Conformal transformations and doubling of the particle states

A. I. Machavariania,b

a Joint Institute for Nuclear Research, Moscow Region 141980 Dubna, Russia

b High Energy Physics Institute of Tbilisi State University, University Str. 9

###### Abstract

The 6D and 5D representations of the four-dimensional (4D) interacting fields
in the Heisenberg picture and the corresponding equations of motion are
studied using equivalence of the conformal transformations of the four-
momentum $q_{\mu}$ ($q^{\prime}_{\mu}=q_{\mu}+h_{\mu}$,
$q^{\prime}_{\mu}=\Lambda^{\nu}_{\mu}q_{\nu}$, $q^{\prime}_{\mu}=\lambda
q_{\mu}$ and $q^{\prime}_{\mu}=-M^{2}q_{\mu}/q^{2}$) and the corresponding
rotations on the 6D cone $\kappa_{A}\kappa^{A}=0$ $(A=\mu;5,6\equiv
0,1,2,3;5,6)$, where $q_{\mu}=M\ \kappa_{\mu}/(\kappa_{5}+\kappa_{6})$ and $M$
is the scale parameter. The 4D reduction of the 6D fields on the cone
$\kappa_{A}\kappa^{A}=0$ is unambiguously fulfilled by the intermediate 5D
projection into two 5D hyperboloids $q_{\mu}q^{\mu}+q_{5}^{2}=M^{2}$ and
$q_{\mu}q^{\mu}-q_{5}^{2}=-M^{2}$ in order to cover the whole domains
$-\infty<q_{\mu}q^{\mu}<\infty$ and $q_{5}^{2}\geq 0$. The resulting 5D and 4D
fields in the coordinate space consist of two parts $\varphi_{1}(x,x_{5})$,
$\varphi_{2}(x,x_{5})$ and $\Phi_{1}(x)=\varphi_{1}(x,x_{5}=0)$,
$\Phi_{2}(x)=\varphi_{2}(x,x_{5}=0)$, where the Fourier conjugate of
$\varphi_{1}(x,x_{5})$ and $\varphi_{2}(x,x_{5})$ are defined on the
hyperboloids $q_{\mu}q^{\mu}+q_{5}^{2}=M^{2}$ and
$q_{\mu}q^{\mu}-q_{5}^{2}=-M^{2}$ respectively. Consequently, the 4D reduction
of the 6D fields generate two kinds of the 5D and 4D fields
$\varphi_{\pm}=\varphi_{1}\pm\varphi_{2}$ and
$\varphi_{\pm}(x,x_{5}=0)=\Phi_{\pm}(x)=\Phi_{1}(x)\pm\Phi_{2}(x)$ with the
same quantum numbers but with the different masses and the sources. This
doubling of the 4D fields $\Phi_{\pm}=\Phi_{1}\pm\Phi_{2}$ can be applied for
unified description of the interacting electron and muon fields, $\pi$ and
$\pi(1300)$-mesons, $N$ and $N(1440)$-nucleons and other particles with the
same quantum numbers but different masses and interactions.

Introduction

The 5D extension of the 4D relativistic theories is the fruitful method that
has a long history. The Kaluza-Klein theory and their generalizations for the
gauge transformations [1, 2, 3, 4] allow to unify the electromagnetic and
gravitation theories. In the traditional Kaluza-Klein theory all partial
derivatives with respect to fifth coordinates have been equated to zero and
the extra spatial dimension was compacted to a small size circle. The rigorous
mathematical approach for the $N+1$ and $N$ dimensional manifolds (see ch. 2.2
in [3]) allow to embed the 4D equation of motion with the sources into the 5D
equation without sources. In the recent 5D field theoretical formulations [5,
6] the extra fifth dimension is required to solve the problems of the
renormalizable $SO(10)$ grand unification theories with the breakdown of the
gauge coupling. In this approach the fifth dimension enable to reproduce the
fermion generations, quark mass hierarchy, flavor mixing and Cabbibbo-
Kabayashi-Maskawa matrices and it is argued, that in virtue of the no go
theorems it is not possible to achieve these results in the 4D space. Other
kind of the 5D relativistic field theories were performed within the invariant
time method [7, 8, 9], where the fifth coordinate is the proper time
$x_{5}^{2}=x_{o}^{2}-{\bf x}^{2}$. In these theories $x_{5}$ is an auxiliary
variable and the sought 4D wave functions and fields are reproduced through
the 5D wave functions and fields via the boundary conditions for $x_{5}=0$ or
$x_{5}=\sqrt{t^{2}-{\bf x}^{2}}$ and the evolution over the fifth coordinates
were often described through the equation for the first derivatives of the
scalar and fermion fields $i\partial\phi/\partial x_{5}$ and
$i\partial\psi/\partial x_{5}$.

The general scheme for the 5D extensions of the 4D relativistic theories and
the 4D reductions of the 5D relativistic formulations presents the conformal
group of the 4D transformations that can be unambiguously represent through
the rotations on the 6D cone. In particular, the conformal transformations of
the four coordinate $x_{\mu}$ consists of the following independent motions
$x^{\prime}_{\mu}=x_{\mu}+a_{\mu}$,
$x^{\prime}_{\mu}=\Lambda^{\nu}_{\mu}x_{\nu}$, $x^{\prime}_{\mu}=\lambda
x_{\mu}$ and $x^{\prime}_{\mu}=-\ell^{2}x_{\mu}/x^{2}$ which can be performed
through the rotations on the 6D cone
$\xi_{A}\xi^{A}\equiv\xi_{\mu}\xi^{\mu}+\xi_{5}\xi^{5}-\xi_{6}\xi^{6}=0$ [10],
where $A=0,1,2,3;5,6\equiv\mu;5,6$, $x_{\mu}=\xi_{\mu}/\xi_{+}$,
$\xi_{\pm}=(\xi_{5}\pm\xi_{6})/\ell$ and $\ell$ is the dimension parameter.
The one-to-one relationship between the 4D conformal transformations and 6D
rotations allow to construct the one-to-one relationship between an
interacting 4D Heisenberg field $\Phi(x)$ and the corresponding 6D field
$\varsigma(\xi)\equiv\varsigma(\xi_{0},\xi_{1},\xi_{2},\xi_{3};\xi_{5},\xi_{6})$
as $\Phi(x)=\Bigl{[}\varsigma(x,\xi_{+},\xi_{-})\Bigr{]}_{\xi_{A}\xi^{A}=0}$
with the fixed scale parameter $\xi_{+}$ and
${\xi_{-}}/\xi_{+}=x^{2}/\ell^{2}$ [10]-[13]. The other 4D reduction of
$\varsigma(\xi)$ was used in the manifestly conformal invariant formulation
[14]-[24], where the homogeneity of the conformal invariant 6D fields
$\Bigl{[}\varsigma(\xi)\Bigr{]}_{\xi_{A}\xi^{A}=0}=\Bigl{[}\varsigma(x,\xi_{+},\xi_{A}\xi^{A}=0)$
over the scale variable $\xi_{+}$ is required, i.e.
$\varsigma(x,\xi_{+},\xi_{A}\xi^{A}=0)={\xi_{+}}^{d}{\phi}(x,\xi_{A}\xi^{A}=0)$
and the 4D conformal invariant field is
$\Phi(x)\equiv{\phi}(x,\xi_{A}\xi^{A}=0)$.

Location of $\varsigma(\xi)$ on the 6D cone $\xi_{A}\xi^{A}=0$ impose
additional condition by conformal transformations. For instance, the Fourier
transformation of an arbitrary field $\varsigma(\xi)$ on the 6D cone produces
the following condition

$\biggl{(}{{\partial^{2}}\over{\partial{\kappa}^{\mu}\partial{\kappa}_{\mu}}}+{{\partial^{2}}\over{\partial{\kappa}^{5}\partial{\kappa}_{5}}}-{{\partial^{2}}\over{\partial{\kappa}^{6}\partial{\kappa}_{6}}}\biggr{)}\int
d^{6}\xi
e^{i\kappa_{A}\xi^{A}}\delta\Bigl{(}\xi_{o}^{2}-\xi_{1}^{2}-\xi_{2}^{2}-\xi_{3}^{2}+\xi_{5}^{2}-\xi_{6}^{2}\Bigr{)}\varsigma(\xi)=0,$
$None$

where $\kappa_{A}$ are Fourier conjugate to $\xi_{A}$ and for derivation of
(I.1a) the condition $(\xi_{A}\xi^{A})\delta(\xi_{A}\xi^{A})=0$ was used. The
intermediate 5D projection of the 6D fields and the condition (I.1a) determine
the corresponding 5D fields and the 5D condition.

In order to obtain the 5D and 6D extensions of the 4D equations of motion it
is convenient to consider the conformal transformations of the four momentum
$q_{\mu}$ ($q^{\prime}_{\mu}=q_{\mu}+h_{\mu}$,
$q^{\prime}_{\mu}=\Lambda^{\nu}_{\mu}q_{\nu}$, $q^{\prime}_{\mu}=\lambda
q_{\mu}$ and $q^{\prime}_{\mu}=-M^{2}q_{\mu}/q^{2}$). The 6D representation of
the conformal transformations for the independent four components of the
momentum $q_{\mu}$ are similar with the conformal transformations in the
coordinate space. The principal difference between the conformal
transformations in the coordinate and momentum space is in the translation. In
the next section it is shown that the translation of the four momentum
$q^{\prime}_{\mu}=q_{\mu}+h_{\mu}$ for the Fourier conjugate of $\Phi(x)$
produces the gauge transformation
$\Phi^{\prime}(x)=e^{ih_{\mu}x^{\mu}}\Phi(x)$. According to the Dirac
geometrical model [10]-[18], each of the conformal transformations in the
momentum space is unambiguously determined via the appropriate 6D rotation
with the invariant 6D form

$\kappa_{A}\kappa^{A}\equiv\kappa_{\mu}\kappa^{\mu}+\kappa_{5}^{2}-\kappa_{6}^{2}=0,$
$None$

where the four momentum $q_{\mu}$ ($\mu=0,1,2,3$) is defined as
$q_{\mu}=\kappa_{\mu}/\kappa_{+}$ and $M$ is a scale parameter. The 6D cone
(I.2a) and the corresponding surface

$q_{\mu}q^{\mu}+M^{2}{{\kappa_{-}}\over{\kappa_{+}}}=0,\ \ \ with\ \ \
\kappa_{\pm}={{\kappa_{5}\pm\kappa_{6}}\over M}$ $None$

are invariant under any combination of the conformal transformations of
$q_{\mu}$.

In analogy with (I.1a) the conformal transformations of a 4D field $\Phi(x)$
in the momentum space can be performed via the 6D rotations of the
corresponding 6D field operator $\varsigma(\kappa)$ which is embedded into the
cone (I.2a). Therefore, location of $\varsigma(\kappa)$ on the same 6D cone
(I.2a) before and after the conformal transformations in the momentum space
impose the condition

$\biggl{(}{{\partial^{2}}\over{\partial{\xi}^{\mu}\partial{\xi}_{\mu}}}+{{\partial^{2}}\over{\partial{\xi}^{5}\partial{\xi}_{5}}}-{{\partial^{2}}\over{\partial{\xi}^{6}\partial{\xi}_{6}}}\biggr{)}\int{{d^{6}\kappa}\over{(2\pi)^{4}}}e^{i\kappa_{A}\xi^{A}}\delta\Bigl{(}\kappa_{\mu}\kappa^{\mu}+\kappa_{5}^{2}-\kappa_{6}^{2}\Bigr{)}\varsigma(\kappa)=0.$
$None$

This paper deals with consistency of the usual 4D equations of motion for 4D
interacting Heisenberg field $\Phi(x)$ and boundary conditions and constrains
for the 5D and 6D representations of $\Phi(x)$ which follows from the
conformal group of the transformations in the momentum space. Two particular
features generate the special interest to the conformal transformations in the
momentum space[20]. First, the observables of the particle interactions, like
the cross sections and polarizations are determined in the momentum space.
Secondly, the accuracy of the measurement of the particle coordinates is in
principle restricted by the Compton length of this particle. Moreover,
determination of the coordinate of the conformal invariant massless particles
produces additional essential troubles (see [19] ch. 20 and [26]). The
conformal transformations of the fields and the corresponding equations of
motion in the momentum space were considered in ref. [11, 13, 20, 25], where
the conformal transformations were performed in the configuration space and
followed relations in the momentum space were obtained using the Fourier
transformation.

The 4D reduction of the 6D operators
$\varsigma(\kappa)\equiv\varsigma(q,\kappa_{+},\kappa_{-})$ generates the 5D
operators as the intermediate 5D projections. There exists only two 5D De
Sitter spaces with the constant curvature which have the invariant forms
$q_{\mu}q^{\mu}\pm q_{5}^{2}\mp M^{2}=0$ ($q_{5}^{2}\geq 0$) of the $O(2,3)$
and $O(1,4)$ rotational groups [18, 24, 28]. The single 5D hyperboloid is not
enough for reproduction of the whole values of $-\infty<q^{2}<\infty$.
Therefore, we use the domains from the both 5D hyperboloids which are
connected by inversion $q^{\prime}_{\mu}=-M^{2}q_{\mu}/q^{2}$. Thus for the
intermediate 5D projections of the 6D cone (I.2a) we shall use the invariant
forms

$q_{\mu}q^{\mu}+q_{5}^{2}=M^{2}\ \ \ with\ \ \ {q_{5}^{2}\over
M^{2}}={{\kappa_{-}}\over{\kappa_{+}}}+1,\ \ \ and\ \ q_{5}^{2}\geq 0$ $None$

$q_{\mu}q^{\mu}-q_{5}^{2}=-M^{2}\ \ with\ \ \ {q_{5}^{2}\over
M^{2}}=-{{\kappa_{-}}\over{\kappa_{+}}}+1\ \ \ and\ \ q_{5}^{2}\geq 0.$ $None$

In (I.3a,b) the fifth variable $q_{5}^{2}$ is positive $0\leq
q_{5}^{2}`\infty$. Consequently, for the positive $q^{2}\equiv
q_{\mu}q^{\mu}\geq 0$ the corresponding four-momenta $q_{\mu}$ are distributed
between the domains $0\leq q^{2}\leq M^{2}$ and $q^{2}>M^{2}$ on the
hyperboloids (I.3a) and (I.3b) respectively. These domains are connected by
inversion $q^{\prime}_{\mu}=-M^{2}q_{\mu}/q^{2}$. The values of the $q^{2}$ on
the hyperboloids (I.3a) and (I.3b) are also connected through the reflection
$q^{2}\longleftrightarrow-q^{2}$. Therefore, for the negative $q^{2}<0$ one
has $-M^{2}<q^{2}<0$ on the hyperboloid (I.3b) and $-\infty<q^{2}<-M^{2}$ is
on the hyperboloid (I.3a). More detailed the distributions of $q^{2}$ on the
hyperboloids (I.3a) and (I.3b) are listed in Table 1 of Section 2.

In order to determine the 5D and 4D projections of the 6D fields
$\delta(\kappa_{A}\kappa^{A})\varsigma(\kappa)$ (I.2a) we shall introduce the
following 5D fields $\varphi_{1}(x,x_{5})$ and $\varphi_{2}(x,x_{5})$

$\varphi_{1}(x,x_{5})=\int{{d^{4}q}\over{(2\pi)^{4}}}dq_{5}^{2}e^{-iqx-
iq^{5}x_{5}}\delta(q^{2}+q_{5}^{2}-M^{2})\Bigl{[}\theta(q^{2})\theta(M^{2}-q^{2})+\theta(-q^{2})\theta(-M^{2}-q^{2})]\Bigr{]}\phi(q,q_{5}^{2}),$
$None$

$\varphi_{2}(x,x_{5})=\int{{d^{4}q}\over{(2\pi)^{4}}}dq_{5}^{2}e^{-iqx-
iq^{5}x_{5}}\delta(q^{2}-q_{5}^{2}+M^{2})\Bigl{[}\theta(q^{2})\theta(-M^{2}+q^{2})+\theta(-q^{2})\theta(M^{2}+q^{2})\Bigr{]}\phi(q,q_{5}^{2}).$
$None$

where

$\phi(q,q_{5}^{2})={{M^{2}}\over
2}\int\kappa_{+}^{3}d\kappa_{+}\theta(\kappa_{+})\varsigma(q,q_{5}^{2},\kappa_{+}).$
$None$

For the sake of simplicity the scale variable $\kappa_{+}$ (I.2b) is taken in
positive i.e. $\phi(\kappa)=\theta(\kappa_{+})\phi(\kappa)$, where
$\theta(\kappa_{+})=1$ for $\kappa_{+}>0$ and $\theta(\kappa_{+})=0$ for
$\kappa_{+}<0$.

