# Convergence of quantum electrodynamics on the Poincaré group

V. V. Varlamov Siberian State Industrial University, Kirova 42, Novokuznetsk
654007, Russia

###### Abstract

Extended particles are considered in terms of the fields on the Poincaré
group. Dirac like wave equations for extended particles of any spin are
defined on the various homogeneous spaces of the Poincaré group. Free fields
of the spin 1/2 and 1 (Dirac and Maxwell fields) are considered in detail on
the eight-dimensional homogeneous space, which is equivalent to a direct
product of Minkowski spacetime and two-dimensional complex sphere. It is shown
that a massless spin-1 field, corresponding to a photon field, should be
defined within principal series representations of the Lorentz group.
Interaction between spin-1/2 and spin-1 fields is studied in terms of a
trilinear form. An analogue of the Dyson formula for $S$-matrix is introduced
on the eight-dimensional homogeneous space. It is shown that in this case
elements of the $S$-matrix are defined by convergent integrals.

Keywords: extended particles, fields on the Poincaré group, homogeneous
spaces, wave equations, quantum electrodynamics
PACS numbers: 02.30.Gp, 02.60.Lj, 03.65.Pm, 12.20.-m

##  1 Introduction

It is well-known that the representation of a particle as an idealized point
has been used in physics for a long time. Historically, such a description
follows from classical (celestial) mechanics, where the distances between
explored objects are much greater than their sizes. However, in quantum field
theory a point-like particle description meets with some conceptual
difficulties. For example, ultra-violet divergences follow directly from the
point-like description. Refusing from the point-like idealization, we come to
_extended objects_.

Our consideration based on the concept of generalized wave functions
introduced by Ginzburg and Tamm in 1947 [24], where the wave function depends
both coordinates $x_{\mu}$ and additional internal variables $u_{\mu}$ which
describe spin of the particle, $\mu=0,1,2,3$. In 1955, Finkelstein showed [18]
that elementary particles models with internal degrees of freedom can be
described on manifolds larger then Minkowski spacetime (homogeneous spaces of
the Poincaré group). The quantum field theories on the Poincaré group were
discussed in the papers [38, 35, 12, 5, 36, 61, 39, 16, 25, 29]. A
consideration of the field models on the homogeneous spaces leads naturally to
a generalization of the concept of wave function (fields on the Poincaré
group). The general form of these fields relates closely with the structure of
the Lorentz and Poincaré group representations [20, 43, 11, 25] and admits the
following factorization $f(x,{\bf z})=\phi^{n}({\bf z})\psi_{n}(x)$, where
$x\in T_{4}$ and $\phi^{n}({\bf z})$ form a basis in the representation space
of the Lorentz group. At this point, four parameters $x^{\mu}$ correspond to
position of the point-like object, whereas remaining six parameters ${\bf
z}\in\operatorname{{\bf Spin}}_{+}(1,3)$ define orientation in quantum
description of orientable (extended) object [26, 27] (see also [33]). It is
obvious that the point-like object has no orientation, therefore, orientation
is an intrinsic property of the extended object. Taking it into account, we
come to consideration of physical quantity as an extended object, the
generalized wave function of which is described by the field

$\boldsymbol{\psi}(\boldsymbol{\alpha})\equiv\langle
x,g\,|\boldsymbol{\psi}\rangle$

on the homogeneous space of some orthogonal group $\operatorname{SO}(p,q)$,
where $x\in T_{n}$ (position) and $g\in\operatorname{{\bf Spin}}_{+}(p,q)$
(orientation), $n=p+q$. So, in [54, 55] Segal and Zhou proved convergence of
quantum field theory, in particular, quantum electrodynamics, on the
homogeneous space $R^{1}\times S^{3}$ of the conformal group
$\operatorname{SO}(2,4)$, where $S^{3}$ is the three-dimensional real sphere.

On the other hand, measurements in quantum field theory lead to extended
objects. As is known, loop divergences emerging in the Green functions in
quantum field theory originate from correspondence of the Green functions to
_unmeasurable_ (and hence unphysical) point-like quantities. This is because
no physical quantity can be measured in a point, but in a region, the size of
which (or ‘diameter’111For example, this diameter can be defined by a
characteristic rest-frame kernel dimension $\sim\;\hbar/mc$ (Compton
wavelength). The magnitude of the ‘kernel’ radius of extended particles was
first proposed by de Broglie [15], see also [60, 52]. of the extended object)
is constrained by the resolution of measuring equipment [2]. Ordinary quantum
field theory defines the field function $\phi(x)$ as a scalar product of the
state vector of the system and the state vector corresponding to the
localization at the point:

$\phi(x)\equiv\langle x|\phi(x)\rangle.$

This field describes the point-like object. On the other hand, _resolution-
dependent fields_

$\phi_{a}(x)\equiv\langle x,a;g|\phi(x)\rangle,$

introduced in the work [2], describe in essence extended objects, here
$\langle x,a;g|$ is the bra-vector corresponding to localization of the
measuring device around the point with the spatial resolution $a$, $g$ labels
the apparatus function of the equipment (an aperture). It is easy to see that
the resolution-dependent fields have the same mathematical structure as the
fields on the Poincaré group (more generally, _fields on the groups_).

The present paper is organized as follows. Basic facts concerning fields on
the Poincaré group are considered in the section 2. Dirac-like wave equations
for extended objects are presented in the section 3. Solutions for the fields
$\boldsymbol{\psi}(\boldsymbol{\alpha})=\langle
x,\mathfrak{g}\,|\boldsymbol{\psi}\rangle$ of spin 1/2 and 1 (Dirac and
Maxwell fields) are given in terms of associated hyperspherical functions
defined on the two-dimensional complex sphere, where
$\mathfrak{g}\in\operatorname{{\bf Spin}}_{+}(1,3)$. Interaction between these
fields is introduced in the section 4 in terms of a trilinear form.

##  2 Fields on the Poincaré group

As it mentioned above, Ginzburg and Tamm proposed to consider new internal
continuous variables $u_{\mu}$. It allows one to generalize the Klein-Gordon
wave equation. The Ginzburg-Tamm equation has the form

$\left[\Box-m^{2}+\frac{\beta}{2}M_{\mu\nu}M_{\mu\nu}\right]\Psi(x_{\rho},u_{\gamma})=0,$
(1)

where

$M_{\mu\nu}=-i\left(u_{\mu}\frac{\partial}{\partial
u_{\nu}}-u_{\nu}\frac{\partial}{\partial
u_{\mu}}\right)\quad(\mu,\nu=0,1,2,3)$

are rotation generators of the four-dimensional space, the constant $\beta$ is
analogous to the momentum $J$ in the rotator hamiltonian ${\cal
H}=(2J)^{-1}L_{ik}L_{ik}$, where $L_{ik}$ are generators of the Lorentz group,
$m^{2}$ is a constant. The wave function $\Psi$ depends on coordinates
$x_{\mu}$ of mass centre of the particle and internal variables $u_{\mu}$. The
4-vector $u_{\mu}$ is a 4-vector of relative motion of the structural
component of the particle around its mass centre. The wave function $\Psi$ can
be factored:

$\Psi(x_{\mu},u_{\mu})=\Psi(x_{\mu})\Phi(u_{\mu}).$

Then the Ginzburg-Tamm equation is decomposed in the following two equations:

$(\Box-m^{2}+\lambda_{1}\beta)\Psi(x_{\mu})=0,\quad
L_{1}\Phi(u_{\mu})=\lambda_{1}\Phi(u_{\mu}).$ (2)

At this point, the function $\Phi(u_{\mu})$ is an eigenfunction of the
operator $L_{1}$,

$L_{1}=M_{\mu\nu}M_{\mu\nu}/2,\quad L_{1}\Phi=\lambda_{1}\Phi.$

The first equation from (2) has the following solution:

$\Psi(x_{\mu})=C\exp(-im_{0}t),$

where

$m^{2}_{0}=m^{2}-\lambda_{1}\beta.$

Thus, a mass spectrum, described by the equation (1), is defined by an
eigenvalue of the operator $L_{1}$. The mass operator $L_{1}$ in spherical
coordinates on the one-sheeted hyperboloid $H^{3}$,

$\begin{array}[]{ccl}u_{0}&=&r\sinh\chi;\\\
u_{1}&=&r\cosh\chi\sin\theta\cos\phi;\end{array}\quad\begin{array}[]{ccl}u_{2}&=&r\cosh\chi\sin\theta\sin\phi;\\\
u_{3}&=&r\cosh\chi\cos\theta,\end{array}$

is defined by an expression

$L_{1}=\frac{1}{r}\frac{\partial}{\partial
r}\left(r^{3}\frac{\partial}{\partial
r}\right)-\frac{1}{\cosh^{2}\chi}\frac{\partial}{\partial\chi}\left(\cosh^{2}\chi\frac{\partial}{\partial\chi}\right)-\\\
-\frac{1}{\cosh^{2}\chi}\left[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right)+\frac{1}{\sin^{2}\theta}\frac{\partial^{2}}{\partial\phi^{2}}\right]-\left(r\frac{\partial}{\partial
r}\right)^{2}-2r\frac{\partial}{\partial r}.$ (3)

_The hyperboloid $H^{3}$ is a homogeneous space of the Lorentz group_.
Solutions of the Ginzburg-Tamm equation have the form of the fields on the
Poincaré group:

$\Psi(x_{\mu},u_{\mu})=C\exp(-im_{0}t)Y_{l,m}(\theta,\phi)P^{j}_{l}(\tanh\chi),$

where $Y_{l,m}(\theta,\phi)$ is a spherical function, and
$P^{j}_{l}(\tanh\chi)$ is a Legendre polynomial.

Further, fields on the Poincaré group present itself a natural generalization
of the concept of wave function. These fields (generalized wave functions)
were introduced independently by several authors [24, 8, 70, 56] mainly in
connection with constructing relativistic wave equations. In essence, this
generalization consists in replacing the Minkowski space by a larger space on
which the Poincaré group acts. If this action is to be transitive, one is lead
to consider the homogeneous spaces of the Poincaré group. All the homogeneous
spaces of this type were listed by Finkelstein [18] and by Bacry and Kihlberg
[6] and the fields on these spaces were considered in the works [38, 7, 45,
34, 35, 61, 25].

A homogeneous space ${\cal M}$ of a group $G$ has the following properties:
a) It is a topological space on which the group $G$ acts continuously, that
is, let $y$ be a point in ${\cal M}$, then $gy$ is defined and is again a
point in ${\cal M}$ ($g\in G$).
b) This action is transitive, that is, for any two points $y_{1}$ and $y_{2}$
in ${\cal M}$ it is always possible to find a group element $g\in G$ such that
$y_{2}=gy_{1}$.
There is a one-to-one correspondence between the homogeneous spaces of $G$ and
the coset spaces of $G$. Let $H_{0}$ be a maximal subgroup of $G$ which leaves
the point $y_{0}$ invariant, $gy_{0}=y_{0}$, $g\in H_{0}$, then $H_{0}$ is
called the stabilizer of $y_{0}$. Representing now any group element of $G$ in
the form $g=g_{c}g_{0}$, where $g_{0}\in H_{0}$ and $g_{c}\in G/H_{0}$, we see
that, by virtie of the transitivity property, any point $y\in{\cal M}$ can be
given by $y=g_{c}g_{0}y_{0}=g_{c}y$. Hence it follows that the elements
$g_{c}$ of the coset space give a parametrization of ${\cal M}$. The mapping
${\cal M}\leftrightarrow G/H_{0}$ is continuous since the group multiplication
is continuous and the action on ${\cal M}$ is continuous by definition. The
stabilizers $H$ and $H_{0}$ of two different points $y$ and $y_{0}$ are
conjugate, since from $H_{0}g_{0}=g_{0}$, $y_{0}=g^{-1}y$, it follows that
$gH_{0}g^{-1}y=y$, that is, $H=gH_{0}g^{-1}$.

Returning to the Poincaré group ${\cal P}$, we see that the enumeration of the
different homogeneous spaces ${\cal M}$ of ${\cal P}$ amounts to an
enumeration of the subgroups of ${\cal P}$ up to a conjugation. Following to
Finkelstein, we require that ${\cal M}$ always contains the Minkowski space
$\bb R^{1,3}$ which means that four parameters of ${\cal M}$ can be denoted by
$x\,(x^{\mu})$. This means that the stabilizer $H$ of a given point in ${\cal
M}$ can never contain an element of the translation subgroup of ${\cal P}$.
Thus, the stabilizer must be a subgroup of the homogeneous Lorentz group
$\mathfrak{G}_{+}$.

In such a way, studying different subgroups of $\mathfrak{G}_{+}$, we obtain a
full list of homogeneous spaces ${\cal M}={\cal P}/H$ of the Poincaré group.
In the present paper we restrict ourselves by a consideration of the following
four homogeneous spaces:

$\displaystyle{\cal M}_{10}$ $\displaystyle=$ $\displaystyle\bb
R^{1,3}\times\mathfrak{L}_{6},\quad H=0;$ $\displaystyle{\cal M}_{8}$
$\displaystyle=$ $\displaystyle\bb R^{1,3}\times S^{2},\quad H=\Omega_{\psi};$
$\displaystyle{\cal M}_{7}$ $\displaystyle=$ $\displaystyle\bb R^{1,3}\times
H^{3},\quad H=\operatorname{SU}(2);$ $\displaystyle{\cal M}_{6}$
$\displaystyle=$ $\displaystyle\bb R^{1,3}\times S^{2},\quad
H=\\{\Omega_{\psi},\Omega_{\tau},\Omega_{\epsilon}\\}.$

Hence it follows that a group manifold of the Poincaré group, ${\cal
M}_{10}=\bb R^{1,3}\times\mathfrak{L}_{6}$, is a maximal homogeneous space of
${\cal P}$, $\mathfrak{L}_{6}$ is a group manifold of the Lorentz group. The
fields on the manifold ${\cal M}_{10}$ were considered by Lurçat [38]. These
fields depend on all the ten parameters of ${\cal P}$:

$\boldsymbol{\psi}(\boldsymbol{\alpha})=\langle
x,\mathfrak{g}\,|\boldsymbol{\psi}\rangle=\psi(x_{0},x_{1},x_{2},x_{3})\psi(\mathfrak{g}_{1},\mathfrak{g}_{2},\mathfrak{g}_{3},\mathfrak{g}_{4},\mathfrak{g}_{5},\mathfrak{g}_{6}),$

where an explicit form of $\psi(x)$ is given by the exponentials, and the
functions $\psi(\mathfrak{g})$ are expressed via the generalized
hyperspherical functions $\mathfrak{M}^{l}_{mn}(\mathfrak{g})$. As is known,
the universal covering of the proper Poincaré group is isomorphic to a
semidirect product $\operatorname{SL}(2;\bb C)\odot T_{4}$ or
$\operatorname{{\bf Spin}}_{+}(1,3)\odot T_{4}$. Since the group $T_{4}$ is
Abelian, then all its representations are one-dimensional. Thus, all the
finite-dimensional representations of the proper Poincaré group in essence are
equivalent to the representations $\mathfrak{C}$ of the group
$\operatorname{{\bf Spin}}_{+}(1,3)$. In the case of finite-dimensional
representations of $\operatorname{{\bf Spin}}_{+}(1,3)$,

$T_{\mathfrak{g}}q(\xi,\overline{\xi})=(\gamma\xi+\delta)^{l_{0}+l_{1}-1}\overline{(\gamma\xi+\delta)}^{l_{0}-l_{1}+1}q\left(\frac{\alpha\xi+\beta}{\gamma\xi+\delta};\frac{\overline{\alpha\xi+\beta}}{\overline{\gamma\xi+\delta}}\right),$
(4)

basic functions (matrix elements) of the finite-dimensional representation of
${\cal P}$ have the form

$t^{l}_{mn}(\boldsymbol{\alpha})=e^{-imx}\mathfrak{M}^{l}_{mn}(\mathfrak{g}),$

where

$\mathfrak{M}^{l}_{mn}(\mathfrak{g})=e^{-im\varphi^{c}}\mathfrak{Z}^{l}_{mn}(\cos\theta^{c})e^{-in\psi^{c}}.$

_Hyperspherical functions_ $\mathfrak{Z}^{l}_{mn}(\cos\theta^{c})$ can be
expressed via the hypergeometric functions [65]. So, at $m\geq n$ and $m\geq
k,\;k\geq n$ we have

$\mathfrak{Z}^{l}_{mn}(\cos\theta^{c})=i^{m-n}\sqrt{\frac{\Gamma(l+m+1)\Gamma(l-n+1)}{\Gamma(l-m+1)\Gamma(l+n+1)}}\cos^{2l}\frac{\theta}{2}\cosh^{2l}\frac{\tau}{2}\times\\\
\times\sum^{l}_{k=-l}\tan^{m-k}\frac{\theta}{2}\tanh^{k-n}\frac{\tau}{2}\times\\\
\times\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx<EMAIL_ADDRESS>\lx@intercol\hfil\begin{array}[]{c}m-k+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{-\tan^{2}\frac{\theta}{2}}\right)$}\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx<EMAIL_ADDRESS>\lx@intercol\hfil\begin{array}[]{c}k-n+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{\tanh^{2}\frac{\tau}{2}}\right)$}.$
(5)

