# Structure Group and Fermion-Mass-Term in General Nonlocality

Lei Han, Hai-Jun Wang  Center for Theoretical Physics and School of Physics,
Jilin University, Changchun 130012, China

###### Abstract

In our previous work [J. Math. Phys. 49, 033513 (2008)] two problems remain to
be resolved. One is that we lack a minimal group to replace GL(4,C), the other
is that the Equation of Motion (EoM) for fermion has no mass term. After
careful investigation we find these two problems are linked by conformal
group, a subgroup of GL(4,C) group. The Weyl group, a subgroup of conformal
group, can bring about the running of mass, charge etc. while making it
responsible for the transformation of interaction vertex. However, once
concerning the generation of the mass term in EoM, we have to resort to the
whole conformal group, in which the generators $K_{\mu}$ play a crucial role
in making vacuum vary from space-like (or light-cone-like)to time-like.
Physically the starting points are our previous conclusion,
$\vec{E}^{2}-\vec{B}^{2}\neq 0$ for massive bosons, and the two-photon process
yielding $e^{+}e^{-}$ pair. Finally we get to the conclusion that the mass
term of strong interaction is linearly relevant to (chromo-)magnetic flux as
well as angular momentum.

##  I Introduction

In our previous paper WA08 , a complex-geometry approach to quantum field
theory was proposed, aiming at describing nonlocal interaction (in the low-
energy limit) through curvature effects in certain kinds of complex space. In
such approach, the field equations for interacting fermions and bosons are
obtained based on the assumption that there always exists a complex reference
frame for a fermion in which it behaves as a plane wave, i.e. as a free (non-
interacting) particle. This assumption is regarded as ”principle of
nonlocality”——but actually it is a ”complexification” of the Einstein’s
Equivalence Principle (EEP). The equation of motion (EoM) for fermions is
derived by demanding that the fermions move just along a ”geodesic line” in
the complex space. And by a transformation of observers, we conclude that it
is just the quadratic form of the standard Dirac equation, yet without the
mass term. The field equation for bosons is also derived, in which the mass of
boson occurs automatically and is proportional to $\vec{E}^{2}-\vec{B}^{2}$.
The conclusion holds apparently for photon since $\vec{E}^{2}-\vec{B}^{2}=0$
there. The theory might be useful in studying non-perturbative dynamical
systems, such as hadron spectra or scattering between hadrons. In such a
theory, a few problems remain to resolve, one of which is that the structure
group GL(4,C) is too large. Another is that a mass term is missing while
deriving EoM for fermions. In this paper we try to analyze and resolve these
two problems consistently.

In our model, there are two kinds of curving——hyperbolic curving and elliptic
curving. They are represented by two different structure groups GL(n,C) and
U(n,C) WA08 , and in this paper we are mainly concerned about the group
GL(n,C). The 4-dimension Conformal Group is a promising mini-group for our
model, in which the scaling transformation is a typical one in generators,
nevertheless is seldom used independent of other generators Yu13 . The algebra
structure of conformal group has been investigated thoroughly from different
aspects Cartan37 ; Ma99 ; Duval0911 ; Henkel94 ; Henkel02 ; Henkel03 ;
Jackiw72 ; Hor10 , and its applications to quantum fields have also been
widely considered. However none of the applications is satisfactory because
hitherto no other perfect quantum system than photon field Bate1910 ; Cunn1910
has been found so that the corresponding Lagrangian is conformally invariant,
unless the masses of involved particles are null Kast08 ; Yu13 ; Jose88 ;
Gross70 ; Dirac35 ; Lus75 . Furthermore, according to Noether’s theorem if a
Lagrangian is conformally invariant, then the trace of the energy-momentum
tensor should be null Kast08 ; Yu13 . Other efforts were also experienced to
search for invariant fermion equation or scattering amplitude Dirac35 ;
Gross70 ; Mack69 , and even to apply it to nonlocal action Ryder74 ; Whe4549 .
In recent years there have been surges using conformal invariance accompanied
with chiral symmetry breaking in studying hadron-level strong interaction Ma13
. So far it can almost be concluded that the conformal invariance is
unreachable to known material fields, by which we may suspect the
applicability of conformal group. However from another point of view, the
perturbative renormalization results Wang11 ; Su99 ; Wang04 ; Cal70 ; Syman70
shed much light on the relationship between the physical constants and scale
transformation, that is, some of the constants vary regularly with the energy
scale. It follows that the scale transformation can cause the variation of
coupling constants and masses. That suggests that scaling transformation might
be helpful in generating mass term in EoM mathematically. Whereas after
delicate analysis, we find out that in our model, without the help of other
generators of conformal group, the mass term cannot be generated solely by
scaling transformation.

We analyze the mass term problem from two respects, the physical respect and
math respect. On physical side, the mass generation ascribes to dynamical
chiral symmetry breaking (DCSB) and Higgs mechanism. The effect of these two
mechanisms is to turn vacuum or a massless particle from space-like (or light-
cone like) to time-like, and thus makes particle massive. In view of our
former result that a massive boson meets the condition
$\vec{E}^{2}-\vec{B}^{2}\neq 0$, and of the Higgs mechanism that demands
bosons occur first and then decay into fermions, we employ a shortcut process
$\gamma+\gamma\longrightarrow e^{+}e^{-}$ Piaz09 ; Jiang11 ; Drei12 ; Tito12 ;
Bakm08 ; Li12 ; Brodsky06 ; Kohl14 ; Blinne14 ; Jans13 ; LiA14 ; Fedotov13  to
analyze the transition from space-like (or light-cone like) to time-like. In
the process we perceive, at least equivalently, one photon is stimulated by
the other then becomes time-like and succeedingly decays into electron pairs.
The stimulation might lead to $\vec{E}^{2}-\vec{B}^{2}\neq 0$. Such processes
are truly relevant to our consideration due to the fact that Higgs boson was
first evidenced by decaying into two-photon resonance. Such seemingly rough
physics consideration is helpful in our treating on math side, where we will
note only particular generators of conformal group are responsible for
shifting space-time or energy-momentum from space-like to time-like Ryder74A ,
which is not able to be realized solely by the scaling transformation of
conformal group. With the above cognition we find out the answer to mass term
problem points to one conjecture in our original paper, i.e an integral which
we had not carried out.

The structure of this paper is as follows. In sect. II we introduce two
representations of 4-dimensional conformal group. The focus is on how to
derive the spinor representation, which is actually employed as our nonlocal
structure group. In section III we link the two representations of conformal
group by physical interaction vertices in the conventional sense, especially
we elucidate the function of generators $K_{\mu}$ on chiral-like vertex
$\gamma_{\mu}(1-\gamma_{5})$, which is useful for our further discussion on
mass term in EoM. Section IV is dedicated to elaborating the mathematical
processes of massless particles (bosons and fermions) getting massive and to
deriving the mass term. Then follows the conclusions and discussions. In
appendix A. we explain why a massive fermion spinor cannot be generated solely
by the scaling transformation. The scaling transformation just transforms a
spinor from helicity representation to spinor representation.

##  II The Spatial and Spinor Representations for Conformal Group

In this section we separately present the spatial/operator representation and
unitary/spinor representation of the 4-dimensional Conformal group Budi79 ;
Yu13 , including their generators and commutations among them. The spatial
representation is mainly referencing to that of Ref. Budi79A ; Budi79  and the
unitary representation is derived by applying Cartan method Cartan37  to
$SO(6)-SU(4)$ transform. The unitary representation is the focus of this
section.

We start with the null vector space (Euclidean space, with the first and the
sixth imaginary),

$\sum_{a=1}^{6}\eta_{a}^{2}=0\text{ .}$ (1)

reserving which gives the popular definition of conformal group Cartan37 . A
special expression of the differential forms in 4-dimension spatial
representation can be derived directly from the above equation. In derivation
we need to apply the following variables Budi79

$x_{\mu}=\frac{\eta_{\mu}}{K}\text{ , where }K=\eta_{5}+i\,\eta_{6}\text{ ,
where }\mu=1,2,3,4$ (2)

together with the differential form

$\frac{\partial}{\partial\eta_{a}}=\frac{1}{K}\\{[\delta_{a\mu}-(\delta_{a5}+i\delta_{a6})x_{\mu}]\frac{\partial}{\partial
x_{\mu}}+(\delta_{a5}+i\delta_{a6})K\frac{\partial}{\partial K}\\}\text{ ,
where }a=1,2,\cdots,6$ (3)

to the definition of 6-dimensional angular-momentum

$M_{ab}=i(\eta_{a}\frac{\partial}{\partial\eta_{b}}-\eta_{b}\frac{\partial}{\partial\eta_{a}})\text{,
where }a,b=1,2,\cdots,6\text{ .}$ (4)

Then one gets the following generators for conformal group Budi79  [of which
in eq. (56)]

$\displaystyle D$ $\displaystyle=$ $\displaystyle
iM_{56}=-(\eta_{5}\frac{\partial}{\partial\eta_{6}}-\eta_{6}\frac{\partial}{\partial\eta_{5}})=i(x_{\mu}\frac{\partial}{\partial
x_{\mu}}-K\frac{\partial}{\partial K})\text{,}$ $\displaystyle P_{\mu}$
$\displaystyle=$ $\displaystyle M_{5\mu}+iM_{6\mu}=i\frac{\partial}{\partial
x_{\mu}}\text{ ,
}K_{\mu}=M_{5\mu}-iM_{6\mu}=i\\{-x^{2}\frac{\partial}{\partial
x_{\mu}}+2x_{\mu}x_{\nu}\frac{\partial}{\partial
x_{\nu}}-2K\,x_{\mu}\frac{\partial}{\partial K}\\}\text{,}$ (5)

The projected form (making $K$ as constant boundary of Minkowski spaceMa99 )
shifting to Minkowski convention then is

$\displaystyle D$ $\displaystyle=$ $\displaystyle
i\,x_{\mu}\frac{\partial}{\partial x_{\mu}}\text{,
}M_{\mu\nu}=i(x_{\mu}\frac{\partial}{\partial
x^{\nu}}-x_{\nu}\frac{\partial}{\partial x^{\mu}})\text{,}$ $\displaystyle
P_{\mu}$ $\displaystyle=$ $\displaystyle i\frac{\partial}{\partial
x^{\mu}}\text{ , }K_{\mu}=-i(x^{2}\frac{\partial}{\partial
x^{\mu}}-2x_{\mu}x^{\nu}\frac{\partial}{\partial x^{\nu}})\text{,}$ (6)

where $M_{\mu\nu}$ represent the components of conventional angular momentum
in 4-dimension. The corresponding commutation relation can be obtained by
direct computation,

$\displaystyle[M_{\mu\nu},M_{\rho\sigma}]$ $\displaystyle=$ $\displaystyle
i(g_{\nu\rho}M_{\mu\sigma}+g_{\mu\sigma}M_{\nu\rho}-g_{\mu\rho}M_{\nu\sigma}-g_{\nu\sigma}M_{\mu\rho}),$
$\displaystyle[M_{\mu\nu},P_{\rho}]$ $\displaystyle=$ $\displaystyle
i(g_{\nu\rho}P_{\mu}-g_{\mu\rho}P_{\nu}),$ $\displaystyle[D,P_{\mu}]$
$\displaystyle=$ $\displaystyle-iP_{\mu}\text{, }[D,K_{\mu}]=iK_{\mu},$
$\displaystyle[D,M_{\mu\nu}]$ $\displaystyle=$ $\displaystyle 0$
$\displaystyle\ \ [M_{\mu\nu},K_{\rho}]$ $\displaystyle=$ $\displaystyle
i(g_{\nu\rho}K_{\mu}-g_{\mu\rho}K_{\nu})$ (7) $\displaystyle\
[P_{\mu},K_{\rho}]$ $\displaystyle=$
$\displaystyle-2\,i\,(g_{\mu\rho}\,D+M_{\mu\rho})$ (8)