The Fourier conjugate of $\varphi_{1}(x,x_{5})$ and $\varphi_{2}(x,x_{5})$ are
located into hyperboloids (I.3a) and (I.3b) respectively. Therefore they
satisfy the conditions

$\biggl{(}{{\partial^{2}}\over{\partial x^{\mu}\partial
x_{\mu}}}+{{\partial^{2}}\over{\partial x^{5}\partial
x_{5}}}+M^{2}\biggr{)}\varphi_{1}(x,x_{5})=0,\ \ \ \ \ \ \ \ \
\biggl{(}{{\partial^{2}}\over{\partial x^{\mu}\partial
x_{\mu}}}-{{\partial^{2}}\over{\partial x^{5}\partial
x_{5}}}-M^{2}\biggr{)}\varphi_{2}(x,x_{5})=0.$ $None$

The fields $\varphi_{1}$ (I.4a) and $\varphi_{2}$ (I.4b) produce two
independent 5D fields

$\varphi_{+}(x,x_{5})=\varphi_{1}(x,x_{5})+\varphi_{2}(x,x_{5});\ \ \
\varphi_{-}(x,x_{5})=\varphi_{1}(x,x_{5})-\varphi_{2}(x,x_{5}),$ $None$

which Fourier conjugate are defined in the whole domains
$(-\infty<q_{\mu}<+\infty)$ and $(-\infty<q^{2}<+\infty)$.

The usual boundary condition for the 5D fields $\varphi_{\pm}(x,x_{5})$ at
$x_{5}=0$ allows to get the 4D fields $\Phi_{\pm}(x)$

$\Phi_{\pm}(x)=\varphi_{\pm}(x,x_{5}=0).$ $None$

According to (I.4a,b), (I.7) and (I.8) the Fourier conjugate of
$\Phi_{\pm}(x)$

$\Phi_{\pm}(q)=\int d^{4}xe^{iqx}\Phi_{\pm}(x)$ $None$

have the following structure

$\Phi_{\pm}(q)=\sum_{N=I,III}\Phi_{N}(q)\pm\sum_{N=II,IV}\Phi_{N}(q);$ $None$

where

$\Phi_{I}(q)=\theta(q^{2})\theta(M^{2}-q^{2})\phi(q,q^{2}_{5}=M^{2}-q^{2});\ \
\
\Phi_{II}(q)=\theta(q^{2})\theta(-M^{2}+q^{2})\phi(q,q^{2}_{5}=M^{2}+q^{2});$
$\Phi_{III}(q)=\theta(-q^{2})\theta(-M^{2}-q^{2})\phi(q,q^{2}_{5}=M^{2}-q^{2});\
\ \
\Phi_{IV}(q)=\theta(-q^{2})\theta(M^{2}+q^{2})\phi(q,q^{2}_{5}=M^{2}+q^{2}),$
$None$

where the lower index $I,\ II,\ III$ and $IV$ of $\Phi(q)$ corresponds to the
domains of $q^{2}$ which are listed in Table 1 of Section 2. The details of
the relationship between the 4D and 5D scalar field are given in in Section 3.

The equations (I.4a,b)-(I.10) presents the relationship between the 4D, 5D and
6D fields $\delta(\kappa_{A}\kappa^{A})\varsigma(\kappa)$. The 4D interacting
Heisenberg fields $\Phi_{\pm}(x)$ and their 6D representations
$\delta(\kappa_{A}\kappa^{A})\varsigma(\kappa)$ are not invariant under the
conformal transformations. Nevertheless, location of
$\delta(\kappa_{A}\kappa^{A})\varsigma(\kappa)$ on the 6D cone (I.2a) and the
corresponding location of the Fourier conjugate of $\varphi_{\pm}(x,x_{5})$ on
the hyperboloids (I.3a,b) produce the conditions (I.1b) and (I.6) for any 6D
field and its 5D projections. The consistency of these conditions with the
equation of motion for $\varphi_{\pm}(x,x_{5})$ and $\Phi_{\pm}(x)$ and the
boundary conditions (I.8) are considered in Sect. 4 and 5.

The intermediate projection of the 6D field
$\delta(\kappa_{A}\kappa^{A})\varsigma(\kappa)$ on the invariant forms
(I.3a,b) of the $O(2,3)$ and $O(1,4)$ subgroups of the conformal group
$O(2,4)$ need to introduce the two independent 5D fields $\varphi_{+}$ and
$\varphi_{-}$ (I.7) which are constructed from the same parts $\varphi_{1}$
and $\varphi_{2}$ (I.4a,b). The invariant forms (I.3a,b) of the $O(2,3)$ and
$O(1,4)$ form the definition area of $\varphi_{1}$ and $\varphi_{2}$ and
correspondingly of the field $\Phi_{N}(q)$ in (I.10) and (I.11). The 4D fields
$\Phi_{+}$ and $\Phi_{-}$ (I.10) have the same quantum numbers. But $\Phi_{+}$
and $\Phi_{-}$ can have the different masses and the different sources. Other
details of the inversion and related constructions of the 4D fields are given
in the next Section.

The reduction formulas (I.5) of the 6D field $\varsigma(\kappa)$ on the cone
(I.2a)
$\varsigma(\kappa)\equiv\varsigma(\kappa_{\mu},\kappa_{+},\kappa_{A}\kappa^{A}\neq
0)$ $\equiv\varsigma(q,q_{5}^{2},\kappa_{+})$ differ from the reduction
formula in the manifestly covariant formulation [14]-[24], where the
homogeneity of
$\varsigma(\kappa)=\phi_{(}q_{\mu},\kappa_{+},\kappa_{A}\kappa^{A}=0)$ over
the scale variable $\kappa_{+}$ is required, i.e.
$\phi(q_{\mu},\kappa_{+},\kappa_{A}\kappa^{A}=0)={\kappa_{+}}^{d}{\Phi}(q_{\mu}),$
where $d$ is the scale dimension of $\phi(\kappa)$. In order to reproduce this
property in the present approach one can use an additional condition in (I.5)

$\varsigma(q,q_{5}^{2},\kappa_{+})=\delta(\kappa_{+}-{\cal
M})\phi(q,q_{5}^{2},{\cal M});\ \ \ \ \ \ or\ \ \ \ \ \ \
\varsigma(q,q_{5}^{2},\kappa_{+})=\theta(\kappa_{+}-{\cal
M})\phi(q,q_{5}^{2},{\cal M})$ $None$

where ${\cal M}$ is a fixed scale parameter.

The 5D quantum field theory with the invariant forms
$q_{\mu}q^{\mu}+q_{5}^{2}=M^{2}$ or $q_{\mu}q^{\mu}-q_{5}^{2}=-M^{2}$
separately was firstly studied in refs. [27, 28, 29], where $M$ was
interpreted as the fundamental (maximal) mass and its inverse $1/M$ as the
fundamental (minimal) length [30, 31]. The conformal transformations in the
momentum space for the complete fields $\varphi_{+}=\varphi_{1}+\varphi_{2}$
was suggested in [32], where $M$ is determined via $m_{\pi}$ and $m_{Higgs}$
according to the chiral symmetry breaking mechanism within the 5D chiral
models. In the present paper is studied coupling between the 5D and 4D
equations of motion for the fields $\varphi_{+}$, $\Phi_{+}$ and
$\varphi_{-}$, $\Phi_{-}$. Besides the present paper contains more general and
self-consistent formulation of translations and inversions in the 4D momentum
space for the 4D charged and neutral fields.

In Section 1 the conformal transformations in the 4D momentum space and the
corresponding transformations of the interacting fields $\Phi(x)$ are
considered. The domains of the variables $q^{2}=q_{\mu}q^{\mu}$ and
$q^{2}_{5}$ (I.3a,b) are determined in Sect. 2. The 4D reduction of the 5D
fields and the related projections and convolution formulas are given in Sect.
3. In Sections 4 and 5 the 4D and 5D equations of motion for the scalar fields
and the corresponding constrains for $x_{5}$ are considered. Sections 6 is
devoted to the 5D and 4D Lagrangians. The 4D and 5D equations of motion for
the fermion fields with the electromagnetic interaction and the constrains for
the fifth coordinates $x_{5}$ are considered in Sect. 7 and 8. In Section 9
the 5D extension of the standard $SU(2)\times U(1)$ model for the electron and
muon fields is given as an example of the suggested doubling for the fermion
states. Besides in this Section is shortly discussed consistency of the
present scheme and the purely 5D models [5, 6] of the grand unification
theory. The generalized translations in the momentum space as the gauge
transformations are considered in Sect 10. The summary is given in Sect. 11.

1\. Conformal transformations in the 4D momentum space.

Conformal transformations of the four-momentum $q_{\mu}$ $(\mu=0,1,2,3)$
consists of

$translations\hfill{q_{\mu}\longrightarrow q_{\mu}^{\prime}=q_{\mu}+h_{\mu},\
\ \ \ \ (1.1a)}$

$rotations\hfill{q_{\mu}\longrightarrow
q_{\mu}^{\prime}=\Lambda_{\mu}^{\nu}q_{\nu},\ \ \ \ \ (1.1b)}$

$dilatation\hfill{q_{\mu}\longrightarrow q_{\mu}^{\prime}=e^{\lambda}\
q_{\mu},\ \ \ \ \ (1.1c)}$

$and\ inversion\hfill{q_{\mu}\longrightarrow
q_{\mu}^{\prime}=-M^{2}q_{\mu}/q^{2},\ \ \ \ \ (1.1d)}$

where $M$ is a mass parameter that insures the correct dimension of $q_{\mu}$.
Translations and inversions form

$special\ conformal\ transformation\hfill{q_{\mu}\longrightarrow
q_{\mu}^{\prime}=\frac{\Large{q_{\mu}-{\hbar}_{\mu}q^{2}/M^{2}}}{\Large{1-2q_{\nu}{\hbar}^{\nu}/M^{2}+{\hbar}^{2}q^{2}/M^{4}}}.\
\ \ \ \ (1.1e)}$

$q_{\mu}$ in (1.1a)-(1.1e) is off mass shell, i.e. $q_{o}$ is an independent
variable and $q_{o}\neq\sqrt{{\bf q}^{2}+m^{2}}$.

According to the Dirac geometrical model [10], transformations (1.1a)-(1.1e)
are equivalent to the rotations on the 6D cone
$\kappa^{2}\equiv\kappa_{A}\kappa^{A}=0$ (I.2a) with the metric tensor
$g_{AB}=diag(+1,-1,-1,-1,+1,-1)$. In particular, translation (1.1a) and the
special conformal translation are generated by the combinations of the
rotations in the planes $(\mu,5)$ and $(\mu,6)$, dilatation is obtained via
the rotation in the plane (5,6) and inversion follows from transposition of
the of the variables $\kappa^{\prime}_{5}=\kappa_{5}$ and
$\kappa_{6}^{\prime}=-\kappa_{6}$

$translation:\ \ \ \ \ \ \ \ \ \ \ \ \ \
\kappa_{\mu}^{\prime}=\kappa_{\mu}+h_{\mu}\kappa_{+};\ \ \ \ \
\kappa_{+}^{\prime}=\kappa_{+};\ \ \ \ \
\kappa_{-}^{\prime}=-{{2h_{\mu}\kappa^{\mu}}\over{\kappa_{+}}}-{{\kappa_{\mu}\kappa^{\mu}}\over{\kappa_{+}}}$
$None$ $rotation:\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \kappa_{\mu}^{\prime}=\Lambda_{\mu}^{\nu}\kappa_{\nu};\ \ \ \ \
\kappa_{+}^{\prime}=\kappa_{+};\ \ \ \ \ \kappa_{-}^{\prime}=\kappa_{-},$
$None$ $dilatation:\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \kappa_{\mu}^{\prime}=\kappa_{\mu};\ \ \ \ \
\kappa_{+}^{\prime}=e^{-\lambda}\kappa_{+};\ \ \ \ \
\kappa_{-}^{\prime}=e^{\lambda}\kappa_{-},$ $None$ $inversion:\ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\kappa_{\mu}^{\prime}=\kappa_{\mu};\ \ \ \ \ \kappa_{+}^{\prime}=\kappa_{-};\
\ \ \ \ \kappa_{-}^{\prime}=\kappa_{+},$ $None$

where

$q_{\mu}={{\kappa_{\mu}}\over{\kappa_{+}}};\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\kappa_{\pm}={{\kappa_{5}\pm\kappa_{6}}\over M};\ \ \ \ \ \ \ \ \
\mu=0,1,2,3.$ $None$

The equivalence of the 4D and 6D conformal transformations implies that
translation, rotation, dilatation and inversion of the 4D field $\Phi(q)$
(I.9) are unambiguously (isomorphic) determined through the corresponding 6D
rotations of the 6D field
$\varsigma(\kappa)\equiv\varsigma(\kappa_{\mu};\kappa_{+},\kappa_{-})$

$\Phi(q_{\mu}^{\prime}=q_{\mu}+h_{\mu})\Longleftrightarrow\varsigma(\kappa_{\mu}=\kappa_{\mu}+h_{\mu}\kappa_{+},\kappa_{+},\kappa_{-}^{\prime}=\kappa_{-}-\frac{2h_{\mu}\kappa^{\mu}}{\kappa_{+}}-\frac{\kappa_{\mu}\kappa^{\mu}}{\kappa_{+}})$
$None$

,

$\Phi(q_{\mu}^{\prime}=\Lambda_{\mu}^{\nu}q_{\nu})\Longleftrightarrow\varsigma(\kappa_{\mu}^{\prime}=\Lambda_{\mu}^{\nu}\kappa_{\nu},\kappa_{+},\kappa_{-})$
$None$
$\Phi(q_{\mu}^{\prime}={\lambda}q_{\nu})\Longleftrightarrow\varsigma(\kappa_{\mu},e^{-\lambda}\kappa_{+},e^{\lambda}\kappa_{-})$
$None$
$\Phi(q_{\mu}^{\prime}=-q_{\mu}/q^{2})\Longleftrightarrow\varsigma(\kappa_{\mu},\kappa_{-},\kappa_{+})$
$None$

These relationships between the 4D and 6D operators $\Phi(q)$ and
$\varsigma(\kappa)$ is achieved in (I.1b) through
$\delta(\kappa_{A}\kappa^{A})$.