There also exist three expressions of the hypergeometric type for the
functions $\mathfrak{Z}^{l}_{mn}(\cos\theta^{c})$ with the index values $m\geq
n,\;m\geq k,\;n\geq k$, $n\geq m,\;k\geq m,\;n\geq k$ and $n\geq m,\;k\geq
m,\;k\geq n$. For the unitary representations, that is, in the case of
principal series representations of the group $\operatorname{SO}_{0}(1,3)$,
there exists an analogue of the formula (4),

$V_{a}f(z)=(a_{12}z+a_{22})^{\frac{\lambda}{2}+i\frac{\rho}{2}-1}\overline{(a_{12}z+a_{22})}^{-\frac{\lambda}{2}+i\frac{\rho}{2}-1}f\left(\frac{a_{11}z+a_{21}}{a_{12}z+a_{22}}\right),$
(6)

where $f(z)$ is a measurable function of the Hilbert space $L_{2}(Z)$,
satisfying the condition $\int|f(z)|^{2}dz<\infty$, $z=x+iy$. A totality of
all representations $a\rightarrow T^{\alpha}$, corresponding to all possible
pairs $\lambda$, $\rho$, is called a principal series of representations of
the group $\operatorname{SO}_{0}(1,3)$ and denoted as
$\mathfrak{S}_{\lambda,\rho}$. At this point, a comparison of (6) with the
formula (4) for the spinor representation $\mathfrak{S}_{l}$ shows that the
both formulas have the same structure; only the exponents at the factors
$(a_{12}z+a_{22})$, $\overline{(a_{12}z+a_{22})}$ and the functions $f(z)$ are
different. In the case of spinor representations the functions $f(z)$ are
polynomials $p(z,\bar{z})$ in the spaces $\operatorname{Sym}_{(k,r)}$, and in
the case of a representation $\mathfrak{S}_{\lambda,\rho}$ of the principal
series $f(z)$ are functions from the Hilbert space $L_{2}(Z)$. Further, we
know that a representation $S_{l}$ of the group $\operatorname{SU}(2)$ is
realized in terms of the functions
$t^{l}_{mn}(u)=e^{-im\varphi}P^{l}_{mn}(\cos\theta)e^{-in\psi}$ [68]. It is
well known that the representation $S_{k}$ of $\operatorname{SU}(2)$ is
contained in $\mathfrak{S}_{\lambda,\rho}$ not more then one time [43]. At
this point, $S_{k}$ is contained in $\mathfrak{S}_{\lambda,\rho}$, when
$\frac{\lambda}{2}$ is one from the numbers $-k,-k+1,\ldots,k$. Matrix
elements of the principal series representations of the group
$\operatorname{SO}_{0}(1,3)$ have the form [65]

$\mathfrak{M}^{-\frac{1}{2}+i\rho,l_{0}}_{mn}(\mathfrak{g})=e^{-m(\epsilon+i\varphi)-n(\varepsilon+i\psi)}\mathfrak{Z}^{-\frac{1}{2}+i\rho,l_{0}}_{mn}=e^{-m(\epsilon+i\varphi)-n(\varepsilon+i\psi)}\times\\\\[5.69046pt]
\sum^{l_{0}}_{t=-l_{0}}i^{m-t}\sqrt{\Gamma(l_{0}-m+1)\Gamma(l_{0}+m+1)\Gamma(l_{0}-t+1)\Gamma(l_{0}+t+1)}\times\\\
\cos^{2l_{0}}\frac{\theta}{2}\tan^{m-t}\frac{\theta}{2}\times\\\\[5.69046pt]
\sum^{\min(l_{0}-m,l_{0}+t)}_{j=\max(0,t-m)}\frac{i^{2j}\tan^{2j}\frac{\theta}{2}}{\Gamma(j+1)\Gamma(l_{0}-m-j+1)\Gamma(l_{0}+t-j+1)\Gamma(m-t+j+1)}\times\\\\[5.69046pt]
\sqrt{\Gamma(\tfrac{1}{2}+i\rho-n)\Gamma(\tfrac{1}{2}+i\rho+n)\Gamma(\tfrac{1}{2}+i\rho-t)\Gamma(\tfrac{1}{2}+i\rho+t)}\cosh^{-1+2i\rho}\frac{\tau}{2}\tanh^{n-t}\frac{\tau}{2}\times\\\\[5.69046pt]
\sum^{\infty}_{s=\max(0,t-n)}\frac{\tanh^{2s}\frac{\tau}{2}}{\Gamma(s+1)\Gamma(\frac{1}{2}+i\rho-
n-s)\Gamma(\frac{1}{2}+i\rho+t-s)\Gamma(n-t+s+1)},$ (7)

where $l_{0}=\left|\frac{\lambda}{2}\right|$ and $\frac{\lambda}{2}$ is one
from the numbers $-k,-k+1,\ldots,k$. It is obvious that
$\mathfrak{M}^{-\frac{1}{2}+i\rho,l_{0}}_{mn}(\mathfrak{g})$ cannot be
attributed as matrix elements to single irreducible representation. From the
latter expression it follows that relativistic spherical functions
$f(\mathfrak{g})$ of the principal series can be defined by means of the
function

$\mathfrak{M}^{-\frac{1}{2}+i\rho,l_{0}}_{mn}(\mathfrak{g})=e^{-m(\epsilon+i\varphi)}\mathfrak{Z}^{-\frac{1}{2}+i\rho,l_{0}}_{mn}(\cos\theta^{c})e^{-n(\varepsilon+i\psi)},$
(8)

where

$\mathfrak{Z}^{-\frac{1}{2}+i\rho,l_{0}}_{mn}(\cos\theta^{c})=\sum^{l_{0}}_{t=-l_{0}}P^{l_{0}}_{mt}(\cos\theta)\mathfrak{P}^{-\frac{1}{2}+i\rho}_{tn}(\cosh\tau).$

In turn, the functions
$\mathfrak{M}^{-\frac{1}{2}+i\rho,l_{0}}_{mn}(\mathfrak{g})$ can be expressed
via the hypergeometric functions. So, at $m\geq t,\;t\geq n$ we have [65]

$\mathfrak{M}^{-\frac{1}{2}+i\rho,l_{0}}_{mn}(\mathfrak{g})=i^{m-n}e^{-m(\epsilon+i\varphi)-n(\varepsilon+i\psi)}\sqrt{\frac{\Gamma(l_{0}+m+1)\Gamma(i\rho-n+\frac{1}{2})}{\Gamma(l_{0}-m+1)\Gamma(i\rho+n+\frac{1}{2})}}\times\\\
\cos^{2l_{0}}\frac{\theta}{2}\cosh^{-1+2i\rho}\frac{\tau}{2}\sum^{l_{0}}_{t=-l_{0}}\tan^{m-t}\frac{\theta}{2}\tanh^{t-n}\frac{\tau}{2}\times\\\
\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx<EMAIL_ADDRESS>\lx@intercol\hfil\begin{array}[]{c}m-t+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{-\tan^{2}\frac{\theta}{2}}\right)$}\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx<EMAIL_ADDRESS>\lx@intercol\hfil\begin{array}[]{c}t-n+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{\tanh^{2}\frac{\tau}{2}}\right)$}.$
(9)

There also exist three expressions of the hypergeometric type for the
functions $\mathfrak{M}^{-\frac{1}{2}+i\rho,l_{0}}_{mn}(\mathfrak{g})$ with
the index values $m\geq t,\;n\geq t$, $t\geq m,\;n\geq t$ and $t\geq m,\;t\geq
n$.

The following eight-dimensional homogeneous space ${\cal M}_{8}=\bb
R^{1,3}\times S^{2}$ is a direct product of the Minkowski space $\bb R^{1,3}$
and the complex two-sphere $S^{2}_{c}$. In this case the stabilizer $H$
consists of the subgroup $\Omega^{c}_{\psi}$ of the diagonal matrices
$\begin{pmatrix}e^{\frac{i\psi^{c}}{2}}&0\\\
0&e^{-\frac{i\psi^{c}}{2}}\end{pmatrix}$. Bacry and Kihlberg [6] claimed that
the space ${\cal M}_{8}$ is the most suitable for a description of both half-
integer and integer spins. The fields, defined in ${\cal M}_{8}$, depend on
the eight parameters of ${\cal P}$:

$\langle
x,\varphi^{c},\theta^{c}|\boldsymbol{\psi}\rangle=\psi(x)\psi(\varphi^{c},\theta^{c})=\psi(x_{0},x_{1},x_{2},x_{3})\psi(\varphi,\epsilon,\theta,\tau),$
(10)

where the functions $\psi(\varphi^{c},\theta^{c})$ are expressed via the
associated hyperspherical functions defined on the surface of the complex two-
sphere $S^{2}_{c}$. The sphere $S^{c}_{2}$ can be constructed from the
quantities $z_{k}=x_{k}+iy_{k}$, $\overset{\ast}{z}_{k}=x_{k}-iy_{k}$
$(k=1,2,3)$ as follows:

$S^{c}_{2}:\;z^{2}_{1}+z^{2}_{2}+z^{2}_{3}={\bf x}^{2}-{\bf y}^{2}+2i{\bf
x}{\bf y}=r^{2}.$ (11)

The complex conjugate (dual) sphere $\dot{S}^{c}_{2}$ is

$\dot{S}^{c}_{2}:\;\overset{\ast}{z}_{1}{}^{2}+\overset{\ast}{z}_{2}{}^{2}+\overset{\ast}{z}_{3}{}^{2}={\bf
x}^{2}-{\bf y}^{2}-2i{\bf x}{\bf y}=\overset{\ast}{r}{}^{2}.$ (12)

For more details about the two-dimensional complex sphere see [30, 31, 59].
The surface of $S^{c}_{2}$ is homeomorphic to the space of the pairs
$(z_{1},z_{2})$, where $z_{1}$ and $z_{2}$ are the points of a _complex
projective line_ , $z_{1}\neq z_{2}$. This space is a homogeneous space of the
Lorentz group [21]. In Euler parametrization we have
$(z_{1},z_{2})\sim(\varphi^{c},\theta^{c})=(\varphi-i\epsilon,\theta-i\tau)$.
It is well-known that both quantities ${\bf x}^{2}-{\bf y}^{2}$, ${\bf x}{\bf
y}$ are invariant with respect to the Lorentz transformations, since a surface
of the complex sphere is invariant (Casimir operators of the Lorentz group are
constructed from such quantities). Moreover, since the real and imaginary
parts of the complex two-sphere transform like the electric and magnetic
fields, respectively, the invariance of ${\bf z}^{2}\sim({\bf E}+i{\bf
B})^{2}$ under proper Lorentz transformations is evident. At this point, the
quantities ${\bf x}^{2}-{\bf y}^{2}$, ${\bf x}{\bf y}$ are similar to the well
known electromagnetic invariants $E^{2}-B^{2}$, ${\bf E}{\bf B}$. This
intriguing relationship between the Laplace-Beltrami operators, Casimir
operators of the Lorentz group and electromagnetic invariants
$E^{2}-B^{2}\sim{\bf x}^{2}-{\bf y}^{2}$, ${\bf E}{\bf B}\sim{\bf x}{\bf y}$
leads naturally to a Riemann-Silberstein representation of the electromagnetic
field (see, for example, [69, 57, 10]). Further, associated hyperspherical
functions

$\mathfrak{M}^{m}_{l}(\varphi^{c},\theta^{c},0)=e^{-im\varphi^{c}}\mathfrak{Z}^{m}_{l}(\cos\theta^{c}),$
(13)

where

$\mathfrak{Z}^{m}_{l}(\cos\theta^{c})=\sum^{l}_{k=-l}P^{l}_{mk}(\cos\theta)\mathfrak{P}^{k}_{l}(\cosh\tau),$

are defined on the surface of the two-dimensional complex sphere (11). In
turn, the functions
$\mathfrak{M}^{\dot{m}}_{\dot{l}}(\dot{\varphi}^{c},\theta^{c},0)=e^{i\dot{m}\dot{\varphi}^{c}}\mathfrak{Z}^{\dot{m}}_{\dot{l}}(\cos\dot{\theta}^{c})$
are defined on the surface of the dual sphere (12). Explicit expressions of
the hypergeometric type for the functions
$\mathfrak{Z}^{m}_{l}(\cos\theta^{c})$ and
$\mathfrak{Z}^{\dot{m}}_{\dot{l}}(\cos\dot{\theta}^{c})$ follow directly from
the previous expressions (5) and (9) at $n=0$. In the case of the principal
series representations of the group $\operatorname{SO}_{0}(1,3)$ we have

$\mathfrak{Z}^{m}_{-\frac{1}{2}+i\rho,l_{0}}(\cos\theta^{c})=\sum^{l_{0}}_{t=-l_{0}}P^{l_{0}}_{mt}(\cos\theta)\mathfrak{P}^{t}_{-\frac{1}{2}+i\rho}(\cosh\tau),$
(14)

where $\mathfrak{P}^{t}_{-\frac{1}{2}+i\rho}(\cosh\tau)$ are conical functions
(see [9]).

In turn, a seven-dimensional homogeneous space ${\cal M}_{7}=\bb R^{1,3}\times
H^{3}$ is a direct product of $\bb R^{1,3}$ and a three-dimensional timelike
(two-sheeted) hyperboloid $H^{3}$. The stabilizer $H$ consists of the subgroup
of three-dimensional rotations, $\operatorname{SU}(2)$. Quantum field theory
on the space ${\cal M}_{7}$ was studied by Boyer and Fleming [12]. They showed
that the fields built over ${\cal M}_{7}$ are in general nonlocal and become
local only when the finite dimensional representations of the Lorentz group
are used. It is easy to see that the fields $\boldsymbol{\psi}\in{\cal M}_{7}$
depend on the seven parameters of the Poincaré group:

$\langle
x,\tau,\epsilon,\varepsilon|\boldsymbol{\psi}\rangle=\psi(x_{0},x_{1},x_{2},x_{3})\psi(\tau,\epsilon,\varepsilon),$

where the functions $\psi(\tau,\epsilon,\varepsilon)$ are expressed via
$e^{-(m\epsilon+n\varepsilon)}\mathfrak{P}^{l}_{mn}(\cosh\tau)$ in the case of
finite dimensional representations, and also via
$e^{-(m\epsilon+n\varepsilon)}\mathfrak{P}^{-\frac{1}{2}+i\rho}_{mn}(\cosh\tau)$
in the case of principal series of unitary representations, and via
$e^{-(m\epsilon+n\varepsilon)}\mathfrak{P}^{l,\pm}_{mn}(\cosh\tau)$ in the
case of the discrete representation series of the subgroup
$\operatorname{SU}(1,1)$. The spherical function
$\mathfrak{P}^{l}_{mn}(\cosh\tau)$ on the group $\operatorname{SU}(1,1)$ has
the form

$\mathfrak{P}^{l}_{mn}(\cosh\tau)=\sqrt{\Gamma(l-n+1)\Gamma(l+n+1)\Gamma(l-m+1)\Gamma(l+m+1)}\times\\\
\cosh^{2l}\frac{\tau}{2}\tanh^{n-m}\frac{\tau}{2}\times\\\
\sum^{\min(l-n,l+m)}_{s=\max(0,m-n)}\frac{\tanh^{2s}\frac{\tau}{2}}{\Gamma(s+1)\Gamma(l-n-s+1)\Gamma(n-m+s+1)\Gamma(l+m-s+1)}.$
(15)

Further, a six-dimensional space ${\cal M}_{6}=\bb R^{1,3}\times S^{2}$ is a
minimal homogeneous space of the Poincaré group, since the real two-sphere
$S^{2}$ has a minimal possible dimension among the homogeneous spaces of the
Lorentz group. In this case, the stabilizer $H$ consists of the subgroup
$\Omega^{c}_{\psi}$ and the subgroups $\Omega_{\tau}$ and $\Omega_{\epsilon}$
formed by the matrices
$\begin{pmatrix}\cosh\frac{\tau}{2}&\sinh\frac{\tau}{2}\\\
\sinh\frac{\tau}{2}&\cosh\frac{\tau}{2}\end{pmatrix}$ and
$\begin{pmatrix}e^{\frac{\epsilon}{2}}&0\\\
0&e^{-\frac{\epsilon}{2}}\end{pmatrix}$, respectively. Field models on the
homogeneous space ${\cal M}_{6}$ have been considered in the works [36, 39,
16]. In the paper [16] Drechsler considered the real two-sphere as a ‘spin
shell’ $S^{2}_{r=2s}$ of radius $r=2s$, where
$s=0,\frac{1}{2},1,\frac{3}{2},\ldots$. The fields, defined on ${\cal M}_{6}$,
depend on the six parameters of ${\cal P}$:

$\langle
x,\varphi,\theta|\boldsymbol{\psi}\rangle=\psi(x_{0},x_{1},x_{2},x_{3})\psi(\varphi,\theta),$

where the functions $\psi(\varphi,\theta)$ are expressed via the generalized
spherical functions of the type $e^{-im\varphi}P^{l}_{m0}(\cos\theta)$ or via
the Wigner $D$-functions, here

$P^{l}_{mn}(\cos\theta)=e^{-i(m\varphi+n\psi)}i^{m-n}\sqrt{\Gamma(l-m+1)\Gamma(l+m+1)\Gamma(l-n+1)\Gamma(l+n+1)}\times\\\
\cos^{2l}\frac{\theta}{2}\tan^{m-n}\frac{\theta}{2}\times\\\
\sum^{\min(l-n,l+n)}_{j=\max(0,n-m)}\frac{i^{2j}\tan^{2j}\frac{\theta}{2}}{\Gamma(j+1)\Gamma(l-m-j+1)\Gamma(l+n-j+1)\Gamma(m-n+j+1)}$
(16)

is a spherical function on the group $\operatorname{SU}(2)$.