Before using Cartan method to achieve its unitary representation, let’s review
first the steps of Cartan method with $SO(3)-SU(2)$ mapping as an example
Cartan37 [of which in pp. 41-48]. To keep the invariance of
$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=0$, one defines the matrix

$X=\left(\begin{array}[]{cc}x_{3}&x_{1}-i\,x_{2}\\\
x_{1}+i\,x_{2}&-x_{3}\end{array}\right)\text{ .}$ (9)

The trace Tr$(X^{\dagger}X)$ is $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}$. With $U$ as
an element of $SU(2)$ group, we define

$X^{\prime}=U^{-1}XU\text{ ,}$ (10)

immediately we have

$Tr(X^{\prime\dagger}X^{\prime})=Tr(X^{\dagger}X)\text{ ,}$ (11)

thus $SU(2)$ group keeps the trace invariant, and by this way the group also
keeps the metric $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}$. With the knowledge that the
$SO(3)$ group directly reserves the metric $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}$, we
conclude that Cartan matrix $X$ acts as a mapping between $SO(3)$ and $SU(2)$.
By the Cartan Matrix $X$, one can define spinor
$\left(\begin{array}[]{c}\xi_{0}\\\ \xi_{1}\end{array}\right)$ by

$X\,\left(\begin{array}[]{c}\xi_{0}\\\ \xi_{1}\end{array}\right)=0\text{ ,}$
(12)

with the solution $\xi_{0}=\pm\sqrt{\frac{x_{1}-i\,x_{2}}{2}}$ and
$\xi_{1}=\pm\sqrt{\frac{-x_{1}-i\,x_{2}}{2}}$, and the reverse yields

$\displaystyle x_{1}$ $\displaystyle=$ $\displaystyle\xi_{0}^{2}-\xi_{1}^{2}$
$\displaystyle x_{2}$ $\displaystyle=$ $\displaystyle
i\,(\xi_{0}^{2}+\xi_{1}^{2})$ $\displaystyle x_{3}$ $\displaystyle=$
$\displaystyle-2\xi_{0}\xi_{1}\text{ ,}$ (13)

which automatically satisfies $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=0$ from which we
can define the spinor reversely.

From the above Cartan matrix $X$ we can extract the Pauli matrices
$\sigma_{1}$, $\sigma_{2}$, $\sigma_{3}$ separately from the coefficients of
$x_{1}$, $x_{2}$, $x_{3}$. Meanwhile Pauli matrices $\sigma_{1}$,
$\sigma_{2}$, $\sigma_{3}$ act as the generators of $SU(2)$ group mentioned
above. Furthermore it is easy to test that $SU(2)$ group reserves the metric

$\mid\xi_{0}\mid^{2}+\mid\xi_{1}\mid^{2}=\Xi^{\dagger}\Xi\text{ .}$ (14)

And coincidentally the $n-$vectors form (defined in eq. (35)) based on Pauli
matrices don’t generate new matrices, neither the multiplications nor the
commutations among them, because they themselves are closed. Now in what
follows we would find the corresponding Cartan matrix from $SO(6)$ to
$SU(4)$/$SU(2,2)$, namely the spinor representation for 4-dimension Conformal
group.

To achieve its unitary/spinor representation in 4-dimension, mimicking the
relationship between the metric $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}$ and that in
Eq. (14), we shall associate the metric in Eq. (1) with the invariant
quadratic form

$\mid z_{1}\mid^{2}+\mid z_{2}\mid^{2}+\mid z_{3}\mid^{2}+\mid
z_{4}\mid^{2}=Z^{\dagger}Z\text{ ,}$ (15)

by the following matrix Este64 ,

$A=\left(\begin{array}[]{cccc}0&x_{1}+i\,x_{2}&x_{3}+i\,x_{4}&x_{5}+i\,x_{6}\\\
-(x_{1}+i\,x_{2})&0&x_{5}-i\,x_{6}&-x_{3}+i\,x_{4}\\\
-(x_{3}+i\,x_{4})&-x_{5}+i\,x_{6}&0&x_{1}-i\,x_{2}\\\
-(x_{5}+i\,x_{6})&x_{3}-i\,x_{4}&-x_{1}+i\,x_{2}&0\end{array}\right)\text{ .}$
(16)

Count the degrees of freedom of the groups conserving separately Eq. (1) and
Eq. (15), one finds they are both 15. Next we only need to extract the
coefficients before $x_{i}$’s to get the unitary matrices as generators of
$SU(4)$, just like the method used in three dimension example Eq (7-13). If we
want to get the generators of $SU(2,2)$ we need only to change the signs
before $x_{1}$ and $x_{2}$ and those ahead of corresponding matrices, which
would change the eqs. (1) and (15) to

$-x_{1}^{2}-x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}=0\ \text{ .}$
(17)

and

$-\mid z_{1}\mid^{2}-\mid z_{2}\mid^{2}+\mid z_{3}\mid^{2}+\mid
z_{4}\mid^{2}=Z^{\dagger}Z\text{ .}$ (18)

the latter falls into Dirac spinor like

$\tilde{\psi}=(z_{1},z_{2},z_{3},z_{4})\text{ .}$

It can be examined that the matrix $A$ in Eq. (16) meets the invariant
expression

$Tr(A^{\dagger}A)=4(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2})$
(19)

just like the above 3-dimension example, while the $SU(4)$ group keeps the
above trace
$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}=constant$, it
simultaneously reserves the metric Eq. (15). The above method of linking real
metric to a matrix is closely analogous to the Cartan method of constructing a
spinor representation in any real space. Actually, the true spinor space for
4-d conformal group following Cartan method should be of 8-dimension instead
of 4-dimension Cartan37 [of which in pp. 88-89]. In what follows we would take
over the process of deriving all of the $n$-vectors along the Cartan method
Cartan37 [of which in pp.81-83], though we work in 4-dimension rather than
8-dimension. First we extract the matrices before $x_{i}$’s in Eq. (16) ,
$i.e.1-$vectors,

$\displaystyle B_{1}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}i\,\sigma_{2}&0\\\
0&i\,\sigma_{2}\end{array}\right)\text{,}\;B_{2}=\left(\begin{array}[]{cc}-\,\sigma_{2}&0\\\
0&\,\sigma_{2}\end{array}\right)\text{ ,}$ (24) $\displaystyle B_{3}$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{cc}0&\,\sigma_{3}\\\
-\,\,\sigma_{3}&0\end{array}\right)\text{,}\;B_{4}=\left(\begin{array}[]{cc}0&i\,I\\\
-i\,I&\,0\end{array}\right)\text{ ,}$ (29) $\displaystyle B_{5}$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{cc}0&\,\sigma_{1}\\\
-\,\,\sigma_{1}&0\end{array}\right)\text{,}\;B_{6}=\left(\begin{array}[]{cc}0&\,-\sigma_{2}\\\
-\,\,\sigma_{2}&0\end{array}\right)\text{ .}$ (34)

where $\sigma_{i}$’s are Pauli matrices. The definition of $k$-vector is

$B_{k-vector}=\sum_{P}(-1)^{P}B_{n_{1}}B_{n_{2}}\cdots B_{n_{k}}\text{ ,}$
(35)

where $P$ denotes different permutations. Apply the above formula to 2-vector,
and use the corresponding subscripts to denote the $1$-vectors involved, then

$B_{12}=B_{1}B_{2}-B_{2}B_{1}=\left(\begin{array}[]{cc}i\,\sigma_{2}&0\\\
0&i\,\sigma_{2}\end{array}\right)\left(\begin{array}[]{cc}-\,\sigma_{2}&0\\\
0&\,\sigma_{2}\end{array}\right)-\left(\begin{array}[]{cc}-\,\sigma_{2}&0\\\
0&\,\sigma_{2}\end{array}\right)\left(\begin{array}[]{cc}i\,\sigma_{2}&0\\\
0&i\,\sigma_{2}\end{array}\right)=0\text{ .}$

Similarly, let’s exhaust all possibilities, then obtain other nontrivial
2-vectors

$\displaystyle B_{13}$ $\displaystyle=$ $\displaystyle
2\left(\begin{array}[]{cc}0&-\,\sigma_{1}\\\
\,\,\sigma_{1}&0\end{array}\right)\text{ ,
}B_{15}=2\left(\begin{array}[]{cc}0&\,\sigma_{3}\\\
\,-\,\sigma_{3}&0\end{array}\right)\text{, }$ (40) $\displaystyle B_{35}$
$\displaystyle=$ $\displaystyle 2i\left(\begin{array}[]{cc}\,\sigma_{2}&0\\\
0&\,\sigma_{2}\end{array}\right)\text{,
}B_{36}=2i\left(\begin{array}[]{cc}\,\sigma_{1}&0\\\
0&\,-\sigma_{1}\end{array}\right)\text{, }$ (45) $\displaystyle B_{46}$
$\displaystyle=$ $\displaystyle-2i\left(\begin{array}[]{cc}\,\sigma_{2}&0\\\
0&\,-\sigma_{2}\end{array}\right)\text{,
}B_{24}=2i\left(\begin{array}[]{cc}0&\,\sigma_{2}\\\
\,\,\sigma_{2}&0\end{array}\right)\text{ ,}$ (50) $\displaystyle\text{
}B_{23}$ $\displaystyle=$
$\displaystyle-2i\left(\begin{array}[]{cc}0&\,\sigma_{1}\\\
\,\,\sigma_{1}&0\end{array}\right)\text{. }$ (53)

We note that the new ones which are independent of $B_{i}$’s are just $B_{23}$
and $B_{36}$. The same line can be followed to carry out the 3-vectors.
Ignoring the repeating ones, we find the new 3-vectors independent of both
1-vectors and 2-vectors are

$\displaystyle B_{123}$ $\displaystyle\sim$
$\displaystyle\left(\begin{array}[]{cc}0&\,\sigma_{3}\\\
\,\,\sigma_{3}&0\end{array}\right)\text{,
}B_{134}\sim\left(\begin{array}[]{cc}\,\sigma_{1}&0\\\
0&\,\sigma_{1}\end{array}\right)\text{, }$ (58) $\displaystyle B_{145}$
$\displaystyle\sim$ $\displaystyle\left(\begin{array}[]{cc}\,\sigma_{3}&0\\\
0&\,\sigma_{3}\end{array}\right)\text{,
}B_{245}\sim\left(\begin{array}[]{cc}\,\sigma_{3}&0\\\
0&-\,\sigma_{3}\end{array}\right)\text{, }$ (63) $\displaystyle B_{345}$
$\displaystyle\sim$ $\displaystyle\left(\begin{array}[]{cc}0&\,\sigma_{2}\\\
\,-\,\sigma_{2}&0\end{array}\right)\text{,
}B_{146}\sim\left(\begin{array}[]{cc}I&\,0\\\ 0&-I\end{array}\right)\text{, }$
(68) $\displaystyle B_{124}$ $\displaystyle\sim$
$\displaystyle\left(\begin{array}[]{cc}0&\,I\\\
\,\,I&0\end{array}\right)\text{ .}$ (71)

Computing the 4-vectors and the higher ones would not give new independent
matrices. Finally, we can rearrange all above k-vector-produced matrices as
follows Este64 ,

