number, whereas in
the latter case it can be associated with a complex angle.
In the case of the nonrelativistic coulomb interaction, the
Bohr formula for the energy levels of the hydrogen atom
result when the parameter is set equal to a negative inte-
ger, which is also the index of the Laguerre polynomials
that represent the radial component of the wave function.
All this can be obtained without the usual procedure of
expanding the radial wave function in a series and ter-
minating it at a certain point to obtain a quantum condi-
tion.
Rather, if the energy becomes positive, the potential
complex, or the potential energy greater than the total
energy, quantization does not occur, and the parameter
represents an angular point whose homologue is branch
point. The new point is the generalization of the angular
point to complex values. This is not unlike the generali-
zation of the scattering amplitude to make it a function of
the angular momentum. In order to make the scattering
amplitude a function of the angular momentum, it had to
be made complex and continuous so that one could take
advantage of Poincaré’s theorem which says that if a
parameter in a differential equation, the angular momen-
tum, or the wavenumber, appears only in the analytic
function in some domain of the parameter, and if some
other domain a solution of the equation is defined by a
boundary condition that is independent of the parameter,
then this solution is analytic in the parameter in the do-
main formed from the intersection of the two domains. In
other words, Poincaré had shown that, under suitable
conditions, the smooth solution to the differential equa-
tion could be made an analytic function of the parameters
of that equation by allowing them to become complex
and continuous, instead of real and discrete.
The reason for naming the trajectories after Regge [1]
was that he brought Poincaré’s theorem to the attention
of high energy physicists. Instead of confining his atten-
tion to integer angular momenta, Regge transformed the
scattering amplitude so that it became a function of the,
continuous, angular momenta. In order to do so, he had
to allow it to become complex. Regge’s idea was not new,
it had already been used by Poincaré himself, and
Nicholson in 1910, to describe the bending of electro-
magnetic waves by a sphere. Sommerfeld and Watson
used it to describe the propagation of radio waves on the
surface of the earth, and their scattering from various
potentials. It has become known as the Sommerfeld-
Watson representation.
The new, unphysical, regions provided proving grounds
for speculative high energy physics. The passage from a
real to a complex parameter is not nearly as radical as
that from a discrete to a continuous one, or from a posi-
tive to a negative one. How does one define negative
angular momenta? The angular momentum is represented
in the equation for the energy as a centrifugal repulsion.
At a constant attractive potential, the only way the en-
ergy could be made more negative, thereby allowing
more bound states to be formed, is to convert a centrifu-
gal repulsion into a centripetal attraction by allowing the
angular momentum to become negative. The limit occurs
where the two indicial solutions to the Schrödinger equa-
tion with a centrifugal term coincide. One is called the
regular solution because it goes to zero at the origin,
while the other is the irregular solution because it blows
up there. Conventionally, the latter solution is rejected
because any admixture of the two would not lead to a
unique solution. However this is incorrect because both
indicial exponents determine how the ratio of the solu-
tions, which is an automorphic function, transform.
An automorphic function is a periodic function under
the group of linear (fractional) substitutions. When Po-
incaré came on the scene in 1880 the only two periodic
functions that were known were the trigonometrical and
elliptical functions. By cutting and pasting edges of the
fundamental region together one could get solid figures
with different amount of holes, or genus. Trigonometric
functions had no holes, elliptic functions, one hole be-
longing to a torus, and Poincaré wondered if there were
automorphic functions with a greater number of holes.
Any given point of the fundamental region would be
transformed into the same point in an adjacent funda-
mental region by the linear fractional transformation. It
would not connect points in the same fundamental region,
for, otherwise, it would not be “fundamental”.
For instance, if the angular momentum is negative and
in the interval 1
0, 2


, the plane would be tessellated




by crescents formed from the intersection of nonconcen-
tric circles whose angle would be the degeneracy of
states. It is precisely in the unphysical region that the
greater than unity cosine has become a hyperbolic cosine
with a complex angle. The crescent of the plane with a
given angle will be successively transformed by the
fractional linear transformation ultimately returning to
itself. Thus, the entire plane is divided into portions equal
in number of the periodic order of the substitution.
When two particle collide there is a scattering angle,

, whose cosine resides between and 11

in the

B. H. LAVENDA 952
physical region. However, by allowing cos

to go to
either plus or minus infinity, enables one to consider in-
finite momentum transfer. Such large momentum transfer
occur over extremely small distances. Thus, large cos