An interacting scalar field $\Phi(x)$ is usually decomposed in the positive
and in the negative frequency parts in the 3D Fock space

$\Phi(x)=\int{{d^{3}p}\over{(2\pi)^{3}2\omega_{{\bf p}}}}\Bigl{[}a_{{\bf
p}}(x_{0})e^{-ipx}+{b^{+}}_{{\bf p}}(x_{0})e^{ipx}\Bigr{]};\ \ \
p_{o}\equiv\omega_{{\bf p}}=\sqrt{{\bf p}^{2}+m^{2}},$ $None$

where in the asymptotic regions $a_{{\bf p}}(x_{0})$ and ${b^{+}}_{{\bf
p}}(x_{0})$ transforms into particle (antiparticle) annihilation (creation)
operators. On the other hand, $\Phi(x)$ can be decomposed in the 4D momentum
space as

$\Phi(x)=\int{{d^{4}q}\over{(2\pi)^{4}}}\Bigl{[}{\Phi^{(+)}}(q)e^{-iqx}+{\Phi^{(-)}}^{+}(q)e^{iqx}\Bigr{]};\
\ \ or\ \ \ \Phi(x)=\int{{d^{4}q}\over{(2\pi)^{4}}}\Phi(q)e^{-iqx}.$ $None$

where

$\Phi(q)=\Phi^{(+)}(q)+{\Phi^{(-)}}^{+}(-q)$ $None$

Comparison of (1.5) and (1.6a) gives

${{e^{-i\omega_{\bf p}x_{o}}}\over{2\omega_{\bf p}}}a_{\bf
p}(x_{o})=\int{{dq_{o}}\over{2\pi}}{\Phi^{(+)}}(q_{o},{\bf
p})e^{-iq_{o}x_{o}}$ $None$

and

${{e^{i\omega_{\bf p}x_{o}}}\over{2\omega_{\bf p}}}{b^{+}}_{\bf
p}(x_{0})=\int{{dq_{o}}\over{2\pi}}{\Phi^{(-)}}^{+}(q_{o},{\bf
p})e^{iq_{o}x_{o}}.$ $None$

The field operators $a_{{\bf p}}(x_{0})$ and ${b^{+}}_{{\bf p}}(x_{o})$ are
simply determined via the corresponding source operator $\partial a_{{\bf
p}}(x_{0})/\partial{x_{0}}=i\int d^{3}xe^{ipx}j(x)$, where
$\Bigl{(}{{\partial^{2}}/{\partial{x_{\mu}}\partial{x^{\mu}}}}+m^{2}\Bigr{)}\Phi(x)=j(x)$.
Moreover, these operators determine the transition ${\cal S}$-matrix

${\cal S}_{mn}\equiv<out;{\bf p^{\prime}}_{1},...,{\bf p^{\prime}}_{m}|{\bf
p}_{1},...,{\bf p}_{n};in>=\prod_{i=1}^{m}\Bigl{[}\int
d{{x^{0}}^{\prime}}_{i}{{d}\over{d{{x^{\prime}}^{0}}_{i}}}\Bigr{]}$
$\prod_{j=1}^{n}\Bigl{[}\int
d{x^{0}}_{j}{{d}\over{d{x^{0}}_{j}}}\Bigr{]}<0|T\Bigl{(}a_{{\bf
p^{\prime}}_{m}}({x^{0}}^{\prime}_{m}),...,a_{{\bf
p^{\prime}}_{1}}({x^{0}}^{\prime}_{1})a_{{\bf
p}_{n}}^{+}({x^{0}}_{n}),...,a_{{\bf p}_{1}}^{+}({x^{0}}_{1})\Bigr{)}|0>.$
$None$

The translation (1.1a) for the 4D field in the momentum space
$\Phi^{\prime}(q)=\Phi(q+h)$ generates the corresponding gauge transformation
for $\Phi(x)$. In particular, for the complex charged field the four-momentum
translation (1.1a) produces the simplest gauge transformation

${\Phi}^{\prime}(x)=e^{ih_{\mu}x^{\mu}}\Phi(x);\ \ \ \ \
i{{\partial}\over{\partial{x_{\mu}^{\prime}}}}=i{{\partial}\over{\partial{x_{\mu}}}}+h_{\mu},$
$None$

where generally $h_{\mu}$ is a complex constant.

For the changeless real fields $\Phi(x)$ the gauge transformation (1.9) is
consistently defined for the pure imaginary $h_{\mu}=ir_{\mu}$ with the real
$r_{\mu}$. In particular, for the real scalar fields we get

${\Phi}^{\prime}(x)=e^{-r_{\mu}x^{\mu}}\Phi(x);\ \ \ \ \
{{\partial}\over{\partial{x_{\mu}^{\prime}}}}={{\partial}\over{\partial{x_{\mu}}}}+r_{\mu},$
$None$

The generalization of the gauge transformations (1.10) for the real scalar
fields within the nonlinear $\sigma$ model was performed in [36, 16]. These
formulation we shall consider at end of Sect. 10.

The transformation of $\Phi(x)$ under the rotation and dilatation of $q_{\mu}$
(1.1b,c) can be reproduced through the Fourier transformations in (1.6a) using
the inverse rotation and dilatation of $x_{\mu}$ in $exp(-iqx)$.

The doubling of the 5D fields $\varphi_{\pm}(x,x_{5})$ in (I.4a,b) and (I.7)
is generated by the intermediate projections onto domains of the 5D
hyperboloids (I.3a,b). These domains cover unambiguously the whole area of
$q_{\mu}$ and $q^{2}$ and cover the whole domain of the variables of the
fields $\Phi_{\pm}(q)$ (I.10). Inversion
$q_{\mu}^{\prime}=-M^{2}q_{\mu}/q^{2}$ transform the domains of the variables
of $\varphi_{1}(q,q^{2}_{5})$ into the domain of the variables of
$\varphi_{1}(q,q^{2}_{5})$ and vice versa. But inversion transforms also these
fields, i.e. inversion replaces $\varphi_{1}(x,x_{5})$ (1.4a) and
$\varphi_{2}(x,x_{5})$ (I.4b) with the $\varphi_{2}^{(I)}$ and
$\varphi_{1}^{(I)}$

$\varphi_{1}{\stackrel{{\scriptstyle
inversion}}{{\Longleftrightarrow}}}\varphi^{(I)}_{2}\ \ \ i.e.\ \ \
\varphi_{+}{\stackrel{{\scriptstyle
inversion}}{{\Longleftrightarrow}}}\varphi^{(I)}_{+};\ \ \
\varphi_{-}{\stackrel{{\scriptstyle
inversion}}{{\Longleftrightarrow}}}-\varphi^{(I)}_{-},$ $None$

. where the upper index (I) denotes the inversion of the corresponding
operator. One needs to introduce this index because the 4D equation of motion
for the massive particles are not invariant under the inversion. For instance,
the equation of motion $(q^{2}-m^{2})\Phi(q)=j(q)$ after inversion transforms
into new type equation $(M^{2}-q^{2}\
m^{2}/M^{2})\phi^{(I)}(q)=q^{2}j^{(I)}(q)/M^{2}$.

2\. Domains of $q^{2}$ and $q^{2}_{5}$

The invariant form $\kappa_{A}\kappa^{A}=0$ (I.2a) of the $O(2,4)$ group can
be represented for $q^{2}$ as

$q^{2}+M^{2}{{\kappa_{-}}\over{\kappa_{+}}}=0,$ $None$

where

$q_{\mu}={{\kappa_{\mu}}\over{\kappa_{+}}};\ \ \ \
\kappa_{\pm}={{\kappa_{5}\pm\kappa_{6}}\over{M}}.$ $None$

It is convenient to use the auxiliary fifth momentum $q_{5}^{2}$ instead of
$\kappa_{5}/\kappa_{6}$ in (2.1a). This procedure implies a projection of the
6D rotational invariant form $\kappa_{A}\kappa^{A}=0$ into corresponding 5D
forms. In order to cover unambiguously the whole domain $-\infty\leq
q^{2}\equiv q_{\mu}q^{\mu}\leq\infty$ we have distributed $q^{2}$ and
corresponding $q_{5}^{2}$ between the domains of the two 5D hyperboloids

$q^{2}+q_{5}^{2}=M^{2}\ \ \ \ \ with\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
q_{5}^{2}=M^{2}{{2\kappa_{5}}\over{\kappa_{5}+\kappa_{6}}},$ $None$

and

$q^{2}-q_{5}^{2}=-M^{2}\ \ \ \ \ with\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
q_{5}^{2}=M^{2}{{2\kappa_{6}}\over{\kappa_{5}+\kappa_{6}}}.$ $None$

The hyperboloids (2.2a,b) presents the simple intermediate 5D projection of
the 6D cone (I.2a) into the 4D momentum space with only one auxiliary variable
$q_{5}^{2}$.

$\kappa_{+}$ and $q^{2}_{5}$ in hyperboloids (2.2a,b) are defined in positive.
Consequently, $\kappa_{5}$ on the hyperboloid (2.2a) and $\kappa_{6}$ in
(2.2b) are also positive. The domains of the variables $q^{2}$, $q^{2}_{5}$,
$\kappa_{5}$ and $\kappa_{6}$ defined on the 6D cone (2.1a) and on the
corresponding 5D hyperboloids $q^{2}\pm q_{5}^{2}=\pm M^{2}$ (2.2a,b) are
listed in Table 1. The border points $q^{2}=0$, $q^{2}=M^{2}$ and
$q^{2}=-M^{2}$ are included in the domain ${\bf I}$ and ${\bf IV}$.

It must be emphasized, that the domain $M^{2}<q_{5}^{2}<2M^{2}$ of the
variables of $\phi(q,q_{5}^{2})$ is excluded by construction of
$\varphi_{1,2}(x,x_{5})$ (I.4a,b). Correspondingly, the variables $\kappa_{5}$
and $\kappa_{6}$ cover also only the part of the 6D cone
$\kappa_{A}\kappa^{A}=0$ (I.2a). The principal restriction for the auxiliary
variables $\kappa_{5}$, $\kappa_{6}$ and $q_{5}^{2}$ is that the corresponding
$q^{2}$ must cover the whole domain $(-\infty,+\infty)$. This property allows
to construct unambiguously the 4D field
$\Phi_{\pm}(x)=\varphi_{\pm}(x,x_{5}=0)$.

Table 1 Domains of $q^{2}$, $q_{5}^{2}$, $\kappa_{5}$ and $\kappa_{6}$ placed
on the hyperboloids $q^{2}\pm q_{5}^{2}=\pm M^{2}$ (2.2a,b) and on the surface
(2.1a).

| I | II | III | IV
---|---|---|---|---
| $q^{2}+q_{5}^{2}=M^{2}$ | $q^{2}-q_{5}^{2}=-M^{2}$ | $q^{2}+q_{5}^{2}=M^{2}$ | $q^{2}-q_{5}^{2}=-M^{2}$
$q^{2}$ | $0\leq q^{2}═\leq═M^{2}$ | $M^{2}<q^{2}<\infty$ | $-\infty<q^{2}<-M^{2}$ | $-M^{2}\leq q^{2}<0$
$q_{5}^{2}$ | $0\leq q_{5}^{2}═\leq M^{2}$ | $2M^{2}<q_{5}^{2}<\infty$ | $2M^{2}<q_{5}^{2}<\infty$ | $0\leq q_{5}^{2}<M^{2}$
$\kappa_{5}\&\kappa_{6}$ | $0\leq\kappa_{5}\leq\kappa_{6}$ |  $\kappa_{6}>0$; $\kappa_{5}<0$; $\kappa_{5}+\kappa_{6}>0$ |  $\kappa_{5}>0$; $\kappa_{6}<0$; $\kappa_{5}+\kappa_{6}>0$ | $0\leq\kappa_{6}<\kappa_{5}$

The hyperboloids $q^{2}+q^{2}_{5}=M^{2}$ and $q^{2}-q^{2}_{5}=-M^{2}$ and the
corresponding domains ${\bf I,III}$ and ${\bf II,IV}$ transforms also into
each other by reflection $q^{2}\longleftrightarrow-q^{2}$ which is generated
by transposition of the variables $\kappa_{5}$ and $\kappa_{6}$

$q^{2}\longleftrightarrow-q^{2}\ \ \ \ \ \ \ \ \kappa_{5}=\kappa_{6},\ \ \
\kappa_{6}=\kappa_{5};\ \ \ \ \ \kappa_{+}=\kappa_{+},\ \ \
\kappa_{-}=-\kappa_{-}.$ $None$

The similar transpositions of the hyperboloids $q^{2}+q^{2}_{5}=M^{2}$ and
$q^{2}-q^{2}_{5}=-M^{2}$ produces inversion $q_{\mu}^{I}=-M^{2}q_{\mu}/q^{2}$
(1.1d) which is generated by transposition of the variables $\kappa_{+}$ and
$\kappa_{-}$ according to (1.2d).

The choice of the 5D hyperboloids is not unique. For instance, instead of the
two 5D hyperboloids (2.2a,b) one can take other hyperboloids $q^{2}\pm
q_{5}^{2}=M^{2}$ with $q_{5}^{2}={{\pm
2M^{2}\kappa_{5}}/{\kappa_{5}+\kappa_{6}}}$, where $\kappa_{+}$ is fixed
$\kappa_{+}={\cal M}/M$ and ${\cal M}$ is a mass parameter. But these domains
of $q^{2}$ are not symmetric under the reflection and inversion of $q^{2}$
unlike the domains in Table 1. Therefore we do not consider them.

An other choice of the intermediate 5D projections presents the stereographic
projection, where $\kappa_{6}=-1$, and the auxiliary momenta
$Q_{\mu}=q_{\mu}/(1-q^{2})$ and $Q_{4}=(1+q^{2})/(1-q^{2})$, i.e.
$q_{\mu}=Q_{\mu}/(1+Q_{4})$, $q^{2}=(Q_{4}-1)/(1+Q_{4})$ are introduced (see,
for example, eq. (13.43) in [7]). This choice of the variables require only
one 5D hyperboloid $Q^{2}-Q_{4}^{2}=-1$ with the 5 auxiliary variables
$Q_{\mu}$ and $Q_{4}$ for the intermediate 5D projections. Certainly, one can
represent this projection via the considered projections on the hyperboloids
$q^{2}\pm q_{5}^{2}=\pm M^{2}$ (2.2a,b) where only one auxiliary variable
$q^{2}_{5}$ is used.

3\. 4D reductions of the 5D fields.

The 5D fields $\varphi_{\pm}=\varphi_{1}\pm\varphi_{2}$ (I.7) consist of the
two parts $\varphi_{1}$ (I.4a) and $\varphi_{1}$ (I.4b). which satisfy the
conditions (I.6), because their Fourier conjugates are defined on the 5D
hyperboloids $q^{2}\pm q^{2}_{5}=\pm M^{2}$ (I.3a,b). For $\varphi_{\pm}$
these conditions can be represented as

${{\partial^{2}\varphi_{\pm}(x,x_{5})}\over{\partial x^{\mu}\partial
x_{\mu}}}+\Bigl{(}{{\partial^{2}}\over{\partial x^{5}\partial
x_{5}}}+M^{2}\Bigr{)}\varphi_{\mp}(x,x_{5})=0.$ $None$

Integration over $q_{5}^{2}$ in (I.4a,b) yields

$\varphi_{\pm}(x,x_{5})=\int{{d^{4}q}\over{(2\pi)^{4}}}e^{-iqx}\Bigl{[}\phi(q,Q^{2}_{1})\Lambda_{1}(q^{2})e^{-iQ_{1}x_{5}}\pm\phi(q,Q^{2}_{2})\Lambda_{2}(q^{2})e^{-iQ_{2}x_{5}}\Bigr{]}.$
$None$

The expression (3.2) can be represented via the 5D convolution formula

$\varphi_{\pm}(x,x_{5})=\int d^{5}y\phi(x-y,x_{5}-y_{5}){\cal
P}_{\pm}(y,y_{5}),$ $None$

where $\phi(x,x_{5})$ is the Fourier conjugate of the full 5D function
$\phi(q,q_{5}^{2})$ in (I.4a,b) and

$\Lambda_{a}(q^{2})=\left\\{\begin{array}[]{ll}\theta(q^{2})\theta(M^{2}-q^{2})+\theta(-q^{2})\theta(-M^{2}-q^{2})&\mbox{if
$a=1$};\\\
\theta(q^{2})\theta(-M^{2}+q^{2})+\theta(-q^{2})\theta(M^{2}+q^{2})&\mbox{if
$a=2$},\end{array}\right.$ $None$

$Q_{a}^{2}=\left\\{\begin{array}[]{ll}M^{2}-q^{2}&\mbox{if $a=1$};\\\
M^{2}+q^{2}&\mbox{if $a=2$},\end{array}\right.\ \ \ and\ \ \
Q_{a}=\sqrt{Q_{a}^{2}}.$ $None$