$\displaystyle U_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\begin{array}[]{cc}\,\sigma_{i}&0\\\
0&\,\sigma_{i}\end{array}\right)$ (74) $\displaystyle V_{\mu}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\left(\begin{array}[]{cc}\,\sigma_{\mu}&0\\\
0&\,-\sigma_{\mu}\end{array}\right)$ (77) $\displaystyle W_{\mu}$
$\displaystyle=$
$\displaystyle\frac{i}{2}\left(\begin{array}[]{cc}\,0&\sigma_{\mu}\\\
\sigma_{\mu}&\,0\end{array}\right)$ (80) $\displaystyle Y_{\mu}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(\begin{array}[]{cc}\,0&\sigma_{\mu}\\\
-\sigma_{\mu}&\,0\end{array}\right)$ (83)

where $\sigma_{i}$, $i=1,2,3$ are normal Pauli matrices and
$\sigma_{0}=\left(\begin{array}[]{cc}1&0\\\ 0&\,1\end{array}\right)$. The
convention can be changed from Minkowski to Euclidean spaces while
alternatively requiring $\sigma_{\mu}^{2}=-1$, i.e. making $\sigma_{0}=i$ and
replacing definition of $\sigma_{i}$ by those in Este64 . The route of
inquiring the concrete matrices following Cartan method as above could be a
shortcut that rarely mentioned in literature. It is can be checked that the
commutations among $U_{i}$, $V_{\mu}$, $W_{\mu}$, $Y_{\mu}$ are just those for
conformal group Budi79 ; Budi79A , accordingly the mapping from these matrices
to differential forms turns out to be

$\displaystyle U_{i}$ $\displaystyle\leftrightarrow$
$\displaystyle\gamma_{i}\gamma_{j}\longrightarrow
i(x_{j}\frac{\partial}{\partial x^{k}}-x_{k}\frac{\partial}{\partial
x^{j}})\longrightarrow M_{jk}$ $\displaystyle W_{i}$
$\displaystyle\leftrightarrow$
$\displaystyle\gamma_{0}\gamma_{i}\longrightarrow
i(x_{i}\frac{\partial}{\partial x^{0}}-x_{0}\frac{\partial}{\partial
x^{i}})\longrightarrow M_{0k}$ $\displaystyle W_{0}$
$\displaystyle\leftrightarrow$ $\displaystyle\gamma_{5}\longrightarrow
i\,x_{\mu}\frac{\partial}{\partial x_{\mu}}\longrightarrow D$ $\displaystyle
V_{\mu}+Y_{\mu}$ $\displaystyle\leftrightarrow$
$\displaystyle\gamma_{\mu}(1-\gamma_{5})\longrightarrow
i\frac{\partial}{\partial x^{\mu}}\longrightarrow P_{\mu}$ $\displaystyle
V_{\mu}-Y_{\mu}$ $\displaystyle\leftrightarrow$
$\displaystyle\gamma_{\mu}(1+\gamma_{5})\longrightarrow-i(\frac{1}{2}x_{\nu}x^{\nu}\frac{\partial}{\partial
x_{\mu}}-x_{\mu}x_{\nu}\frac{\partial}{\partial x_{\nu}})\text{
}\longrightarrow K_{\mu}\text{.}$ (84)

We use $\longrightarrow$ to represent the accurate mappings and
$\leftrightarrow$ the equivalence. After examining the commutations by
computer, we have to extend the matrix of operator $D$ to be $1+\gamma_{5}$,
due to the one between $P_{\mu}$ and $K_{\mu}$.

##  III The physical relationship between the two representations of conformal
group

In this section we use scaling transformation as paradigm to investigate the
affection of conformal group on interaction vertex. Enlightened by applying
Lorentz transformation to Dirac equation, we first try to link physically the
spatial form of scaling transformation with its spinor/unitary form, the
former representing the realistic expansions and contractions (dilatation and
shrinkage) of space-time, the latter representing the intrinsic freedom
closely analogous to spin angular momentum. Then we will generalize the
results from scaling transformation to other generators of conformal group. By
this way we analyze the structure of conformal group and its relationship with
the vertices of conventional quantum field theory, and judge the concrete
physical significance of every block of the conformal group.

As for Lorentz transformation, the transformation matrix
$\left(\Lambda_{\;\,\mu}^{\nu}\right)$ for
$j^{\mu}(y)=\bar{\psi}(y)\gamma^{\mu}\psi(y)$ corresponds to a complex
transformation $S$ for $\psi(y)$ so that the effect of the transformed result
$\bar{\psi}(y)S^{-1}\gamma^{\mu}S\psi(y)$ is equivalent to
$\bar{\psi}(y)\Lambda_{\;\,\nu}^{\mu}\gamma^{\nu}\psi(y)$. Referencing the
case of Lorentz transformation, our goal in this section is to search for the
corresponding vertex-form $\Gamma^{\mu}$ so that it links with transformation
$S^{\prime}$ by
$S^{\prime-1}\Gamma^{\mu}S^{\prime}=\Lambda_{\;\,\nu}^{\prime\mu}\Gamma^{\nu}$,
where $S^{\prime}=e^{\frac{u}{2}(1+\gamma_{5})}$, $1+\gamma_{5}$ is the spinor
representation of the scaling operator $D$, and
$\Lambda_{\;\,\nu}^{\prime\mu}$ represent tensor’s components of scaling
transformation. The similar method was used in a previous paper WA08B , but
for totally different motivation.

Usually we perform the spatial Lorentz transformation on the vectors $A_{\mu}$
and $\gamma^{\mu}$. Obviously this combination brings about invariant
formalism like $A^{\nu}(q^{2})\bar{\psi}(p)\gamma_{\nu}\psi(p^{\prime})$. We
follow the convention that the same set of $\\{\gamma^{\mu}\\}$ is used in
different coordinate systems, which naturally yields an equivalence
transformation $S$ satisfying Mandl10 ; Feynman62

$S^{-1}\gamma^{\mu}S=\Lambda_{\;\nu}^{\mu}\gamma^{\nu}\,=\gamma^{\prime\mu},$
(85)

where $\Lambda_{\;\nu}^{\mu}$ stand for the tensors’ components of the Lorentz
transformation. Substituting Eq.(85) into
$A_{\mu}(x)\bar{\psi}(x)\gamma^{\mu}\psi(x)$ yields

$A_{\mu}^{\prime}(y)\bar{\psi}^{\prime}(y)S^{-1}\gamma^{\mu}S\psi^{\prime}(y)=A_{\mu}^{\prime}(y)\bar{\psi}(y)\gamma^{\prime\mu}\psi(y)\text{\thinspace.}$
(86)

While looking for $\Gamma^{\mu}$ we would follow the same convention as that
in the above paragraph, i.e., in different coordinate system we use the same
set of $\\{\Gamma^{\mu}\\}$. Then analogously, we use the form of the above
formula Eq. (85) for scaling transformation as

$S^{\prime-1}\Gamma^{\mu}S^{\prime}=\Lambda_{\;\,\nu}^{\prime\mu}\Gamma^{\nu}\text{\thinspace,}$
(87)

where formally we have used $\Lambda_{\;\,\nu}^{\prime\mu}$ to represent the
scaling transformation to every coordinate component Dirac35 ; Gross70 ;
Mack69 [of which eq.(2)] instead of using the usual form $e^{-\alpha}$Budi79A
. Different from the operator $\mu\frac{\text{d}}{\text{d}\mu}$ appearing in
renormalization group equation Su99 ; Cal70 ; Syman70

$\mu\frac{\text{d}\Lambda_{R}}{\text{d}\mu}+\gamma_{F}\,\Lambda_{R}=0\text{
,}$ (88)

where $\gamma_{F}$ is the anomalous scaling dimension defined by

$\gamma_{F}=\mu\frac{\text{d}}{\text{d}\mu}\ln Z_{F}\text{ ,}$

here the operator $D$ has the usual form $D=i\,x^{\nu}\partial_{\nu}$ , being
a hermit one. With the relation
$e^{-i\,\alpha\,D}p_{\mu}e^{i\,\alpha\,D}=e^{-\alpha}p_{\mu}$, i.e.
$[D,\,p_{\mu}]=-i\,p_{\mu}$Budi79A , we have

$(\Gamma^{\mu}p_{\mu})_{scaling\text{
}transform}^{\prime}=S^{\prime-1}\Gamma^{\mu}S^{\prime}\Lambda_{\;\,\mu}^{\prime\nu}\,p_{\nu}\text{\thinspace}=S^{\prime-1}\Gamma^{\mu}S^{\prime}\;e^{-i\,\alpha\,D}p_{\mu}e^{i\,\alpha\,D}\text{.}$
(89)

Now let’s submit $S^{\prime}=e^{\frac{u}{2}\gamma_{5}}$ obtained from the last
section (henceforth we use $e^{\frac{u}{2}\gamma_{5}}$ instead of
$e^{\frac{u}{2}(1+\gamma_{5})}$ as scaling transformation while no confusion
occurs), where $u$ is the infinitesimal parameter. Formally we get

$\displaystyle
S^{\prime-1}\Gamma^{\mu}S^{\prime}\Lambda_{\;\,\mu}^{\prime\nu}\,p_{\nu}$
$\displaystyle=$
$\displaystyle\,e^{-\frac{u}{2}\gamma_{5}}\Gamma^{\mu}e^{\frac{u}{2}\gamma_{5}}(p_{\mu})_{scaling\text{
}transform}^{\prime}$ (90) $\displaystyle=$
$\displaystyle\,e^{-\frac{u}{2}\gamma_{5}}\Gamma^{\mu}e^{\frac{u}{2}\gamma_{5}}e^{-i\,\alpha\,D}p_{\mu}e^{i\,\alpha\,D}$
$\displaystyle=$ $\displaystyle
e^{-\frac{u}{2}\gamma_{5}}\Gamma^{\mu}e^{\frac{u}{2}\gamma_{5}}e^{-\alpha}p_{\mu}\text{
}$ $\displaystyle\doteq$
$\displaystyle\,e^{-\frac{u}{2}\gamma_{5}}\Gamma^{\mu}e^{\frac{u}{2}\gamma_{5}}p_{\mu}(1-\alpha)\text{
.}$

From the experience of calculating $\gamma-$matrix and the following relations

$e^{-\frac{u}{2}\gamma_{5}}\gamma^{\mu}e^{\frac{u}{2}\gamma_{5}}\simeq(1-\frac{u}{2}\gamma_{5})\gamma^{\mu}(1+\frac{u}{2}\gamma_{5})\simeq\gamma^{\mu}+u\,\gamma^{\mu}\gamma_{5}\,,$
(91)

$e^{-\frac{u}{2}\gamma_{5}}\gamma^{\mu}\gamma_{5}e^{\frac{u}{2}\gamma_{5}}\simeq(1-\frac{u}{2}\gamma_{5})\gamma^{\mu}\gamma_{5}(1+\frac{u}{2}\gamma_{5})\simeq\gamma^{\mu}\gamma_{5}+u\,\gamma^{\mu}\,,$
(92)

$e^{-\frac{u}{2}\gamma_{5}}\gamma^{\mu}(1\pm\gamma_{5})e^{\frac{u}{2}\gamma_{5}}\simeq(1-\frac{u}{2}\gamma_{5})\gamma^{\mu}(1\pm\gamma_{5})(1+\frac{u}{2}\gamma_{5})\simeq(1\pm
u\,)\gamma^{\mu}(1\pm\gamma_{5})\,,$ (93)

we find out a possible form of $\Gamma^{\mu}$

$\Gamma^{\mu}=\gamma^{\mu}(1\pm\gamma_{5})\text{ while }\alpha\sim u\text{ .}$
(94)

The transformation in Eqs. (91, 92, 93) can be replaced by
$e^{-\frac{u}{2}(1+\gamma_{5})\,}$ and $e^{\frac{u}{2}(1+\gamma_{5})}$, the
results would be the same obviously. The coefficients $(1\pm u\,)$ of Eq. (93)
can be contracted now to be $1$ with coefficients $(1\mp u\,)$ that come from
the transformation of $p_{\mu}$. And we note that the infinitesimal parameters
$u$ and $\alpha$ are not independent. By this way we set up the relationship
between the operator $D$ and $S^{\prime}=e^{\frac{u}{2}(1+\gamma_{5})}$
directly.