is adapted to the study of strongly scattered waves that
can bind and resonate.
Poles can be expanded in a bilinear series of the pro-
ducts of Legendre functions of the first and second kinds.
Only when the pole is within a given ellipse will the
series converge. The ellipse is determined by the trace of
the monodromy matrix which is a hyperbolic cosine with
a complex argument. The hyperbolic cosine of the real
component is the semi-major axis of the ellipse, while
the imaginary component is the eccentric angle of the
ellipse. In the case of coulomb scattering the real com-
ponent is the ratio of the charges to the velocity of the
incoming particle so that the ellipse will be larger the
smaller the velocity of the incoming particle. This is
referred to as the classical region. As the velocity in-
creases we are transformed into the relativistic region
with a decrease in the size of the ellipse.
What is a complex angle? In optics, complex angles
arise when a refractive wave does not penetrate into the
second medium, but rather, propagates parallel to the
surface. The system is then said to suffer total internal
reflection. There is no energy flow across the surface,
and at the angle of incidence there is total reflection. For
angles of incidence greater than the critical value, the
angle of reflection becomes complex with a pure imagi-
nary cosine meaning the wave is attenuated exponentially
beyond the interface.
The three poles of the second order differential equ-
ation are associated with the thresholds of particle crea-
tion in high energy physics [2]. Their residues are given
by the angles of a triangle which tessellate the complex
plane. The angles themselves may be complex like the
argument of the hyperbolic cosine above. In order to
tessellate the plane it must reproduce itself by rotating
about any given axis. When the angles become complex,
it will not only rotate the sides of the triangle but will
also deform them. The fact that any two angles are
complex conjugates, related to source and sink, will
render the sum of the angles real but, may not be equal to
.

For Regge trajectories these angles represent the com-
plex angular momentum. Below threshold the imaginary
component of the angle momentum vanishes for there is
no state to decay into so the resonance width is zero.
Regge gave an interesting interpretation to the imaginary
component of the angular momentum. Just as the longer
the time the smaller the uncertainty in energy, so that
long time uncertainty is related to small resonance widths,
imaginary angular momentum is related to change in
angle through which the particle orbits during the course
of a resonance. For extremely long resonances, the angle
of orbit is large until it becomes permanent in a bound
state.
Triangle functions which tessellate the complex plane
are described by automorphic functions which represent
solid figures. The automorphic functions are the inverses
of the quotients of the two independent solutions to a
second order differential equation. These quotients can
be moved around the complex plane by linear fractional
transformations which delineate the fundamental regions.
Schwarz showed how these automorphic functions map
one complex plane onto another, just like the hyperbolic
cosine maps ellipses in one plane onto circles in another.
The angles are either the interior, or exterior, angles of
the triangle which is the fundamental region. In the case
of large, but real, cos

, the automorphic function is
none other than the expression for the Legendre function
of the second kind at a large value of its order, the
angular momentum [3]. It tessellates the surface of a
sphere with triangles whose bases lie on the equator of
the sphere, each angle being radians, and one vertex
at the north pole whose angle is proportional to the diffe-
rence between the order of the Legendre function and the
angular momentum of the Regge trajectory. The solid is
a double pyramid, or a dihedron, whose triangles have
sums greater than


radians, and, therefore, belong to
elliptic geometry.
In fact, this automorphic function has been proposed
as a partial wave scattering amplitude. Another proposal
was made by Veneziano [4] who showed that the Euler
beta integral satisfies the duality principle of high energy
physical where the scattering amplitude remains the same
under the exchange of total energy and momentum
transfer. Experimentally, this is achieved by replacing the
particle with its anti-particle. It so happens that the beta
integral is the automorphic function of Schwarz for tri-
angle tessellations. The angles are the Regge trajectories
which become complex above threshold. In fact, a beta
integral with real arguments could not represent a com-
plex scattering amplitude for it would be physically
measureable being the distance between the angles in the
triangle. The Veneziano model, which has served as the
impetus of string theories, was found to be wanting in the
hard sphere limit because it did not reflect the granular,
or parton-like, behavior observed in deep inelastic scat-
tering experiments.
What is the use of automorphic functions in high en-
ergy particle physics? First, it provides restrictions on the
nature of the complex Regge trajectories and on the na-
ture of the potentials. The potentials must be real for
bound states, complex, or imaginary, for resonances.
Second, the possibility of their being a complementarity
between continuous groups in quantized systems and
discrete groups with a continuous range of non-quantized

B. H. LAVENDA 953
parameters for bound states and resonances. Third, the
spectrum of resonances that lie along a Regge trajectory
is likened to the nesting and proliferation of circles when
the number of generators is increased.
2. Nonrelativistic Coulomb Interaction
The confluent hypergeometric equation arises from the
confluence of two singularities in Riemann’s hyper-
geometric equation leaving the regular and irregular sin-
gularities at 0 and ∞. It can therefore describe an infi-
nite-range potential like the coulomb potential for which
it is given by1
2
2
dd
10
d d
ca
rrr r



 



21,c
1,ai
. (1)
The parameters for the coulomb interaction are
(2)
and




(3)
where is the total angular momentum,
22
Z
Ze E

  is the coulomb parameter which is
negative if the charges
Z
e and
Z
e

E
1cm 
an


n
0E
are opposite, and
is the energy of the incoming particle in units
.
The parameter a determines the nature of the tra-
jectories. If , the system has bound states where
is the radial quantum number, and the total energy,
. This specifies the Kummer function

as a
Laguerre polynomial of index . Specifying the Regge
trajectory (3) avoids the introduction of a series ex-
pansion in the Schrödinger equation, and imposing a cut-
off. When
n
0