The operators

${\cal P}_{\pm}(x,x_{5})={\cal P}_{1}(x,x_{5})\pm{\cal P}_{2}(x,x_{5})$ $None$

consist of the two parts that are placed onto hyperboloids
$q^{2}+q_{5}^{2}=M^{2}$ and $q^{2}-q_{5}^{2}=-M^{2}$

${\cal P}_{a}(x,x_{5})=\left\\{\begin{array}[]{ll}\int
dq_{5}^{2}e^{-iq_{5}x_{5}}{{d^{4}q}/{(2\pi)^{4}}}e^{-iqx}\Lambda_{1}(q^{2})\delta(q^{2}+q_{5}^{2}-M^{2})&\mbox{if
$a=1$};\\\ \int
dq_{5}^{2}e^{-iq_{5}x_{5}}{{d^{4}q}/{(2\pi)^{4}}}e^{-iqx}\Lambda_{2}(q^{2})\delta(q^{2}-q_{5}^{2}+M^{2})&\mbox{if
$a=2$},\end{array}\right.$ $None$

that satisfy the orthogonality and completeness conditions at $x_{5}=0$

$\int d^{4}y{\cal P}_{a}(x-y,0){\cal P}_{b}(y,0)=\delta_{ab}{\cal
P}_{a}(x,0);$ $None$

${\cal P}_{+}(x,0)=\delta^{(4)}(x)$ $None$

The relation (3.3) presents the projections of the complete 5D field
$\phi(x,x_{5})$ onto the two independent 5D fields $\varphi_{\pm}(x,x_{5})$
which for $x_{5}=0$ produce the 4D fields $\Phi_{\pm}(x)$ (I.8). The Fourier
conjugate $\Phi_{\pm}(q)$ (I.9) and the Fourier conjugate of the 5D fields
$\varphi_{\pm}$ (3.3) consist of the same four parts that are given in (I.10)
and (I.11). Therefore from (3.3) we get

$\Phi_{\pm}(x)=\varphi_{\pm}(x,x_{5}=0)=\int{{d^{4}q}\over{(2\pi)^{4}}}e^{-iqx}\Bigl{[}\phi(q,q_{5}^{2}=M^{2}-q^{2})\Lambda_{1}(q^{2})\pm\phi(q,q_{5}^{2}=M^{2}+q^{2})\Lambda_{2}(q^{2})\Bigr{]},$
$None$

The equations (I.5) and (3.9) presents the 4D reduction of the 6D field
$\varsigma(\kappa,\kappa_{-},\kappa_{+})$ and 5D field $\phi(q,q^{2}_{5})$
onto the 4D field $\Phi_{\pm}(q)$ (I.10).

It must be noted that $\phi(q,M^{2}<q^{2}_{5}<2M^{2})$ does not contribute in
the 4D fields $\Phi_{\pm}(x)$ because the region $M^{2}<q^{2}_{5}<2M^{2}$ is
excluded from the domains in Table 1.

(3.3) and (3.8) allow to represent (3.9) m as the 4D convolution formula

$\varphi_{\pm}(x,x_{5})=\int d^{4}y\Phi_{\pm}(x-y){\cal P}_{+}(y,x_{5}).$
$None$

that determines the two 5D fields $\varphi_{\pm}$ via the corresponding 4D
fields $\Phi_{\pm}$.

4\. 4D and 5D equations of motion.

According to the boundary condition (3.9) and the convolution formula (3.10)
the 5D and 4D scalar fields $\varphi_{\pm}$ and $\Phi_{\pm}$ satisfy the
similar equation of motion

$\biggl{(}{{\partial^{2}}\over{\partial x^{\mu}\partial
x_{\mu}}}+m_{\pm}^{2}\biggr{)}\varphi_{\pm}(x,x_{5})=j_{\pm}(x,x_{5}),$ $None$

$\biggl{(}{{\partial^{2}}\over{\partial x^{\mu}\partial
x_{\mu}}}+m_{\pm}^{2}\biggr{)}\Phi_{\pm}(x)=J_{\pm}(x)$ $None$

where

$J_{\pm}(x)=j_{\pm}(x,x_{5}=0).$ $None$

The 5D source operators $j_{\pm}(x,x_{5})$ in (4.1a) consist of the products
of the 5D fields $\varphi_{\pm}$ and their derivatives. But the Fourier
conjugate of the multiplication of $\varphi_{\pm}(x,x_{5})$ are not placed on
the hyperboloids $q^{2}\pm q_{5}^{2}=\pm M^{2}$. The source operators
$j_{\pm}(x,x_{5})$ in (4.1a) must be placed on the hyperboloids $q^{2}\pm
q_{5}^{2}=\pm M^{2}$ as well as $\varphi_{\pm}(x,x_{5})$ in (3.1).
Consequently, $j_{\pm}(x,x_{5})$ must satisfy the additional 5D constrains in
analogy with (3.1) for $\varphi_{\pm}(x,x_{5})$. In order to obtain these
conditions for $j_{\pm}$ we combine the equations (4.1a,b) and the conditions
(3.1) as

${{\partial^{2}j_{\pm}(x,x_{5})}\over{\partial x^{\mu}\partial
x_{\mu}}}+\Bigl{(}{{\partial^{2}}\over{\partial x^{5}\partial
x_{5}}}+M^{2}\Bigr{)}j_{\mp}(x,x_{5})=$ $\Bigl{(}{{\partial^{2}}\over{\partial
x_{\mu}\partial x^{\mu}}}{{\partial^{2}}\over{\partial x_{5}\partial
x^{5}}}-{{\partial^{2}}\over{\partial x_{5}\partial
x^{5}}}{{\partial^{2}}\over{\partial x_{\mu}\partial
x^{\mu}}}\Bigr{)}\varphi_{\mp}(x,x_{5})+(m_{\pm}^{2}-m_{\mp}^{2}){{\partial^{2}\varphi_{\pm}(x,x_{5})}\over{\partial
x_{\mu}\partial x^{\mu}}}.$ $None$

For the independent variables $x_{\mu}$ and $x_{5}$ the operator
${{\partial^{2}}/{\partial x_{\mu}\partial x^{\mu}}}$ and
${{\partial^{2}}/{\partial x_{5}\partial x^{5}}}$ commute. Therefore we get

${{\partial^{2}{{\widetilde{j}}_{\pm}(x,x_{5})}\over{\partial x^{\mu}\partial
x_{\mu}}}}+\Bigl{(}{{\partial^{2}}\over{\partial x^{5}\partial
x_{5}}}+M^{2}\Bigr{)}{\widetilde{j}}_{\mp}(x,x_{5})=0,$ $None$

where

$\widetilde{j}_{\pm}(x,x_{5})=j_{\pm}(x,x_{5})-m^{2}_{\pm}\varphi_{\pm}(x,x_{5})$
$None$

Thus $\widetilde{j}_{\pm}$ satisfy also the same sourceless equation (3.1).
Using (4.1a) one can rewrite (4.4a) as

${{\partial^{2}\over{\partial x^{\nu}\partial
x_{\nu}}}}{{\partial^{2}{{\varphi}_{\pm}(x,x_{5})}\over{\partial
x^{\mu}\partial x_{\mu}}}}+{{\partial^{2}\over{\partial x^{\nu}\partial
x_{\nu}}}}\Bigl{(}{{\partial^{2}}\over{\partial x^{5}\partial
x_{5}}}+M^{2}\Bigr{)}{\varphi}_{\mp}(x,x_{5})=0,$ $None$

which indicates that the Fourier conjugate of
${{\partial^{2}{{\varphi}_{\pm}(x,x_{5})}/{\partial x^{\mu}\partial
x_{\mu}}}}$ and ${\widetilde{j}}_{\pm}$ in (4.1a) are placed on the
hyperboloids $q^{2}\pm q_{5}^{2}=\pm M^{2}$. Therefore, in analogy with
(I.4a,b) we get

$j_{1}(x,x_{5})=\int{{d^{4}q}\over{(2\pi)^{4}}}dq_{5}^{2}e^{-iqx-
iq^{5}x_{5}}\delta(q^{2}+q_{5}^{2}-M^{2})\Bigl{[}\theta(q^{2})\theta(M^{2}-q^{2})+\theta(-q^{2})\theta(-M^{2}-q^{2})]\Bigr{]}{\cal
J}(q,q_{5}^{2}),$ $None$

$j_{2}(x,x_{5})=\int{{d^{4}q}\over{(2\pi)^{4}}}dq_{5}^{2}e^{-iqx-
iq^{5}x_{5}}\delta(q^{2}-q_{5}^{2}+M^{2})\Bigl{[}\theta(q^{2})\theta(-M^{2}+q^{2})+\theta(-q^{2})\theta(M^{2}+q^{2})\Bigr{]}{\cal
J}(q,q_{5}^{2}).$ $None$

where ${\cal J}$ denote the full 5D source operator

${\cal J}(x,x_{5})=\int{{d^{5}q}\over{(2\pi)}}e^{-iqx-iq_{5}x_{5}}{\cal
J}(q.q_{5}^{2})$ $None$

Afterwards the 5D sources ${j}_{\pm}$ can be represented as

${{j}}_{\pm}(x,x_{5})=\int{{d^{4}qe^{-iqx}}\over{(2\pi)^{4}}}\Bigl{[}{\cal
J}(q,Q^{2}_{1})\Lambda_{1}(q^{2})e^{-iQ_{1}x_{5}}\pm{\cal
J}(q,Q^{2}_{2})\Lambda_{2}(q^{2})e^{-iQ_{2}x_{5}}\Bigr{]},$ $None$

that in analogy with (3.3) and (3.10) allow to construct the 5D sources
${{j}}_{\pm}(x,x_{5})$ through the 4D sources $J_{\pm}(x)$ or the full 5D
sources ${\cal J}$ using the following convolution formulas

${{j}}_{\pm}(x,x_{5})=\int d^{4}yJ_{\pm}(x-y){\cal P}_{+}(y,x_{5}),$ $None$

$j_{\pm}(x,x_{5})=\int d^{5}y{\cal J}(x-y,x_{5}-y_{5}){\cal
P}_{\pm}(y,y_{5}).$ $None$

According to (4.6a) the sources $J_{\pm}(x)$ and their Fourier conjugate
$J_{\pm}(q)$ consist of the four parts as well as the 4D field $\Phi_{\pm}(q)$
in (I.10)

$J_{\pm}(q)=\sum_{N=I,III}J_{N}(q)\pm\sum_{N=II,IV}J_{N}(q),$ $None$

where

$J_{I}(q)=\theta(q^{2})\theta(M^{2}-q^{2}){\cal J}(q,q^{2}_{5}=M^{2}-q^{2});\
\ \ \ \ J_{III}(q)=\theta(-q^{2})\theta(-M^{2}-q^{2}){\cal
J}(q,q^{2}_{5}=M^{2}-q^{2});$ $None$
$J_{II}(q)=\theta(q^{2})\theta(-M^{2}+q^{2}){\cal
J}(q,q^{2}_{5}=M^{2}+q^{2});\ \ \ \ \
J_{IV}(q)=\theta(-q^{2})\theta(M^{2}+q^{2}){\cal J}(q,q^{2}_{5}=M^{2}+q^{2}),$
$None$

where the lower index $N=I,II,III,IV$ in ${{J}_{N}}$ indicates the domains of
$q^{2}$ in Table 1.

In (4.6a) ${{j}}_{\pm}(x,x_{5})$ is constructed through the 4D sources
$J_{\pm}(q)$ (4.8a,b) that are defined via the 4D fields $\Phi_{\pm}$, i.e.
(4.6a) allow to determine the 5D source $j_{\pm}$ through the 4D fields
$\Phi_{\pm}$. In (4.6b) ${{j}}_{\pm}(x,x_{5})$ is constructed is constructed
via the 5D sources ${\cal J}(x,x_{5})$ (4.5c) which consists of the products
of the complete 5D fields $\phi(x,x_{5})$. The 5D source ${\cal J}$ (4.5c) can
be determined via the fields $\phi$ according to the 5D extension of the
Klein-Gordon equation

$\biggl{(}{{\partial^{2}}\over{\partial x^{\mu}\partial
x_{\mu}}}+{\widetilde{m}}^{2}\biggr{)}\phi(x,x_{5})={\cal J}(x,x_{5}).$ $None$

In the present formulation the terms with $\partial\varphi_{\pm}/\partial
x_{5}$ are determined through the constrains which ensure the consistency of
the condition (3.1) and the equation of motion (4.1a). These constrains are
considered in the next section. In (4.9) the projections of
$\partial\phi_{\pm}/\partial x_{5}$ can be included in ${\cal J}$. But these
terms must be consistent with the constrains for $\partial\varphi/\partial
x_{5}$.

The projections of the 5D field $\phi$ with mass ${\widetilde{m}}$ and source
${\cal J}$ into two 5D fields $\varphi_{\pm}$ with the two different masses
$m_{\pm}$ and the two sources $j_{\pm}$ are performed in (3.3) and (4.6c)
using the projection operators ${\cal P}_{\pm}$ (3.6a). In these projections
only the parts of the complete 5D field and its source are used for
construction of the 5D fields $\varphi_{\pm}$ and $j_{\pm}$ because the parts
of these fields in the domains $N=I,II,III,IV$ in Table 1 cover only the part
of the full 5D space.

The input 5D fields $\phi$, ${\cal J}$ and their projections $\varphi_{\pm}$,
$j_{\pm}$ can be constructed within the various relativistic invariant time
models [8, 9].

It must be noted, that $\partial\varphi_{\pm}(x,x_{5})/\partial x_{5}$,
$j_{\pm}(x,x_{5})$ and $m^{2}_{\pm}$ satisfy an additional conditions that can
be obtained combining (4.1a) and (3.1)

$M^{2}\biggl{[}1+\Bigl{(}{1\over M}{{\partial}\over{\partial
x_{5}}}\Bigr{)}^{2}\biggr{]}\varphi_{\pm}=m_{\mp}^{2}\varphi_{\mp}-j_{\mp}.$
$None$

The conditions (4.10) presents the relationship between $j_{\pm}$, $m_{\pm}$
and the operators $\biggl{[}1+\Bigl{(}{1\over M}{{\partial}/{\partial
x_{5}}}\Bigr{)}^{2}\biggr{]}\varphi_{\pm}$. The factorization of
$\biggl{[}1+\Bigl{(}{1\over M}{{\partial}/{\partial
x_{5}}}\Bigr{)}^{2}\biggr{]}$ allows to determine
$\partial\varphi_{\pm}/\partial x_{5}$ through $j_{\pm}(x,x_{5})$,
$m^{2}_{\pm}$ and vice versa. This problem will be considered in the next
section.

5\. Constrains for ${{\partial\varphi_{\pm}}/{\partial x^{5}}}$.

In order to obtain the constrains for ${{\partial\varphi_{\pm}}/{\partial
x^{5}}}$ we shall factorize (4.10). The general form of the sought linear
conditions for ${{\partial\varphi_{\pm}}/{\partial x_{5}}}$ of the scalar
charged fields are

${i\over M}{{\partial\varphi_{\pm}}\over{\partial
x_{5}}}=\alpha_{\pm}\varphi_{+}+\beta_{\pm}\varphi_{-}+C_{\pm},$ $None$

where $C_{\pm}$ consist of the products of $\varphi_{\pm}$. The Fourier
conjugate of these products are not located on the hyperboloids $q^{2}\pm
q_{5}^{2}=\pm M^{2}$. Therefore, $C_{\pm}$ are defined as

$C_{\pm}(x,x_{5})=\int{\widetilde{\cal C}}_{\pm}(x-y,x_{5}-y_{5})d^{5}y{\cal
P}_{+}(y,y_{5}).$ $None$

Substituting (5.1) in (4.10) one obtains

$m_{+}^{2}=-2M^{2}\alpha_{+}{{1-\alpha_{+}^{2}}\over{\beta_{+}}}\ \ \ \ \
j_{+}=M^{2}\biggl{[}{{1-\alpha_{+}^{2}}\over{\beta_{+}}}C_{+}+\alpha_{+}C_{-}+{i\over
M}{{\partial C_{-}}\over{\partial x_{5}}}\biggr{]};$ $None$
$m^{2}_{-}=-2M^{2}\alpha_{+}\beta_{+};\ \ \ \ \
j_{-}=M^{2}\biggl{[}\alpha_{+}C_{+}+\beta_{+}C_{-}+{i\over M}{{\partial
C_{+}}\over{\partial x_{5}}}\biggr{]}.$ $None$

where are using the conditions

$\alpha_{+}^{2}+\alpha_{-}\beta_{+}=1;\ \ \ \ \ \ \ \ \ \ \
\beta_{-}^{2}+\alpha_{-}\beta_{+}=-1$ $None$

that insures reproduction of the mass terms in (4.10).