Further, let ${\cal H}$ be a Hilbert space and let $U(g)$ be a square
integrable representation of a locally compact Lie group $G$ acting
transitively on ${\cal H}$, $\forall\phi\in{\cal H}$, $g\in
G:\,U(g)\phi\in{\cal H}$. Let there exist such a vector
$\mid\psi\rangle\in{\cal H}$ that satisfies the admissibility condition:

$C_{\psi}=||\psi||^{-2}\int_{g\in G}|\langle\mid
U(g)\psi\rangle|^{2}d\mu(g)\;<\;\infty,$

where $d\mu(g)$ is the left-invariant Haar measure on $G$. Then, any vector
$|\phi\rangle\in{\cal H}$ can be represented in the form

$|\phi\rangle=C^{-1}_{\psi}\int_{G}\mid U(g)\psi\rangle d\mu(g)\langle\psi
U^{\ast}(g)\mid\phi\rangle.$

At the restriction of $G$ on the affine subgroup $x^{\prime}=ax+b$ (the group
of translations and dilatations of the real axis) a harmonic analysis on the
group $G$ is reduced to the wavelet-transformation:

$\displaystyle\phi(x)$ $\displaystyle=$
$\displaystyle\frac{1}{C_{\psi}}\int\frac{1}{a^{d}}\psi\left(\frac{x-b}{a}\right)\phi_{a}(b)\frac{dad^{d}b}{a},$
$\displaystyle\phi_{a}(b)$ $\displaystyle=$
$\displaystyle\int\frac{1}{a^{d}}\overline{\psi\left(\frac{x-b}{a}\right)}\phi(x)d^{d}x,$
$\displaystyle C_{\psi}$ $\displaystyle=$
$\displaystyle\int^{\infty}_{0}|\widetilde{\psi}(ak)|^{2}\frac{da}{a}\;<\;\infty.$
(17)

At this restriction the field $\boldsymbol{\psi}(\boldsymbol{\alpha})=\langle
x,g\,|\boldsymbol{\psi}\rangle$ on the group $G$ is reduced to the field
$\boldsymbol{\psi}(\boldsymbol{\alpha})=\langle
x,a\,|\boldsymbol{\psi}\rangle\sim\langle x,a;g|\phi(x)\rangle$ on the affine
group, where $a$ is a scale factor which can be associated with the resolution
of measuring equipment [2]. It should be noted that further restriction of $G$
on the translation subgroup leads to the restriction of
$\boldsymbol{\psi}(\boldsymbol{\alpha})=\langle
x,g\,|\boldsymbol{\psi}\rangle$ to the field $\boldsymbol{\psi}(x)=\langle
x\,|\boldsymbol{\psi}\rangle$ describing the point-like object, and the
wavelet-transformation in this case is reduced to the Fourier-transformation.
Thus, the affine group is a minimal group which can be used for description of
the extended object. The restriction of $G$ to the affine subgroup is
equivalent formally to the Faddeev-Popov method considering translation and
dilatation invariance in quantum field calculations related with the
renormalization group, at this point, $a=1/\Lambda$, where $\Lambda$ is a cut-
off parameter. Moreover, in the framework of wavelet-analysis, loop Green
functions emerging in $\phi^{4}$-model, are free from ultra-violet divergences
[1, 2].

##  3 Free fields on the two-dimensional complex sphere

We will start with a more general homogeneous space of the group ${\cal P}$,
${\cal M}_{10}=\bb R^{1,3}\times\mathfrak{L}_{6}$ (group manifold of the
Poincaré group). Let $\mathcal{L}(\boldsymbol{\alpha})$ be a Lagrangian on the
group manifold ${\cal M}_{10}$ (in other words,
$\mathcal{L}(\boldsymbol{\alpha})$ is a 10-dimensional point function), where
$\boldsymbol{\alpha}$ is the parameter set of this group. Then an integral for
$\mathcal{L}(\boldsymbol{\alpha})$ on some 10-dimensional volume $\Omega$ of
the group manifold we will call an action on the Poincaré group:

$A=\int\limits_{\Omega}d\boldsymbol{\alpha}\mathcal{L}(\boldsymbol{\alpha}),$

where $d\boldsymbol{\alpha}$ is a Haar measure on the group ${\cal P}$.

Let $\boldsymbol{\psi}(\boldsymbol{\alpha})=\langle
x,\mathfrak{g}\,|\boldsymbol{\psi}\rangle$ be a function on the group manifold
${\cal M}_{10}$ (now it is sufficient to assume that
$\boldsymbol{\psi}(\boldsymbol{\alpha})$ is a square integrable function on
the Poincaré group) and let

$\frac{\partial\mathcal{L}}{\partial\boldsymbol{\psi}}-\frac{\partial}{\partial\boldsymbol{\alpha}}\frac{\partial\mathcal{L}}{\partial\frac{\partial\boldsymbol{\psi}}{\partial\boldsymbol{\alpha}}}=0$
(18)

be Euler-Lagrange equations on ${\cal M}_{10}$ (more precisely speaking, the
equations (18) act on the tangent bundle $T{\cal
M}_{10}=\underset{\boldsymbol{\alpha}\in{\cal
M}_{10}}{\cup}T_{\boldsymbol{\alpha}}{\cal M}_{10}$ of the manifold ${\cal
M}_{10}$, see [4]). Let us introduce a Lagrangian
$\mathcal{L}(\boldsymbol{\alpha})$ depending on the field function
$\boldsymbol{\psi}(\boldsymbol{\alpha})$ as follows

$\mathcal{L}(\boldsymbol{\alpha})=-\frac{1}{2}\left(\boldsymbol{\psi}^{\ast}(\boldsymbol{\alpha})B_{\mu}\frac{\partial\boldsymbol{\psi}(\boldsymbol{\alpha})}{\partial\boldsymbol{\alpha}_{\mu}}-\frac{\partial\boldsymbol{\psi}^{\ast}(\boldsymbol{\alpha})}{\partial\boldsymbol{\alpha}_{\mu}}B_{\mu}\boldsymbol{\psi}(\boldsymbol{\alpha})\right)-\kappa\boldsymbol{\psi}^{\ast}(\boldsymbol{\alpha})B_{11}\boldsymbol{\psi}(\boldsymbol{\alpha}),$

where $B_{\nu}$ ($\nu=1,2,\ldots,10$) are square matrices. The number of rows
and columns in these matrices is equal to the number of components of
$\boldsymbol{\psi}(\boldsymbol{\alpha})$, $\kappa$ is a non-null real
constant.

Further, if $B_{11}$ is non-singular, then we can introduce the matrices

$\Gamma_{\mu}=B^{-1}_{11}B_{\mu},\quad\mu=1,2,\ldots,10,$

and represent the Lagrangian $\mathcal{L}(\boldsymbol{\alpha})$ in the form

$\mathcal{L}(\boldsymbol{\alpha})=-\frac{1}{2}\left(\overline{\boldsymbol{\psi}}(\boldsymbol{\alpha})\Gamma_{\mu}\frac{\partial\boldsymbol{\psi}(\boldsymbol{\alpha})}{\partial\boldsymbol{\alpha}_{\mu}}-\frac{\overline{\boldsymbol{\psi}}(\boldsymbol{\alpha})}{\partial\boldsymbol{\alpha}_{\mu}}\Gamma_{\mu}\boldsymbol{\psi}(\boldsymbol{\alpha})\right)-\kappa\overline{\boldsymbol{\psi}}(\boldsymbol{\alpha})\boldsymbol{\psi}(\boldsymbol{\alpha}),$
(19)

where

$\overline{\boldsymbol{\psi}}(\boldsymbol{\alpha})=\boldsymbol{\psi}^{\ast}(\boldsymbol{\alpha})B_{11}.$

Varying independently $\psi(x)$ and $\overline{\psi}(x)$, we obtain from (19)
in accordance with (18) the following equations:

$\begin{array}[]{ccc}\Gamma_{i}\dfrac{\partial\psi(x)}{\partial
x_{i}}+\kappa\psi(x)&=&0,\\\
\Gamma^{T}_{i}\dfrac{\partial\overline{\psi}(x)}{\partial
x_{i}}-\kappa\overline{\psi}(x)&=&0.\end{array}\quad(i=0,\ldots,3)$ (20)

The matrix $\Gamma_{0}$ in equations (20) can be written in the form (see [20,
3, 48])

$\Gamma_{0}=\text{diag}\left(C^{0}\otimes I_{1},C^{1}\otimes
I_{3},\ldots,C^{s}\otimes I_{2s+1},\ldots\right)$ (21)

for integer spin and

$\Gamma_{0}=\text{diag}\left(C^{\frac{1}{2}}\otimes
I_{2},C^{\frac{3}{2}}\otimes I_{4},\ldots,C^{s}\otimes I_{2s+1},\ldots\right)$
(22)

for half-integer spin, where $C^{s}$ is a spin block. If the spin block
$C^{s}$ has non-null roots, then the particle possesses the spin $s$. The spin
block $C^{s}$ in (21)–(22) consists of the elements
$c^{s}_{\boldsymbol{\tau}\boldsymbol{\tau}^{\prime}}$, where
$\boldsymbol{\tau}_{l_{1},l_{2}}$ and
$\boldsymbol{\tau}_{l^{\prime}_{1},l^{\prime}_{2}}$ are interlocking
irreducible representations of the Lorentz group, that is, such
representations, for which $l^{\prime}_{1}=l_{1}\pm\frac{1}{2}$,
$l^{\prime}_{2}=l_{2}\pm\frac{1}{2}$. At this point, the block $C^{s}$
contains only the elements
$c^{s}_{\boldsymbol{\tau}\boldsymbol{\tau}^{\prime}}$ corresponding to such
interlocking representations $\boldsymbol{\tau}_{l_{1},l_{2}}$,
$\boldsymbol{\tau}_{l^{\prime}_{1},l^{\prime}_{2}}$ which satisfy the
conditions

$|l_{1}-l_{2}|\leq s\leq l_{1}+l_{2},\quad|l^{\prime}_{1}-l^{\prime}_{2}|\leq
s\leq l^{\prime}_{1}+l^{\prime}_{2}.$

The two most full schemes of the interlocking irreducible representations of
the Lorentz group for integer and half-integer spins are shown on the Fig. 1
and Fig. 2.

$\dgARROWLENGTH=0.5em\dgHORIZPAD=1.7em\dgVERTPAD=2.2ex\begin{diagram}$

Fig. 1: Interlocking representation scheme for the fields of integer spin
(Bose-scheme).

$\dgARROWLENGTH=0.5em\dgHORIZPAD=1.7em\dgVERTPAD=2.2ex\begin{diagram}$

Fig. 2: Interlocking representation scheme for the fields of half-integer spin
(Fermi-scheme).

As follows from Fig. 1 the simplest field is the scalar field

$(0,0).$

This field is described by the Fock-Klein-Gordon equation. In its turn, the
simplest field from the Fermi-scheme (Fig. 2) is the electron-positron
(spinor) field corresponding to the following interlocking scheme:

$\dgARROWLENGTH=2.5em\begin{diagram}.$

This field is described by the Dirac equation. Further, the next field from
the Bose-scheme (Fig. 1) is a photon field (Maxwell field) defined within the
interlocking scheme

$\dgARROWLENGTH=2.5em\begin{diagram}.$

This interlocking scheme leads to the Maxwell equations. The fields
$(1/2,0)\oplus(0,1/2)$ and $(1,0)\oplus(0,1)$ (Dirac and Maxwell fields) are
particular cases of fields $\boldsymbol{\psi}(\boldsymbol{\alpha})=\langle
x,\mathfrak{g}\,|\boldsymbol{\psi}\rangle$ of the type $(l,0)\oplus(0,l)$,
where $\mathfrak{g}\in\operatorname{{\bf Spin}}_{+}(1,3)$. Wave equations for
such fields and their general solutions were found in the works [62, 63, 64].

It is easy to see that the interlocking scheme, corresponding to the Maxwell
field, contains the field of tensor type:

$\left(\frac{1}{2},\frac{1}{2}\right).$

Further, the next interlocking scheme (see Fig. 2)

$\dgARROWLENGTH=2.5em\begin{diagram},$

corresponding to the Pauli-Fierz equations [19], contains a chain of the type

$\dgARROWLENGTH=2.5em\begin{diagram}.$

In such a way we come to wave equations for the fields
$\boldsymbol{\psi}(\boldsymbol{\alpha})=\langle
x,\mathfrak{g}\,|\boldsymbol{\psi}\rangle$ of tensor type
$(l_{1},l_{2})\oplus(l_{2},l_{1})$. Wave equations for such fields and their
general solutions were found in the work [67].

Further, varying independently $\psi(\mathfrak{g})$ and
$\overline{\psi}(\mathfrak{g})$ one gets from (19) the following equations:

$\begin{array}[]{ccc}\Gamma_{k}\dfrac{\partial\psi(\mathfrak{g})}{\partial\mathfrak{g}_{k}}+\kappa\psi(\mathfrak{g})&=&0,\\\
\Gamma^{T}_{k}\dfrac{\overline{\psi}(\mathfrak{g})}{\partial\mathfrak{g}_{k}}-\kappa\overline{\psi}(\mathfrak{g})&=&0,\end{array}\quad(k=1,\ldots,6)$
(23)

where

$\psi(\mathfrak{q})=\begin{pmatrix}\psi(\mathfrak{q})\\\
\dot{\psi}(\mathfrak{q})\end{pmatrix},\quad\Gamma_{k}=\begin{pmatrix}0&\Lambda^{\ast}_{k}\\\
\Lambda_{k}&0\end{pmatrix}.$

Non-zero elements of the matrices $\Lambda_{k}$ and $\Lambda^{\ast}_{k}$ have
the form (for more details see [62])