, the solution to Equation (1) reduces to
a product of an exponential function and a hyperbolic
Bessel function.
The confluent hypergeometric Equation, (1), can be
easily converted into the Schrödinger equation,
22
22
d11 41 0,
4 drr


 



ir (4)
by the substitution

11d
2
ecr r




21
0r
1,
. , where


Alternatively, if we didn’t know Equation (1), we could
transform (4) into it by inverting the substitution. The
indicial equation as is
A
rBr 



(5)
where the constant
A
is conventionally set equal to
zero in order for

not to diverge at the origin. How-
ever, at 1
2


r
both terms give the same dependence
upon . It is precisely at 1
2

B
 where the regular,
, and irregular,
A
, solutions in Equation (5), coincide.
There is no reason to constrain the angular momentum to
positive integral or semi-integral values since we are
considering “elementary” and “composite” particles,
which may be stable or unstable.
We may look for a solution to Equation (1) in the form
of a Laplace transform [6],
 
2
1
ed,
r
r



  

 
(6)
for the (normalized) wavenumber, . Introducing it into
(1) results in
 
 
2
1
2
1
2
1
d
1ed
d
1e
d1d,
d
r
r
ca
ca








 



 


 









after an integration by parts has been performed. If the
limits 1

and 2

can be chosen so as to make the
integrated part to vanish, then




will satisfy

d10.
d
ca 



 



1
11,
ca
a
C  



C
 
1
11ed,
ca
ar
rC

The solution to this first order equation is
(7)
where is a constant of integration. In view of the
Laplace transform, (6), we find
 





(8)

where, if the contour is not closed the integrand in (8)
is required to have the same value at the endpoints.
In contrast to the original confluent hypergeometric
Equation (1), which has branch points at 0 and

, we
have added an additional branch point at 1 by con-
sidering the wave number. This branch point may be
thought of as placing a bound,
1This is formerly identical with Equation (26) on page 52 in Ref. [5].
However, the definition of

2,pE
p on the momentum by the square root of twice the
total energy, like the maximum momentum of a Fermi
gas of elementary particles at absolute zero [8].

there is real. The same expression can
be found in Ref. [6]. But then the quantization condition an


is
complex [6, Equation (27) p. 156]. Since
E
2 is imaginary for
bounded states, and not the velocity , the quantization condition is
real. Rather, in Ref. [7] the condition is taken as the condi-
tion of the poles of the -matrix, giving the position of the
v
an


S n

Re
gg
e
p
ole.

B. H. LAVENDA 954
The difference between (2) and (3) determines the
second angle as
 
1,i a  
y y
ca (9)
which is the complex conjugate of (3). We will appreci-
ate that attractive and repulsive coulomb potentials al-
ways appear as complex conjugates, or, equivalently, for
every source there is a sink.
If 1 and 2 are any two particular solutions to the
hypergeometric equation, then the Wronskian is given by
[9]

1
1,
cab
rr


2
12
21
dd
dd
c
yy
yyC
rr


where C is a constant. Dividing both sides by 2
y
, it be-
comes the derivative of the ratio 12 of the two
particular solutions. Now, any other two solutions, say
and can be expressed as linear combinations of
and , viz.,
wyy
1
y2 y
1
y2 y
112
yyy


212
yyy



so that the quotient of the new solutions, 12 is
related to the quotient of the old, , by a linear
fractional transformation
wyy


w
.
w
ww







(10)
When the quotient of the solutions is inverted, we get a
function automorphic with respect to a certain group of
the linear fractional transformations. For the hyper-
geometric equation, the group of automorphisms are tri-
angular tessellations of the unit disc.
Our interest will be focused on the momentum space at
in (6). Since the hypergeometric equation with the
coefficient ,
0r
0b
2
2
11
d





22
dd
0,
1d
YY
aa







(11)
has one solution,

1
11 d,
a
t t


20
,, a Yac Ct




 (12)
which is an incomplete beta function,

,;Baa

, if the
constant of integration is set equal to unity. The second
solution
12
,YwY

1
w 

(13)
is given by an automorphic function .
Dividing the Wronskian,

1
11,
ca



2
2
Y

12
21
dd
dd
a
YY
YY C



1
1
2
2
1,
a
a
wC Y





by , it becomes the derivative of the ratio, viz.,
which has the Schwarzian derivative,



2
22
22
3
,: 2
11
11 ,
1 221
ww
www
aa
aa



 







 

(14)
where the prime now stands for the derivative with
respect to

.
With the transformation,
 ln1 ln 1
e,
ca
YY
 

0,YIY

(11) can be converted into


 (15)


2,Iw
where

, with the Schwarzian derivative
given by (14). The Schwarzian derivative has a long and
glorius history dating back to Lagrange’s investigations
on stereographic projection used in map making [10].
The third angle can be read off from the Schwarzian,
(14), and is