From (5.4) follows that $\alpha_{+}=\beta_{-}$ and the condition
$m^{2}_{\pm}>0$ requires that $-1<\alpha_{+}<1$. If $\alpha_{+}=-\beta_{-}$,
then $m_{+}=m_{-}=0$.

The relations (5.3a,b) determine $m^{2}_{\pm}$ through the three parameters
$M^{2}$, $\alpha_{+}$ and $\beta_{+}$. The same parameters, $C_{\pm}$ and
${i/M}\ {{\partial C_{\pm}}/{\partial x_{5}}}$ determine the sources
$j_{\pm}$.

For the neutral particles the analogue of (5.1) is

${1\over M}{{\partial\varphi_{\pm}}\over{\partial
x_{5}}}=\alpha_{\pm}\varphi_{+}+\beta_{\pm}\varphi_{-}+C_{\pm},$ $None$

Substituting this condition in (4.10) we get

$m_{+}^{2}=-2M^{2}\alpha_{+}{{1+\alpha_{+}^{2}}\over{\beta_{+}}}\ \ \ \ \
j_{+}=M^{2}\biggl{[}{{1+\alpha_{+}^{2}}\over{\beta_{+}}}C_{+}+\alpha_{+}C_{-}+{1\over
M}{{\partial C_{-}}\over{\partial x_{5}}}\biggr{]};$ $None$
$m^{2}_{-}=2M^{2}\alpha_{+}\beta_{+};\ \ \ \ \
j_{-}=M^{2}\biggl{[}\alpha_{+}C_{+}+\beta_{+}C_{-}+{1\over M}{{\partial
C_{+}}\over{\partial x_{5}}}\biggr{]},$ $None$

where the following conditions for the parameters are used

$\alpha_{+}^{2}+\alpha_{-}\beta_{+}=-1;\ \ \ \ \ \ \ \ \ \ \
\beta_{-}^{2}+\alpha_{-}\beta_{+}=-1.$ $None$

The ratio $m_{+}^{2}/m_{-}^{2}=-(\alpha_{+}^{2}+1)/\beta_{+}^{2}$ in this case
is negative. Therefore, the constrains (5.5) can not generate the positive
$m_{+}^{2}$ and $m_{-}^{2}$ simultaneously for the neutral fields.

The positive masses $m_{\pm}^{2}\geq 0$ for the neuthral particles in (4.10)
can be reproduced using the other kind of the constrains

$\varphi_{-}=\alpha\varphi_{+}+{\em G}(\varphi_{+},\varphi_{-}),$ $None$

where ${\em G}$ does not contain the linear over $\varphi_{\pm}$ terms. For
the sake of simplicity the dependence over ${\partial\varphi_{\pm}}/{\partial
x_{\mu}}$ and ${\partial\varphi_{\pm}}/{\partial x_{5}}$ in ${\em G}$ (5.7a)
are omitted.

Acting with $\biggl{[}1+\Bigl{(}{1\over M}{{\partial}/{\partial
x_{5}}}\Bigr{)}^{2}\biggr{]}$ in (5.7a) and using the conditions

$m_{+}^{2}=\alpha^{2}m_{-}^{2};\ \ \ \ \ {\em
F}=M^{-2}\biggl{[}1+\alpha{{m_{-}^{2}}\over{M^{2}}}+\Bigl{(}{1\over
M}{{\partial}\over{\partial x_{5}}}\Bigr{)}^{2}\biggr{]}^{-1}{\em G}$ $None$

one determine $j_{-}$ via $j_{+}$ as

$\alpha j_{-}-j_{+}={\em F}.$ $None$

In particular, for the $\Phi_{+}^{4}$ model with the 4D source
$J_{+}=a\Phi_{+}^{2}+b\Phi_{+}^{3}$ and mass $m_{+}$ the relations (5.7a,b,c)
allow to reproduce the 5D equations (4.1a) with the 5D source operator
$j_{\pm}$ and the masses $m_{\pm}^{2}$. These 5D equation generates the 4D
equation of motion with $J_{-}=(J_{+}+{\em F})/\alpha$ and the real mass
$m_{-}=m_{+}/\alpha$.

For the charged fields it is convenient to use the following constrains

$\biggl{[}1-{i\over M}{{\partial}\over{\partial
x_{5}}}\biggr{]}\varphi_{\pm}={\em C}_{\pm}.$ $None$

where ${\em C}_{\pm}$ depends generally on the products of $\varphi_{\pm}^{+}$
and $\varphi_{\pm}$.

$M^{2}\biggl{[}1+{i\over M}{{\partial}\over{\partial x_{5}}}\biggr{]}{\em
C}_{\mp}=m_{\pm}^{2}\varphi_{\pm}-j_{\pm}.$ $None$

Present formulation has in common with the other 5D field-theoretical
approaches within the relativistic invariant time method [7, 8, 9] with
$x_{5}^{2}=x_{0}^{2}-{\bf x}^{2}\equiv x_{\mu}x^{\mu}$. Unlike these 5D
formulations our approach based on the invariance of the 6D and 5D forms
(I.2a,b) and (I.3a,b) under the conformal transformations in the momentum
space. The main difference between the our conditions for
$\partial\varphi_{\pm}/\partial x_{5}$ (4.10), (5.1) and (5.7c) and the
evaluation-type equations over the $x_{5}$ other authors is that in the
present formulation $\partial\varphi_{+}/\partial x_{5}$ is determined through
the source and mass of the coupling field $j_{-}$ and $m_{-}$. Nevertheless,
one can use the 5D models in [7, 8, 9] for input 5D field $\phi(q,q^{2}_{5})$
in (I.4a,b) and input 5D and 4D source operators
$J_{\pm}(x)=j_{\pm}(x,x_{5}=0)$ in (4.1,b).

6\. The 5D and 4D Lagrangians.

In this section we shall consider the 5D Lagrangians ${{\cal
L}}_{\pm}(x,x_{5})$ for the two interacted scalar 5D fields $\varphi_{\pm}$
with the same quantum numbers, but with the different masses $m_{\pm}$ and
sources $j_{\pm}(x,x_{5})$. These Lagrangians must reproduce the 4D
Lagrangians $L_{\pm}(x)$ at $x_{5}=0$ and the 5D equations of motion (4.1a).
The sought Lagrangian can be represented as

${{\cal L}}_{\pm}(x,x_{5})=({{\cal L}}_{\pm})_{o}(x,x_{5})+({{\cal
L}}_{\pm})_{INT}(x,x_{5})+({{\cal L}}_{\pm})^{C}(x,x_{5})$ $None$

where $({{\cal L}}_{\pm})_{o}$ and $({{\cal L}}_{\pm})_{INT}$ denote the non-
interacted and interacted parts of these Lagrangians. The third term $({{\cal
L}}_{\pm})^{C}$ reproduce the constrains for the
$\partial\varphi_{\pm}/\partial x_{5}$ in (5.1), (5.5), (5.7a) and (5.8a,b).
Any one of these constrains together with the 5D equation of motion (4.1a)
generate the conditions (3.1). In particular, $({{\cal L}}_{\pm})^{C}$ for the
constrains (5.8a,b) are

${\cal L}_{\pm}^{C}=M^{2}|{i\over M}{{\partial\varphi_{\pm}}\over{\partial
x_{5}}}-\varphi_{\pm}+{\em C}_{\pm}|^{2}+M^{2}|{i\over M}{{\partial{\em
C}_{\pm}}\over{\partial x_{5}}}+{\em
C}_{\pm}-{{m_{\mp}^{2}\varphi_{\mp}-j_{\mp}}\over{M^{2}}}|^{2},$ $None$

where $\delta{\cal L}_{\pm}^{C}/\delta{\em C}_{\pm}$ and $\delta{\cal
L}_{\pm}^{C}/\delta[{{\partial{\em C}_{\pm}}/{\partial x_{5}}}]$ reproduce
(5.8a) and (5.8b) correspondingly.

The 5D Lagrangians (6.1) can be simply constructed starting from the two 4D
Lagrangians $L_{\pm}(x)$ for the two interacted scalar fields $\Phi_{\pm}(x)$

$L_{\pm}(x)=(L_{\pm})_{o}(x)+({L}_{\pm})_{INT}(x),$ $None$

where the non-interacting part $(L_{\pm})_{o}$

$({{L}}_{\pm})_{o}(x)={{\partial{\Phi_{\pm}}^{+}}\over{\partial
x_{\mu}}}{{\partial{\Phi_{\pm}}}\over{\partial
x^{\mu}}}-m_{\pm}^{2}{\Phi_{\pm}}^{+}{\Phi_{\pm}}$ $None$

determines the non-interacted part of the 5D Lagrangians (6.1)

$({{\cal L}}_{\pm})_{o}(x,x_{5})={{\partial{\varphi_{\pm}}^{+}}\over{\partial
x_{\mu}}}{{\partial{\varphi_{\pm}}}\over{\partial
x^{\mu}}}-m_{\pm}^{2}{\varphi_{\pm}}^{+}{\varphi_{\pm}}.$ $None$

The 5D source $j_{\pm}$ in (4.1a) can be constructed from the 4D source
$J_{\pm}$ (4.1b) using the convolution formula (4.6b). The sought 5D
Lagrangian $({{\cal L}}_{\pm})_{INT}$ consists of the fields $\varphi_{\pm}$.
The replacement of the 4D fields $\Phi_{\pm}$ in $(L_{\pm})_{INT}$ with the 5D
fields $\varphi_{\pm}$ give

$({{\cal L}}_{\pm})_{INT}(x,x_{5})=\int d^{4}y({\sf
L}_{\pm})_{INT}(x-y,x_{5}){\cal P}_{+}(y,x_{5}),\ \ \ where\ \ \ ({\sf
L}_{\pm})_{INT}(x,x_{5})=({L}_{\pm})_{INT}\Bigl{(}\Phi_{\pm}\Longleftrightarrow\varphi_{\pm}\Bigr{)}.$
$None$

The 5D sources ${j}_{\pm}$ satisfy the Euler-Lagrange equations

${j}_{\pm}(x,x_{5})={{\partial{({\cal
L}_{\pm})_{INT}}}\over{\partial{\varphi_{\pm}}^{+}}}-{{d}\over{dx_{\mu}}}\biggl{(}{{\partial{({\cal
L}_{\pm})_{INT}}}\over{\partial{{[\partial{\varphi_{\pm}}^{+}}/{\partial
x^{\mu}}]}}}\biggr{)},$ $None$

The other kind of the 5D sources $j_{\pm}$ can be obtained from the general 5D
Lagrangian ${\ell}(\phi,\phi^{+})$ for the 5D scalar field $\phi(q,q_{5}^{2})$
that parts are used in (I.4a,b) and (3.2) for reproduction of $\varphi_{\pm}$

${\ell}(\phi,\phi^{+})={{\partial{\phi}^{+}}\over{\partial
x_{\mu}}}{{\partial{\phi}}\over{\partial
x^{\mu}}}+{\widetilde{m}}^{2}\phi^{+}\phi+{\ell}_{INT}(\phi,\phi^{+}),$ $None$

where the terms with $\partial\phi/\partial x_{5}$ are included in
${\ell}_{INT}$. The Lagrangian (6.7) generate the 5D equation of motion (4.9).
In order to reproduce $j_{\pm}$ from ${\ell}(\phi,\phi^{+})$ (6.7) one must
determine $\partial\phi/\partial x_{5}$ according to constrains for
$\partial\varphi_{\pm}/\partial x_{5}$ that are given in $({{\cal
L}}_{\pm})^{C}$ (6.1). For this aim we shall consider the projection of the
equation of motion (4.9) according to (3.3) and (4.6c)

$\int
d^{5}y\Bigr{[}{{\partial^{2}\phi(x-y,x_{5}-y_{5})}\over{\partial(x-y)^{\mu}\partial(x-y)_{\mu}}}-{{\cal
J}}(x-y,x_{5}-y_{5})+{\widetilde{m}}^{2}\phi(x-y,x_{5}-y_{5})\Bigl{]}{\cal
P}_{\pm}(y,y_{5})=0$ $None$

This equation has the form of the equation (4.1a) if

$\int d^{5}y{{\cal J}}(x-y,x_{5}-y_{5}){\cal
P}_{\pm}(y,y_{5})=j_{\pm}(x,x_{5})-\Bigl{(}m^{2}_{\pm}-{\widetilde{m}}^{2}\Bigr{)}\varphi_{\pm}(x,x_{5})$
$None$

The sources $j_{\pm}$ in (6.8a,b) are defined in (4.6c) via the projections of
${\cal J}$. The complete 5D source ${\cal J}$ consists of the products of the
fields $\phi$ and their derivatives. Therefore, ${\cal J}$ can not be reduced
to the combination of the fields $\varphi_{\pm}$ that are the parts of
$\phi(x,x_{5})$. Thus in opposite to $j_{\pm}$ constructed from the Lagrangian
(6.5), the projections of ${\cal J}$ in (6.8a,b) does not satisfy the equation
(6.6) and the Lagrangian . ${\ell}(\phi,\phi^{+})$ (6.7) can not be reduced to
the Lagrangians (6.1) $({\cal L}_{\pm}$. The consistency conditions of the
projections of ${\cal J}$ in (6.8a,b) in the equation of motion (4.1a) and the
condition (3.1) can be obtained using the constrains for
$\partial\varphi/\partial x_{5}$ considered in the previous section.

The third form of the 5D Lagrangians (6.1) reproduce exactly the constrains
(3.1) and (5.8a,b) with the a priory given source $j_{\pm}$

${\cal L}(x,x_{5})={{\partial{\varphi_{+}}^{+}}\over{\partial
x_{\mu}}}{{\partial{\varphi_{+}}}\over{\partial
x^{\mu}}}+M^{2}{\varphi_{-}}^{+}{\varphi_{+}}+({{\cal
L}}_{+})^{C}+{{\partial{\varphi_{-}}^{+}}\over{\partial
x_{\mu}}}{{\partial{\varphi_{-}}}\over{\partial
x^{\mu}}}+M^{2}{\varphi_{+}}^{+}{\varphi_{-}}+({{\cal L}}_{-})^{C}.$ $None$

where the Lagrangians $({{\cal L}}_{\pm})^{C}$ generate the conditions
(5.8a,b) and variation over ${\varphi_{\pm}}^{+}$ and
${{\partial{\varphi_{\pm}}^{+}}/{\partial x_{\mu}}}$ produces (3.1). The
combination of (3.1) and (5.8a.b) gives the equation of motion (4.1a).

7\. The 4D and 5D equation of motion for the fermion fields with the
electromagnetic interactions.