And we know $S^{\prime}=e^{\frac{u}{2}(1+\gamma_{5})}$ is acting on Dirac
spinor as expected. The same transformation holds for vertex
$\Gamma^{\mu}\,A_{\mu}$, as well as $\Gamma^{\mu}\,p_{\mu}$. The resultant
vertex is different from that of Ref. Mack69  due to the choice of
$\gamma_{5}$, since we have followed the convention of Quantum Field Theory.
In fact we have extended the transformation, the interaction vertex and spinor
space simultaneously, and these elements can be extended further while we
involving more generators of conformal group.

What if we perform the scaling transformation $S^{\prime}$ succeedingly $N$
times upon the vector vertex-form $\gamma^{\mu}$. Different from Eqs. (91, 92,
93), now we employ the following formulism without approximation

$(e^{-\frac{u}{2}(1+\gamma_{5})})^{N}\gamma^{\mu}(e^{\frac{u}{2}(1+\gamma_{5})})^{N}=\gamma^{\mu}\cosh
Nu+\gamma^{\mu}\gamma_{5}\sinh Nu\,\text{ ,}$ (95)

from which one notes that the vector vertex arrives at its limits
$\gamma^{\mu}(1\pm\gamma_{5})$ only if $\frac{\cosh Nu}{\sinh
Nu}\rightarrow\pm 1$, i.e. $Nu$ $\rightarrow\pm\infty$. $Nu$
$\rightarrow\pm\infty$ means one carrying out enough steps of inflating or
shrinking transformation. We call such states involving interaction vertices
$\gamma^{\mu}(1\pm\gamma_{5})$ as extreme states, which evolve from the
interaction vertex $\gamma^{\mu}$ after the scale constantly changing. And the
variation of coupling constant is assumed to be absorbed into coupling
constant. It turns out that such scaling transformation doesn’t conserve the
vector-dominant interaction, or alternatively, the transformation tends to
transform the relating spinor from a normal one to a chiral one.

We have derived the above limits with the relation
$e^{-i\,\alpha\,D}p_{\mu}e^{i\,\alpha\,D}=e^{-\alpha}p_{\mu}$, i.e.
$[D,\,p_{\mu}]=-i\,p_{\mu}$Budi79A , The $P_{\mu}$ plays the role of the
production operator or annihilation operator of $D$ from the point of view of
quantum mechanics. Analogously, we may notice another similar relationship
among conformal group by the commutation
$[M_{\mu\nu},K_{\rho}]=i(g_{\nu\rho}K_{\mu}-g_{\mu\rho}K_{\nu})$, however we
fail to get the variation of vertex $\gamma^{\rho}(1\pm\gamma_{5})$ with the
transformation $e^{\frac{u}{2}\gamma_{\mu}\gamma_{\nu}}$ (operation of
$M_{\mu\nu}$ in spinor space). So the analog cannot be followed by any other
commutations of conformal group. The details are as follows. While the angular
momentum transformation is written to be
$S=e^{\frac{u}{2}\gamma_{\mu}\gamma_{\nu}}$, its effect on vector vertex
$\gamma^{\mu}$ [we label it by $\Gamma^{\mu}$]

$\displaystyle\Gamma^{\mu\prime}$ $\displaystyle=$ $\displaystyle
e^{-\frac{u}{2}\gamma_{\rho}\gamma_{\sigma}}\Gamma^{\mu}e^{\frac{u}{2}\gamma_{\rho}\gamma_{\sigma}}\approx\left(1-\frac{u}{2}\gamma_{\rho}\gamma_{\sigma}\right)\Gamma^{\mu}\left(1+\frac{u}{2}\gamma_{\rho}\gamma_{\sigma}\right)\approx\gamma^{\mu}+u\left(g_{\rho}^{\mu}\gamma_{\sigma}-g_{\sigma}^{\mu}\gamma_{\rho}\right)$
$\displaystyle P_{\mu}^{\prime}$ $\displaystyle=$ $\displaystyle
e^{-iaM_{\rho\sigma}}P_{\mu}e^{iaM_{\rho\sigma}}=P_{\mu}+ia\left[P_{\mu},M_{\rho\sigma}\right]=P_{\mu}-a\left(g_{\mu\rho}P_{\sigma}-g_{\mu\sigma}P_{\rho}\right)\text{
.}$ (96)

Combining the above transformations yields

$\displaystyle\left(\Gamma^{\mu}P_{\mu}\right)^{\prime}$ $\displaystyle=$
$\displaystyle\left[\gamma^{\mu}+u\left(g_{\rho}^{\mu}\gamma_{\sigma}-g_{\sigma}^{\mu}\gamma_{\rho}\right)\right]\left[P_{\mu}-a\left(g_{\mu\rho}P_{\sigma}-g_{\mu\sigma}P_{\rho}\right)\right]$
$\displaystyle=$
$\displaystyle\Gamma^{\mu}P_{\mu}+u\left(g_{\rho}^{\mu}\gamma_{\sigma}-g_{\sigma}^{\mu}\gamma_{\rho}\right)P_{\mu}-a\gamma^{\mu}\left(g_{\mu\rho}P_{\sigma}-g_{\mu\sigma}P_{\rho}\right)$
$\displaystyle=$
$\displaystyle\Gamma^{\mu}P_{\mu}+\left(u+a\right)\left(\gamma_{\sigma}P_{\rho}-\gamma_{\rho}P_{\sigma}\right)\text{
,}$ (97)

when $u\sim-a$, one obtains
$\left(\Gamma^{\mu}P_{\mu}\right)^{\prime}=\Gamma^{\mu}P_{\mu}$. If replacing
$\gamma^{\mu}$ axil-vector vertex yeilds $\gamma^{\mu}(1\pm\gamma_{5})$, we
would get the same results

$\displaystyle\Gamma^{\mu\prime}$ $\displaystyle=$ $\displaystyle
e^{-\frac{u}{2}\gamma_{\rho}\gamma_{\sigma}}\Gamma^{\mu}e^{\frac{u}{2}\gamma_{\rho}\gamma_{\sigma}}\approx\left(1-\frac{u}{2}\gamma_{\rho}\gamma_{\sigma}\right)\Gamma^{\mu}\left(1+\frac{u}{2}\gamma_{\rho}\gamma_{\sigma}\right)\approx\Gamma^{\mu}+u\left(g_{\rho}^{\mu}\Gamma_{\sigma}-g_{\sigma}^{\mu}\Gamma_{\rho}\right)$
$\displaystyle P_{\mu}^{\prime}$ $\displaystyle=$ $\displaystyle
e^{-iaM_{\rho\sigma}}P_{\mu}e^{iaM_{\rho\sigma}}=P_{\mu}+ia\left[P_{\mu},M_{\rho\sigma}\right]=P_{\mu}-a\left(g_{\mu\rho}P_{\sigma}-g_{\mu\sigma}P_{\rho}\right)\text{
.}$ (98)

Combining the above transformations yields

$\displaystyle\left(\Gamma^{\mu}P_{\mu}\right)^{\prime}=\Gamma^{\mu}P_{\mu}+\left(u+a\right)\left(\Gamma_{\sigma}P_{\rho}-\Gamma_{\rho}P_{\sigma}\right)\text{
.}$ (99)

As for the kinetic term of an extended particle in the extreme state where the
scale transformation is repeated infinite times, the momentum becomes light-
cone like and the kinetic mass tends to zero since,

$m_{kinetic}^{2}=(\Gamma_{\mu}p^{\mu})(\Gamma_{\nu}p^{\nu})=\gamma_{\mu}(1-\gamma_{5})p^{\mu}\gamma_{\nu}(1-\gamma_{5})p^{\nu}=p^{2}(1+\gamma_{5})(1-\gamma_{5})=0\text{
,}$ (100)

here the $m^{2}=0$ may just have comparable meaning while its momentum is very
large and its mass can be ignored approximately, we call such mass as
kinematic mass. So we conclude the extreme vertices
$\gamma^{\mu}(1\pm\gamma_{5})$ give rise to no mass, and to make a fermion
massive, we have to use the group method to make the interaction vertex leave
the form $\gamma^{\mu}(1\pm\gamma_{5})$. However since that the limits
$\gamma^{\mu}(1\pm\gamma_{5})$ of interaction form are invariant under
transformation $e^{\frac{u}{2}(1+\gamma_{5})}$, it is impossible for
$e^{\frac{u}{2}(1+\gamma_{5})}$ to make the extreme states
$\gamma^{\mu}(1\pm\gamma_{5})$ leave its form. To make a fermion leave this
state to get mass, it is necessary to consider further the other generators of
conformal group. Whereas according to our knowledge of commutations among the
generators of conformal group, there is no much room to choose another
generator. For the vertex $\gamma^{\mu}(1-\gamma_{5})$ we may choose the
generators $K_{\mu}$, otherwise for $\gamma^{\mu}(1+\gamma_{5})$ we may choose
$P_{\mu}$. For example, for $\Gamma^{\mu}=\gamma^{\mu}(1-\gamma_{5})$ we have,

$\displaystyle\Gamma^{\mu\prime}$ $\displaystyle=$ $\displaystyle
e^{\frac{u}{2}\gamma_{\rho}\left(1+\gamma_{5}\right)}\gamma^{\mu}\left(1-\gamma_{5}\right)e^{\frac{u}{2}\gamma_{\rho}\left(1+\gamma_{5}\right)}$
$\displaystyle=$
$\displaystyle[1+\frac{u}{2}\gamma_{\rho}\left(1+\gamma_{5}\right)]\gamma^{\mu}\left(1-\gamma_{5}\right)[1-\frac{u}{2}\gamma_{\rho}\left(1+\gamma_{5}\right)]$
$\displaystyle=$
$\displaystyle\gamma^{\mu}\left(1-\gamma_{5}\right)+\frac{u}{2}\left[\gamma_{\rho}\left(1+\gamma_{5}\right),\gamma^{\mu}\left(1-\gamma_{5}\right)\right]-\frac{u^{2}}{4}\gamma_{\rho}\left(1+\gamma_{5}\right)\gamma^{\mu}\left(1-\gamma_{5}\right)\gamma_{\rho}\left(1+\gamma_{5}\right)$
$\displaystyle=$
$\displaystyle\gamma^{\mu}\left(1-\gamma_{5}\right)+2u\left[g_{\rho}^{\mu}\left(1-\gamma_{5}\right)-\gamma^{\mu}\gamma_{\rho}\right]-u^{2}\gamma_{\rho}\gamma^{\mu}\gamma_{\rho}\left(1+\gamma_{5}\right)\text{
,}$ (101)

when $u\longrightarrow 0$,
$\Gamma^{\mu\prime}\longrightarrow\left(\gamma^{\mu}+2ug_{\rho}^{\mu}\right)\left(1-\gamma_{5}\right)-2u\gamma^{\mu}\gamma_{\rho}$
and when $u\longrightarrow\infty$,
$\Gamma^{\mu\prime}\longrightarrow-u^{2}\gamma_{\rho}\gamma^{\mu}\gamma_{\rho}\left(1+\gamma_{5}\right)$.
Then combine the above equation with the spatial transformation
$P_{\mu}^{\prime}=e^{-iaK_{\rho}}P_{\mu}e^{iaK_{\rho}}\approx\left[1-iaK_{\rho}\right]P_{\mu}\left[1+iaK_{\rho}\right]=P_{\mu}+ia\left[P_{\mu},K_{\rho}\right]=P_{\mu}+2a\left(g_{\mu\rho}D-M_{\mu\rho}\right)$,
we have

$\displaystyle\left(\Gamma^{\mu}P_{\mu}\right)^{\prime}$
$\displaystyle\approx$
$\displaystyle\left[\left(\gamma^{\mu}+2ug_{\rho}^{\mu}\right)\left(1-\gamma_{5}\right)-2u\gamma^{\mu}\gamma_{\rho}\right]\left[P_{\mu}+2a\left(g_{\mu\rho}D-M_{\mu\rho}\right)\right]$
$\displaystyle\approx$
$\displaystyle\Gamma^{\mu}P_{\mu}+2u\left[g_{\rho}^{\mu}\left(1-\gamma_{5}\right)-\gamma^{\mu}\gamma_{\rho}\right]P_{\mu}+2a\gamma^{\mu}\left(1-\gamma_{5}\right)\left(g_{\mu\rho}D-M_{\mu\rho}\right)\text{
.}$ (102)

The generator $P_{\mu}$ would have similar effect on vertex
$\gamma^{\mu}(1+\gamma_{5})$. The generator $K_{\mu}(P_{\mu})$ first makes the
bosons getting mass by transmitting space-time from space-like to time-like,
then it makes fermions (with chiral interaction vertex, meaning the fermions
have no dynamical mass) leave the vertex, i.e. makes fermions massive.