1,c

 (16)
which is the original angle of the crescent, having branch
points at 0 and

. This is due to the centrifugal poten-
tial in the Schrödinger equation. The coulomb potential
introduces the complex conjugate angles, and
aa,
which make the interaction independent of whether it is
attractive or repulsive since they appear symmetrically. If
we adhere to the triangle representation, the requirement
that the angles be less than limits  to the closed 
interval 1,0
2


. In this interval, centrifugal repulsion






2
1r becomes centrifugal “attraction”.
For 1
2, the sum of the angles of the triangle,


1
2i






, 1
2i





, and 0 is . [11] The analytic 


function

,;Baa

given by (12) for maps the
upper half-plane
1C


0 

a
onto the interior of a half-
strip formed by the two base angles and a

corre-
sponding to the points 01
  and in the


-
plane. The distance between the two vertices in the -
plane is
B


11
1
0
11
1d22
cosh
a
a
ttt ii




 

 





(17)

B. H. LAVENDA 955
In general, the amplitude will be given by the com-
plete Beta function




11
.
21
ii
,;1=Baa






(18)
otentials. In the attractive case, there
is an infinite number of Reggie poles determin
poles in the numerator of (18) [7], i.e.,
The beta function is symmetric with respect to attract-
tive and repulsive p
ed by the
2
of the numerator
8) are equasite so
alyticity of thde. In the
tial, with Eation (18) tells us there there
pole
esp n
t
po
1.
2
ZZ en
E

  (19)
These are bound states with 0E. In the case of
resonances, 0E, the imaginary parts
of (1
an
poten
wi
l and oppo
e amplitu
0, Equ
to
as to preserve the
case of a repulsive
ll again be an infinity of Regge s. However, these
will not corrod to bound states or resonances be-
cause the Regge poles are restricted to the left half of the
 plane. This information is contained in the amplitude,
and it is equivalentwo S-matrix elements, one being
the inverse of the other [7].
Parenthetically, we would like to point out that the
Regge pole behavior for an infinite-range potential is
quite different than that for short range ones. A Regge
trajectory for a short-range tential would have the form
 
1,E EE

  (20)
where the prime stands for differentiation. The intercept
of 1 is called for by the form of the angular momen-
tum term

1 in the Schrödinger equation. The fact
that the cross-sections do not vanish asym
the energy, but increase slowly with it is at
r, a
gula
ptotically with
tributed to an
exchange of a Reggeon whose intercept is +1 [12]. How-
eve positive intercept cannot be interpreted as origina-
ting in the anr momentum. This is substantiated by
the fact that only negative intercepts give rise to the con-
servation of angular momentum in hyperbolic space [13].
In Equation (20) there would be no violent jump in the
trajectories as the energy passes through zero. In the case
0E the poles are complex, but there is no violation of
analyticity since they are complex conjugates. In the
asymptotic case of large angular momentum the ampli-
tude (18) will be modulated by oscillations, viz.,

2
1
,;1 e.
i
i
Baa i




 



 

The Schwarz-Christoffel transform remains valid even
when one of the vertices of the triangle coincides with
the point at infinity. The lengths of the sides from either
vertex to the vertex at infinity are infinite, so that
 

11
22
0
,;1 d
ii
i
Baattt




 



(21)
3. Generators from Monodromy Relations
If
2
0
lim ,I

0
1d
1
tt
ttt



 

will map the upper half-plane

0   onto the interior
of the “half-strip” shown in Figure 1.

(22)


the indicial equation for the branch point 0 is


nn  10.


The two unequal roots, 1 n and 2 n, are the exponents
e integrals of the equation
1
1
n
Y 
of th



so that ratio of the solutions is
2
2,
n
Y  
12
12
nn
wYY 



quadratic form is
2
14 ,a  
we have
 (23)
Since the discriminant of the
2
2
11
4
a
I




in view of (22).
en as
1i
w



The ratio of the two solutions, (23), can be writt


 (24)
ing to Riemann, the exponents of the indicial
equation,

Accord

1,2
11,
2
na (25)
Figure 1. The half-strip obtained by the conformal mapping
Equation (21).

B. H. LAVENDA 956
“completely determine the periodicity of the function” [14],
in reference to the solutions of the hypergeometric equa-
tion. Equation (25) shows that the difference in roots of
the indicial equation, or exponents as they are commonly
referred to, is the angle at the singular point. The mono-
dromy relations imply that the roots to the indicial equ-
ations are rational numbers, which, in turn imply elliptic
generators. Here, they are generalized to complex num-
bers so as to allow for loxodromic generators, and take as
a definition of monodromy as the invariance of auto-
morphic functions, or functions inverse to the quotint of
two independent solutions of a differential equation,
 
e
under a certain group of transformations.
A circuit around in the positive direction that
returns 1 Y as 1
2
1
ein Y
, while it returns 2
Y as 2
2
2
ein Y
,
is performed by
1
2
2
2
e0
.
0e
in
in