The 4D and 5D equation of motion for the two fermion fields
$\Psi_{+}(x)=\psi_{+}(x,x_{5}=0)$ and $\Psi_{-}(x)=\psi_{-}(x,x_{5}=0)$ with
the same quantum numbers but with the different masses $m_{\pm}$ and source
operators can be represented in analogy with the scalar fields. In order to
place the Fourier conjugate of the 5D Dirac fields
$\psi_{1,2}(x,x_{5})=1/2\Bigl{(}\psi_{\pm}(x,x_{5})\pm\psi_{\pm}(x,x_{5})\Bigr{)}$
on the hyperboloids $q^{2}\pm q_{5}^{2}=\pm M^{2}$ the fields
$\psi_{+}(x,x_{5})$ and $\psi_{-}(x,x_{5})$ must satisfy the condition (3.1)

${{\partial^{2}\psi_{\pm}(x,x_{5})}\over{\partial x^{\mu}\partial
x_{\mu}}}+\Bigl{(}{{\partial^{2}}\over{\partial x^{5}\partial
x_{5}}}+M^{2}\Bigr{)}\psi_{\mp}(x,x_{5})=0.$ $None$

Similarly with $\varphi_{\pm}$ in (I.10), we divide the Fourier conjugate of
$\psi_{\pm}(x,x_{5})$ into four parts defined in the domains I,II,III and IV
in Table 1

$\psi_{I}(q,q_{5}^{2})=\theta(q^{2})\theta(M^{2}-q^{2})\theta(M^{2}-q_{5}^{2})\Upsilon(q,q_{5}^{2});\
\ \
\psi_{II}(q,q_{5}^{2})=\theta(q^{2})\theta(-M^{2}+q^{2})\theta(q_{5}^{2}-2M^{2})\Upsilon(q,q^{2}_{5});$
$\psi_{III}(q,q_{5}^{2})=\theta(-q^{2})\theta(-M^{2}-q^{2})\theta(q_{5}^{2}-2M^{2})\Upsilon(q,q^{2}_{5});\
\ \
\psi_{IV}(q,q_{5}^{2})=\theta(-q^{2})\theta(M^{2}+q^{2})\theta(M^{2}-q_{5}^{2})\Upsilon(q,q_{5}^{2}),$
$None$

Then as in (3.2) for $\varphi_{\pm}(x,x_{5})$, for $\psi_{\pm}(x,x_{5})$ we
have

$\psi_{\pm}(x,x_{5})=\int{{d^{4}q}\over{(2\pi)^{4}}}e^{-iqx}\biggl{[}\sum_{N=I,III}\psi_{N}(q,Q_{1}^{2})e^{-iQ_{1}x_{5}}\pm\sum_{N=II,IV}\psi_{N}(q,Q_{2}^{2})e^{-iQ_{2}x_{5}}\biggr{]},$
$None$

and $\psi_{\pm}(x,x_{5})$ satisfy automatically the condition (7.1).

For $x_{5}=0$ $\psi_{\pm}$ (7.3) generate the physical 4D fields
$\Psi_{\pm}(x)$

$\Psi_{\pm}(x)=\psi_{\pm}(x,x_{5}=0)=\int{{d^{4}q}\over{(2\pi)^{4}}}e^{-iqx}\biggl{[}\sum_{N=I,III}\Psi_{N}(q)\pm\sum_{N=II,IV}\Psi_{N}(q)\biggr{]},$
$None$

where

$\Psi_{I}(q)=\psi_{I}(q,q^{2}_{5}=M^{2}-q^{2});\ \ \
\Psi_{II}(q)=\psi_{II}(q,q^{2}_{5}=M^{2}+q^{2});$

$\Psi_{III}(q)=\psi_{III}(q,q^{2}_{5}=M^{2}-q^{2});\ \ \
\Psi_{IV}(q)=\psi_{IV}(q,q^{2}_{5}=M^{2}+q^{2})$ $None$

and

$\Psi_{\pm}(q)=\sum_{N=I,III}\Psi_{N}(q)\pm\sum_{N=II,IV}\Psi_{N}(q).$ $None$

According to (7.4) and (7.5a,b) the Fourier conjugate of the 5D fields
$\psi_{\pm}(x,x_{5})$ (7.3) contains the same four parts as the Fourier
conjugate of the 4D fields $\Psi_{\pm}(x)$. Therefore, one can represent the
$\psi_{\pm}(x,x_{5})$ through $\Psi(x)$ as

$\psi_{\pm}(x,x_{5})=\int d^{4}y\Psi_{\pm}(x-y){\cal P}_{+}(y,x_{5}).$ $None$

The convolution formula (7.6) is similar to (3.10) for the scalar fields
$\varphi_{\pm}(x,x_{5})$.

The equation of motion for the 5D fields $\psi_{\pm}$ can be derived using the
gauge transformation of the 5D Dirac equations for the two non-interacting
fields $\psi_{\pm}^{(o)}$ with the different masses $m_{\pm}$

$\Bigl{(}i\gamma_{\mu}{{\partial}\over{\partial
x_{\mu}}}-m_{\pm}\biggr{)}\psi_{\pm}^{(o)}(x,x_{5})=0,$ $None$

where

$\psi_{\pm}^{(o)}(x,x_{5})=\int
d^{4}y\exp{\Bigl{(}ie\Lambda(x-y,x_{5})\Bigr{)}}\psi_{\pm}(x-y,x_{5}){\cal
P}_{+}(y,x_{5});$ $None$
$ie{A}^{\mu}(x,x_{5})=\exp{\Bigl{(}-ie\Lambda(x,x_{5})\Bigr{)}}{{\partial}\over{\partial
x_{\mu}}}\exp{\Bigl{(}ie\Lambda(x,x_{5})\Bigr{)}}.$ $None$

Substituting (7.8a,b) into (7.7) we obtain

$\int
d^{4}ye^{ie\Lambda(x-y,x_{5})}\biggl{(}i\gamma_{\mu}{{\partial}\over{\partial
x_{\mu}}}-e\gamma_{\mu}{A}^{\mu}(x-y,x_{5})-m_{\pm}\biggr{)}\psi_{\pm}(x-y,x_{5}){\cal
P}_{+}(y,x_{5})=0.$ $None$

According to (3.8) the projection operator ${\cal P}_{+}(y,x_{5})$ for
$x_{5}=0$ transforms into $\delta^{4}(y)$. Therefore, for $x_{5}=0$ the 5D
non-local equations (7.9) generate the usual 4D Dirac equation for the fermion
field with the electromagnetic interaction

$\biggl{(}i\gamma_{\mu}{{\partial}\over{\partial x_{\mu}}}-e\gamma_{\mu}{\sf
A}^{\mu}(x)-m_{\pm}\biggr{)}\Psi_{\pm}(x)=0.$ $None$

where ${\sf A}^{\mu}(x)={A}^{\mu}(x,x_{5}=0)$,
$\Psi_{\pm}^{(o)}(x)=\psi_{\pm}^{(o)}(x,x_{5}=0)$ are the asymptotic
${}^{\prime\prime}in^{\prime\prime}$ or ${}^{\prime\prime}out^{\prime\prime}$
fields which satisfy the equations

$\Bigl{(}i\gamma_{\mu}{{\partial}\over{\partial
x_{\mu}}}-m_{\pm}\biggr{)}\Psi_{\pm}^{(o)}(x)=0;$ $None$

$\Psi_{\pm}^{(o)}(x)=\exp{\Bigl{(}ie\Lambda(x,0)\Bigr{)}}\Psi_{\pm}(x);\ \ \ \
\ \ ie{\sf
A}^{\mu}(x)=\exp{\Bigl{(}-ie\Lambda(x,0)\Bigr{)}}{{\partial}\over{\partial
x_{\mu}}}\exp{\Bigl{(}ie\Lambda(x,0)\Bigr{)}}.$ $None$

The Fourier conjugate of the 5D gauge fields ${A}^{\mu}(x,x_{5})$ are embedded
on the hyperboloids $q^{2}\pm q_{5}^{2}=\pm M^{2}$ and they satisfy the
condition (3.1) As well as the scalar fields $\varphi_{\pm}$. Due to gauge
invariance one can use any combination of ${A}^{\mu}_{+}$ and ${A}^{\mu}_{-}$
in the gauge transformations (7.8b). Therefore, in (7.8a,b)-(7.11a,b) and in
the following formulas the lower indexes ${\pm}$ of $\Lambda$, ${\sf A}^{\mu}$
and ${A}^{\mu}$ are omitted.

The 5D representations of the interacted and non-interacted 5D fields
$\psi_{\pm}$ and $\psi_{\pm}^{(o)}$ are single-valued determined through the
their 4D reductions $\Psi_{\pm}$ and $\Psi_{\pm}^{(o)}$ according to (7.6).
Therefore $\psi_{\pm}^{(o)}$ satisfies also the condition (7.1)

${{\partial^{2}\psi_{\pm}^{(o)}(x,x_{5})}\over{\partial x^{\mu}\partial
x_{\mu}}}+\Bigl{(}{{\partial^{2}}\over{\partial x^{5}\partial
x_{5}}}+M^{2}\Bigr{)}\psi_{\mp}^{(o)}(x,x_{5})=0.$ $None$

The gauge transformation (7.8a) in (7.12) together with (7.1) yields

$\eta_{\pm}(x,x_{5})=-\eta_{\mp}^{(5)}(x,x_{5}),$ $None$

where

$\eta_{\pm}(x,x_{5})=\biggl{(}ie{{\partial}\over{\partial
x^{\mu}}}{A}^{\mu}(x,x_{5})+ie{A}^{\mu}(x,x_{5}){{\partial}\over{\partial
x^{\mu}}}-e^{2}{A}^{\mu}(x,x_{5}){A}_{\mu}(x,x_{5})\biggr{)}\psi_{\pm}(x,x_{5}),$
$None$

and the auxiliary source operator $\eta_{\pm}^{(5)}$ is defined via ${A}^{5}$
as

$\eta_{\pm}^{(5)}(x,x_{5})=\biggl{(}ie{{\partial}\over{\partial
x^{5}}}{A}^{5}(x,x_{5})+ie{A}^{5}(x,x_{5}){{\partial}\over{\partial
x^{5}}}-e^{2}{A}^{5}(x,x_{5}){A}_{5}(x,x_{5})\biggr{)}\psi_{\pm}(x,x_{5}),$
$None$

$ie{A}^{5}(x,x_{5})=\exp{\Bigl{(}-ie\Lambda(x,x_{5})\Bigr{)}}{{\partial}\over{\partial
x_{5}}}\exp{\Bigl{(}ie\Lambda(x,x_{5})\Bigr{)}}.$ $None$

Condition (7.13) allows to define ${A}^{5}$ through $\eta_{\pm}$ and
$\psi_{\pm}$.

The complete 5D fields $\Upsilon(x,x_{5})$ and their Fourier conjugate
$\Upsilon(q,q_{5}^{2})$ in (7.2) can be defined in the relativistic invariant
time models [8, 9]. The projections of $\Upsilon(x,x_{5})$ and
$\Upsilon^{(o)}(x,x_{5})$ onto hyperboloids $q^{2}\pm q_{5}^{2}=\pm M^{2}$ can
be determined also via the 5D convolution formulas

$\psi_{\pm}(x,x_{5})=\int d^{5}y{\Upsilon}(x-y,x_{5}-y_{5}){\cal
P}_{\pm}(y,y_{5});\ \ \ \psi_{\pm}^{(o)}(x,x_{5})=\int
d^{5}y{\Upsilon}^{(o)}(x-y,x_{5}-y_{5}){\cal P}_{\pm}(y,y_{5}),$ $None$

8\. Constrains for $\partial\psi_{\pm}/\partial x_{5}$.

The consistency condition between the 5D Dirac equations (7.9) and the
constrains (7.1) can be represent through the linear equations for
$\partial\psi_{\pm}/\partial x_{5}$. For this aim it is convenient to rewrite
the (7.9) in the form of the Klein-Gordon equations. Action of the operator
$i\gamma_{\mu}{{\partial}/{\partial x_{\mu}}}+m_{\pm}$ on (7.7) produces the
following Klein-Gordon equations

$\biggl{(}{{\partial^{2}}\over{\partial x^{\mu}\partial
x_{\mu}}}+m_{\pm}^{2}\biggr{)}\psi_{\pm}^{(o)}(x,x_{5})=0.$ $None$

Then the gauge transformation (7.8a) yields

$\int
d^{4}ye^{ie\Lambda(x-y,x_{5})}\biggl{[}\Bigl{(}{{\partial^{2}}\over{\partial
x_{\mu}\partial
x^{\mu}}}-m_{\pm}^{2}\Bigr{)}\psi_{\pm}(x-y,x_{5})-\eta_{\pm}(x-y,x_{5})\biggr{]}{\cal
P}_{+}(y,x_{5})=0,$ $None$

where $\eta_{\pm}(x,x_{5})$ is defined in (7.14a).

One can replace (7.1) with

$\int
d^{4}ye^{ie\Lambda(x-y,x_{5})}\biggl{[}{{\partial^{2}\psi_{\pm}(x-y,x_{5})}\over{\partial
x_{\mu}\partial x^{\mu}}}+\Bigl{(}M^{2}+{{\partial^{2}}\over{\partial
x_{5}\partial x^{5}}}\Bigr{)}\psi_{\mp}(x-y,x_{5})\biggr{]}{\cal
P}_{+}(y,x_{5})=0.$ $None$

Afterwards we obtain the following consistency condition of (8.3) and (8.2)
for the 5D Dirac fields

$\int
d^{4}ye^{ie\Lambda(x-y,x_{5})}\Biggl{\\{}\biggl{[}M^{2}+\Bigl{(}{{\partial}\over{\partial
x_{5}}}\Bigr{)}^{2}\biggr{]}\psi_{\pm}(x-y,x_{5})-m_{\mp}^{2}\psi_{\mp}(x-y,x_{5})+\eta_{\mp}(x-y,x_{5})\Biggr{\\}}{\cal
P}_{+}(y,x_{5})=0.$ $None$

According to (8.4) $\Bigr{[}M^{2}+\bigl{(}\partial/{\partial
x_{5}}\bigr{)}^{2}\Bigr{]}\psi_{+}$ is determined by the source and mass of
the $\psi_{-}$ fields in (8.2) and vice versa. Unlike (4.10) the condition
(8.4) is given in the form of the 4D convolution formula.

One can factorise (8.4) in the same way as (4.10) using the constrain for
$\partial\ \psi_{\pm}/\partial x_{5}$. In particular,

${i\over M}{{\partial\over{\partial
x_{5}}}\psi_{\pm}}=\alpha_{\pm}\psi_{+}+\beta_{\pm}\psi_{-}+C_{\pm},$ $None$

where $C_{\pm}$ are defined onto hyperboloids $q^{2}\pm q_{5}^{2}=\pm M^{2}$
according to (5.2) and $C_{\pm}$ does not contain the linear terms over
$\psi_{\pm}$.

The constrains (8.5) allow to construct the relationship between the masses
$m_{\pm}^{2}$, sources $j_{\pm}$ and the parameters $M^{2}$, $\alpha_{+}$,
$\beta_{+}$ and $C_{\pm}$ in the same way as in (5.3a,b). In particular, the
choice of the parameters according to (5.4) gives

${{m_{+}^{2}}\over{m_{-}^{2}}}={{1-\alpha_{+}^{2}}\over{\beta_{+}^{2}}};\ \ \
\ \ {{m_{+}^{2}m_{-}^{2}}\over{4M^{2}}}=\alpha_{+}^{2}(1-\alpha_{+}^{2});\ \ \
\ \ \alpha_{+}^{2}={1\over 2}\pm{1\over
2}\sqrt{1-{{m_{+}^{2}m_{-}^{2}}\over{M^{4}}}}$ $None$

The ratio $m_{+}^{2}/m_{-}^{2}$ and the product $m_{+}^{2}m_{+}^{2}/4M^{4}$ in
(8.6) are positive if

$0<\alpha_{+}^{2}<1;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
{{m_{+}^{2}m_{+}^{2}}\over{M^{4}}}\leq 1.$ $None$

Therefore, the constrain (8.5) generate the positive masses $m_{+}^{2}$ and
$m_{-}^{2}$ simultaneously if $M^{2}\geq m_{+}m_{-}$.

9\. 5D extension of the standard $SU(2)\times U(1)$ model for the electron and
muon.