${\Lambda_{1}:\quad\left\\{\begin{array}[]{ccc}a^{k^{\prime}k}_{l-1,l,m-1,m}&=&-\frac{c_{l-1,l}}{2}\sqrt{(l+m)(l+m-1)},\\\
a^{k^{\prime}k}_{l,l,m-1,m}&=&\frac{c_{ll}}{2}\sqrt{(l+m)(l-m+1)},\\\
a^{k^{\prime}k}_{l+1,l,m-1,m}&=&\frac{c_{l+1,l}}{2}\sqrt{(l-m+1)(l-m+2)},\\\
a^{k^{\prime}k}_{l-1,l,m+1,m}&=&\frac{c_{l-1,l}}{2}\sqrt{(l-m)(l-m-1)},\\\
a^{k^{\prime}k}_{l,l,m+1,m}&=&\frac{c_{ll}}{2}\sqrt{(l+m+1)(l-m)},\\\
a^{k^{\prime}k}_{l+1,l,m+1,m}&=&-\frac{c_{l+1,l}}{2}\sqrt{(l+m+1)(l+m+2)}.\end{array}\right.}$
(24)
${\Lambda_{2}:\quad\left\\{\begin{array}[]{ccc}b^{k^{\prime}k}_{l-1,l,m-1,m}&=&-\frac{ic_{l-1,l}}{2}\sqrt{(l+m)(l+m-1)},\\\
b^{k^{\prime}k}_{l,l,m-1,m}&=&\frac{ic_{ll}}{2}\sqrt{(l+m)(l-m+1)},\\\
b^{k^{\prime}k}_{l+1,l,m-1,m}&=&\frac{ic_{l+1,l}}{2}\sqrt{(l-m+1)(l-m+2)},\\\
b^{k^{\prime}k}_{l-1,l,m+1,m}&=&-\frac{ic_{l-1,l}}{2}\sqrt{(l-m)(l-m-1)},\\\
b^{k^{\prime}k}_{l,l,m+1,m}&=&-\frac{ic_{ll}}{2}\sqrt{(l+m+1)(l-m)},\\\
b^{k^{\prime}k}_{l+1,l,m+1,m}&=&\frac{ic_{l+1,l}}{2}\sqrt{(l+m+1)(l+m+2)}.\end{array}\right.}$
(25)
${\Lambda_{3}:\quad\left\\{\begin{array}[]{ccc}c^{k^{\prime}k}_{l-1,l,m}&=&c^{k^{\prime}k}_{l-1,l}\sqrt{l^{2}-m^{2}},\\\
c^{k^{\prime}k}_{l,l,m}&=&c^{k^{\prime}k}_{ll}m,\\\
c^{k^{\prime}k}_{l+1,l,m}&=&c^{k^{\prime}k}_{l+1,l}\sqrt{(l+1)^{2}-m^{2}}.\end{array}\right.}$
(26)
${\Lambda^{\ast}_{1}:\quad\left\\{\begin{array}[]{ccc}d^{\dot{k}^{\prime}\dot{k}}_{\dot{l}-1,\dot{l},\dot{m}-1,\dot{m}}&=&-\frac{c_{\dot{l}-1,\dot{l}}}{2}\sqrt{(\dot{l}+\dot{m})(\dot{l}-\dot{m}-1)},\\\
d^{\dot{k}^{\prime}\dot{k}}_{\dot{l},\dot{l},\dot{m}-1,\dot{m}}&=&\frac{c_{\dot{l}\dot{l}}}{2}\sqrt{(\dot{l}+\dot{m})(\dot{l}-\dot{m}+1)},\\\
d^{\dot{k}^{\prime}\dot{k}}_{\dot{l}+1,\dot{l},\dot{m}-1,\dot{m}}&=&\frac{c_{\dot{l}+1,\dot{l}}}{2}\sqrt{(\dot{l}-\dot{m}+1)(\dot{l}-\dot{m}+2)},\\\
d^{\dot{k}^{\prime}\dot{k}}_{\dot{l}-1,\dot{l},\dot{m}+1,\dot{m}}&=&\frac{c_{\dot{l}-1,\dot{l}}}{2}\sqrt{(\dot{l}-\dot{m})(\dot{l}-\dot{m}-1)},\\\
d^{\dot{k}^{\prime}\dot{k}}_{\dot{l},\dot{l},\dot{m}+1,\dot{m}}&=&\frac{c_{\dot{l}\dot{l}}}{2}\sqrt{(\dot{l}+\dot{m}+1)(\dot{l}-\dot{m})},\\\
d^{\dot{k}^{\prime}\dot{k}}_{\dot{l}+1,\dot{l},\dot{m}+1,\dot{m}}&=&-\frac{c_{\dot{l}+1,\dot{l}}}{2}\sqrt{(\dot{l}+\dot{m}+1)(\dot{l}+\dot{m}+2)}.\end{array}\right.}$
(27)
${\Lambda^{\ast}_{2}:\quad\left\\{\begin{array}[]{ccc}e^{\dot{k}^{\prime}\dot{k}}_{\dot{l}-1,\dot{l},\dot{m}-1,\dot{m}}&=&-\frac{ic_{\dot{l}-1,\dot{l}}}{2}\sqrt{(\dot{l}+\dot{m})(\dot{l}-\dot{m}-1)},\\\
e^{\dot{k}^{\prime}\dot{k}}_{\dot{l},\dot{l},\dot{m}-1,\dot{m}}&=&\frac{ic_{\dot{l}\dot{l}}}{2}\sqrt{(\dot{l}+\dot{m})(\dot{l}-\dot{m}+1)},\\\
e^{\dot{k}^{\prime}\dot{k}}_{\dot{l}+1,\dot{l},\dot{m}-1,\dot{m}}&=&\frac{ic_{\dot{l}+1,\dot{l}}}{2}\sqrt{(\dot{l}-\dot{m}+1)(\dot{l}-\dot{m}+2)},\\\
e^{\dot{k}^{\prime}\dot{k}}_{\dot{l}-1,\dot{l},\dot{m}+1,\dot{m}}&=&\frac{-ic_{\dot{l}-1,\dot{l}}}{2}\sqrt{(\dot{l}-\dot{m})(\dot{l}-\dot{m}-1)},\\\
e^{\dot{k}^{\prime}\dot{k}}_{\dot{l},\dot{l},\dot{m}+1,\dot{m}}&=&\frac{-ic_{\dot{l}\dot{l}}}{2}\sqrt{(\dot{l}+\dot{m}+1)(\dot{l}-\dot{m})},\\\
e^{\dot{k}^{\prime}\dot{k}}_{\dot{l}+1,\dot{l},\dot{m}+1,\dot{m}}&=&-\frac{ic_{\dot{l}+1,\dot{l}}}{2}\sqrt{(\dot{l}+\dot{m}+1)(\dot{l}+\dot{m}+2)}.\end{array}\right.}$
(28)
${\Lambda^{\ast}_{3}:\quad\left\\{\begin{array}[]{ccc}f^{\dot{k}^{\prime}\dot{k}}_{\dot{l}-1,\dot{l},\dot{m}}&=&c^{\dot{k}^{\prime}\dot{k}}_{\dot{l}-1,\dot{l}}\sqrt{\dot{l}^{2}-\dot{m}^{2}},\\\
f^{\dot{k}^{\prime}\dot{k}}_{\dot{l}\dot{l},\dot{m}}&=&c^{\dot{k}^{\prime}\dot{k}}_{\dot{l}\dot{l}}\dot{m},\\\
f^{\dot{k}^{\prime}\dot{k}}_{\dot{l}+1,\dot{l},\dot{m}}&=&c^{\dot{k}^{\prime}\dot{k}}_{\dot{l}+1,\dot{l}}\sqrt{(\dot{l}+1)^{2}-\dot{m}^{2}}.\end{array}\right.}$
(29)

In general, the matrix $\Lambda_{3}$ must be a reducible representation of the
proper Lorentz group $\mathfrak{G}_{+}$, and can always be written in the form

$\Lambda_{3}=\begin{bmatrix}\Lambda^{l_{1}}_{3}&&&&\\\
&\Lambda^{l_{2}}_{3}&&\text{\huge 0}&\\\ &&\Lambda^{l_{3}}_{3}&&\\\
&\text{\huge 0}&&\ddots&\\\ &&&&\Lambda^{l_{n}}_{3}\end{bmatrix},$ (30)

where $\Lambda^{l_{i}}_{3}$ is a spin block (the matrix
$\overset{\ast}{\Lambda}_{3}$ has the same decompositions). It is obvious that
the matrices $\Lambda_{1}$, $\Lambda_{2}$ and $\overset{\ast}{\Lambda}_{1}$,
$\overset{\ast}{\Lambda}_{2}$ admit also the decompositions of the type (30)
by definition. If the spin block $\Lambda^{l_{i}}_{3}$ has non-null roots,
then the particle possesses the spin $l_{i}$. Corresponding to the
decomposition (30), the wave function also decomposes into a direct sum of
component wave functions which we write

$\boldsymbol{\psi}=\psi_{l_{1}m_{1}}+\psi_{l_{2}m_{2}}+\psi_{l_{3}m_{3}}+\ldots.$

According to a de Broglie theory of fusion [14], interlocking representations
give rise to indecomposable relativistic wave equations. Otherwise, we have
decomposable equations. As is known, the indecomposable wave equations
correspond to composite particles. A relation between indecomposable wave
equations and composite particles will be studied in a separate work.

Returning to the equations (23), we see that the first equation can be written
in the form

$\displaystyle\sum^{3}_{j=1}\Lambda^{\ast}_{j}\frac{\partial\dot{\psi}}{\partial\widetilde{a}_{j}}+i\sum^{3}_{j=1}\Lambda^{\ast}_{j}\frac{\partial\dot{\psi}}{\partial\widetilde{a}^{\ast}_{j}}+\kappa^{c}\psi$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\sum^{3}_{j=1}\Lambda_{j}\frac{\partial\psi}{\partial
a_{j}}-i\sum^{3}_{j=1}\Lambda_{j}\frac{\partial\psi}{\partial
a^{\ast}_{j}}+\dot{\kappa}^{c}\dot{\psi}$ $\displaystyle=$ $\displaystyle 0,$
(31)

where $\mathfrak{g}_{1}=a_{1}$, $\mathfrak{g}_{2}=a_{2}$,
$\mathfrak{g}_{3}=a_{3}$, $\mathfrak{g}_{4}=ia_{1}$,
$\mathfrak{g}_{5}=ia_{2}$, $\mathfrak{g}_{6}=ia_{3}$,
$a^{\ast}_{1}=-i\mathfrak{g}_{4}$, $a^{\ast}_{2}=-i\mathfrak{g}_{5}$,
$a^{\ast}_{3}=-i\mathfrak{g}_{6}$, and $\widetilde{a}_{j}$,
$\widetilde{a}^{\ast}_{j}$ are the parameters corresponding to the dual basis.
In essence, the equations (31) are defined in a three-dimensional complex
space $\bb C^{3}$. In turn, the space $\bb C^{3}$ is isometric to a
6-dimensional bivector space $\bb R^{6}$ (a parameter space of the Lorentz
group [47]). The bivector space $\bb R^{6}$ is a tangent space of the group
manifold $\mathfrak{L}_{6}$ of the Lorentz group, that is, the manifold
$\mathfrak{L}_{6}$ in the each its point is equivalent locally to the space
$\bb R^{6}$. Thus, for all $\mathfrak{g}\in\mathfrak{L}_{6}$ we have
$T_{\mathfrak{g}}\mathfrak{L}_{6}\simeq\bb R^{6}$. There exists a close
relationship between the metrics of the Minkowski spacetime $\bb R^{1,3}$ and
the metrics of $\bb R^{6}$ (see Appendix). In the case of $\bb R^{1,3}$ with
the metric tensor

$g_{\alpha\beta}=\begin{pmatrix}-1&0&0&0\\\ 0&-1&0&0\\\ 0&0&-1&0\\\
0&0&0&1\end{pmatrix}$

in virtue of (86) for the metric tensor of $\bb R^{6}$ we obtain

$g_{ab}=\begin{bmatrix}-1&0&0&0&0&0\\\ 0&-1&0&0&0&0\\\ 0&0&-1&0&0&0\\\
0&0&0&1&0&0\\\ 0&0&0&0&1&0\\\ 0&0&0&0&0&1\end{bmatrix},$ (32)

where the order of collective indices in $\bb R^{6}$ is $23\rightarrow 0$,
$10\rightarrow 1$, $20\rightarrow 2$, $30\rightarrow 3$, $31\rightarrow 4$,
$12\rightarrow 5$. The system (31) in the bivector space is written as
follows:

$\sum_{i}\left[g^{-}_{i1}\Lambda_{1}T^{-1}_{\mathfrak{g}}\frac{\partial\psi^{\prime}}{\partial
a^{\prime}_{i}}+g^{-}_{i2}\Lambda_{2}T^{-1}_{\mathfrak{g}}\frac{\partial\psi^{\prime}}{\partial
a^{\prime}_{i}}+g^{-}_{i3}\Lambda_{3}T^{-1}_{\mathfrak{g}}\frac{\partial\psi^{\prime}}{\partial
a^{\prime}_{i}}-\right.\\\
\left.-ig^{-}_{i1}\Lambda_{1}T^{-1}_{\mathfrak{g}}\frac{\partial\psi^{\prime}}{\partial
a^{\ast}_{i}{}^{\prime}}-ig^{-}_{i2}\Lambda_{2}T^{-1}_{\mathfrak{g}}\frac{\partial\psi^{\prime}}{\partial
a^{\ast}_{i}{}^{\prime}}-ig^{-}_{i3}\Lambda_{3}T^{-1}_{\mathfrak{g}}\frac{\partial\psi^{\prime}}{\partial
a^{\ast}_{i}{}^{\prime}}\right]+\kappa^{c}T^{-1}_{\mathfrak{g}}\psi^{\prime}=0,\\\
\sum_{i}\left[g^{+}_{i1}\Lambda^{\ast}_{1}\overset{\ast}{T}_{\mathfrak{g}}\\!\\!{}^{-1}\frac{\partial\dot{\psi}^{\prime}}{\partial\widetilde{a}^{\prime}_{i}}+g^{+}_{i2}\Lambda^{\ast}_{2}\overset{\ast}{T}_{\mathfrak{g}}\\!\\!{}^{-1}\frac{\partial\dot{\psi}^{\prime}}{\partial\widetilde{a}^{\prime}_{i}}+g^{+}_{i3}\Lambda^{\ast}_{3}\overset{\ast}{T}_{\mathfrak{g}}\\!\\!{}^{-1}\frac{\partial\dot{\psi}^{\prime}}{\partial\widetilde{a}^{\prime}_{i}}+\right.\\\
\left.+ig^{+}_{i1}\Lambda^{\ast}_{1}\overset{\ast}{T}_{\mathfrak{g}}\\!\\!{}^{-1}\frac{\partial\dot{\psi}^{\prime}}{\partial\widetilde{a^{\ast}}_{i}{}^{\prime}}+ig^{+}_{i2}\Lambda^{\ast}_{2}\overset{\ast}{T}_{\mathfrak{g}}\\!\\!{}^{-1}\frac{\partial\dot{\psi}^{\prime}}{\partial\widetilde{a^{\ast}}_{i}{}^{\prime}}+ig^{+}_{i3}\Lambda^{\ast}_{3}\overset{\ast}{T}_{\mathfrak{g}}\\!\\!{}^{-1}\frac{\partial\dot{\psi}^{\prime}}{\partial\widetilde{a^{\ast}}_{i}{}^{\prime}}\right]+\dot{\kappa}^{c}\overset{\ast}{T}_{\mathfrak{g}}\\!\\!{}^{-1}\dot{\psi}^{\prime}=0,$
(33)

where

$\displaystyle\frac{\partial}{\partial a_{1}}$ $\displaystyle=$
$\displaystyle-\frac{\sin\varphi^{c}}{r\sin\theta^{c}}\frac{\partial}{\partial\varphi}+\frac{\cos\varphi^{c}\cos\theta^{c}}{r}\frac{\partial}{\partial\theta}+\cos\varphi^{c}\sin\theta^{c}\frac{\partial}{\partial
r},$ (34) $\displaystyle\frac{\partial}{\partial a_{2}}$ $\displaystyle=$
$\displaystyle\frac{\cos\varphi^{c}}{r\sin\theta^{c}}\frac{\partial}{\partial\varphi}+\frac{\sin\varphi^{c}\cos\theta^{c}}{r}\frac{\partial}{\partial\theta}+\sin\varphi^{c}\sin\theta^{c}\frac{\partial}{\partial
r},$ (35) $\displaystyle\frac{\partial}{\partial a_{3}}$ $\displaystyle=$
$\displaystyle-\frac{\sin\theta^{c}}{r}\frac{\partial}{\partial\theta}+\cos\theta^{c}\frac{\partial}{\partial
r}.$ (36) $\displaystyle\frac{\partial}{\partial a^{\ast}_{1}}$
$\displaystyle=$ $\displaystyle i\frac{\partial}{\partial
a_{1}}=-\frac{\sin\varphi^{c}}{r\sin\theta^{c}}\frac{\partial}{\partial\epsilon}+\frac{\cos\varphi^{c}\sin\theta^{c}}{r}\frac{\partial}{\partial\tau}+i\cos\varphi^{c}\sin\theta^{c}\frac{\partial}{\partial
r},$ (37) $\displaystyle\frac{\partial}{\partial a^{\ast}_{2}}$
$\displaystyle=$ $\displaystyle i\frac{\partial}{\partial
a_{2}}=\frac{\cos\varphi^{c}}{r\sin\theta^{c}}\frac{\partial}{\partial\epsilon}+\frac{\sin\varphi^{c}\cos\theta^{c}}{r}\frac{\partial}{\partial\tau}+i\sin\varphi^{c}\sin\theta^{c}\frac{\partial}{\partial
r},$ (38) $\displaystyle\frac{\partial}{\partial a^{\ast}_{3}}$
$\displaystyle=$ $\displaystyle i\frac{\partial}{\partial
a_{3}}=-\frac{\sin\theta^{c}}{r}\frac{\partial}{\partial\tau}+i\cos\theta^{c}\frac{\partial}{\partial
r}.$ (39) $\displaystyle\frac{\partial}{\partial\widetilde{a}_{1}}$
$\displaystyle=$
$\displaystyle-\frac{\sin\dot{\varphi}^{c}}{r^{\ast}\sin\dot{\theta}^{c}}\frac{\partial}{\partial\varphi}+\frac{\cos\dot{\varphi}^{c}\cos\dot{\theta}^{c}}{r^{\ast}}\frac{\partial}{\partial\theta}+\cos\dot{\varphi}^{c}\sin\dot{\theta}^{c}\frac{\partial}{\partial
r^{\ast}},$ (40) $\displaystyle\frac{\partial}{\partial\widetilde{a}_{2}}$
$\displaystyle=$
$\displaystyle\frac{\cos\dot{\varphi}^{c}}{r^{\ast}\sin\dot{\theta}^{c}}\frac{\partial}{\partial\varphi}+\frac{\sin\dot{\varphi}^{c}\cos\dot{\theta}^{c}}{r^{\ast}}\frac{\partial}{\partial\theta}+\sin\dot{\varphi}^{c}\sin\dot{\theta}^{c}\frac{\partial}{\partial
r^{\ast}},$ (41) $\displaystyle\frac{\partial}{\partial\widetilde{a}_{3}}$
$\displaystyle=$
$\displaystyle-\frac{\sin\dot{\theta}^{c}}{r^{\ast}}\frac{\partial}{\partial\theta}+\cos\dot{\theta}^{c}\frac{\partial}{\partial
r^{\ast}}.$ (42)
$\displaystyle\frac{\partial}{\partial\widetilde{a}^{\ast}_{1}}$
$\displaystyle=$
$\displaystyle-i\frac{\partial}{\partial\widetilde{a}_{1}}=\frac{\sin\dot{\varphi}^{c}}{r^{\ast}\sin\dot{\theta}^{c}}\frac{\partial}{\partial\epsilon}-\frac{\cos\dot{\varphi}^{c}\cos\dot{\theta}^{c}}{r^{\ast}}\frac{\partial}{\partial\tau}-i\cos\dot{\varphi}^{c}\sin\dot{\theta}^{c}\frac{\partial}{\partial
r^{\ast}},$ (43)
$\displaystyle\frac{\partial}{\partial\widetilde{a}^{\ast}_{2}}$
$\displaystyle=$
$\displaystyle-i\frac{\partial}{\partial\widetilde{a}_{2}}=-\frac{\cos\dot{\varphi}^{c}}{r^{\ast}\sin\dot{\theta}^{c}}\frac{\partial}{\partial\epsilon}-\frac{\sin\dot{\varphi}^{c}\cos\dot{\theta}^{c}}{r^{\ast}}\frac{\partial}{\partial\tau}-i\sin\dot{\varphi}^{c}\sin\dot{\theta}^{c}\frac{\partial}{\partial
r^{\ast}},$ (44)
$\displaystyle\frac{\partial}{\partial\widetilde{a}^{\ast}_{3}}$
$\displaystyle=$
$\displaystyle-i\frac{\partial}{\partial\widetilde{a}_{3}}=\frac{\sin\dot{\theta}^{c}}{r^{\ast}}\frac{\partial}{\partial\tau}-i\cos\dot{\theta}^{c}\frac{\partial}{\partial
r^{\ast}}$ (45)

are derivatives defined on the spheres (11) and (12). Solutions of wave
equations for extended object are found in the form of series in associated
hyperspherical functions which defined on the spheres $S^{2}_{c}$ and
$\dot{S}^{2}_{c}$:

$\displaystyle\psi^{k}_{lm;\dot{l}\dot{m}}$ $\displaystyle=$
$\displaystyle\boldsymbol{f}^{l_{0}}_{lmk}(r)\mathfrak{M}^{l_{0}}_{mn}(\varphi,\epsilon,\theta,\tau,0,0),$
$\displaystyle\psi^{\dot{k}}_{\dot{l}\dot{m};lm}$ $\displaystyle=$
$\displaystyle\boldsymbol{f}^{\dot{l}_{0}}_{\dot{l}\dot{m}\dot{k}}(r^{\ast})\mathfrak{M}^{\dot{l}_{0}}_{\dot{m}\dot{n}}(\varphi,\epsilon,\theta,\tau,0,0),$

where $l_{0}\geq l$, $-l_{0}\leq m$, $n\leq l_{0}$ and
$\dot{l}_{0}\geq\dot{l}$, $-\dot{l}_{0}\leq\dot{m}$, $\dot{n}\leq\dot{l}_{0}$.

_We claim that a system of Dirac like wave equations_

${\left\\{\begin{array}[]{cll}\Gamma_{i}\dfrac{\partial\psi(x)}{\partial
x_{i}}+\kappa\psi(x)&=&0,\\\
\Gamma^{T}_{i}\dfrac{\partial\overline{\psi}(x)}{\partial
x_{i}}-\kappa\overline{\psi}(x)&=&0,\quad(i=0,\ldots,3);\\\
\Gamma_{k}\dfrac{\partial\psi(\mathfrak{g})}{\partial\mathfrak{g}_{k}}+\kappa\psi(\mathfrak{g})&=&0,\\\
\Gamma^{T}_{k}\dfrac{\overline{\psi}(\mathfrak{g})}{\partial\mathfrak{g}_{k}}-\kappa\overline{\psi}(\mathfrak{g})&=&0\quad(k=1,\ldots,6)\end{array}\right.}$
(46)

_describes extended particles of any spin on the group manifold ${\cal
M}_{10}=\bb R^{1,3}\times\mathfrak{L}_{6}$ of the Poincaré group ${\cal P}$_
(in particular, on the homogeneous space ${\cal M}_{8}=\bb R^{1,3}\times
S^{2}$). In its turn, the Ginzburg-Tamm system of Klein-Gordon like wave
equations (2) describes extended particles on the homogeneous space ${\cal
M}=\bb R^{1,3}\times H^{3}$ of ${\cal P}$, where $H^{3}$ is a 3-dimensional
one-sheeted hyperboloid. A relation between the systems (46) and (2) is
similar to a relation between usual Dirac and Klein-Gordon wave equations for
the point-like particles. Namely, equations (46) (differential equations of
the first order) can be understood as ‘square roots’ of the equations (2)
(differential equations of the second order).

###  3.1 The Dirac field

In accordance with the general Fermi-scheme (Fig. 1) of the interlocking
representations of $\mathfrak{G}_{+}$ the field $(1/2,0)\oplus(0,1/2)$ is
defined within the following chain:

$\dgARROWLENGTH=2.5em\begin{diagram}.$

We start with the Lagrangian (19) on the group manifold ${\cal M}_{10}$:

$\mathcal{L}(\boldsymbol{\alpha})=-\frac{1}{2}\left(\overline{\boldsymbol{\psi}}(\boldsymbol{\alpha})\Gamma_{\mu}\frac{\partial\boldsymbol{\psi}(\boldsymbol{\alpha})}{\partial
x_{\mu}}-\frac{\partial\overline{\boldsymbol{\psi}}(\boldsymbol{\alpha})}{\partial
x_{\mu}}\Gamma_{\mu}\boldsymbol{\psi}(\boldsymbol{\alpha})\right)-\\\
-\frac{1}{2}\left(\overline{\boldsymbol{\psi}}(\boldsymbol{\alpha})\Upsilon_{\nu}\frac{\partial\boldsymbol{\psi}(\boldsymbol{\alpha})}{\partial\mathfrak{g}_{\nu}}-\frac{\partial\overline{\boldsymbol{\psi}}(\boldsymbol{\alpha})}{\partial\mathfrak{g}_{\nu}}\Upsilon_{\nu}\boldsymbol{\psi}(\boldsymbol{\alpha})\right)-\kappa\overline{\boldsymbol{\psi}}(\boldsymbol{\alpha})\boldsymbol{\psi}(\boldsymbol{\alpha}),$
(47)

where $\boldsymbol{\psi}(\boldsymbol{\alpha})=\psi(x)\psi(\mathfrak{g})$
($\mu=0,1,2,3,\;\nu=1,\ldots,6$), and

$\gamma_{0}=\begin{pmatrix}\sigma_{0}&0\\\
0&-\sigma_{0}\end{pmatrix},\;\;\gamma_{1}=\begin{pmatrix}0&\sigma_{1}\\\
-\sigma_{1}&0\end{pmatrix},\;\;\gamma_{2}=\begin{pmatrix}0&\sigma_{2}\\\
-\sigma_{2}&0\end{pmatrix},\;\;\gamma_{3}=\begin{pmatrix}0&\sigma_{3}\\\
-\sigma_{3}&0\end{pmatrix},$ (48)
$\Upsilon_{1}=\begin{pmatrix}0&\Lambda^{\ast}_{1}\\\
\Lambda_{1}&0\end{pmatrix},\quad\Upsilon_{2}=\begin{pmatrix}0&\Lambda^{\ast}_{2}\\\
\Lambda_{2}&0\end{pmatrix},\quad\Upsilon_{3}=\begin{pmatrix}0&\Lambda^{\ast}_{3}\\\
\Lambda_{3}&0\end{pmatrix},$ (49)
$\Upsilon_{4}=\begin{pmatrix}0&i\Lambda^{\ast}_{1}\\\
i\Lambda_{1}&0\end{pmatrix},\quad\Upsilon_{5}=\begin{pmatrix}0&i\Lambda^{\ast}_{2}\\\
i\Lambda_{2}&0\end{pmatrix},\quad\Upsilon_{6}=\begin{pmatrix}0&i\Lambda^{\ast}_{3}\\\
i\Lambda_{3}&0\end{pmatrix},$ (50)

where $\sigma_{i}$ are the Pauli matrices, and the matrices $\Lambda_{j}$ and
$\Lambda^{\ast}_{j}$ are derived from (24)–(26) and (27)–(29) at $l=1/2$:

$\displaystyle\Lambda_{1}=\frac{1}{2}c_{\frac{1}{2}\frac{1}{2}}\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\quad\Lambda_{2}=\frac{1}{2}c_{\frac{1}{2}\frac{1}{2}}\begin{pmatrix}0&-i\\\
i&0\end{pmatrix},\quad\Lambda_{3}=\frac{1}{2}c_{\frac{1}{2}\frac{1}{2}}\begin{pmatrix}1&0\\\
0&-1\end{pmatrix},$
$\displaystyle\Lambda^{\ast}_{1}=\frac{1}{2}\dot{c}_{\frac{1}{2}\frac{1}{2}}\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\quad\Lambda^{\ast}_{2}=\frac{1}{2}\dot{c}_{\frac{1}{2}\frac{1}{2}}\begin{pmatrix}0&-i\\\
i&0\end{pmatrix},\quad\Lambda^{\ast}_{3}=\frac{1}{2}\dot{c}_{\frac{1}{2}\frac{1}{2}}\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}.$ (51)

It is easy to see that these matrices coincide with the Pauli matrices
$\sigma_{i}$ when $c_{\frac{1}{2}\frac{1}{2}}=2$. The Dirac bispinor
$\psi=(\psi_{1},\psi_{2},\dot{\psi}_{1},\dot{\psi}_{2})^{T}$ is defined on
${\cal M}_{8}=\bb R^{1,3}\times S^{2}$ by the following components [63]:

$\displaystyle\psi^{l}_{1n}(\boldsymbol{\alpha})=\psi^{+}_{1}(x)\psi^{l}_{1n}(\mathfrak{g})=u_{1}({\bf
p})e^{-ipx}\boldsymbol{f}^{l}_{\frac{1}{2},\frac{1}{2}}(\mbox{\rm
Re}\,r)\mathfrak{M}^{l}_{\frac{1}{2},n}(\varphi,\epsilon,\theta,\tau,0,0),$
$\displaystyle\psi^{l}_{2n}(\boldsymbol{\alpha})=\psi^{+}_{2}(x)\psi^{l}_{2n}(\mathfrak{g})=\pm
u_{2}({\bf p})e^{-ipx}\boldsymbol{f}^{l}_{\frac{1}{2},\frac{1}{2}}(\mbox{\rm
Re}\,r)\mathfrak{M}^{l}_{-\frac{1}{2},n}(\varphi,\epsilon,\theta,\tau,0,0),$
$\displaystyle\dot{\psi}^{\dot{l}}_{1\dot{n}}(\boldsymbol{\alpha})=\psi^{-}_{1}(x)\dot{\psi}^{\dot{l}}_{1\dot{n}}(\mathfrak{g})=\mp
v_{1}({\bf
p})e^{ipx}\boldsymbol{f}^{\dot{l}}_{\frac{1}{2},-\frac{1}{2}}(\mbox{\rm
Re}\,r^{\ast})\mathfrak{M}^{\dot{l}}_{\frac{1}{2},\dot{n}}(\varphi,\epsilon,\theta,\tau,0,0),$
$\displaystyle\dot{\psi}^{\dot{l}}_{2\dot{n}}(\boldsymbol{\alpha})=\psi^{-}_{2}(x)\dot{\psi}^{\dot{l}}_{2\dot{n}}(\mathfrak{g})=v_{2}({\bf
p})e^{ipx}\boldsymbol{f}^{\dot{l}}_{\frac{1}{2},-\frac{1}{2}}(\mbox{\rm
Re}\,r^{\ast})\mathfrak{M}^{\dot{l}}_{-\frac{1}{2},\dot{n}}(\varphi,\epsilon,\theta,\tau,0,0),$

where

$\displaystyle u_{1}({\bf
p})=\left(\frac{E+m}{2m}\right)^{1/2}\begin{pmatrix}1\\\ 0\\\
\frac{p_{z}}{E+m}\\\ \frac{p_{+}}{E+m}\end{pmatrix},\quad u_{2}({\bf
p})=\left(\frac{E+m}{2m}\right)^{1/2}\begin{pmatrix}0\\\ 1\\\
\frac{p_{-}}{E+m}\\\ \frac{-p_{z}}{E+m}\end{pmatrix},$ $\displaystyle
v_{1}({\bf
p})=\left(\frac{E+m}{2m}\right)^{1/2}\begin{pmatrix}\frac{p_{z}}{E+m}\\\
\frac{p_{+}}{E+m}\\\ 1\\\ 0\end{pmatrix},\quad v_{2}({\bf
p})=\left(\frac{E+m}{2m}\right)^{1/2}\begin{pmatrix}\frac{p_{-}}{E+m}\\\
\frac{-p_{z}}{E+m}\\\ 0\\\ 1\end{pmatrix},$

here $p_{\pm}=p_{x}\pm ip_{y}$. Radial functions have the form

$\boldsymbol{f}^{l}_{\frac{1}{2},\frac{1}{2}}(\mbox{\rm
Re}\,r)=C_{1}\sqrt{\kappa^{c}\dot{\kappa}^{c}}\mbox{\rm
Re}\,rJ_{l}\left(\sqrt{\kappa^{c}\dot{\kappa}^{c}}\mbox{\rm
Re}\,r\right)+C_{2}\sqrt{\kappa^{c}\dot{\kappa}^{c}}\mbox{\rm
Re}\,rJ_{-l}\left(\sqrt{\kappa^{c}\dot{\kappa}^{c}}\mbox{\rm Re}\,r\right),$
$\boldsymbol{f}^{\dot{l}}_{\frac{1}{2},-\frac{1}{2}}(\mbox{\rm
Re}\,r^{\ast})=\frac{C_{1}}{2}\sqrt{\frac{\dot{\kappa}^{c}}{\kappa^{c}}}\mbox{\rm
Re}\,r^{\ast}J_{l+1}\left(\sqrt{\kappa^{c}\dot{\kappa}^{c}}\mbox{\rm
Re}\,r^{\ast}\right)-\frac{C_{2}}{2}\sqrt{\frac{\dot{\kappa}^{c}}{\kappa^{c}}}\mbox{\rm
Re}\,r^{\ast}J_{-l-1}\left(\sqrt{\kappa^{c}\dot{\kappa}^{c}}\mbox{\rm
Re}\,r^{\ast}\right),$

where $J_{l}\left(\sqrt{\kappa^{c}\dot{\kappa}^{c}}\mbox{\rm Re}\,r\right)$
are the Bessel functions of half-integer order, and

$\displaystyle l=\frac{1}{2},\;\frac{3}{2},\;\frac{5}{2},\ldots;$
$\displaystyle\dot{l}=\frac{1}{2},\;\frac{3}{2},\;\frac{5}{2},\ldots;$
$\mathfrak{M}_{l}^{\pm\frac{1}{2}}(\varphi,\epsilon,\theta,\tau,0,0)=e^{\mp\frac{1}{2}(\epsilon+i\varphi)}\mathfrak{Z}_{l}^{\pm\frac{1}{2}}(\theta,\tau),$
$\mathfrak{Z}_{l}^{\pm\frac{1}{2}}(\theta,\tau)=\cos^{2l}\frac{\theta}{2}\cosh^{2l}\frac{\tau}{2}\sum^{l}_{k=-l}i^{\pm\frac{1}{2}-k}\tan^{\pm\frac{1}{2}-k}\frac{\theta}{2}\tanh^{-k}\frac{\tau}{2}\times\\\
\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx<EMAIL_ADDRESS>\lx@intercol\hfil\begin{array}[]{c}\pm\frac{1}{2}-k+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{i^{2}\tan^{2}\frac{\theta}{2}}\right)$}\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx<EMAIL_ADDRESS>\lx@intercol\hfil\begin{array}[]{c}-k+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{\tanh^{2}\frac{\tau}{2}}\right)$},$
(52)
$\mathfrak{M}_{\dot{l}}^{\pm\frac{1}{2}}(\varphi,\epsilon,\theta,\tau,0,0)=e^{\mp\frac{1}{2}(\epsilon-i\varphi)}\mathfrak{Z}_{\dot{l}}^{\pm\frac{1}{2}}(\theta,\tau),$
$\mathfrak{Z}_{\dot{l}}^{\pm\frac{1}{2}}(\theta,\tau)=\cos^{2\dot{l}}\frac{\theta}{2}\cosh^{2\dot{l}}\frac{\tau}{2}\sum^{\dot{l}}_{\dot{k}=-\dot{l}}i^{\pm\frac{1}{2}-\dot{k}}\tan^{\pm\frac{1}{2}-\dot{k}}\frac{\theta}{2}\tanh^{-\dot{k}}\frac{\tau}{2}\times\\\
\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx<EMAIL_ADDRESS>\lx@intercol\hfil\begin{array}[]{c}\pm\frac{1}{2}-\dot{k}+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{i^{2}\tan^{2}\frac{\theta}{2}}\right)$}\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx<EMAIL_ADDRESS>\lx@intercol\hfil\begin{array}[]{c}-\dot{k}+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{\tanh^{2}\frac{\tau}{2}}\right)$}$
(53)

are the associated hyperspherical functions defined on the spheres $S^{c}_{2}$
and $\dot{S}^{c}_{2}$. General solutions are found in the form of generalized
Fourier integrals