The monodromy theorem asserts that any global analy-
tic function can be continued along all curves in a simply
connected region that determines a single-valued analytic
function on every sheet of the branch points, one sheet
per branch point. The monodromy matrix about


,
2
2
e0
,
0e
i
i








 (26)

has unittermind trace denant, a


2cos2,
so that it is an elliptic transformatio Rather, around
0  , the monodromy matrix,
Tr
n.


e0
,
0e
ii
ii












 (27)
also has unit determinant, but trace,


Tr 2cos ,i (28)
so that it is a loxodromic generator
around 1
 , where

,
0e
ii  



 (29)
ose
 
. This is also true

e0
ii   




wh trace is


cos .i   (30)
The product of any two matrices, or thei
yield the third, or its inverse. Stated slightly differently, a
points in a positive
(anti-clockwise) direction will give a
third branch point in the negative (clo
atrices are abelian since they have the same
fixed points, 0 and
Tr 2
r inverses, will
circuit around two of the branch
circuit around the
ckwise) direction,
since  is the unit matrix.
The m

.  and  are loxodromic be-
cause their traces, (28) and (30), are com
Loxodromic transformations do not leave the disc in-
sfer the inside of the disc to the outside
The generators, (27), (29), and (26
points 0 and ∞ in common. Conjugation with the Cayley
m
plex and 2.
variant: they tran
of its inverse.
), have the fixed
apping,

,
zi
zzi

  (31)
carries these fixed points to 1 and 1, respectively,
while se, 1 its inver

, takes them to i and i

, re-
sp
ca

 
1
cosh si
sinh cosh
ii
ii i













s to
ww
carries points in the interior of w I to the exterior of
w
ectively. For instance, the conjugate generator of (27),
 
 
1
coshsinh ,
sinh cosh
ii
ii




 








(32)
rries the fixed points 0 and ∞ to −1 and 1, respectively,
while those of the inverse conjugation,


nh , i  
(33)
carry the fixed point i and i, respectively.
The loxodromic transformation, (32), pairs the iso-
metric circle, I, with its inverse,  I. That is, 

I. The fixed points of (32), 1 and 1, lie i
w
n w I and

I, respectively. With 0

, (32) is pure stretching,
pushing points from 1, the source, to 1, the sink.

The cyclic group consists of one generator, and apply-
ing it n times gives


 
coshsinh ,
sinh cosh
nnn
nn









 (34)
for 0

. Whereas the isometric circle of (32) is

 
sinhcosh 1,  z

 (35)
the isometric circle of (34) is


sinhcosh 1.nz n   (36)
he isometric circle, (35), has its center at T


coth



,
and radius


11sinRh



. The isomec circle, (36tri),
on the other hand, has its center at

coth n



, and
radius


1sinh
n
Rn . Since

sinh sinhn




  ,
it follows that 1n RR. And since this is true for any
1n, the isometric circles will be neste
another, becoming ever smaller until the limit point is
hed.
xposed
d inside one
reac
The loodromic generator, (33), can be decom

B. H. LAVENDA 957
into a product,



 
 
sinh cosh
coshsinh cos
i
ii i
i









 
sinhcoshsin cosi









sin





cosh sinhii  






of hyp
with the same fi nts. The isometric circles of


erbolic, , and elliptic, , transformations,
xed

poi

s inverse, 1
,

and it
sinhcosh 1,iz  
have their centers on the imaginary axis at

0cothzi


, and radius

1sinh

. Part of the
ental
on for the grted by  [15]. In other
words, it does not detic transformation
Since the isometric circle is defined by

plane exterior to these two circles is the fundam
oup genera
pend on the ellip
regi
.

21,




sinh coshiz
 

and that of its inverse by

 
2
1nh  
2
si cosh
sinh cosh
iz
iz



 

,
,
whatever is inside the isometric circle of 

sinhcosh 1,iz  
i.e., 1

 is outside of its inverse, because 11




,
and vice versa.
Let  and  be the generators whic
ometric circles, w I with its inverse,
h pair off the
is w

I, and v I
, respectandwith v

Iively. The matrices 

will
al to one anot

be externher provided cosh
, and
2 2.
pairs circles
coth

The matrix, (32), has fixed p

oints 1
with centers,

radii, , and the same

sinh

, on the real will be
located in the circle on the negative axis, while the
attracting fixed point +1, which is a sink, will be located
axis. The fixed point 1
in the circle on the positive axis, since

1
ta 1
2
 nh


for whatever value

happenbe.
rtherm the iso
s to
Fuore, let  be associated with-
metric circle
z
I. If Iternal to one an-
ot en
v
her th
and w