##  IV Generating fermion mass term with the help of conformal group

Conventionally only linearly flat unitary-complex-space is used in quantum
theory. In our model we have instead tried to express some of the effects,
such as interaction or symmetries breaking etc., by curved complex-space in
the strong interaction or strong correlation regime. Once a space is curved,
particles become living in a larger space with larger symmetry group. Here we
find that the space-curving can also be helpful in understanding the mass
generation. Mathematically, we explain the mass generation in two steps. The
first is how a boson becomes massive with the help of conformal
transformation, the second is how a massless fermion gets massive from the
massive boson. Coincidentally, we find the two steps could be ascribed to the
same generators of conformal group. Furthermore, we will discuss the first
step by photons pair production Piaz09 ; Jiang11 ; Drei12 ; Tito12 ; Bakm08 ;
Li12 ; Brodsky06 ; Kohl14 ; Blinne14 ; Jans13 ; LiA14 ; Fedotov13 ,
$\gamma+\gamma\longrightarrow e^{+}e^{-}$. The previous works on photons pair
production mainly focus on calculating the amplitude in semi-classical level.
In this paper we don’t get involved in the concrete calculation of the
amplitude of the process, but instead focus on how the process could be in
accordance with the inequality $\vec{E}^{2}-\vec{B}^{2}\neq 0$.

To interpret the first step, let’s carry some simple calculation of energy-
momentum of classical electromagnetic field. It is well known that the energy
density of electromagnetic field is

$\varpi=\frac{1}{2}(\vec{E}^{2}+\vec{B}^{2})\text{ ,}$

and the momentum of electromagnetic fields is Poynting vector

$\vec{S}=\vec{E}\times\vec{B}\text{ ,}$

using the formula

$(\vec{a}\times\vec{b})\times\vec{c}=(\vec{a}\cdot\vec{c})\vec{b}-(\vec{b}\cdot\vec{c})\vec{a}\text{
,}$

we obtain

$\displaystyle\varpi^{2}-\vec{S}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}(\vec{E}^{2}+\vec{B}^{2})^{2}-(\vec{E}\times\vec{B})^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{4}(\vec{E}^{2}+\vec{B}^{2})^{2}-(\vec{E}\times\vec{B})\cdot(\vec{E}\times\vec{B})$
$\displaystyle=$
$\displaystyle\frac{1}{4}(\vec{E}^{2}+\vec{B}^{2})^{2}-(\vec{E}\times\vec{B}\times\vec{E})\cdot\vec{B}$
$\displaystyle=$
$\displaystyle\frac{1}{4}(\vec{E}^{2}+\vec{B}^{2})^{2}-[(\vec{E}\cdot\vec{E})\vec{B}-(\vec{E}\cdot\vec{B})\vec{E}]\cdot\vec{B}$
$\displaystyle=$
$\displaystyle\frac{1}{4}(\vec{E}^{4}+2\vec{E}^{2}\vec{B}^{2}+\vec{B}^{4})-(\vec{E}\cdot\vec{E})(\vec{B}\cdot\vec{B})$
$\displaystyle=$
$\displaystyle\frac{1}{4}(\vec{E}^{4}-2\vec{E}^{2}\vec{B}^{2}+\vec{B}^{4})$
$\displaystyle=$ $\displaystyle\frac{1}{4}(\vec{E}^{2}-\vec{B}^{2})^{2}\text{
,}$

which to some extent mimics the general relationship of energy and momentum
$p_{0}^{2}-\vec{p}^{2}=m^{2}$. And the result $\vec{E}^{2}-\vec{B}^{2}\neq 0$
corresponds to boson mass according to our conclusion of previous paper. We
note that the free electromagnetic field satisfies
$\varpi^{2}-\vec{S}^{2}=\frac{1}{4}(\vec{E}^{2}-\vec{B}^{2})^{2}=0$ since
$\mid\vec{E}\mid=\mid\vec{B}\mid$ in nature unit. And if the photons
participate in the reaction $\gamma+\gamma\longrightarrow e^{+}e^{-}$, then
maybe a photon meets $\vec{E}^{2}-\vec{B}^{2}\neq 0$, subsequently
$(\vec{E}^{2}-\vec{B}^{2})^{2}>0$ and thus $\varpi^{2}-\vec{S}^{2}>0$ , which
is the typical time-like relationship of energy-momentum. This suggests that
somehow a photon has gained its mass by the stimulation of another photon. In
view of the above analysis, we regard one ultra-high-energy (laser) photon
stimulates another light-cone photon to make it transit to time-likeKohl14 ;
Blinne14 ; Jans13 ; LiA14 ; Fedotov13 . We find the conformal group can be
responsible for such transition from space-(light-cone-)like to time-like .

The second step is to transform a massless state ”like neutrino” to a massive
one ”like electron” explaA . Since even for a massive boson it can decay into
many massless energetic fermions, at least from mathematics it is possible.
With the analysis of the last section, we note that the massless state seems
to occur effectively at the limits of very high energy while the mass of
fermion is very small comparing with the kinetic energy. At the limits, the
fermions interact with each other by the vertex
$\gamma^{\mu}(1\pm\gamma_{5})$, coincidentally the neutrinos pick the
$\gamma^{\mu}(1-\gamma_{5})$. So the interaction vertex deviating from the
form $\gamma^{\mu}(1-\gamma_{5})$ is a sign of ”neutrino” getting its mass.
With the analysis of the previous section, we know that the dilatation
transformation cannot play the role in varying the interaction vertex, since
$e^{\frac{u}{2}(1+\gamma_{5})}$ reserves the vertex
$\gamma^{\mu}(1-\gamma_{5})A_{\mu}$ [$e^{\frac{u}{2}(1+\gamma_{5})}$ can only
ex-change the spinor between helicity representation and spin representation.
See Appendix A.]. Neither can the momentum transformation
$M_{\mu\nu}=e^{\frac{u}{2}\gamma_{\mu}\gamma_{\nu}}$ change the vertex
$\gamma^{\mu}(1-\gamma_{5})$, as evidenced in the last section. So let’s turn
to the transformation $e^{\frac{u}{2}K_{\nu}}$ , while the
$e^{\frac{u}{2}K_{\nu}}$ keeps directly the vertex
$\gamma^{\mu}(1-\gamma_{5})$, it does not keep the whole form
$\gamma^{\mu}(1-\gamma_{5})A_{\mu}$ invariant, nor does the generator
$P_{\mu}$ keep the whole form $\gamma^{\mu}(1+\gamma_{5})A_{\mu}$. Thus we
choose $K_{\mu}$ as a candidate to make $\gamma^{\mu}(1-\gamma_{5})A_{\mu}$
leave $\gamma^{\mu}(1-\gamma_{5})$ and begin to run. It runs to normal
interaction vertex $\gamma^{\mu}$, as shown in Eq. (102) , which is for
massive fermions.

Now let’s come back to the equation of our model. In our previous paper WA08 ,
we have laid two ways of generating fermion mass term in EoM, which are just
the possibilities from the viewpoint of mathematics. Now with the deep insight
of physics, we recognize that the second one must be the candidate for the
mass term, i.e. calculating a remaining integral in the equation. Now we will
treat the candidate term $-i\int
A_{\nu}\partial_{\lambda}\,\partial^{\nu}\psi\,dx^{\lambda}$ in the EoM.
Considering the hermit of the operators $i\partial_{\lambda}$,
$\,i\partial^{\nu}$, and apart from the coefficients, we have

$\displaystyle\ \ \ \ \int
A_{\nu}\partial_{\lambda}\,\partial^{\nu}\psi\,dx^{\lambda}$ $\displaystyle=$
$\displaystyle\int
dx^{\lambda}\partial_{\lambda}\,(\partial^{\nu}A_{\nu})\psi\,$
$\displaystyle=$ $\displaystyle\int(dt\frac{\partial}{\partial
t}-d\vec{x}\cdot\vec{\nabla})\,(\frac{\partial}{\partial
t}A_{0}-\vec{\nabla}\cdot\vec{A})\psi\,\text{ .}$ (103)

If we omit the term $\vec{\nabla}\cdot\vec{A}$ by considering the Columb gauge
as used previously WA08 , then the above equation yeilds

$\displaystyle\ \ \ \ \int(dt\frac{\partial}{\partial
t}-d\vec{x}\cdot\vec{\nabla})\,(\frac{\partial}{\partial t}A_{0})\psi$
$\displaystyle=$ $\displaystyle\int[dt\frac{\partial^{2}}{\partial
t^{2}}A_{0}-\frac{\partial}{\partial t}(d\vec{x}\cdot\vec{\nabla}A_{0})]\psi$
$\displaystyle=$ $\displaystyle\int[dt\frac{\partial^{2}}{\partial
t^{2}}A_{0}-d\vec{x}\cdot\frac{\partial}{\partial
t}(\vec{E}+\frac{\partial}{\partial t}\vec{A})]\psi$ $\displaystyle=$
$\displaystyle\int[dt\frac{\partial^{2}}{\partial
t^{2}}A_{0}-d\vec{x}\cdot\frac{\partial^{2}}{\partial
t^{2}}\vec{A}-d\vec{x}\cdot(\vec{\nabla}\times\vec{B}-\vec{J})]\psi$
$\displaystyle=$ $\displaystyle\int[dt\frac{\partial^{2}}{\partial
t^{2}}A_{0}-d\vec{x}\cdot\frac{\partial^{2}}{\partial
t^{2}}\vec{A}-d\vec{x}\cdot(\vec{\nabla}\times\vec{\nabla}\times\vec{A}-\vec{J})]\psi$
$\displaystyle=$ $\displaystyle\int[dt\frac{\partial^{2}}{\partial
t^{2}}A_{0}-d\vec{x}\cdot\frac{\partial^{2}}{\partial
t^{2}}\vec{A}-d\vec{x}\cdot(-\vec{\nabla}^{2}\vec{A}-\vec{J})]\psi$
$\displaystyle=$ $\displaystyle\int[dt\frac{\partial^{2}}{\partial
t^{2}}A_{0}-d\vec{x}\cdot(\frac{\partial^{2}}{\partial
t^{2}}\vec{A}-\vec{\nabla}^{2}\vec{A})+d\vec{x}\cdot\vec{J}]\psi$
$\displaystyle=$ $\displaystyle\int[dt\frac{\partial^{2}}{\partial
t^{2}}A_{0}-d\vec{x}\cdot\Box\vec{A}+d\vec{x}\cdot\vec{J}]\psi$
$\displaystyle=$ $\displaystyle\int[dt\frac{\partial^{2}}{\partial
t^{2}}A_{0}+m^{2}d\vec{x}\cdot\vec{A}+d\vec{x}\cdot\vec{J}]\psi$
$\displaystyle=$ $\displaystyle(\lambda_{1}m^{2}\phi+\lambda_{2}L)\psi\text{
,}$ (104)

where we propose $\psi$ is almost flat (thus unchanged) under perturbative
condition, $\phi$ is a flux relating to concrete situation, and $L$ is an
angular moment. The imaginary number $i$ is absorbed by $d\vec{x}$ since we
have carried the conformal transformation operated by $K_{\mu}$, which makes
space-time varying from space-like to time-like, i.e. isotropically
interpreted by $x_{\mu}\rightarrow ix_{\mu}$ Ryder74 . In the last two
equalities of eq. (45) we have used the equation referring to (9.27) of
previous paper WA08