$\displaystyle\psi_{1}(\boldsymbol{\alpha})$ $\displaystyle=$
$\displaystyle\sum^{\infty}_{l=\frac{1}{2}}\boldsymbol{f}^{l}_{\frac{1}{2},\frac{1}{2}}(\mbox{\rm
Re}\,r)\sum^{l}_{n=-l}\int\limits_{T_{4}}u_{1}({\bf
p})e^{ipx}\alpha^{\frac{1}{2}}_{l,n}\mathfrak{M}^{l}_{\frac{1}{2},n}(\varphi,\epsilon,\theta,\tau,0,0)d^{4}x,$
$\displaystyle\psi_{2}(\boldsymbol{\alpha})$ $\displaystyle=$
$\displaystyle\pm\sum^{\infty}_{l=\frac{1}{2}}\boldsymbol{f}^{l}_{\frac{1}{2},\frac{1}{2}}(\mbox{\rm
Re}\,r)\sum^{l}_{n=-l}\int\limits_{T_{4}}u_{2}({\bf
p})e^{ipx}\alpha^{-\frac{1}{2}}_{l,n}\mathfrak{M}^{l}_{-\frac{1}{2},n}(\varphi,\epsilon,\theta,\tau,0,0)d^{4}x,$
$\displaystyle\dot{\psi}_{1}(\boldsymbol{\alpha})$ $\displaystyle=$
$\displaystyle\mp\sum^{\infty}_{\dot{l}=\frac{1}{2}}\boldsymbol{f}^{\dot{l}}_{\frac{1}{2},-\frac{1}{2}}(\mbox{\rm
Re}\,r^{\ast})\sum^{\dot{l}}_{\dot{n}=-\dot{l}}\int\limits_{T_{4}}v_{1}({\bf
p})e^{-ipx}\alpha^{\frac{1}{2}}_{\dot{l},\dot{n}}\mathfrak{M}^{\dot{l}}_{\frac{1}{2},\dot{n}}(\varphi,\epsilon,\theta,\tau,0,0)d^{4}x,$
$\displaystyle\dot{\psi}_{2}(\boldsymbol{\alpha})$ $\displaystyle=$
$\displaystyle\sum^{\infty}_{\dot{l}=\frac{1}{2}}\boldsymbol{f}^{\dot{l}}_{\frac{1}{2},-\frac{1}{2}}(\mbox{\rm
Re}\,r^{\ast})\sum^{\dot{l}}_{\dot{n}=-\dot{l}}\int\limits_{T_{4}}v_{2}({\bf
p})e^{-ipx}\alpha^{-\frac{1}{2}}_{\dot{l},\dot{n}}\mathfrak{M}^{\dot{l}}_{-\frac{1}{2},\dot{n}}(\varphi,\epsilon,\theta,\tau,0,0)d^{4}x,$

where

$\displaystyle\alpha^{\pm\frac{1}{2}}_{l,n}$ $\displaystyle=$
$\displaystyle\frac{(-1)^{n}(2l+1)(2\dot{l}+1)}{32\pi^{4}\boldsymbol{f}^{l}_{\frac{1}{2},\frac{1}{2}}(\mbox{\rm
Re}\,a)}\int\limits_{S^{2}_{c}}\int\limits_{T_{4}}F_{\pm\frac{1}{2}}(\boldsymbol{\alpha})e^{-ipx}\mathfrak{M}^{l}_{\pm\frac{1}{2},n}(\varphi,\epsilon,\theta,\tau,0,0)d^{4}xd^{4}\mathfrak{g},$
$\displaystyle\alpha^{\pm\frac{1}{2}}_{\dot{l},\dot{n}}$ $\displaystyle=$
$\displaystyle\frac{(-1)^{\dot{n}}(2l+1)(2\dot{l}+1)}{32\pi^{4}\boldsymbol{f}^{\dot{l}}_{\frac{1}{2},-\frac{1}{2}}(\mbox{\rm
Re}\,a^{\ast})}\int\limits_{S^{2}_{c}}\int\limits_{T_{4}}\dot{F}_{\pm\frac{1}{2}}(\boldsymbol{\alpha})e^{ipx}\mathfrak{M}^{\dot{l}}_{\pm\frac{1}{2},\dot{n}}(\varphi,\epsilon,\theta,\tau,0,0)d^{4}xd^{4}\mathfrak{g}.$

###  3.2 The Maxwell field

In accordance with the general Bose-scheme of the interlocking representations
of $\mathfrak{G}_{+}$ (see Fig. 1), the field $(1,0)\oplus(0,1)$ is defined
within the following interlocking scheme:

$\dgARROWLENGTH=2.5em\begin{diagram}.$

We start with the Lagrangian (19) on the group manifold ${\cal M}_{10}$. Let
us rewrite (19) in the form

$\mathcal{L}(\boldsymbol{\alpha})=-\frac{1}{2}\left(\overline{\boldsymbol{\phi}}(\boldsymbol{\alpha})\Gamma_{\mu}\frac{\partial\boldsymbol{\phi}(\boldsymbol{\alpha})}{\partial
x_{\mu}}-\frac{\partial\overline{\boldsymbol{\phi}}(\boldsymbol{\alpha})}{\partial
x_{\mu}}\Gamma_{\mu}\boldsymbol{\phi}(\boldsymbol{\alpha})\right)-\\\
-\frac{1}{2}\left(\overline{\boldsymbol{\phi}}(\boldsymbol{\alpha})\Upsilon_{\nu}\frac{\partial\boldsymbol{\phi}(\boldsymbol{\alpha})}{\partial\mathfrak{g}_{\nu}}-\frac{\partial\overline{\boldsymbol{\phi}}(\boldsymbol{\alpha})}{\partial\mathfrak{g}_{\nu}}\Upsilon_{\nu}\boldsymbol{\phi}(\boldsymbol{\alpha})\right),$
(54)

where $\boldsymbol{\phi}(\boldsymbol{\alpha})=\phi(x)\phi(\mathfrak{g})$
($\mu=0,1,2,3,\;\nu=1,\ldots,6$), and

$\Gamma_{0}=\begin{pmatrix}0&I\\\
I&0\end{pmatrix},\;\;\Gamma_{1}=\begin{pmatrix}0&-\alpha_{1}\\\
\alpha_{1}&0\end{pmatrix},\;\;\Gamma_{2}=\begin{pmatrix}0&-\alpha_{2}\\\
\alpha_{2}&0\end{pmatrix},\;\;\Gamma_{3}=\begin{pmatrix}0&-\alpha_{3}\\\
\alpha_{3}&0\end{pmatrix},$ (55)
$\Upsilon_{1}=\begin{pmatrix}0&\Lambda^{\ast}_{1}\\\
\Lambda_{1}&0\end{pmatrix},\quad\Upsilon_{2}=\begin{pmatrix}0&\Lambda^{\ast}_{2}\\\
\Lambda_{2}&0\end{pmatrix},\quad\Upsilon_{3}=\begin{pmatrix}0&\Lambda^{\ast}_{3}\\\
\Lambda_{3}&0\end{pmatrix},$ (56)
$\Upsilon_{4}=\begin{pmatrix}0&i\Lambda^{\ast}_{1}\\\
i\Lambda_{1}&0\end{pmatrix},\quad\Upsilon_{5}=\begin{pmatrix}0&i\Lambda^{\ast}_{2}\\\
i\Lambda_{2}&0\end{pmatrix},\quad\Upsilon_{6}=\begin{pmatrix}0&i\Lambda^{\ast}_{3}\\\
i\Lambda_{3}&0\end{pmatrix},$ (57)

where

$\alpha_{1}=\begin{pmatrix}0&0&0\\\ 0&0&i\\\
0&-i&0\end{pmatrix},\quad\alpha_{2}=\begin{pmatrix}0&0&-i\\\ 0&0&0\\\
i&0&0\end{pmatrix},\quad\alpha_{3}=\begin{pmatrix}0&i&0\\\ -i&0&0\\\
0&0&0\end{pmatrix},$ (58)

and the matrices $\Lambda_{j}$ and $\Lambda^{\ast}_{j}$ are derived from
(24)–(26) and (27)–(29) at $l=1$:

$\Lambda_{1}=\frac{c_{11}}{\sqrt{2}}\begin{pmatrix}0&1&0\\\ 1&0&0\\\
0&1&0\end{pmatrix},\quad\Lambda_{2}=\frac{c_{11}}{\sqrt{2}}\begin{pmatrix}0&-i&0\\\
i&0&-i\\\ 0&i&0\end{pmatrix},\quad\Lambda_{3}=c_{11}\begin{pmatrix}1&0&0\\\
0&0&0\\\ 0&0&-1\end{pmatrix}.$ (59)
$\Lambda^{\ast}_{1}=\frac{\sqrt{2}}{2}\dot{c}_{11}\begin{pmatrix}0&1&0\\\
1&0&1\\\
0&1&0\end{pmatrix},\quad\Lambda^{\ast}_{2}=\frac{\sqrt{2}}{2}\dot{c}_{11}\begin{pmatrix}0&-i&0\\\
i&0&-i\\\
0&i&0\end{pmatrix},\quad\Lambda^{\ast}_{3}=\dot{c}_{11}\begin{pmatrix}1&0&0\\\
0&0&0\\\ 0&0&-1\end{pmatrix}.$ (60)

At this point, electromagnetic field should be defined in the Riemann-
Silberstein representation [69, 57, 10]. The Riemann-Silberstein (Majorana-
Oppenheimer) representation considered during long time by many authors [40,
46, 28, 42, 51, 41, 13, 23]. The interest to this formulation of
electrodynamics has grown in recent years [32, 58, 22, 17]. One of the main
advantages of this approach lies in the fact that Dirac and Maxwell fields are
derived similarly from the Dirac-like Lagrangians. These fields have the
analogous mathematical structure, namely, they are the functions on the
Poincaré group. This circumstance allows us to consider the fields
$(1/2,0)\oplus(0,1/2)$ and $(1,0)\oplus(0,1)$ on an equal footing, from the
one group theoretical viewpoint.

In 1949, Newton and Wigner [44] showed that for the photon there exist no
localized states. Therefore, a massless field $(1,0)\oplus(0,1)$ should be
considered within the unitary infinite-dimensional representation of the
Lorentz group. In accordance with the Naimark theorem [43], the representation
$S_{k}$ of the subgroup $\operatorname{SU}(2)$ is contained in
$\mathfrak{S}_{\lambda,\rho}$ not more then one time. Matrix elements of the
representation $\mathfrak{S}_{\lambda,\rho}$ are defined by the functions (7).
_We suppose that the photon field is described within the infinite-dimensional
representation $\mathfrak{S}_{\lambda,\rho}$_. This representation contains a
finite-dimensional representation associated with the field
$(1,0)\oplus(0,1)$.

Since longitudinal solutions
$\phi^{-\frac{1}{2}+i\rho,l_{0}}_{0,n}(\boldsymbol{\alpha})$ and
$\dot{\phi}^{-\frac{1}{2}-i\rho,l_{0}}_{0,\dot{n}}(\boldsymbol{\alpha})$ do
not contribute to a real photon due to their transversality conditions, then
particular solutions for the Majorana-Oppenheimer bispinor
$\boldsymbol{\phi}=(\phi_{1},\phi_{2},\phi_{3},\dot{\phi}_{1},\dot{\phi}_{2},\dot{\phi}_{3})^{T}$
on ${\cal M}_{8}$ are defined by the following expressions [64]:

$\phi^{-\frac{1}{2}+i\rho,l_{0}}_{\pm
1,n}(\boldsymbol{\alpha})=\phi_{\pm}({\bf k};{\bf
x},t)\phi^{-\frac{1}{2}+i\rho,l_{0}}_{\pm 1,n}(\mathfrak{g})=\\\
\left\\{2(2\pi)^{3}\right\\}^{-\frac{1}{2}}\begin{pmatrix}\varepsilon_{\pm}({\bf
k})\\\ \varepsilon_{\pm}({\bf k})\end{pmatrix}\exp[i({\bf k}\cdot{\bf
x}-\omega t)]\boldsymbol{f}^{-\frac{1}{2}+i\rho,l_{0}}_{1,\pm
1}(r)\mathfrak{M}_{\pm
1,n}^{-\frac{1}{2}+i\rho,l_{0}}(\varphi,\epsilon,\theta,\tau,0,0),$ (61)
$\dot{\phi}^{-\frac{1}{2}-i\rho,\dot{l}_{0}}_{\pm
1,\dot{n}}(\boldsymbol{\alpha})=\phi^{\ast}_{\pm}({\bf k};{\bf
x},t)\dot{\phi}^{-\frac{1}{2}-i\rho,\dot{l}_{0}}_{\pm
1,\dot{n}}(\mathfrak{g})=\\\
\left\\{2(2\pi)^{3}\right\\}^{-\frac{1}{2}}\begin{pmatrix}\varepsilon^{\ast}_{\pm}({\bf
k})\\\ \varepsilon^{\ast}_{\pm}({\bf k})\end{pmatrix}\exp[-i({\bf k}\cdot{\bf
x}-\omega t)]\boldsymbol{f}^{-\frac{1}{2}-i\rho,\dot{l}_{0}}_{1,\pm
1}(r^{\ast})\mathfrak{M}_{\pm
1,\dot{n}}^{-\frac{1}{2}-i\rho,\dot{l}_{0}}(\varphi,\epsilon,\theta,\tau,0,0),$
(62)

where

$\displaystyle\varepsilon_{\pm}({\bf k})$ $\displaystyle=$
$\displaystyle\left\\{2|{\bf
k}|^{2}(k^{2}_{1}+k^{2}_{2})\right\\}^{-\frac{1}{2}}\begin{bmatrix}-k_{1}k_{3}\pm
ik_{2}|{\bf k}|\\\ -k_{2}k_{3}\mp ik_{1}|{\bf k}|\\\
k^{2}_{1}+k^{2}_{2}\end{bmatrix}$

are the polarization vectors of a photon. Radial functions have the form

$\displaystyle\boldsymbol{f}^{-\frac{1}{2}+i\rho,l_{0}}_{1,1}(r)$
$\displaystyle=$ $\displaystyle
C\sqrt{r}+\sqrt{\left(l_{0}+i\rho+\frac{1}{2}\right)\left(\frac{l_{0}}{2}+i\frac{\rho}{2}+\frac{5}{4}\right)}r,$
$\displaystyle\boldsymbol{f}^{-\frac{1}{2}-i\rho,\dot{l}_{0}}_{1,1}(r^{\ast})$
$\displaystyle=$
$\displaystyle\dot{C}\sqrt{r^{\ast}}+\sqrt{\left(\dot{l}_{0}-i\rho+\frac{3}{2}\right)\left(\frac{\dot{l}_{0}}{2}-i\frac{\rho}{2}+\frac{7}{4}\right)}r^{\ast},$

and

$\mathfrak{M}^{-\frac{1}{2}+i\rho,l_{0}}_{\pm
1,n}(\varphi,\epsilon,\theta,\tau,0,0)=e^{\mp(\epsilon+i\varphi)}\mathfrak{Z}^{-\frac{1}{2}+i\rho,l_{0}}_{\pm
1,n}(\theta,\tau),$ $\mathfrak{Z}^{-\frac{1}{2}+i\rho,l_{0}}_{\pm
1,n}(\theta,\tau)=\cos^{2l_{0}}\frac{\theta}{2}\cosh^{-1+2i\rho,l_{0}}\frac{\tau}{2}\sum^{l_{0}}_{k=-l_{0}}i^{\pm
1-k}\tan^{\pm 1-k}\frac{\theta}{2}\tanh^{n-k}\frac{\tau}{2}\times\\\
\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx@intercol\hfil\begin{array}[]{c}\pm
<EMAIL_ADDRESS>\lx@intercol\hfil\begin{array}[]{c}\pm
1-k+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{i^{2}\tan^{2}\frac{\theta}{2}}\right)$}\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx<EMAIL_ADDRESS>\lx@intercol\hfil\begin{array}[]{c}n-k+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{\tanh^{2}\frac{\tau}{2}}\right)$},$
(63) $\mathfrak{M}^{-\frac{1}{2}-i\rho,l_{0}}_{\pm
1,\dot{n}}(\varphi,\epsilon,\theta,\tau,0,0)=e^{\mp(\epsilon-i\varphi)}\mathfrak{Z}^{-\frac{1}{2}-i\rho,l_{0}}_{\pm
1,\dot{n}}(\theta,\tau),$ $\mathfrak{Z}^{-\frac{1}{2}-i\rho,l_{0}}_{\pm
1,\dot{n}}(\theta,\tau)=\cos^{2l_{0}}\frac{\theta}{2}\cosh^{-1-2i\rho,l_{0}}\frac{\tau}{2}\sum^{l_{0}}_{\dot{k}=-l_{0}}i^{\pm
1-\dot{k}}\tan^{\pm
1-\dot{k}}\frac{\theta}{2}\tanh^{\dot{n}-\dot{k}}\frac{\tau}{2}\times\\\
\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx@intercol\hfil\begin{array}[]{c}\pm
<EMAIL_ADDRESS>\lx@intercol\hfil\begin{array}[]{c}\pm
1-\dot{k}+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{i^{2}\tan^{2}\frac{\theta}{2}}\right)$}\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx<EMAIL_ADDRESS>\lx@intercol\hfil\begin{array}[]{c}\dot{n}-\dot{k}+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{\tanh^{2}\frac{\tau}{2}}\right)$}$
(64)

are associated hyperspherical functions defined on the spheres $S^{2}_{c}$ and
$\dot{S}^{2}_{c}$. General solutions are found via the following generalized
Fourier integrals:

$\boldsymbol{\phi}_{\pm
1}(\boldsymbol{\alpha})=\left\\{2(2\pi)^{3}\right\\}^{-\frac{1}{2}}\sum^{\infty}_{l_{0}=1}\boldsymbol{f}^{-\frac{1}{2}+i\rho,l_{0}}_{1,\pm
1}(r)\sum^{l_{0}}_{n=-l_{0}}\int\limits_{T_{4}}\begin{pmatrix}\varepsilon_{\pm}({\bf
k})\\\ \varepsilon_{\pm}({\bf k})\end{pmatrix}e^{ikx}\alpha^{\pm
1}_{l_{0},n}\mathfrak{M}_{\pm
1,n}^{-\frac{1}{2}+i\rho,l_{0}}(\varphi,\epsilon,\theta,\tau,0,0)d^{4}x,$
$\dot{\boldsymbol{\phi}}_{\pm
1}(\boldsymbol{\alpha})=\left\\{2(2\pi)^{3}\right\\}^{-\frac{1}{2}}\sum^{\infty}_{l_{0}=1}\boldsymbol{f}^{-\frac{1}{2}-i\rho,\dot{l}_{0}}_{1,\pm
1}(r^{\ast})\sum^{l_{0}}_{\dot{n}=-l_{0}}\int\limits_{T_{4}}\begin{pmatrix}\varepsilon^{\ast}_{\pm}({\bf
k})\\\ \varepsilon^{\ast}_{\pm}({\bf k})\end{pmatrix}e^{-ikx}\alpha^{\pm
1}_{l_{0},\dot{n}}\mathfrak{M}_{\pm
1,\dot{n}}^{-\frac{1}{2}-i\rho,l_{0}}(\varphi,\epsilon,\theta,\tau,0,0)d^{4}x,$

where

$\displaystyle\alpha^{\pm 1}_{l_{0},n}$ $\displaystyle=$
$\displaystyle\frac{(-1)^{n}(2l_{0}+1)^{2}}{32\pi^{4}\boldsymbol{f}^{-\frac{1}{2}+i\rho,l_{0}}_{1,\pm
1}(a)}\int\limits_{S^{2}_{c}}\int\limits_{T_{4}}F_{\pm
1}(\boldsymbol{\alpha})e^{-ikx}\mathfrak{M}^{-\frac{1}{2}+i\rho,l_{0}}_{\pm
1,n}(\varphi,\epsilon,\theta,\tau,0,0)d^{4}xd^{4}\mathfrak{g},$
$\displaystyle\alpha^{\pm 1}_{l_{0},\dot{n}}$ $\displaystyle=$
$\displaystyle\frac{(-1)^{\dot{n}}(2l_{0}+1)^{2}}{32\pi^{4}\boldsymbol{f}^{-\frac{1}{2}-i\rho,\dot{l}_{0}}_{1,\pm
1}(a^{\ast})}\int\limits_{S^{2}_{c}}\int\limits_{T_{4}}\dot{F}_{\pm
1}(\boldsymbol{\alpha})e^{ikx}\mathfrak{M}^{-\frac{1}{2}-i\rho,l_{0}}_{\pm
1,\dot{n}}(\varphi,\epsilon,\theta,\tau,0,0)d^{4}xd^{4}\mathfrak{g}.$

##  4 Interaction

Up to now we analyze free Dirac and Maxwell fields. Let us consider an
interaction between these fields. As usual, interactions between the fields
are described by an interaction Lagrangian $\mathcal{L}_{I}$. In our case we
take the following Lagrangian [66]:

$\mathcal{L}_{I}(\boldsymbol{\alpha})=\mu(\overline{\boldsymbol{\psi}}(\boldsymbol{\alpha})\sigma^{D}_{(\mu\nu)^{k}}\boldsymbol{\psi}(\boldsymbol{\alpha}))(\Xi^{M}_{(\rho)k}\boldsymbol{\phi}(\boldsymbol{\alpha})),$
(65)

where
$\sigma^{D}_{\mu\nu}=\frac{1}{2}(\Xi^{D}_{\mu}\Xi^{D}_{\nu}-\Xi^{D}_{\nu}\Xi^{D}_{\mu})$
and
$\Xi^{D}=(\Gamma^{D}_{0},\Gamma^{D}_{1},\Gamma^{D}_{2},\Gamma^{D}_{3},\Upsilon^{D}_{1},\Upsilon^{D}_{2},\Upsilon^{D}_{3},\Upsilon^{D}_{4},\Upsilon^{D}_{5},\Upsilon^{D}_{6})$,
$\Xi^{M}=(\Gamma^{M}_{0},\Gamma^{M}_{1},\Gamma^{M}_{2},\Gamma^{M}_{3},\Upsilon^{M}_{1},\Upsilon^{M}_{2},\Upsilon^{M}_{3},\Upsilon^{M}_{4},\Upsilon^{M}_{5},\Upsilon^{M}_{6})$,
here $\Gamma^{D}$ and $\Upsilon^{D}$ are the matrices (48) and (49)–(50), and
$\Gamma^{M}$ and $\Upsilon^{M}$ are the matrices (55) and (56)–(57).

The full Lagrangian of interacting Dirac and Maxwell fields equals to a sum of
the free field Lagrangians and the interaction Lagrangian:

$\mathcal{L}(\boldsymbol{\alpha})=\mathcal{L}_{D}(\boldsymbol{\alpha})+\mathcal{L}_{M}(\boldsymbol{\alpha})+\mathcal{L}_{I}(\boldsymbol{\alpha}),$

where $\mathcal{L}_{D}(\boldsymbol{\alpha})$ and
$\mathcal{L}_{M}(\boldsymbol{\alpha})$ are of the type (47) and (54),
respectively. Or,

$\mathcal{L}(\boldsymbol{\alpha})=-\frac{1}{2}\left(\overline{\boldsymbol{\psi}}(\boldsymbol{\alpha})\Xi^{D}_{\mu}\frac{\partial\boldsymbol{\psi}(\boldsymbol{\alpha})}{\partial\boldsymbol{\alpha}_{\mu}}-\frac{\partial\overline{\boldsymbol{\psi}}(\boldsymbol{\alpha})}{\partial\boldsymbol{\alpha}_{\mu}}\Xi^{D}_{\mu}\boldsymbol{\psi}(\boldsymbol{\alpha})\right)-\\\
-\frac{1}{2}\left(\overline{\boldsymbol{\phi}}(\boldsymbol{\alpha})\Xi^{M}_{\mu}\frac{\partial\boldsymbol{\phi}(\boldsymbol{\alpha})}{\partial\boldsymbol{\alpha}_{\mu}}-\frac{\partial\overline{\boldsymbol{\phi}}(\boldsymbol{\alpha})}{\partial\boldsymbol{\alpha}_{\mu}}\Xi^{M}_{\mu}\boldsymbol{\phi}(\boldsymbol{\alpha})\right)-\\\
-\kappa\overline{\boldsymbol{\psi}}(\boldsymbol{\alpha})\boldsymbol{\psi}(\boldsymbol{\alpha})+\mu(\overline{\boldsymbol{\psi}}(\boldsymbol{\alpha})\sigma^{D}_{(\mu\nu)^{k}}\boldsymbol{\psi}(\boldsymbol{\alpha}))(\Xi^{M}_{(\rho)k}\boldsymbol{\phi}(\boldsymbol{\alpha})).$
(66)

Since the Lagrangian (65) does not contain derivatives on the field functions,
then for a Hamiltonian density we have ${\cal
H}_{I}(\boldsymbol{\alpha})=-\mathcal{L}_{I}(\boldsymbol{\alpha})$.

As is known, in the standard quantum field theory the $S$-matrix is expressed
via the Dyson formula [53]

$S=T\left[\exp\left(-\frac{i}{\hbar c}\int\limits^{+\infty}_{-\infty}{\cal
H}_{I}(x)d^{4}x\right)\right],$ (67)

where $T$ is the time ordering operator.

In our case, the electron-positron and photon fields are defined on the space
${\cal M}_{8}=\bb R^{1,3}\times S^{2}$ which larger then the Minkowski space
$\bb R^{1,3}$. With a view to define a formula similar to the equation (67) it
is necessarily to replace $d^{4}x$ by the following invariant measure on
${\cal M}_{8}$:

$d^{8}\mu=d^{4}xd^{4}\mathfrak{g},$

where

$d^{4}\mathfrak{g}=\sin\theta^{c}d\theta d\tau d\varphi d\epsilon.$

Therefore, an analogue of the Dyson formula (67) on the manifold ${\cal
M}_{8}$ can be written as follows

$S=T\left[\exp\left(-\frac{i}{\hbar
c}\int\limits_{T_{4}}\int\limits_{S^{2}_{c}}{\cal
H}_{I}(\boldsymbol{\alpha})d^{4}xd^{4}\mathfrak{g}\right)\right].$

Let us consider an integral on trilinear form, defining the interaction, over
the two-dimensional complex sphere:

$\int\limits_{S^{2}_{c}}{\cal
H}_{I}(\boldsymbol{\alpha})dS^{2}_{c}\sim\int\limits_{S^{2}_{c}}\mathfrak{M}^{\dot{m}_{e}}_{\dot{l}_{e}}(\varphi,\epsilon,\theta,\tau,0,0)\mathfrak{M}^{m_{e}}_{l_{e}}(\varphi,\epsilon,\theta,\tau,0,0)\mathfrak{M}^{m_{f}}_{-\frac{1}{2}+i\rho}(\varphi,\epsilon,\theta,\tau,0,0)\sin\theta^{c}d\theta^{c}d\varphi^{c},$

that is, to investigate convergence of the integral

$I_{1}=\int\limits^{+\infty}_{-\infty}\int\limits^{+\infty}_{-\infty}e^{-i(m_{e}\varphi^{c}-\dot{m}_{e}\dot{\varphi}^{c}-m_{f}\varphi^{c})}\mathfrak{Z}^{\dot{m}_{e}}_{\dot{l}_{e}}(\cos\dot{\theta}^{c})\mathfrak{Z}^{m_{e}}_{l_{e}}(\cos\theta^{c})\mathfrak{Z}^{m_{f}}_{-\frac{1}{2}+i\rho}(\cos\theta^{c})\sin\theta^{c}d\theta^{c}d\varphi^{c}.$

Here the symbols $l_{e}$, $m_{e}$ correspond to the electron field
$\boldsymbol{\psi}(\mathfrak{g})$, $\dot{l}_{e}$, $\dot{m}_{e}$ correspond to
the positron field $\overline{\boldsymbol{\psi}(\mathfrak{g})}$, and $m_{f}$
corresponds to the photon field $\boldsymbol{\phi}(\mathfrak{g})$. It is
obvious that convergence of the integral

$I_{2}=\int\limits^{+\infty}_{-\infty}e^{-i(m_{e}\varphi^{c}-\dot{m}_{e}\dot{\varphi}^{c}-m_{f}\varphi^{c})}d\varphi^{c}$

is not difficult to investigate. Let us consider the following integral:

$I_{3}=\int\limits^{+\infty}_{-\infty}\mathfrak{Z}^{\dot{m}_{e}}_{\dot{l}_{e}}(\cos\dot{\theta}^{c})\mathfrak{Z}^{m_{e}}_{l_{e}}(\cos\theta^{c})\mathfrak{Z}^{m_{f}}_{-\frac{1}{2}+i\rho}(\cos\theta^{c})\sin\theta^{c}d\theta^{c}.$

Rewriting hyperspherical functions via the hypergeometric functions, we obtain

$I_{3}=\frac{i^{m_{e}+\dot{m}_{e}+m_{f}}}{\Gamma(m_{e}+1)\Gamma(\dot{m}_{e}+1)\Gamma(m_{f}+1)}\sqrt{\frac{\Gamma(l_{e}+m_{e}+1)\Gamma(\dot{l}_{e}+\dot{m}_{e}+1)\Gamma(m_{f}+i\rho+\frac{1}{2})}{\Gamma(l_{e}-m_{e}+1)\Gamma(\dot{l}_{e}-\dot{m}_{e}+1)\Gamma(i\rho-
m_{f}+\frac{1}{2})}}\times\\\
\int\limits^{+\infty}_{-\infty}\sin^{\dot{m}_{e}+m_{e}+m_{f}}\frac{\theta^{c}}{2}\cos^{\dot{m}_{e}+m_{e}+m_{f}}\frac{\theta^{c}}{2}\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx<EMAIL_ADDRESS>\lx@intercol\hfil\begin{array}[]{c}\dot{m}_{e}+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{\sin^{2}\frac{\theta^{c}}{2}}\right)$}\times\\\
\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx<EMAIL_ADDRESS>\lx@intercol\hfil\begin{array}[]{c}m_{e}+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{\sin^{2}\frac{\theta^{c}}{2}}\right)$}\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx<EMAIL_ADDRESS>\lx@intercol\hfil\begin{array}[]{c}m_{f}+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{\sin^{2}\frac{\theta^{c}}{2}}\right)$}\sin\theta^{c}d\theta^{c}.$
(68)

Further, using an explicit expression for the associated hyperspherical
function

$\mathfrak{Z}^{m}_{l}(\cos\theta^{c})=i^{m}\sqrt{\frac{\Gamma(l-m+1)}{\Gamma(l+m+1)}}\cos^{m}\frac{\theta^{c}}{2}\sin^{m}\frac{\theta^{c}}{2}\times\\\
\sum^{l-m}_{j=0}\frac{(-1)^{j}\Gamma(l+m+j+1)}{\Gamma(j+1)\Gamma(m+j+1)\Gamma(l-m-j+1)}\sin^{2j}\frac{\theta^{c}}{2},$
(69)

we rewrite the first two hypergeometric functions in the integral (68). Then

$I_{3}=\frac{i^{m_{e}+\dot{m}_{e}+m_{f}}}{\Gamma(m_{f}+1)}\sqrt{\frac{\Gamma(l_{e}-m_{e}+1)\Gamma(\dot{l}_{e}-\dot{m}_{e}+1)\Gamma(m_{f}+i\rho+\frac{1}{2})}{\Gamma(l_{e}+m_{e}+1)\Gamma(\dot{l}_{e}+\dot{m}_{e}+1)\Gamma(i\rho-
m_{f}+\frac{1}{2})}}\times\\\
\sum^{l_{e}-m_{e}}_{j_{1}=0}\sum^{\dot{l}_{e}-\dot{m}_{e}}_{j_{2}=0}\frac{(-1)^{j_{1}+j_{2}}\Gamma(l_{e}+m_{e}+j_{1}+1)}{\Gamma(j_{1}+1)\Gamma(j_{2}+1)\Gamma(m_{e}+j_{1}+1)\Gamma(\dot{m}_{e}+j_{2}+1)}\times\\\
\int\limits^{+\infty}_{-\infty}\cos^{m_{e}+\dot{m}_{e}+m_{f}}\frac{\theta^{c}}{2}\sin^{m_{e}+\dot{m}_{e}+m_{f}+2j_{1}+2j_{2}}\frac{\theta^{c}}{2}\times\\\
\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx<EMAIL_ADDRESS>\lx@intercol\hfil\begin{array}[]{c}m_{f}+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{\sin^{2}\frac{\theta^{c}}{2}}\right)$}\sin\theta^{c}d\theta^{c}.$
(70)

Making the substitution $z=\cos\theta^{c}$ in the integral

$I_{4}=\frac{1}{2^{m_{e}+\dot{m}_{e}+m_{f}}}\int\limits^{+\infty}_{-\infty}\sin^{m_{e}+\dot{m}_{e}+m_{f}}\theta^{c}\sin^{2j_{1}+2j_{2}}\frac{\theta^{c}}{2}\times\\\
\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx<EMAIL_ADDRESS>\lx@intercol\hfil\begin{array}[]{c}m_{f}+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{\sin^{2}\frac{\theta^{c}}{2}}\right)$}\sin\theta^{c}d\theta^{c},$
(71)

we obtain

$I_{4}=\frac{1}{2^{m_{e}+\dot{m}_{e}+m_{f}+j_{1}+j_{2}}}\int\limits^{+\infty}_{-\infty}(1-z^{2})^{\frac{m_{e}+\dot{m}_{e}+m_{f}}{2}}\left(\frac{1-z}{2}\right)^{j_{1}+j_{2}}\times\\\
\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx<EMAIL_ADDRESS>\lx@intercol\hfil\begin{array}[]{c}m_{f}+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{\frac{1-z}{2}}\right)$}dz.$
(72)