I are ex
z
I isn w I [15, p. 53, Thm 12]. For sup-
pose that the circles are not tangent to oneother, th
if p a point outside of, and not on, , the generator
i
an en
is v
I

will carry the point p into, or on, v

I, say p

with a decrease in length, or at least no change in length.
Since p

is w I,  will transforoutside of m it with
a decreaseine in length. Consequently, the combd opera-
tion, , will transform p with a decrease in length,
implying the p is outside of
z
. And since every point
on or outside of v I is also outside of
z
I, the latter
sting in a
neith their
v
I
must be inside the former.
Each time we add a generator, we get a ne
sting of circles w proliferation [16, p. 170].
The isometric circle will contain three nested circles,
z
I, and
z

Ifor the generator 
 
, and another
for

. This is shown in Figure 2.
Ech of the other three discs will also have three
sted discs. Increasing th generator by e, so that
there are now three generators, or “letters”, there will be
three nested discs in each thormer discs, and so on.
Thus, there would be no limit of an elementary particle,
but, rher, particles withn particles within particles
and so on. There may result inhigh energy collisions
additional particles to those of the compound particle
disintegrating into its compon
a
nee on
ofe f
at i
ent parts because there
m
lativ
tio ometric ci
geneand one for its in-
ve
ay be sufficient energy that can be converted into mat-
ter before disintegrating again into other forms of mat-
ter. Moreover, the reistic phenomenon of pair crea-
n may be related to pairs of isrcles, one for
the rator of the transformation,
rse.
4. The Coulomb Phase Shift as a Projective
Invariant
The absolute conic for elliptic geometry is the null conic.
It is defined in projective coordinates by an equation with
real coefficients, but it is composed exclusively of imagi-
nary points. The secant through the points 1
k and 2 k
join the conic at i

and i. The cross ratio is
2If the inequality becomes an equality, it is treaded by the example Ref.
[16] which is the condition that the four circles are tangent to one an-
other. The trace of the commutator is –2 indicating that the two fixed
p
oints have coalesced into one at the point where the circles touch. Both
groups are Fuchsian since their limit points are either on a line or on a
circle.
igure 2. The nesting of circles in the isometric circle and
its inverse.
F

B. H. LAVENDA 958





12
12
21
2
,;,
1tan e,
1tan
i
kk ii
kiik kk
kiikkk
i
i










1221
1221
1
1
ikk
ikk
 

(37)
tan where tan

21

 , and tan
ii
k

, for
er a real conic with imagi-
1
ik and 2 ik , so that their join cuts the ab-
solute at and 1. The cross ratio is now
1, 2i.
Equivalently, we can consid
nary points,
1


 

12
211221
2
,;1,1
11 1
1tan e,
1tan
i
ikik
ikikk kikk
i
i









which is the complex conjugate of (37). Thus, the eucli-
dean angle,
12 1221
11 1ikikk kikk
 (38)
Figure 3. The real conic, tangents, pole P, and polar.
1
1
0
0
tan ,
j
jj

 
 


  




 (42)
which is none other than a generalized Breit-Wigner
expression [20] in the neighborhood of a resonance where
the angular momentum, j, stands in the for resonance
energy, and 2

1 2
, ;
22
kki iikik
ii
 (39)
ca
ar he
(3any valued
quantities. In order to associatent 12 kk with the
logarithm of the cross ratio, a pure imaginary absolute
constant must be chosen [17].
Conjugacy with respect to a polarity general
pendicularity with respect to an inner product thus allow-
ing euclidean geometry to be defined from affine geo-
metry by singling out a polarity. [18] The imaginary
t poi1 and 1hown in Figure
3.
The coulomase shift,
12
11
ln,;,ln1,1,   
n be expressed in terms of a cross ratio, and, hence, is a
projective inviant. T logarithms of the cross ratios,
(37) and 8), are pure imaginary and m

is the width.
It was Laguerre’s great achievement to define eucli-
dean geometry from affine geometry by singling out a
polarity. The projective invariant (42) is a projective in-
variant in that it expresses an euclidean angle directly as
the logarithm of the cross ratio [18]. It seems odd that the
same name, Laguerre, should be associated with both the
orthogonal polynomials when a is a negative integer in
the an
eometry from affine geometry through a projective
invariant.
5. Relativistic Coulomb Interaction
ts
e a segm
izes per-
points, 1 ik and 2 ik , lie on the polar whose pole, P, is
determined by the point of contact of two tangent lines to
the real conic ants , as s
b ph

, which determines pure-
ly electrostatic, or Rutherford, scattering is also a pro-
jective invariant. It is defined by [19]


0
1
2 2
0
!
ee.
!
ji
i
j
iji
iji
 











 (40)
Transposing and taking the logarithm of both sides give

0
0
1
1
0
1ln
2
tanh .
j
j
j
j
ji
iji
i
j

 



 


 












(41)
The coulomb phase shift ius givy
coulomb interaction and the derivation of euclide
g
The relativistic generalization of the nonrelativistic cou-
lomb interaction is given in this section. By specifying
the coefficienin the confluent hypergeometric equation,
we can obtain Dirac’s expression for the energy of the
hydrogen atom from the Klein-Gordon equation instead
of from the Dirac equation.
The coefficients in the confluent hypergeometric equa-
tion for the relativistic coulomb interaction are