$\vec{A}\cdot\Box\vec{A}=\vec{B}^{2}-\vec{E}\stackrel{{\scriptstyle
d}}{{=}}-m^{2}\vec{A}^{2}\text{ ,}$ (105)

where the equation holds only while the considered transverse filed (Coulomb
gauge) has no source, thus no scalar component Mandl10  (of first chapter).
Now we discuss the mass term problem under same situ, so follows the same
conditions. The conditions seem not so general, however at least they reflect
certain aspects of physics, actually the Coulomb gauge $\nabla\cdot\vec{A}=0$
plus radiation fields method can resolve a wide class of pratical problems.

To understand the $\lambda_{1}$ and $\lambda_{2}$ physically, let’s analyze
the first term and the second term in the last equality of eq. (45). The first
term is more like for hadrons, or for another kind of composite particle. At
first sight, the square of fermion mass being proportional to the square of
boson mass looks strange. However, this might become possible if we look into
Higgs-mechanism Mandl10  (eqs. (19.8) and (19.4)), with relations
$m_{l}=\upsilon g_{l}/\sqrt{2}$ , and $m_{W}=\frac{1}{2}\upsilon g$, where $g$
and $g_{l}$ being adjusting parameters and $\upsilon$ a part of a scalar
fields. The two relations suggest the reasonability of the formula like
$m_{fermion}^{2}\propto m_{boson}^{2}$. On the other hand, we can analyze the
dimension of the first term. Concerning the geometrical phase factor like
$\exp[i\frac{e}{\hbar}\oint\vec{A}\cdot d\vec{r}]$, the dimension of magnetic
flux is $\left[\frac{\hbar}{e}\right]$. And concerning the definition of
magnetic moment $\vec{\mu}=\frac{e}{m_{f}}\vec{S}$ ($\vec{S}$ is the spin,
$m_{f}$ is the fermion mass), the dimension of magnetic moment is
$\left[\frac{\hbar e}{m_{f}}\right]$. Therefore the ratio of the flux and
magnetic moment has dimension like $\left[\frac{\hbar}{e}/(\frac{\hbar
e}{m_{f}})\right]=\left[\frac{m_{f}}{e^{2}}\right]$, i. e. if alternatively we
write the first term as $\lambda_{1}^{\prime}\frac{\phi}{\mu}$ (where $\mu$ is
the absolute value of magnetic moment) then the
$\left[\lambda_{1}^{\prime}=\lambda_{1}m\hbar e\right]$ is the proportional
coeficient of flux and magnetic moment, which is coincident with the view that
this mass term is for a composite particle. We also observe that in natural
unit $\lambda_{1}^{\prime}$ differ from $\lambda_{1}$ in a mass factor, where
$\lambda_{1}$ itself has the dimension reciprocal of magnetic flux. Please
caution that although we constantly mention the electric or magnetic fields,
they are not the true electromagnetic fields. We just employ the analorgy
between Abelian field and non-Abelian field, our interest is non-Abelian
field.

The second term may be for an elementary particle. The term mimics the result
$M^{2}\propto\mu^{2}L+\alpha$ ($\mu^{2}$, $\alpha$ constants) from Regge poles
Mas12 . From this analogy the parameter $\lambda_{2}$ has the dimension of
mass square, since angular momentum $L$ is dimensionless. At present stage we
may further surmise that the first term, which represents the mass of a hadron
fermion and thus is dependent on interaction, may be related to DCSB Chang11 ;
Kor14 . And the second term may possibly link with our suggested process
$\gamma+\gamma\longrightarrow e^{+}e^{-}$, in which the photon in curved
complex space is split into elementary particles. At the same time the
electro-field and magnetic field are wrapped: the electro-field wrapping to
form charges, and the magnetic field wrapping to form spins relating to
masses. Thus both the first and second terms of the equation correspond to
similar pictures, the picture of curved (wrapped) complex-space. This
conclusion may help us make the mass problem calculable, not just as
legitimate parameters confirmed by DCSB and Higgs mechanism. And also we
should caution that here the obtained mass may be just ”effective mass”, since
classically there is no system mimicking the strong interaction and strong
correlation, and all that we have at hand are just the experiences of treating
strong interaction in previous literature Hooft81 ; Bao07 ; Bao13 , which must
be relevant to topology. And mostly topology is relating to magnetic flux.

##  V Conclusions and Discussions

In this paper we have analyzed the effect of conformal group on interaction
vertices, especially on the vector vertex and the chiral vertex, and then
pointed out the relationship between conformal group and generation of mass
term. If only concerning mass-term problem in EoM, the structure group would
not be smaller than conformal group. Based on our analyses, the scaling
transformation allows a special chiral-like vertex
$\gamma^{\mu}(1\pm\gamma_{5})$ to be invariant. And normal vector vertex
becomes running under such scaling transformation. So with the scaling
transformation only, one cannot endow a massless fermion like ”neutrino” with
mass, since scaling transformation happens to keep the chiral interaction
form, and cannot make fermions like ”neutrino” deviate from the vertex form
explaA . So we have to involve other generators of conformal group to
transform the interaction vertex, for example, the generator $K_{\mu}$. And
coincidentally, we find $K_{\mu}$ can also be responsible for transforming
space-time from space-like to time-like, which in physics should occur before
transforming fermions from massless to massive. With the above cognition, we
get the mass term by directly calculating the integral occurred in original
paper, which falls into one of the possibilities of our original surmise.

According to the formula we get, we present a plausible understanding of the
generation of mass term. Since we have obtained the mass by directly
calculating the integral in original equation, we believe the mass term is
generated by the curvature (wrapped) of complex space. The wrapping of complex
space shows its effect by deforming electro-field and magnetic field if we use
the shortcut process $\gamma+\gamma\longrightarrow e^{+}e^{-}$ as paradigm.
The progress implies that the photon in curved complex space is split into
particle pairs, and at the same time the electro-field and magnetic-field of
the photon are wrapped: the electro-field forms charge, and magnetic field
forms spin. This conclusion may help us make the mass problem calculable, not
just as legitimate parameters in DCSB and Higgs mechanism. Of course it is
necessary to confirm physically whether spin or magnetic flux actually holds
mass. And we should caution that here the obtained mass may be just ”effective
mass”, which roots obviously in interaction. We conceive that this result
might be helpful in understanding spectra problem of hadrons, whose mass may
be proportional to (chromo-)magnetic flux. The further work along this line is
in progress.

We have achieved the conclusion that in our theory the minimal group should be
the entire conformal group. On mathematical side the conformal group has been
investigated thoroughly from different aspects Hor10 , and its application to
physics especially to quantum field once was also widely considered. However
the application is not so satisfactory because hitherto no other perfect
quantum system than photon field Bate1910 ; Cunn1910  has been found so that
the corresponding Lagrangian is conformally invariant, unless the mass of the
involved particles are null Kast08 ; Yu13 ; Jose88 ; Gross70 ; Dirac35 ; Lus75
. In our treatment we turn from searching for invariant dynamics to how the
conformal group makes physics quantities running, so as to match the
renormalization results that some of the physics constants vary regularly with
the energy scale, for instance charge, mass, and Green functions. According to
our analysis, the scaling transformation can somehow cause variation of
coupling constants and masses. And in this paper, we have also discussed the
function of generators $K_{\mu}$ and $P_{\mu}$ which conformally change the
interaction vertex from a chiral one to a non-chiral one. The action performed
by generators $P_{\mu}$ or $K_{\mu}$ may be caused by the external non-hermite
stimulation. Basically it is with such stimulations that the value of
$\vec{E}^{2}-\vec{B}^{2}$ changes from $0$ to $\neq 0$. Then by certain
decaying process, fermion mass is generated. In summary, we get to the
conclusion that the running of interaction vertex corresponds to certain kinds
of curving of complex space, which is certainly governed by generators of
conformal group. Conformal group might live for curving, but not for
invariance.

###### Acknowledgements.

I am grateful to Prof. W. Q. Wang and Prof. W. Han for their encouragements.
The Project Sponsored by the Scientific Research Foundation for the Returned
Overseas Chinese Scholars, State Education Ministry and Fundamental Research
Funds for the Central Universities, and Natural Scientific Foundation of China
with granted No.91227114.

##  VI Appendix A. how $e^{\frac{u}{2}\gamma_{5}}$ exchange the fermion
spinors between helicity representation and spin representation

We recognize here that the performance of $e^{\frac{u}{2}\gamma_{5}}$ can only
exchange the fermion spinors between helicity representation and spin
representation, or vice versa. Accordingly such transformation cannot yield
the generation of spinor, or the mass of fermions. In what follows we present
the example of how $e^{\frac{u}{2}\gamma_{5}}$ changes the representation of
spinors.

Despite of the algebric structure, while $m=0$ the spinor could be any
”vector-form” homogenously because the term
$\left(\begin{array}[]{cccc}a&b&c&d\end{array}\right)m\left(\begin{array}[]{c}a\\\
b\\\ c\\\ d\end{array}\right)$ is equal to null. Nevertheless, when $m\neq 0$
, the term spinor(vector) $\left(\begin{array}[]{c}a\\\ b\\\ c\\\
d\end{array}\right)$ cannot be selected arbitrarily, since the condition
$\left(\begin{array}[]{cccc}a&b&c&d\end{array}\right)m\left(\begin{array}[]{c}a\\\
b\\\ c\\\ d\end{array}\right)=m$ should be satisfied. The most simple option
is $\left(\begin{array}[]{c}a\\\ b\\\ c\\\
d\end{array}\right)=\left(\begin{array}[]{c}1\\\ 0\\\ 0\\\
0\end{array}\right)$, or some analog like $\left(\begin{array}[]{c}0\\\ 1\\\
0\\\ 0\end{array}\right)$ can be used. Subsequently we will focus on the
transformation from the special spinor $\left(\begin{array}[]{c}1\\\ 0\\\ 0\\\
0\end{array}\right)$ to its helicity form in arbitrary reference coordinates.