As example of unification of the 4D fields with the same quantum numbers and
different masses and sources, we shall consider the relationship between the
4D electron and muon fields $\Psi_{+}\equiv\Psi_{el}$ and
$\Psi_{-}\equiv\Psi_{muon}$, $\Psi_{+}\equiv\Psi_{el}$ and
$\Psi_{-}\equiv\Psi_{muon}$ according to (7.3)-(7.6). In the standard
Weinberg-Salam $SU(2)\times U(1)$ model [35] we have the following 4D
equations of motion

$\Bigl{(}i\gamma_{\mu}{{\partial}\over{\partial
x_{\mu}}}-m_{el}\Bigr{)}\Psi_{el}={\cal J}_{el};\ \ \ \ \ \ \ \ \ \ \ \ \
\Bigl{(}i\gamma_{\mu}{{\partial}\over{\partial
x_{\mu}}}-m_{muon}\Bigr{)}\Psi_{muon}={\cal J}_{muon};$ $None$ ${\cal
J}_{el}=\Bigl{(}-e\gamma_{\mu}{\sf
A}^{\mu}+{{g^{2}-{g^{\prime}}^{2}}\over{2\sqrt{g^{2}+{g^{\prime}}^{2}}}}\gamma_{\mu}{\sf
Z}^{\mu}{{1+\gamma_{5}}\over 2}+g^{\prime}\gamma_{\mu}{\sf
Z}^{\mu}{{1-\gamma_{5}}\over
2}\Bigr{)}\Psi_{el}+{g\over{\sqrt{2}}}\gamma_{\mu}{\sf
W}^{\mu}{{1+\gamma_{5}}\over 2}\nu_{el},$ $None$ ${\cal
J}_{muon}=\Bigl{(}-e\gamma_{\mu}{\sf
A}^{\mu}+{{g^{2}-{g^{\prime}}^{2}}\over{2\sqrt{g^{2}+{g^{\prime}}^{2}}}}\gamma_{\mu}{\sf
Z}^{\mu}{{1+\gamma_{5}}\over 2}+g^{\prime}\gamma_{\mu}{\sf
Z}^{\mu}{{1-\gamma_{5}}\over
2}\Bigr{)}\Psi_{muon}+{g\over{\sqrt{2}}}\gamma_{\mu}{\sf
W}^{\mu}{{1+\gamma_{5}}\over 2}\nu_{muon},$ $None$

where $\nu_{el}$ and $\nu_{muon}$ denote the corresponding neutrino fields,
${\sf W}^{\mu}$ and ${\sf Z}^{\mu}$ stands for the charged and neutral vector
fields, ${\sf A}^{\mu}$ is the photon field

$g=-{e\over{sin\theta}};\ \ \ \ \ g^{\prime}=-{e\over{cos\theta}};\ \ \ \ \
sin^{2}\theta=0.222\pm 0.011$ $None$

For derivation of (9.1a,b,c) was used the 4D gauge transformation of the
neutrino-electron and neutrino-muon doublets according to (21.3.12)-(21.3.20)
in [35].

$\sum_{\alpha}{\cal A}^{\mu}_{\alpha}(x){\em
t}_{\alpha}+y{B}^{\mu}=\exp{\Bigl{(}\Lambda(x)\Bigr{)}}{{\partial}\over{\partial
x_{\mu}}}\exp{\Bigl{(}\Lambda(x)\Bigr{)}}.$ $None$

${\sf W}^{\mu}={1\over{\sqrt{2}}}\Bigl{(}{\cal A}_{1}^{\mu}+i{\cal
A}_{2}^{\mu}\Bigr{)};\ \ \ {\sf Z}^{\mu}=cos\theta\ {\cal
A}_{3}^{\mu}+sin\theta\ {B}^{\mu};\ \ \ {\sf A}^{\mu}=-sin\theta\ {\cal
A}_{3}^{\mu}+cos\theta\ {B}^{\mu};$ $None$

The direct 5D extension of the 4D Dirac equations (9.1a) can be obtained in
the same way as (7.9) using the 4D gauge transformations (7.8a,b) with the
common parameter $x_{5}$

$\int
d^{4}ye^{ie\Lambda(x-y,x_{5})}\biggl{[}\Bigl{(}i\gamma_{\mu}{{\partial}\over{\partial
x_{\mu}}}-m_{el}\Bigr{)}\psi_{el}(x-y,x_{5})-{\widetilde{\eta}}_{el}(x-y,x_{5})\biggr{]}{\cal
P}_{+}(y,x_{5})=0.$ $None$

$\int
d^{4}ye^{ie\Lambda(x-y,x_{5})}\biggl{[}\Bigl{(}i\gamma_{\mu}{{\partial}\over{\partial
x_{\mu}}}-m_{muon}\Bigr{)}\psi_{muon}(x-y,x_{5})-{\widetilde{\eta}}_{muon}(x-y,x_{5})\biggr{]}{\cal
P}_{+}(y,x_{5})=0,$ $None$

where $\Lambda(x,x_{5}=0)=\Lambda(x)$ For $x_{5}=0$ the 5D fields and sources
in (9.3a,b) are reduced into the 4D fields and sources

$\Psi_{el,}(x)=\psi_{el}(x,x_{5}=0);\ \ \ \ \ \ \ \ \ \
\Psi_{muon}(x)=\psi_{muon}(x,x_{5}=0);$ $None$
${\widetilde{\eta}}_{el}(x,x_{5}=0)={\cal J}_{el}(x);\ \ \ \ \ \ \ \ \ \
{\widetilde{\eta}}_{muon}(x,x_{5}=0)={\cal J}_{muon}(x),$ $None$

where

${\widetilde{\eta}}_{el}(x,x_{5})=\Bigl{(}-e\gamma_{\mu}{A}^{\mu}+{{g^{2}-{g^{\prime}}^{2}}\over{2\sqrt{g^{2}+{g^{\prime}}^{2}}}}\gamma_{\mu}{Z}^{\mu}{{1+\gamma_{5}}\over
2}+g^{\prime}\gamma_{\mu}{Z}^{\mu}{{1-\gamma_{5}}\over
2}\Bigr{)}\psi_{el}+{g\over{\sqrt{2}}}\gamma_{\mu}{W}^{\mu}{{1+\gamma_{5}}\over
2}\nu_{el},$ $None$
${\widetilde{\eta}}_{muon}(x,x_{5})=\Bigl{(}-e\gamma_{\mu}{A}^{\mu}+{{g^{2}-{g^{\prime}}^{2}}\over{2\sqrt{g^{2}+{g^{\prime}}^{2}}}}\gamma_{\mu}{Z}^{\mu}{{1+\gamma_{5}}\over
2}+g^{\prime}\gamma_{\mu}{Z}^{\mu}{{1-\gamma_{5}}\over
2}\Bigr{)}\psi_{muon}+{g\over{\sqrt{2}}}\gamma_{\mu}{W}^{\mu}{{1+\gamma_{5}}\over
2}\nu_{muon},$ $None$

It is easy to check, that for $x_{5}=0$ equation (9.3a,b) transforms into
(9.1a) due to ${\cal P}_{+}(x,x_{5}=0)=\delta(x)$.

In order to place the Fourier conjugate of the 5D sources (9.4a,b) onto
hyperboloids $q^{2}\pm q_{5}^{2}=\pm M^{2}$ we shall use the following
convolution formula

$\eta_{el}(x,x_{5})=\int d^{4}y{\widetilde{\eta}}_{el}(x-y,x_{5}){\cal
P}_{+}(y,x_{5});\ \ \ \eta_{muon}(x,x_{5})=\int
d^{4}y{\widetilde{\eta}}_{muon}(x-y,x_{5}){\cal P}_{+}(y,x_{5}).$ $None$

The exact form of ${\cal P}_{+}$ (3.6a,b) allows to rewrite (9.5a) as

$\eta_{el}(x,x_{5})=\int d^{4}y{\cal J}_{el}(x-y){\cal P}_{+}(y,x_{5});\ \ \
\eta_{muon}(x,x_{5})=\int d^{4}y{\cal J}_{muon}(x-y){\cal P}_{+}(y,x_{5}).$
$None$

$\psi_{el}$ and $\psi_{muon}$ satisfy the condition (7.1). Therefore, there
are two different ways for construction of the 5D fields $\psi_{el}$ and
$\psi_{muon}$

1. If the electron and muon fields are completely independent, then the constrain (7.1) can be reproduced through doubling of the electron and muon fields separately, i.e. through $(\psi_{el})_{\pm}$ and $(\psi_{muon})_{\pm}$, where $(\psi_{el})_{-}$ and $(\psi_{muon})_{-}$ corresponds to the “electron” and “muon“ with the negative or imaginary mass. This doubling of the electron and muon states can be realized via the appropriate constrains (8.5) and the equations of motion (7.9).

2. If the 5D fields $\psi_{el}$ and $\psi_{muon}$ consists of the same parts, then in (7.1) and (7.9) $\psi_{el}\equiv\psi_{+}$ and $\psi_{muon}\equiv\psi_{-}$ and for their 4D reductions we have

$\Psi_{el}(x)=\int{{d^{4}q}\over{(2\pi)^{4}}}e^{-iqx}\biggl{[}\sum_{N=I,III}\Psi_{N}(q)+\sum_{N=II,IV}\Psi_{N}(q)\biggr{]},$
$None$
$\Psi_{muon}(x)=\int{{d^{4}q}\over{(2\pi)^{4}}}e^{-iqx}\biggl{[}\sum_{N=I,III}\Psi_{N}(q)-\sum_{N=II,IV}\Psi_{N}(q)\biggr{]}.$
$None$

The equations (9.6a,b) unify the 4D electron and muon Heisenberg fields which
satisfies the 4D equation of motion (9.1a). Despite of the mixing of the
$\Psi_{el}(x)$ and $\Psi_{muon}(x)$ in (9.6a,b) the 4D equations of motion
(9.1a) are the same as in the standard model [35], where the electron-muon
coupling is strongly suppressed. Therefore, the perturbation series
constructed in the framework of the Weinberg-Salam $SU(2)\times U(1)$ theory
and the perturbation series based on the equation of motion (9.1a,b,c) with
the mixed fields (9.6a,b) coincides.

The common structure of the interacted Heisenberg fields $\Psi_{el}(x)$ and
$\Psi_{muon}(x)$ in (9.6a,b) is important for the theories beyond Weinberg-
Salam model. In particular, within the non-perturbative formulation the
functional integrals with the 4D electron and muon fields $\int{\cal
D}(\Psi_{el}){\cal D}({\overline{\Psi}}_{el})\Bigl{[}...\Bigr{]}$ and
$\int{\cal D}(\Psi_{muon}){\cal
D}({\overline{\Psi}}_{muon})\Bigl{[}...\Bigr{]}$ are strongly correlated due
to (9.6a,b).

Doubling of the electron and muon fields (9.6a,b) can be examined within the
5D invariant time theories [8, 9] which allow to construct the independent 5D
fields $\Upsilon_{el}(x,x_{5})$ and $\Upsilon_{muon}(x,x_{5})$. If the parts
of the $\Upsilon_{el}(q,q^{2}_{5})$ and $\Upsilon_{muon}(q,q_{5}^{5})$ are not
the same as in (9.6a,b), then they are dubled and according to the present
approach appear the fields $(\Upsilon_{el})_{-}(x,x_{5})$ and
$(\Upsilon_{muon})_{-}(x,x_{5})$ with the negative or imaginary mass.

Nowadays unification of the different lepton families is performed within the
5D grand unification models [5]. In these models the fifth dimension is the
indivisible part of the particle interaction, the space-time is not
asymptotically flat, in the extra dimensions enter other than the gravitation
fields and is argued the breakdown of the gauge coupling unification. The
present formulation can be used for the 4D projections’ of the corresponding
5D electron and muon fields $\Upsilon_{el}$ and $\Upsilon_{muon}$. For this
aim we put the Fourier conjugate of the complete 5D fields $\Upsilon_{el}$ and
$\Upsilon_{muon}$ on the hyperboloids $q^{2}\pm q_{5}^{2}=\pm M^{2}$ using the
replacement of the integrals $\int d^{4}qdq^{2}_{5}\\{...\\}$ with $\int
d^{4}qdq^{2}_{5}\Bigl{[}\delta(q^{2}+q_{5}^{2}-M^{2})\\{...\\}\pm\delta(q^{2}-q_{5}^{2}+M^{2})\\{...\\}\Bigr{]}$.
Then in analogy with (I.4a,b) and (I.7) we get the 5D fields
$(\psi_{el})_{\pm}(x,x_{5})$ and $(\psi_{muon})_{\pm}(x,x_{5})$. In the low
energy region, where the asymptotically flat space is assumed,
$(\psi_{el})_{\pm}(x,x_{5}=0)$ and $(\psi_{muon})_{\pm}(x,x_{5}=0)$ determine
the 4D fields $(\Psi_{el})_{\pm}(x)$ and $(\Psi_{muon})_{\pm}(x)$. The gauge
invariant 4D fields $(\Psi_{el})_{+}(x)$ and $(\Psi_{el})_{-}(x)$ consist from
the same four parts as well as $(\Psi_{muon})_{+}(x)$ and
$(\Psi_{muon})_{-}(x)$. Thus in addition to $\Psi_{el}(x)$ $\Psi_{muon}(x)$ we
get two other fields $(\Psi_{el})_{-}(x)$ and $(\Psi_{muon})_{-}(x)$ with the
negative or imaginary mass.

10\. Gauge transformation as generalized translation

The generalized translation of the four-momentum
${q^{\mu}}^{\prime}=q^{\mu}-eA^{\mu}(q)$ in the equation of motion can be
performed through the gauge transformations. In particular, the 4D and 5D
gauge transformations (7.11a,b) and (7.8a,b) generates the corresponding
translations of the four momentum $q^{\mu}$ and $i\partial/\partial x^{\mu}$

${q^{\mu}}^{\prime}=q^{\mu}-e{A}^{\mu}_{\pm}(q,q_{5}^{2})\Longleftrightarrow
i{{\partial}\over{\partial{x^{\prime}}^{\mu}}}=i{{\partial}\over{\partial{x}^{\mu}}}-e{A}^{\mu}_{\pm}(x,x_{5})$
$None$

${q^{5}}^{\prime}=q^{5}-e{A}^{5}_{\pm}(q,q_{5}^{2})\Longleftrightarrow
i{{\partial}\over{\partial{x^{\prime}}^{5}}}=i{{\partial}\over{\partial{x}^{5}}}-e{A}^{5}_{\pm}(x,x_{5})$
$None$

with the 5D fields ${A}^{\mu}_{\pm}$, ${A}^{5}_{\pm}$, $\psi_{\pm}$ and
$\psi_{\pm}^{\prime}$.

One can construct ${A}^{\mu}_{\pm}$ and ${A}^{5}_{\pm}$ starting from the 6D
gauge translations

${\kappa_{C}}^{\prime}={\kappa_{C}}-ea_{C}(\kappa)\Longleftrightarrow
i{{\partial}\over{\partial{\xi^{\prime}}^{C}}}=i{{\partial}\over{\partial{\xi}^{C}}}-ea_{C}(\xi)$
$None$

$\psi^{\prime}(\xi)=\exp{\Bigl{(}ie\lambda(\xi)\Bigr{)}}\psi(\xi);\ \ \ \ \ \
\ \ \ \ \ \ \ \ \
a_{C}(\xi)=\exp{\Bigl{(}-ie\lambda(\xi)\Bigr{)}}{{\partial}\over{\partial{\xi}^{C}}}\exp{\Bigl{(}ie\lambda(\xi)\Bigr{)}},$
$None$

where $C\equiv\mu;5,6=0,1,2,3;5,6$ and the Fourier conjugate of
$\psi^{\prime}(\xi)$, $\psi(\xi)$, $\lambda(\xi)$ and $a_{C}(\xi)$ are not
placed on the cone $\kappa_{C}\kappa^{C}=0$.