2
222
11
22
aiEEm 


 





 (43) 
2
2,c
2
11
22





 










 (44)
s then b

B. H. LAVENDA 959
where
Z

 ,

is the fine structure constant, and
we have reinstated the mass m. If 0a and Em,
we have scattering, and the second angle will be

2
222
11 .
22
caiEEm 


 





 (45)
The two base angles, corresponding to the branch
points at 0 and 1 in the
 -plane, are, again, complex
conjugates, the + and – signs, before the energy term (43)
and (45), correspond, respectively, to an attractive and
repulsive coulomb potential.
The third angle is

2
1
12 2 c

 







 
to
2,




 (46)
and in order it be , the second inequality in for

22
11,
4

   (47)
has to be satisfied. The other inequality is the con
that (46) is real. The upper bound converts a repu
centrifugal

wal quantum number. This avoids the
necessity of looking for a series solution to the radial
wave equation, and imposing a cut-off on the
condition that a be equal to a negative integer implies
a-
Equation (48) gives the exact energy
particle bound by a coulomb potential
dition
lsive
force into an attractive force, as can be seen
in the Klein-Gordon equation, (50), below.
Alternatively, for 0a, and Em, the Regge tra-
jectories for the bound states are given by

1,an n

  (48)
here n is the princip
series. The
that the Kummer function becomes a generalized L
guerre polynomial.
levels of a Dirac
,
1
2
n
Em




2
2
1
11








 
22
1
222 nn





 

 








 2









(49)
This is rather surprising since the confluent hyper-
geometric equation with coefficients, (43) and (44), is
completely equivalent to the Klein-Gordon equation


2

22
2
22
1
d1
0.
d
EmE
r
rr





 


 (50)
4


It is commonly believed that (50) describes a spinless
particle in a coulomb field, and is, therefore, not capable
of describing the hydrogen atom since electrons have
spin 1
2 [21].
Now the indicial equation for (50) about the origin has
exponents
2
2
1,2
11
,
22
n 



 








which reduce to 1
n
 (51)

 and

21n in the non-
relativistic limit [cf. Equation (5)]. This would lead us to
consider a solution to the indicial equation with only the
former exponent. However, with 0, the square root
can become complex for 1
2
 , and the solution would
diverge. A further complication is that the exponents,
(51), become complex for 1
2
  and 0. With a
complex exponents, the solutions near the origin would
at [21] contends that the value of
Z
oscille. Bethe that
w
avelength so as to invalidate the solution
ould be required to make the exponents complex would
correspond to atoms whose radii are several times the
Compton w
ln
e,
ir
r
 
(52)
for small
r
. However true thit would still
make the energy levels, (49), complex, again making it
unacceptable. We now address this in some detail.
In the nonrelativistic limit, the of the two solu-
is may be,
ratio
tions is r

, where   . The conformal
mapping between the z a planes is [22]

21
nd r
1
2cot
eip r




1
2cot , i
zi  (53)
ep r
 

which upon solving for r

becomes
1
ln2cot
ee .
rip
zi
zi







(54)
The circles intersect at i

and i, and p is the dis-
tance from the smaller circle to the origin as shown in
Figure 4. These correspond to the branch points 0r

. Now, 1 cot 


si

1
cothpi ip

 and the and r

right-hand de of (54) is the cross ratio,
 







 2
11
co cot
11 e
i
pz
ii
ii







(55)
11
2 cothcoth
ln
eeip iz
rpizi
pizi






t,
11
cot cot
11
pz ii
ii






B. H. LAVENDA 960
Figure 4. Crescent formed from intersecting circles making
an angle λπ.
e pwher cot

 and cotz

.
Taking the logarithm of both sides of (55) gives

2ln,ir   (56)
where the cross ratio is the distance between the points
cos ,s

cos, sin, 0  with respect to
the circnity, and

1, ,0i in
omplane called circular
points at infinity because they lie on the complexification
of every real circle. Both points satisfy the homogeneous
equation,
2
233
0.xEx (57)
By specifying the line at infinity, 30x, the circular
points then satisfy,
22
12
0,xx (58)


in, 0  and
ular points at infi
plex projective

1, ,0i
. They arethe c
22
12 13
22AxBxCx xDx
if 1
A
B. The involution, (58), is elliptic since it has
maginary ciircular points for its fixed elements. Equation
fines a pai 12
0xix. (58) der of imlaaginary pnes,
In the relativistic case, where 1 Z   the in
2dices,
(51), become
1,2 .
2
1
ni

 (59)
The ratio of the two solutions is 2i r

, which is the
crescent problem, but with an imaginary angle. Rotate
the crescent by 2, and the circles intersect at 1

and
1, which lie on the disc. Again specifying the line at
infinity, x30, the c7), reces to
22
0,xx
onic, (5du
12
for 1
(60)
A
B . Equation (60) is a hy involution,
since it has 1 as
perbolic
its fixed elements, and defines a pair
, 0xx.
eted, here p
of real planes
With the cr
12
scent rotapip, w