Or,

$I_{4}=\frac{1}{2^{m_{e}+\dot{m}_{e}+m_{f}+j_{1}+j_{2}}}\sum^{m_{e}+\dot{m}_{e}+m_{f}}_{q=0}(-1)^{q}\frac{(m_{e}+\dot{m}_{e}+m_{f})!}{q!(m_{e}+\dot{m}_{e}+m_{f}-q)!}\times\\\
\int\limits^{+\infty}_{-\infty}z^{k}\left(\frac{1-z}{2}\right)^{j_{1}+j_{2}}\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx<EMAIL_ADDRESS>\lx@intercol\hfil\begin{array}[]{c}m_{f}+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{\frac{1-z}{2}}\right)$}dz.$
(73)

Introducing a new variable $t=(1-z)/2$, we find

$I_{4}=\frac{1}{2^{m_{e}+\dot{m}_{e}+m_{f}+j_{1}+j_{2}}}\sum^{m_{e}+\dot{m}_{e}+m_{f}}_{q=0}(-1)^{q+1}\frac{(m_{e}+\dot{m}_{e}+m_{f})!}{q!(m_{e}+\dot{m}_{e}+m_{f}-q)!}\times\\\
\int\limits^{+\infty}_{-\infty}(1-2t)^{q}t^{j_{1}+j_{2}}\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx<EMAIL_ADDRESS>\lx@intercol\hfil\begin{array}[]{c}m_{f}+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{t}\right)$}dt.$
(74)

Decomposing $(1-2t)^{q}$ via the Newton binomial, we obtain

$I_{4}=\frac{1}{2^{m_{e}+\dot{m}_{e}+m_{f}+j_{1}+j_{2}}}\sum^{m_{e}+\dot{m}_{e}+m_{f}}_{q=0}\sum^{q}_{p=0}(-1)^{q+p+1}\frac{(m_{e}+\dot{m}_{e}+m_{f})!}{(m_{e}+\dot{m}_{e}+m_{f}-q)!}\times\\\
\frac{2^{p}}{p!(k-p)!}\int\limits^{+\infty}_{-\infty}t^{j_{1}+j_{2}+p}\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx<EMAIL_ADDRESS>\lx@intercol\hfil\begin{array}[]{c}m_{f}+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{t}\right)$}dt.$
(75)

With the aim to calculate the latter integral we use the following formula
[49]:

$I_{5}=\int
t^{n}\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx<EMAIL_ADDRESS>\lx@intercol\hfil\begin{array}[]{c}c\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{t}\right)$}dt=\\\
=n!\sum^{n+1}_{k=1}(-1)^{k+1}\frac{(c-k)_{k}t^{n-k+1}}{(a-k+1)!(a-k)_{k}(b-k)_{k}}\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx<EMAIL_ADDRESS>\lx@intercol\hfil\begin{array}[]{c}c-k\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{t}\right)$}.$
(76)

Then

$I_{4}=\frac{1}{2^{m_{e}+\dot{m}_{e}+m_{f}+j_{1}+j_{2}}}\sum^{m_{e}+\dot{m}_{e}+m_{f}}_{q=0}\sum^{q}_{p=0}(j_{1}+j_{2}+p)!\times\\\
\sum^{j_{1}+j_{2}+p}_{k=1}\frac{(-1)^{q+p+k}2^{p}(m_{e}+\dot{m}_{e}+m_{f})!}{p!(k-p)!(m_{e}+\dot{m}_{e}+m_{f}-q)!(m_{f}+i\rho+\frac{3}{2}-k)!}\times\\\
\frac{(m_{f}-k+1)_{k}t^{j_{1}+j_{2}+p-k+1}}{(m_{f}+i\rho-k+\frac{1}{2})_{k}(m_{f}-k+1)_{k}}\times\\\
\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx<EMAIL_ADDRESS>\lx@intercol\hfil\begin{array}[]{c}m_{f}-k+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{t}\right)$}dt.$
(77)

With the aim to investigate the convergence of (68) let us apply the following
asymptotic expansion for the hypergeometric function [9]:

$\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx<EMAIL_ADDRESS>\lx@intercol\hfil\begin{array}[]{c}c\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{t}\right)$}=\frac{\Gamma(c)\Gamma(b-a)}{\Gamma(b)\Gamma(c-a)}(-t)^{-a}\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx<EMAIL_ADDRESS>\lx@intercol\hfil\begin{array}[]{c}1-b+a\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{\frac{1}{t}}\right)$}+\\\
+\frac{\Gamma(c)\Gamma(a-b)}{\Gamma(a)\Gamma(c-b)}(-t)^{-b}\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx<EMAIL_ADDRESS>\lx@intercol\hfil\begin{array}[]{c}1-a+b\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{\frac{1}{t}}\right)$}.$
(78)

Thus,

$I_{3}=\frac{i^{m_{e}+\dot{m}_{e}+m_{f}}}{\Gamma(m_{f}+1)}\sqrt{\frac{\Gamma(l_{e}-m_{e}+1)\Gamma(\dot{l}_{e}-\dot{m}_{e}+1)\Gamma(m_{f}+i\rho+\frac{1}{2})}{\Gamma(l_{e}+m_{e}+1)\Gamma(\dot{l}_{e}+\dot{m}_{e}+1)\Gamma(i\rho-
m_{f}+\frac{1}{2})}}\times\\\
\sum^{l_{e}-m_{e}}_{j_{1}=0}\sum^{\dot{l}_{e}-\dot{m}_{e}}_{j_{2}=0}\frac{(-1)^{j_{1}+j_{2}}\Gamma(l_{e}+m_{e}+j_{1}+1)}{2^{m_{e}+\dot{m}_{e}+m_{f}+j_{1}+j_{2}}\Gamma(j_{1}+1)\Gamma(j_{2}+1)\Gamma(m_{e}+j_{1}+1)\Gamma(\dot{m}_{e}+j_{2}+1)}\times$
(79)
$\frac{\Gamma(\dot{l}_{e}+\dot{m}_{e}+j_{2}+1)}{\Gamma(l_{e}-m_{e}-j_{1}+1)\Gamma(\dot{l}_{e}-\dot{m}_{e}-j_{2}+1)}\times\\\
\sum^{m_{e}+\dot{m}_{e}+m_{f}}_{q=0}\sum^{q}_{p=0}(j_{1}+j_{2}+p)!\sum^{j_{1}+j_{2}+p}_{k=1}\frac{(-1)^{q+p+k}2^{p}(m_{e}+\dot{m}_{e}+m_{f})!}{p!(k-p)!(m_{e}+\dot{m}_{e}+m_{f}-q)!(m_{f}+i\rho+\frac{3}{2}-k)!}\times$
(80)
$\frac{(m_{f}-k+1)_{k}}{(m_{f}+i\rho-k+\frac{1}{2})_{k}(m_{f}-k+1)_{k}}\left[(-1)^{k-m_{f}-i\rho-\frac{1}{2}}\frac{\Gamma(m_{f}-k+1)\Gamma(-2i\rho)}{\Gamma(m_{f}-i\rho-k+\frac{1}{2})\Gamma(\frac{1}{2}-i\rho)}\times\right.\\\
t^{j_{1}+j_{2}+p-m_{f}-i\rho+\frac{1}{2}}\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx<EMAIL_ADDRESS>\lx@intercol\hfil\begin{array}[]{c}2i\rho+1\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{\frac{1}{t}}\right)$}+$
(81)
$\left.(-1)^{k-m_{f}+i\rho-\frac{1}{2}}\frac{\Gamma(m_{f}-k+1)\Gamma(2i\rho)}{\Gamma(m_{f}+i\rho-k+\frac{1}{2})\Gamma(\frac{1}{2}-i\rho)}\times\right.\\\
\left.t^{j_{1}+j_{2}+p-m_{f}+i\rho+\frac{1}{2}}\mbox{${}_{2}F_{1}\left.\\!\\!\left(\\!\\!\begin{array}[]{c}\lx<EMAIL_ADDRESS>\lx@intercol\hfil\begin{array}[]{c}1-2i\rho\end{array}\hfil\lx@intercol\end{array}\\!\\!\right|\displaystyle{\frac{1}{t}}\right)$}\right],$
(82)

where $t=\sin^{2}\frac{\theta^{c}}{2}$. Since $I_{3}\sim 1/t^{M}$ (the
hypergeometric function ${}_{2}F_{1}$ can be written as a power series in
$1/t$) and $M\rightarrow\infty$, then $I_{3}$ converges at
$M>j_{1}+j_{2}+p-m_{f}+i\rho+\frac{1}{2}$ and, therefore, in accordance with
(67) the elements of $S$-matrix are defined by convergent expressions.

##  5 Summary

We have proved convergence of quantum electrodynamics on the Poincaré group in
the case of homogeneous space ${\cal M}_{8}=\bb R^{1,3}\times S^{2}$. We
considered Dirac like equations for extended objects on ${\cal M}_{8}$ and
their particular solutions corresponding to the fields of spin 1/2 and 1. At
this point, the spin-1 field (Maxwell field) is defined within an infinite-
dimensional representation of the Lorentz group. We showed that an analogue of
the Dyson formula for $S$-matrix in the case of ${\cal M}_{8}$ is defined by
convergent integrals. It would be interesting to consider quantum field models
and their convergence on other homogeneous spaces of the Poincaré group, such
as ${\cal M}_{6}=\bb R^{1,3}\times S^{2}$, ${\cal M}_{7}=\bb R^{1,3}\times
H^{3}$ and ${\cal M}_{10}=\bb R^{1,3}\times\mathfrak{L}_{6}$. It would be
interesting also to consider wave equations and field models for the fields
$\boldsymbol{\psi}(\boldsymbol{\alpha})=\langle
x,\mathfrak{q}\,|\boldsymbol{\psi}\rangle$ on the de Sitter group, where $x\in
T_{5}$ and $\mathfrak{q}\in\operatorname{{\bf
Spin}}_{+}(1,4)\simeq\operatorname{Sp}(1,1)$, and for the fields
$\boldsymbol{\psi}(\boldsymbol{\alpha})=\langle
x,\mathfrak{c}\,|\boldsymbol{\psi}\rangle$ on the conformal group, where $x\in
T_{6}$ and $\mathfrak{c}\in\operatorname{{\bf
Spin}}_{+}(2,6)\simeq\operatorname{SU}(2,2)$. Our next paper will be devoted
to these questions.

## Appendix: Spinor groups and bivector spaces

As is known, rotations of pseudo-Euclidean spaces $\bb R^{p,q}$ are defined by
spinor groups [37]

$\operatorname{{\bf
Spin}}(p,q)=\left\\{s\in\boldsymbol{\Gamma}^{+}_{p,q}\;|\;N(s)=\pm
1\right\\},$

where $\boldsymbol{\Gamma}^{+}_{p,q}=\boldsymbol{\Gamma}_{p,q}\cap
C\kern-1.99997pt\ell^{+}_{p,q}$ is a special Lipschitz group, $s\in
C\kern-1.99997pt\ell^{+}_{p,q}$ is an even invertible element of the real
Clifford algebra $C\kern-1.99997pt\ell_{p,q}$. In more detail, the element $s$
is a linear combination of even basic elements, that is,

$s=\sum_{k}a^{i_{1}\ldots i_{2k}}\mbox{\bf e}_{i_{1}\ldots i_{2k}}.$

The condition $N(s)=\pm 1$ means that

$\sum_{k}\sigma(i_{1})\cdots\sigma(i_{2k})\left(a^{i_{1}\ldots
i_{2k}}\right)^{2}=\pm 1.$ (83)

On the other hand, from the fact that the elements
$s\in\boldsymbol{\Gamma}^{+}_{p,q}$ are even products of
$\nu=\sum\nu^{i}\mbox{\bf e}_{i}$, it is easy to derive that coordinates of
$s\in\operatorname{{\bf Spin}}(p,q)$ are related by the following conditions
[50]:

$a^{i_{1}i_{2}\ldots
i_{2k}}(a)^{k-1}=(2k-1)!!a^{[i_{1}i_{2}}a^{i_{2}i_{3}}\cdots
a^{i_{2k-1}i_{2k}]}.$ (84)

Conditions (84) can be rewritten in the form

$\left.\begin{array}[]{rcl}aa^{i_{1}i_{2}i_{3}i_{4}}&=&3!!a^{[i_{1}i_{2}}a^{i_{3}i_{4}]},\\\
aa^{i_{1}i_{2}i_{3}i_{4}i_{5}i_{6}}&=&5!!a^{[i_{1}i_{2}}a^{i_{3}i_{4}i_{5}i_{6}]},\\\
[2]{3}\\\ aa^{i_{1}i_{2}\ldots
i_{2k}}&=&(2k-1)!!a^{[i_{1}i_{2}\ldots}a^{i_{3}i_{4}\ldots
i_{2k-1}i_{2k}]}.\end{array}\right\\}$ (85)

Conditions (83) and (84) express all coordinates of the elements $s$,
belonging to the spinor group $\operatorname{{\bf Spin}}(p,q)$, via $n(n-1)/2$
coordinates $a^{ij}$, where $n=p+q$. The number of the coordinates $a^{ij}$
coincides with the number of parameters of the rotation group of the space
$\bb R^{p,q}$. This fact shows that expressions (83) and (84) form a full
system of the conditions separating the spinor group $\operatorname{{\bf
Spin}}(p,q)$ from the algebra $C\kern-1.99997pt\ell_{p,q}$.

Further, parameters of the rotation group of the space $\bb R^{p,q}$ form a
bivector space $\bb R^{N}$, where $N=\frac{n(n-1)}{2}$. Indeed, Let $\bb
R^{p,q}$ be the $n$-dimensional pseudo-Euclidean space, $p+q=n$. Let us evolve
in $\bb R^{p,q}$ all the tensors satisfying the following two conditions: 1) a
rank of the tensors is even; 2) covariant and contravariant indexes are
divided into separate skewsymmetric pairs. Such tensors can be exemplified by
bivectors (skewsymmetric tensors of the second rank). The set of all bivector
tensor fields in $\bb R^{p,q}$ is called a bivector set, and its
representation in a given point of $\bb R^{p,q}$ is called a local bivector
set. In any tensor from the bivector set we take the each skewsymmetric pair
$\alpha\beta$ as one collective index. At this point, from the two possible
pairs $\alpha\beta$ and $\beta\alpha$ we fix only one, for example,
$\alpha\beta$. The number of all collective indexes is equal to
$N=\frac{n(n-1)}{2}$. In a given point, the bivector set of the space $\bb
R^{p,q}$ with the contravariant components defines in the collective indexes a
vector set, and the each vector of this set has $N$ components. Identifying
these vectors with the points of the $N$-dimensional manifold, we come to an
affine manifold $E^{N}$ if and only if this manifold admits a Klein geometry
with the group

$\displaystyle\eta^{a^{\prime}}=A^{a^{\prime}}_{a}\eta^{a},\quad\eta^{a}=A^{a}_{a^{\prime}}\eta^{a^{\prime}},$
$\displaystyle\det A^{a^{\prime}}_{a}\neq 0,\quad
A^{a}_{b^{\prime}}A^{b^{\prime}}_{c}=\delta^{a}_{c},$

where

$A^{a^{\prime}}_{a}\longrightarrow
A^{\alpha^{\prime}}_{[\alpha}A^{\beta^{\prime}}_{\beta]}.$

Thus, any local bivector set of the space $\bb R^{p,q}$ ($p+q=n$) can be
mapped onto the affine space $E^{N}$ . Therefore, $E^{N}$ is related with the
each point of the space $\bb R^{p,q}$. The space $E^{N}$ is called a bivector
space. It should be noted that the bivector space is a particular case of the
most general mathematical construction called a Grassmannian manifold (a
manifold of $m$-dimensional planes of the affine space). In the case $m=2$ the
manifold of two-dimensional planes is isometric to the bivector space, and the
Grassmann coordinates in this case are called Pluecker coordinates.

The metrization of the bivector space $E^{N}$ is given by the formula (see
[47])

$g_{ab}\longrightarrow g_{\alpha\beta\gamma\delta}\equiv
g_{\alpha\gamma}g_{\beta\delta}-g_{\alpha\delta}g_{\beta\gamma},$ (86)

where $g_{\alpha\beta}$ is a metric tensor of the space $\bb R^{p,q}$, and the
collective indexes are skewsymmetric pairs $\alpha\beta\rightarrow a$,
$\gamma\delta\rightarrow b$. After introduction of $g_{ab}$, the bivector
affine space $E^{N}$ is transformed to a metric space $\bb R^{N}$.

As is known, any transformation from the rotation group of the space $\bb
R^{p,q}$ can be represented via $\frac{n(n-1)}{2}$ transformations in the
planes $(x_{1},x_{2})$, $(x_{1},x_{3})$, $\ldots$, $(x_{p+q},x_{1})$. The full
number of the planes $(x_{i},x_{j})$ is equal to $N=\frac{n(n-1)}{2}$, and the
each plane $(x_{i},x_{j})$ corresponds to the coordinate $a^{ij}$ of the
spinor group $\operatorname{{\bf Spin}}(p,q)$.

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