According to invariance of the 6D cone $\kappa_{C}{\kappa}^{C}=0$ we have

$\kappa_{C}{\kappa}^{C}=\kappa^{\prime}_{C}{\kappa^{\prime}}^{C}=\Bigl{(}\kappa_{C}-ea_{C}(\kappa)\Bigr{)}\Bigl{(}\kappa^{C}-ea^{C}(\kappa)\Bigr{)}=0,$
$None$

where $\kappa_{+}$ is invariant under the translations as it is indicated in
(1.2a). The invariance of $\kappa_{+}$ under the 4D translations (10.2a)
requires that $a_{5}(k)=-a_{6}(k)$. Substituting this condition in (10.3) we
get

$a_{5}(k)={1\over{2M\kappa_{+}}}\Bigl{(}-a_{\nu}(\kappa)\kappa^{\nu}-\kappa^{\nu}a_{\nu}(\kappa)+ea_{\nu}(\kappa)a^{\nu}(\kappa)\Bigr{)};\
\ \ \ \ \ \ \ \
{{\kappa_{-}^{\prime}}\over{\kappa_{+}}}={{\kappa_{-}}\over{\kappa_{+}}}-e{{a_{5}(k)}\over{\kappa_{+}}}.$
$None$

In the considered formulation the 4D reduction of the 6D fields
$a_{\mu}(\kappa)$ and $\psi(\kappa)$ based on the intermediate projections of
the 6D cone $\kappa_{C}{\kappa}^{C}=0$ into 5D hyperboloids $q^{2}\pm
q_{5}^{2}=\pm M^{2}$ (I.3a,b) for the domains I, III and II, IV of the Table
1. The corresponding 5D reductions of the 6D fields $a_{\mu}(\kappa)$ and
$\psi(\kappa)$ are

$\Upsilon(q,q_{5}^{2})={{M^{2}}\over 2}\int
k_{+}^{3}d\kappa_{+}\theta(\kappa_{+})\psi(q,q_{5}^{2},\kappa_{+}),$ $None$

where $\Upsilon(q,q_{5}^{2})$ is Fourier conjugate of $\Upsilon(x,x_{5})$ in
(7.2) and

${\widetilde{A}}^{\mu}(q,q_{5}^{2})={{M^{2}}\over 2}\int
k_{+}^{3}d\kappa_{+}\theta(\kappa_{+})a^{\mu}(q,q_{5}^{2},\kappa_{+});\ \ \ \
\ \ {\widetilde{A}}^{5}(q,q_{5}^{2})={{M^{2}}\over 2}\int
k_{+}^{3}d\kappa_{+}\theta(\kappa_{+})a^{5}(q,q_{5}^{2},\kappa_{+}).$ $None$

The 5D gauge fields ${A}^{\mu}_{\pm}(x,x_{5})$ which Fourier conjugate are
placed on the hyperboloids $q^{2}\pm q_{5}^{2}=\pm M^{2}$ are constructed
through ${\widetilde{A}}^{\mu}$

${A}^{\mu}_{\pm}(x,x_{5})=\int
d^{5}y{\widetilde{A}}^{\mu}(x-y,x_{5}-y_{5}){\cal P}_{\pm}(y,y_{5}),$ $None$

${A}^{5}_{\pm}(x,x_{5})=\int d^{5}y{\widetilde{A}}^{5}(x-y,x_{5}-y_{5}){\cal
P}_{\pm}(y,y_{5}),$ $None$

The relationship between the fifth gauge field ${A}^{5}_{\pm}$ (10.7b) and
${A}^{\mu}_{\pm}$ (10.7a) is given in (7.13) and (7.14a,b).

The 4D gauge field ${\sf A}^{\mu}_{\pm}(x)={A}^{\mu}_{\pm}(x,0)$ are
determined through ${A}^{\mu}_{N}(q,q^{2}_{5})$ in the domains $N=I.II,III,IV$
as

${\sf
A}^{\mu}_{\pm}(x)=\int{{d^{4}q}\over{(2\pi)^{4}}}\exp{(-iqx)}\Bigl{[}{\sf
A}^{\mu}_{I}(q)\pm{\sf A}^{\mu}_{II}(q)+{\sf A}^{\mu}_{III}(q)\pm{\sf
A}^{\mu}_{IV}(q)\Bigr{]}.$ $None$

The convolution formula (10.7) can be represented in the 4D form

${A}^{\mu}_{\pm}(x,x_{5})=\int d^{4}y{\sf A}^{\mu}_{\pm}(x-y){\cal
P}_{+}(y,x_{5}).$ $None$

This representation of ${A}^{\mu}_{\pm}(x,x_{5})$ indicates that the 5D and 4D
fields ${A}^{\mu}_{\pm}(x,x_{5})$ and ${\sf A}^{\mu}_{\pm}(x)$ satisfy the
same 4D equations of motion. On the other hand ${A}^{\mu}_{\pm}(x,x_{5})$
consists of the parts placed on the hyperboloids $q^{2}\pm q_{5}^{2}=\pm
M^{2}$ and satisfy the similar to (3.1) 5D conditions

${{\partial^{2}{A}^{\mu}_{\pm}(x,x_{5})}\over{\partial x^{\mu}\partial
x_{\mu}}}+\Bigl{(}{{\partial^{2}}\over{\partial x^{5}\partial
x_{5}}}+M^{2}\Bigr{)}{A}^{\mu}_{\mp}(x,x_{5})=0,$ $None$

The consistency condition of (10.10) and the corresponding equation of motion
have the same form as (4.10) for the scalar field. Thus the present 5D
formulation allows to construct simultaneously the 4D fields ${\sf
A}^{\mu}_{+}(x)$ and ${\sf A}^{\mu}_{-}(x)$ which consists from the same parts
${\sf A}^{\mu}_{I}$, ${\sf A}^{\mu}_{II}$, ${\sf A}^{\mu}_{III}$ and ${\sf
A}^{\mu}_{IV}$. ${\sf A}^{\mu}_{+}(x)$ and ${\sf A}^{\mu}_{-}(x)$ have the
same quantum numbers, but the different sources. For the photon field ${\sf
A}^{\mu}_{+}(x)$ the role of ${\sf A}^{\mu}_{-}(x)$ can play the Z-boson.

It must be noted that the gauge transformations can be performed also for the
neutral (uncharged) particles. Within the nonlinear $\sigma$ model [16, 36]
for the triplet of the neutral auxiliary pion fields $\pi^{\alpha}$
($\alpha=1,2,3$, $\pi^{\pm}=1/2(\pi^{1}\pm i\pi^{2})$; $\pi^{0}\equiv\pi^{3}$)
is replaced with the interpolating pion field

$\pi^{\alpha}(x)={\cal U}(x)\chi^{\alpha}(x),$ $None$

where in [16, 36] ${\cal
U}(x)=\Bigl{(}1+\chi^{2}(x)/4f_{\pi}^{2}\Bigr{)}^{-1}$, $f_{\pi}=93\ MeV$ is
the pion decay constant and
$\chi^{2}=\sum_{\alpha=1}^{3}\chi^{\alpha}\chi^{\alpha}$. The replacement
(10.11) generates the following transformations

${{\partial}\over{\partial x_{\mu}}}\pi^{\alpha}={\cal
U}\Bigl{[}{{\partial}\over{\partial x_{\mu}}}+{\cal
D}^{\mu}\Bigr{]}\chi^{\alpha};\ \ \ {\cal D}^{\mu}={\cal
U}^{-1}{{\partial}\over{\partial x_{\mu}}}{\cal U}$ $None$

which presents the extension of (7.11a,b) for the neutral fields. The
transformations (10.11) and (10.12) generalise also the gauge transformations
(1.11) for the neutral fields.

11\. Summary

The one-to-one relationship between the 4D and 5D fields and their equations
of motion, established by present article, based on the equivalence of the
conformal transformations of the four momentum $q_{\mu}$ (1.1a)-(1.1e) and the
6D rotations on the cone $\kappa_{A}\kappa^{A}=0$ (1.2a)-(1.2d) and its 5D
projections on the two invariant forms $q^{2}\pm q_{5}^{2}=\pm M^{2}$ of the
$O(2,3)$ and $O(1,4)$ subgroups of the conformal group $O(2,4)$. The 6D cone
$\kappa_{A}\kappa^{A}=0$ and its 5D projections $q^{2}\pm q_{5}^{2}=\pm M^{2}$
are invariant under the 6D rotations and the corresponding 4D conformal
transformations. Consequently, the 4D projection of the 6D field
$\varsigma(\kappa)$ with the intermediate projection on the two hyperboloids
$q^{2}\pm q_{5}^{2}=\pm M^{2}$ determine the two 5D fields
$\varphi_{1}(x,x_{5})$ (I.4a) and $\varphi_{2}(x,x_{5})$ (I.4b). The Fourier
conjugate of these 5D fields are placed on the hyperboloids $q^{2}\pm
q_{5}^{2}=\pm M^{2}$ before and after conformal transformations. Consequently
the 5D fields $\varphi_{+}=\varphi_{1}+\varphi_{2}$ and
$\varphi_{-}=\varphi_{1}-\varphi_{2}$ (I.7) are defined on the whole domain of
$-\infty<q^{2}<\infty$ and reproduce the 4D fields
$\Phi_{\pm}(x)=\varphi_{\pm}(x,x_{5}=0)$. In addition the fields
$\varphi_{\pm}(x,x_{5})$ satisfy the 5D constrains (3.1) that have the form of
the coupled sourceless 5D equations. The 5D equation of motion for
$\varphi_{\pm}$ (4.1a) and their 4D reductions (4.1b) are embedded into these
sourceless coupled 5D conditions (3.1). The consistency conditions of the 5D
equation of motion (4.1a) and the 5D constrains (3.1) generate the constrains
for $\partial\varphi_{\pm}/\partial x_{5}$. The similar sourceless 5D coupling
condition (7.1) are valid for the interacting fermion fields $\psi_{+}$ and
$\psi_{-}$.

This scheme can be used for the 5D extension of the 4D models and for the 4D
reductions of the 5D formulations. The parts of the 5D field
$\phi(q,q_{5}^{2}=M^{2}-q^{2})$ and $\phi(q,q_{5}^{2}=M^{2}+q^{2})$
unambiguously determine the Fourier conjugate of the 4D fields $\Phi_{+}(x)$
and $\Phi_{-}(x)$ (I.10)-(I.11). The same expressions determine also the
Fourier conjugate of the 5D fields $\varphi_{1}$ (I.4a), $\varphi_{2}$ (I.4b)
and $\varphi_{\pm}$ (I.7). These parts of the single 5D field
$\phi(q,q_{5}^{2})$ determine unambigously the two 4D fields $\Phi_{\pm}(x)$
with the same quantum numbers, but with the different masses and sources. And
vice versa, starting from the 4D field $\Phi_{+}(x)$ one can construct
$\Phi_{-}(x)$ and the parts of the 5D field $\phi(q,q_{5}^{2}=M^{2}-q^{2})$
and $\phi(q,q_{5}^{2}=M^{2}+q^{2})$. This doubling of the 4D fields is result
the intermediate projection of the 6D field placed on the 6D cone
$\kappa_{A}\kappa^{A}=0$ into the two 5D fields $\varphi_{\pm}$ embedded into
two invariant forms $q^{2}\pm q_{5}^{2}=\pm M^{2}$. The considered
intermediate 5D projections take into account the symmetry under the inversion
$q^{\prime}_{\mu}=-M^{2}q_{\mu}/q^{2}$ and reflection
${q^{\prime}}^{2}=-q^{2}$ between the domains of these forms. As it is
mentioned in the last two paragraphs of Sect. 2, the stereographic and other
5D projections of the 6D cone $\kappa_{A}\kappa^{A}=0$ can be reproduced
through the considered projections on the hyperboloids $q^{2}\pm q_{5}^{2}=\pm
M^{2}$.

The boundary conditions of the 5D fields $\varphi_{\pm}$ (I.8) and
$\psi_{\pm}$ (7.4) at $x_{5}=0$ can be extended for an arbitrary value
$x_{5}=t_{5}$ if one replaces $x_{5}$ by $x_{5}-t_{5}$ in the definitions of
the 5D fields (I.4a,b) and (i.7). In particular, $t_{5}=\sqrt{x_{0}^{2}-{\bf
x}^{2}}$ in the formulation within the relativistic invariant time theories[8,
9].

The common parts of the 4D interacted fields $\Phi_{+}(q)$ and $\Phi_{-}(q)$
in (I.10)-(I.11) allow to separate the $4!=24$ different fields with the same
quantum numbers and with the different masses and sources. This unification
scheme of the interacting Heisenberg fields can be applied for description of
the nucleons and resonances in the $P11$ states $N(1440)$, $N(1710)$, … [37]
based on an additional dynamical mechanism for the generation of the
resonances. Similarly, one can combine the interacting Heisenberg fields of
the pions and the resonances with the same quantum numbers $\pi(1300)$.

The particle states with the same quantum numbers and the different masses and
sources can be constructed within the various 5D relativistic invariant time
theories [8, 9]. The different 5D fields $\phi_{+}(q,q_{5}^{2})$ and
$\phi_{-}(q,q_{5}^{2})$ in these theories do not have the common parts
$\phi_{+}(q,q_{5}^{2}=M^{2}-q^{2})$ and $\phi_{-}(q,q_{5}^{2}=M^{2}+q^{2})$.
Nevertheless the present formulation requires doubling of the 5D fields, i.e.
instead of the $\phi_{+}(x,x_{5})$ and $\phi_{-}(x,x_{5})$ we get
$(\phi_{+})_{\pm}(x,x_{5})$ and $(\phi_{-})_{\pm}(x,x_{5})$, where
$(\phi_{+})_{-}(x,x_{5})$ and $(\phi_{-})_{-}(x,x_{5})$ must have the negative
or imaginary masses. The relationships between the different masses and
sources for the 5D fields $(\phi_{+})_{\pm}(x,x_{5})$ and
$(\phi_{-})_{\pm}(x,x_{5})$ are determined by constrains for the
$\partial(\phi_{+})_{\pm}/partialx_{5}$ and
$\partial(\phi_{-})_{\pm}/partialx_{5}$. Thus the present approach allows to
get the 4D projections of the 5D fields from [8, 9] for $x_{5}=0$.

It must be noted, that if the 4D fields $\Phi_{\pm}(x)$ are determined through
the parts of the 5D fields $\phi_{+}(q,q^{2}_{5}=M^{2}\mp q^{2})$ and
$\phi_{-}(q,q^{2}_{5}=M^{2}\mp q^{2})$, then the Fourier conjugate of the
observed 4D fields $\Phi_{\pm}(q)$ can have the jump singularities at
$q^{2}=0,\pm M^{2}$.

Presently the different lepton families including the electron and muon fields
$\Upsilon_{el}$ and $\Upsilon_{muon}$ are constructed within the 5D grand
unification theories [5, 6]. In this formulation the 5D electron and muon
fields $\Upsilon_{el}(q,q_{5}^{2}=M^{2}\mp q^{2})$ and
$\Upsilon_{muon}(q,q_{5}^{2}=M^{2}\mp q^{2})$ are independent. The present
scheme of the 4D reduction of the 5D fields allows to get the 4D projections
of these 5D fields with the doubling of the 4D electron and muon fields
$(\Psi_{el})_{\pm}(x)$ and $(\Psi_{muon})_{\pm}(x)$, where $(\Psi_{el})_{-}$
and $(\Psi_{muon})_{-}$, have the negative or imanary masses. These 4D
projections of the 5D fields restore the 4D gauge invariance, because the
gauge transformations are considered as the generalized 4D translation in the
momentum space (see Sect. 10) and translations of the four momentum preserve
invariance of the 6D cone $\kappa_{A}\kappa^{A}=0$ and their projections on
the 5D invariant forms.

Other kind of the unification of the 4D electron and muon fields in this
approach is considered in the Section 9 for the Standard Model [35], where
$\Phi_{el}(q)$ and $\Phi_{muon}(q)$ consist of the same parts (9.6a,b) of the
complete 5D field $\Upsilon(q,q^{2}_{5}=M^{2}\mp q^{2})$. The coupling between
the electron and muon fields are strongly suppressed in the Standard Model.
Therefore, the common structure of the electron and muon fields can not be
observed in the perturbation series of the corresponding 4D equations.

I thank V.G.Kadyshevsky for numerous helpful discussions. I am thankful to A.
Machavariani (junior) and G. Münster for the current interest in this work.

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