, the
ss ratio ocrof the 4 points,

,1,0p
,

,1,0z,


1,1,0,
and


1,1,0 is real and is given by






2
11cot1c 1pz
e



11
2 cothcoth
2ln 1
ee
pz
ir 

1pz1
1 11
11
1111
coth 1
11
oth
pz
pz
 coth 1  









 


where cothp




(61)


 and cothz


. Hence,


ln .ir    (62)
The substitution is hyperbolic s
hich lie
ince it has real fixed
points w on the disc in terms of the homogeneous
coordinates


osh ,sinh,0  and

cosh ,sinh ,0  .
A compariso (56) and (62) shows that whereas in the
fo
c
n
e the lo the
a many-valued quantity, i
the logarithm o
es
e,
i
r
rmer the angle is real whilgarithm ofcross
ratio is pure imaginary, and n
the latter, the angle is imaginary whilef
the cross ratio is real. Thus, in both cas

 (63)
where the condition,




 determines the single-
valued principal value of ln r. Both (55) and (61) ex-
press angles in the form of a projective invariant
the logarithm of both sides of the first equality in (55)
yields
. Taking

11
1lncotcot .
2ri zp


 (64)
ducing (63taking thf both
sides yield
Intro), and e cotangent o

11cotcot1
cot 2cotcot
,
zp
pz


cot








 (65)

which, upon equating arguments, becomes

2.



 (66)
Likewise, taking the logarithm
(61) gives
1
h

of the first equality in
1
lncotcoth .irz p 


 (67)
Introducing (63) and taking the hyperbolic cotangent of
both sides result in

1 cothcoth1
coth coth coth
coth ,
zp
zp













or
(68)
upon equating the arguments give

B. H. LAVENDA 961

.

  (69)
Whereas the Möbius transformation in (65),
,
1
zp
zpz

 (70)
an imaginary
circle, the Möbius transform in (68),
has imaginary fixed points, and is related to
,
zp
z
 (71)
points and is the most general analytic
function that maps the unit circle onto itself
i
1pz
has real fixed
. [23] That is,
for ez

, 1z since

e
iiii
zpp
 
   e e1e1.pp pz   
Consequently, the small r solution to the Klein-Gor
equation, (52), not allow r to be interpre
real, radial coordinate. This is in contrast to the usual
interpretation wher
don
ted as a does
eby the factor
1
22
1
2


 





du



 intro-
ces a fixed branch cut along the real axis from
1
2

  to 1
2

Wh. en 1
takes the cal state 0, and there is no consistent
ant in (55) is pure
imaginary, meaning that we are dealing
substitution, while the angle is
jective invariant in (61) is real, meaning that hyperbolic
inally, all r solution to the non-
relativistic Schrödinger equation and show that if r is
real,
2
  this cut over-
physi
solution. [24]
In summary, the projective invari
with an elliptic
real, whereas the pro-
substitutions map the real circle into itself, while the
angle pure imaginary.
Fwe return to the sm

must be clex. If we agrotate the cre
ce
omp ain s-
nt in Figure 4 so that the vertices are at 1 and 1

,
while keeping p real, we have

11
12 cothcoth
2cot 1
ee

2
e.
1
z
ip
ip
z
r


 




i  

z


 (72)
as a Equation (72) shows that in order to interpret r
real, radial coordinate, the angle

must be complex,
except iit at lim
confo
n the lim. In thit the crescent s pa
degenerates into a half circle, and (72) becomes real.
This is in contrast with rmal analysis which holds
that r is complex and the angle

is real and positive.
The corresponding Möbius transform,
,
1
zip
zipz

 (73)
has fixed points at 1 and 1

, which are the vertices
of the crescent. In the limit as p, (73) becomes an
inversion, 1zz


. This, again, mps circles into circles
for which straight lines are regarded as circles that ass
through the point of infinity. In other words, this
transformation associates points in the interior of a uit
circle with poin exterior to it. S
a
p
n
ts o that it would appear
that classical quantum mechanics emerges when confor-
mality disappears, and the independent coordi
comes real as well as the coefficient in the
geometric equation, (1).
nate be-
hyper-
Equation (73) can be written as
11
,
11
zz
zz
K







where the multiplier of the transform,

1i ip
 
2
shows that the transformation
e,
1
Kip 

is elliptic, i.e., 1K

.
The angle is constrained to the interval 02



, and
is determined by theeight ohe center of the smaller
circle of the crescent from the origin of the r plane.
The transformation (72) will therefore map the upper half
of the z plane onto the interior of the crescent in the r
plane with angles
hf t


.
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B. H. Lavenda, Journal of Modern Physics
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[6] K. Gottfried, “Quantum Mechanics,” Vol. 1, Fundamen-
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[14] J. Gray, “Linear Differential Equations and Group Theory
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