Firstly, we refer to the Dirac equation for a rest particle

$\gamma_{0}\psi=m\psi\text{ ,}$ (106)

whose solution is $\left(\begin{array}[]{c}1\\\ 0\\\ 0\\\ 0\end{array}\right)$
or $\left(\begin{array}[]{c}0\\\ 1\\\ 0\\\ 0\end{array}\right)$ for normal
particles, and $\left(\begin{array}[]{c}0\\\ 0\\\ 1\\\ 0\end{array}\right)$ or
$\left(\begin{array}[]{c}0\\\ 0\\\ 0\\\ 1\end{array}\right)$ for anti-
particles. One also notes that
$\left(\begin{array}[]{cccc}1&0&0&0\end{array}\right)\gamma_{0}\left(\begin{array}[]{c}1\\\
0\\\ 0\\\ 0\end{array}\right)=m$ from Eq. (106). This knowledge legitimates
the writing from the rhs to lhs of Eq. (106). That means we can generate
spinor via mass term

$m=\left(\begin{array}[]{cccc}1&0&0&0\end{array}\right)m\left(\begin{array}[]{c}1\\\
0\\\ 0\\\
0\end{array}\right)=\left(\begin{array}[]{cccc}1&0&0&0\end{array}\right)\gamma_{0}\left(\begin{array}[]{c}1\\\
0\\\ 0\\\ 0\end{array}\right)\text{ .}$ (107)

Assume that the motion of fermion is along z direction, one solution of Dirac
equation $(\not p-m)\psi=0$ in helicity representation is
$\psi=\left(\begin{array}[]{c}sth\\\ 0\\\ sth\\\ 0\end{array}\right)$. Further
the transformation responsible for the variation from
$\left(\begin{array}[]{c}1\\\ 0\\\ 0\\\ 0\end{array}\right)$ to
$\left(\begin{array}[]{c}sth\\\ 0\\\ sth\\\ 0\end{array}\right)$ is
$e^{\frac{u}{2}\gamma_{5}}$, explicitly

$e^{\frac{u}{2}\gamma_{5}}\left(\begin{array}[]{c}1\\\ 0\\\ 0\\\
0\end{array}\right)=\left(\begin{array}[]{c}\cosh\frac{u}{2}\\\ 0\\\
\sinh\frac{u}{2}\\\ 0\end{array}\right)\text{ ,}$ (108)

succeeding transformation $e^{\frac{u}{2}\gamma_{5}}$ could be employed if
necessary. Then the resultant spinor doesn’t satisfy the Eq. (106) any longer,
which matches our common knowledge of quantum field theory.

## References

  * (1) Hai-Jun Wang, J. Math. Phys. 49, 033513 (2008).
  * (2) Yu Nakayama, arXiv:1302.0884 [hep-th].
  * (3) Élie Cartan, _The Theory of Spinors_ , Dover Publications, Inc. 1981. This is a republication of the first version published by Hermann, Paris. 1966.
  * (4) Yufen Liu, Zhongqi Ma, Boyuan Hou , Commun. Theor. Phys.31, 481(1999). and references therein.
  * (5) R. Jackiw, Phys. Today 25(1972)23.
  * (6) M, Henkel, J. Stat. Phys. 75(1994)1023.
  * (7) M. Henkel, Nucl. Phys. B 641(2002)405.
  * (8) M. Henkel and J. Unterberger, Nucl. Phys. B 660(2003)407. (arXiv:hep-th/0302187)
  * (9) P. Horvathy and P. M. Zhang, Eur. Phys. J. C 65(2010)607. (arXiv:0906.3594)
  * (10) C. Duval and P. Horvathy, J. Phys. A 42 (2009) 465206; _ibid_ 44 (2011),335203. and references therein.
  * (11) H. Bateman, Proc. London Math. Soc.8, 223 (1910).
  * (12) E. Cunningham, Proc. London Math. Soc.8, 77 (1910).
  * (13) Joseph Polchinski, Nucl. Phys. B, 303 (1988)226.
  * (14) David J. Gross and J. Wess, Phys. Rev. D 2, 753 (1970).
  * (15) H.A. Kastrup, Annalen der Physik, 17, 631 (2008). also, arXiv:0808.2730 [physics.hist-ph].
  * (16) P. A. M. Dirac, Annals of Mathematics, 37, 429 (1935).
  * (17) M. Lüscher and G.Mack, Commun. math. Phys. 41, 203(1975); G. Mack, Commun. math. Phys. 55, 1 (1977).
  * (18) G. Mack and Abdus Salam, Ann. Phys. 53, 174(1969).
  * (19) L. H. Ryder, J. Phys. A: Math. Nucl. Gen. Vol. 7, 1817 (1974).
  * (20) J. A. Wheeler and R. P. Feynman, Rev. Mod. Phys. 17, 157 (1945); _idid_ , 21, 425 (1949).
  * (21) Yong-Liang Ma, Masayasu Harada, Hyun Kyu Lee, Yongseok Oh, Byung-Yoon Park, and Mannque Rho, Phys. Rev. D 88, 014016(2013); Erratum Phys. Rev. D 88, 079904(2013), and references therein.
  * (22) Ping Wang, Chinese Phys. C 35(2011)223.
  * (23) Jun-Chen Su, Xue-Xi Yi and Ying-Hui Cao, J. Phys. G: Nucl. Part. Phys. 25(1999)2325: eq. (2.19).
  * (24) Jun-Chen Su and Hai-Jun Wang, Phys. Rev. C 70, 044003 (2004).
  * (25) Curtis G. Callan, Jr., Phys. Rev. D 2, 1541 (1970).
  * (26) K. Symanzik, Comm. Math. Phys. 18, 227(1970).
  * (27) A. Di Piazza, E. Lötstedt, A. I. Milstein, and C. H. Keitel, Phys. Rev. Lett. 103, 170403(2009).
  * (28) M. Jiang, W. Su, X. Lu, Z. M. Sheng, Y. T. Li, Y. J. Li, J. Zhang, R. Grobe, and Q. Su, Phys. Rev. A 83, 053402(2011).
  * (29) F. Dreisow, S. Longhi, S. Nolte, A. Tnnermann, and A. Szameit, Phys. Rev. Lett. 109, 110401(2012).
  * (30) A. I. Titov, H. Takabe, B. Kämpfer, and A. Hosaka, Phys. Rev. Lett. 108, 240406(2012).
  * (31) S. Bakmaev, E.A. Kuraev, I. Shapoval, Yu.P. Peresun’ko, Phys. Lett. B 660, 494 (2008).
  * (32) Xiao-Ping Li, Guo-Xi Pei, Shi-Lun Pei, Chinese Physics C 36, 1009 (2012).
  * (33) S. J. Brodsky. Photon-photon collisions—past and future. Acta Physica Polonica B, 37(3), 619 (2006).
  * (34) Christian Kohlfurst, Holger Gies, and Reinhard Alkofer, Phys. Rev. Lett. 112, 050402(2014)
  * (35) Alexander Blinne and Holger Gies, Phys. Rev. D 89, 085001(2014).
  * (36) Martin J. A. Jansen and Carsten Muller, Phys. Rev. A 88, 052125 (2013).
  * (37) Z.L. Li, D. Lu, B.S. Xie, L.B. Fu, J. Liu, and B.F. Shen, Phys. Rev. D 89, 093011(2014)
  * (38) A. M. Fedotov and A. A. Mironov, Phys. Rev. A 88, 062110(2013).
  * (39) L. H. Ryder, J. Phys. A: Math. Nucl. Gen. Vol. 7, 1817 (1974). Under eq.(7) the author mentioned that while sigma less than zero, then the space-like and time-like exchange.
  * (40) P. Budini, Czechoslovak Journal of Physics B 29, 6 (1979).
  * (41) P. Budini, P. Furlan, R. Raczka, IL Nuovo Cimento A 52, 191 (21 Luglio 1979).
  * (42) A. Esteve, P.G. Sona, IL Nuovo Cimento XXXII, 473 (1964).
  * (43) Hai-Jun Wang, J. Math. Phys. 49, 053508 (2008).
  * (44) Richard P. Feynman, Quantum Electrodynamics–A Lecture Note and Preprint Volume, (W. A. Benjamin, Inc. 1962). pp.44, please refer to the _Tenth Lecture_ on Equivalence Transformation.
  * (45) F. Mandl and G. Shaw, Quantum Field Theory, 2nd Edition (John Wiley and Sons Ltd., 2010). page 93, Section 2.4 and Appendix A.7.
  * (46) This is just a metaphor, in our next work it will be shown that we actually want to resolve mass problem of fermion-like hadrons.
  * (47) Pere Masjuan, Enrique Ruiz Arriola, and Wojciech Broniowski, Phys. Rev. D 85, 094006(2012)
  * (48) Lei Chang,Yu-Xin Liu, and Craig D. Roberts, Phys. Rev. Lett. 106, 072001 (2011)
  * (49) N.S.Korchagin, N.I.Kochelev, arXiv:1402.3101v2.
  * (50) G. t Hooft, Nucl. Phys. B 190, 455 (1981).
  * (51) A. D. Bao and S. S. Wu, Int. J. Theor. Phys. 46, (2007)3093.
  * (52) D. Wang, A. D. Bao, J. Mod. Phys. 4 (2013)35.


Aaditya Prakash Kattekola, Sanjana Dontha, Anuradha Sundru 11institutetext:
Department of Electronics & Communication Engineering
National Institute of Technology, Warangal

# Performance Analysis of RIS-Aided NOMA Networks in $\alpha$-$\mu$ &
$\kappa$-$\mu$ Generalized Fading Channel

Aaditya Prakash Kattekola  Sanjana Dontha  Anuradha Sundru

###### Abstract

For forthcoming 5G networks, Non-Orthogonal Multiple Access (NOMA) is a very
promising technique. And in today’s world, Line of Sight communication is
becoming increasingly harder to achieve. Hence, technologies like
Reconfigurable Intelligent Surfaces(RIS) emerge. RIS-aided NOMA network is a
widely researched implementation of RIS. The environment where these networks
are employed are non-homogeneous & non-linear in nature. The effectiveness of
these systems must thus be evaluated using generalized fading channels. In
this paper, the performance of a RIS-aided NOMA is compared with conventional
NOMA. in $\alpha$-$\mu$ & $\kappa$-$\mu$ channels. This paper also shows that
the well-known fading distribution are special cases of these generalized
fading channels, both analytically and through simulation.

###### keywords:

Reconfigurable Intelligent Surfaces, Non-Orthogonal Multiple Access,
Generalized Fading Channels, 5G & Beyond 5G Networks

##  1 Introduction

The transmission of speech and data through a wireless network is done without
the need of cables or wires. Data is sent through electromagnetic signals that
are broadcast from sending facilities to intermediate and end-user devices in
place of a physical link.[7]. The fifth-generation (5G) technological standard
for broadband cellular networks, which cellular phone providers started
rolling out globally in 2019, is the anticipated replacement for the 4G
networks that connect the majority of modern smart phones.[1] For 5G cellular
networks, NOMA was suggested as a potential radio access method. Real-time
power allocation and consecutive interference cancellation methods for NOMA
must be implemented with great computing efficiency in cellular networks.[5].
RIS is a new wireless communication technology that uses a flat surface made
up of numerous small, programmable reflecting elements to manipulate radio
waves in order to improve wireless communication performance. Each element of
the RIS can be controlled individually to change the phase of the reflected
signal, which enables the surface to reflect, refract, or diffract the
incoming signal in a specific direction. Additionally, RIS-assisted NOMA is
viewed as a potential new technology that can reconfigure the environment for
wireless transmission via software-controlled reflection to meet the
requirements of 5G and Beyond-5G communication.[2].

In wireless communication, the transmitted signal undergoes large scale &
small scale effects such as reflection, diffraction etc. Hence, the receiver
receives multiple components of the signal, each having experienced changes in
amplitude, phase and several other properties[6]. Based on the type of
communication, there exist well-known statistical models of these received
signal. Rayleigh fading channel is used when the communication is
predominantly Non Line-Of-Sight (NLOS) [8]. On the other hand, the dominant
signal is a Line-Of-Sight (LOS) signal, Rician fading model is used [9]. Other
channels include Nakagami-m, Hoyt and Weibull. Nakagami-m in particular is
widely used due to its generalized nature. When the Nakagami parameter ’m’ is
set to $1$ & $0.5$, Rayleigh and One-sided Gaussian are obtained respectively.
Yet there exists situations in channel modelling where none of the known
channels seem adequate. [11]

In the context of RIS-aided NOMA (non-orthogonal multiple access) systems the
performance heavily depends on the location and configuration of the RISs.
Therefore, the fading model should consider the impact of the RISs on the
channel. However, the conventional fading models such as Rayleigh, Rician, or
Nakagami-m do not explicitly capture the effect of RISs on the channel [10].
NOMA networks use superposition coding and successive interference
cancellation (SIC) to separate the signals from different users[5]. This
requires a non-linear signal processing operation that can only be applied
under certain conditions. However, these fading channels do not satisfy these
conditions and hence may not be suitable for RIS-aided NOMA systems. Secondly,
the existing RIS-Assisted channel models lack generality . RISs can
significantly modify the channel between the transmitter and the receiver by
reflecting the signal in a specific direction. This can result in a non-line-
of-sight (NLOS) channel that is different from the conventional LOS channel
model. As a result, the conventional fading models such as Rayleigh, Rician,
or Nakagami-m may not accurately capture the characteristics of the RIS-aided
channel[4].

This paper aims to analyse the performance of RIS-assisted NOMA system under
two generalized fading channels. Namely, the $\alpha$-$\mu$ & $\kappa$-$\mu$
fading channels. Particularly, the outage probability of the system is
simulated. In Section 2 a brief description of the system model is given.
Section 3 & Section 4 goes through $\alpha$-$\mu$ & $\kappa$-$\mu$ fading
channels and their numerical simulation.

##  2 System Model

Figure 1: RIS-aided NOMA System

Figure 1 shows the system model. It is believed that the nearby user (U2) is
so close to the base station (BS) that U2 receives all of its signals from the
BS. It is possible to represent the channel between the BS and U2 as Rayleigh
fading. The Nakagami-m fading channel may be used to represent the BS-RIS
connection, which can be either LOS or NLOS. As U1 gets both direct BS signals
and reflected RIS signals, it will be our primary user of interest. The
Nakagami-m fading channel may be used to represent the RIS-U1 connection,
which can also be either a LOS or NLOS link. [3]. The signal received by U1
is:

$h_{1}=h_{u1}d_{1}^{\frac{-\eta_{n}}{2}}+g_{1}\Phi
h_{r}^{H}d_{ris}^{\frac{-\eta_{l}}{2}}d_{r,1}^{\frac{-\eta_{n}}{2}}$ (1)

where $\Phi$ is the phase shift matrix of the RIS which models the effective
phase shift applied by all intelligent surfaces. Mathematically, $\Phi$ is a
diagonal matrix with each element is of form $\beta_{i}e^{j\phi_{i}}$.
$h_{u1}$ denotes the signal received by U1 from BS, $d_{1}$ denotes the
distance between BS and U1, $h_{r}$ denotes the signal received by RIS from
BS, $d_{ris}$ denotes the distance between BS and RIS, and $d_{r,1}$ denotes
the distance between RIS and U1. $\eta_{n}\ \text{and}\ \eta_{l}$ denote the
path loss factor for the BS to U1 and BS-RIS to U1 path, respectively.

##  3 Generalized Fading Channels

###  3.1 $\kappa$-$\mu$ Channel

$\kappa$-$\mu$ fading channel has become popular in analytical research of
wireless communication scenarios. $\kappa$-$\mu$ distribution is suitable to
model LOS environments just like the Nakagami. In non-homogeneous environments
where reflecting & scattering obstacles of different dimensions are present,
$\kappa$-$\mu$ statistics can be used to model the scenario [6].

####  3.1.1 Probability Density Function (PDF) of $\kappa$-$\mu$ Channel

$\displaystyle p_{X}(x)=$ (2)
$\displaystyle\frac{\mu(1+\kappa)^{\frac{\mu+1}{2}}}{\kappa^{\frac{\mu-1}{2}}e^{\mu\kappa}\bar{X}^{\frac{\mu+1}{2}}}$
$\displaystyle
x^{\frac{\mu-1}{2}}e^{-\frac{\mu(1+\kappa)x}{\bar{X}}}I_{\mu-1}\left(2\mu\sqrt{\frac{\kappa(1+\kappa)x}{\bar{X}}}\right)$

####  3.1.2 Cumulative Density Function (CDF) of $\kappa$-$\mu$ Channel

$P_{x}(y)=1-Q_{u}\left(\sqrt{2\kappa\mu},\sqrt{\frac{2\mu(1+\kappa)y}{\bar{X}}}\right)$
(3)

####  3.1.3 Moment Generating Function (MGF) of $\kappa$-$\mu$ Channel

$M_{X}(s)=\left(\frac{\mu(1+\kappa)}{\mu(1+\kappa)+s\bar{X}}\right)^{\mu}e^{\frac{\mu^{2}\kappa(1+\kappa)}{\mu(1+\kappa)+s\bar{X}}-\kappa\mu}$
(4)

Figure 2: The PDF of $\kappa$-$\mu$ for various $\kappa$ & $\mu$

###  3.2 $\alpha$-$\mu$ Channel

$\alpha$-$\mu$ channel is a non-linear Nakagami-m channel. In non-homogeneous
environments that introduce non-linearity in the signal, $\alpha$-$\mu$ fading
channel can be employed [6].

####  3.2.1 Probability Density Function (PDF) of $\alpha$-$\mu$ Channel

$p_{X}(x)=\frac{\alpha\mu^{\mu}x^{\alpha\mu-1}}{\Gamma(\mu)\Omega_{\alpha}^{\alpha\mu}}e^{-\mu\left(\frac{x}{\Omega_{\alpha}}\right)^{\alpha}},\mu,\alpha>0,x\geq
0$ (5)

####  3.2.2 Cumulative Density Function (CDF)of $\alpha$-$\mu$ Channel:

$P_{x}(y)=\frac{\gamma\left(\mu,\mu\left(\frac{y}{\bar{x}}\right)^{\frac{\alpha}{2}}\right)}{\Gamma(\mu)}$
(6)

####  3.2.3 Moment Generating Function (MGF) of $\alpha$-$\mu$ Channel

$\displaystyle M_{x}(s)=$ (7)
$\displaystyle\frac{\alpha\mu^{\mu}}{2\bar{x}^{\frac{\alpha\mu}{2}}}\frac{\sqrt{k}l^{\frac{\alpha\mu-1}{2}}}{(2\pi)^{\frac{l+k-2}{2}}s^{\frac{\alpha\mu}{2}}}$
$\displaystyle
G_{l,k}^{k,l}\left(\left(\frac{\mu}{\bar{x}^{\frac{\alpha}{2}}}\right)^{k}\frac{l^{l}}{s^{l}k^{k}},\begin{array}[]{c}P(l,1-\frac{\alpha\mu}{2})\\\
P(k,0)\end{array}\right)$

Figure 3: The PDF of $\alpha$-$\mu$ for various $\alpha$ & $\mu$

##  4 Simulation & Results

###  4.1 Simulation Setup

The downlink’s carrier frequency and bandwidth are set to BW = 100
Hz.$\mathrm{MHz}$ and 1$\mathrm{GHz}$, respectively. The AWGN’s power is set
to $\sigma^{2}=-174+10\cdot\text{log}_{10}(BW)$ $\mathrm{dB}$m. At the
reference distance, the power attenuation is set to -30 $\mathrm{dB}$, and 1
meter is chosen as the reference distance value.Be aware that the Nakagami
fading parameter identifies the LOS and NLOS connections, where $m_{g1}=1$ and
$m_{g1}>1$ are for NLOS and for LOS links, respectively. The target rates are
$R1$ = 1.5 and $R2$ = 1 $\mathrm{bps}\text{\,}{\mathrm{Hz}}^{-1}$. The paired
NOMA users’ power allocation factors are set to $\alpha^{2}_{2}=0.25\
\text{and}\ \alpha^{2}_{1}=0.75$. The direct BS-user connections and the
reflected BS-RIS-route user’s loss exponents are both set to $\eta_{l}=2.2\
\text{and}\ \eta_{n}=3.5$, respectively. The signal alignment method was used
to determine the phase shift matrix.[3].

###  4.2 Results

Outage probability (OP) is defined as the proboability that the SNR at the
receiver falls below a certain threshold [12]. In this section, OP of a
conventional NOMA and a RIS-assisted NOMA network are compared for performance
analysis. This comparison is followed by the showing that the well-known
channels are shown as special cases of $\alpha$-$\mu$ & $\kappa$-$\mu$ models.
To prove this point further, the OP of a Nakagami-m fading channel is compared
with $\alpha$-$\mu$ & $\kappa$-$\mu$ versions of the Nakagami. Figure 4 shows
that the RIS-assisted NOMA outperforms conventional NOMA. This result is
rational as U1 receives a stronger signal and better directed signal in the
presence of the RIS. It should be noted that, the RIS should be tuned
appropriately.

Figure 4: Outage Probability in a Nakagami-m Fading

The generalized fading channels, as mentioned earlier, can be use to generate
many well-known fading channels. In the case of $\alpha$-$\mu$ , we can
generate Nakagami-m, Rayleigh, and Weibull to name a few. In Figure 6 & Table
2 the specific $\alpha$ and $\mu$ values for these distributions is shown.

Distribution | Alpha($\alpha$) | Mu($\mu$)
---|---|---
Exponential | 1 | 1
Weibull | 3 | 1
Nakagami-m | 2 | 2
Rayleigh | 2 | 1
One-sided Gaussian | 2 | 0.5
Table 1: $\alpha$-$\mu$ values for well-known distributions

Distribution | Kappa($\kappa$) | Mu($\mu$)
---|---|---
Rice | 1 | 1
Nakagami-m | 0 | 2
Rayleigh | 0 | 1
One-sided Gaussian | 0 | 0.5
Table 2: $\kappa$-$\mu$ values for well-known distributions

Similar to $\alpha$-$\mu$ , $\kappa$-$\mu$ distribution can model Nakagami-m,
Rayleigh & Rice distributions. The values for $\kappa$ and $\mu$ to replicate
these fading channels is given in Figure 6 & Table 2.

Figure 5: Outage Probability of
Generalized $\kappa$-$\mu$ channel

Figure 6: Outage Probability of
Generalized $\alpha$-$\mu$ channel

Finally, the OP for the physical model described in section 4.1 is simulated
using Nakagami-m channel and also $\alpha$-$\mu$ & $\kappa$-$\mu$ generated
Nakagami channel. Figure 7 clearly shows that when $alpha=2\ \text{and}\
\kappa=0$ for $\kappa$-$\mu$ and $\alpha$-$\mu$ channels, respectively, they
replicate the OP of a Nakagami-m distribution almost perfectly

Figure 7: Nakagami-m Channel using Generalized $\alpha$-$\mu$ & $\kappa$-$\mu$
Channels

##  5 Conclusion

The RIS-Aided NOMA system is modelled and its performance (in-terms of outage
probabilty) was compared and it was shown that RIS-aided NOMA performance
better for U1. The utility of $\alpha$-$\mu$ & $\kappa$-$\mu$ channels in
generating well-known fading channels was also shown. It was also demonstrated
that these channels can replicate a Nakagami-m channel almost perfectly.

## References

  * [1] College, M.T.E.:  5G wireless network.  https://mtec.ac.in/story/5g-wireless-network (2020) Accessed: 2023-4-7.
  * [2] Ghous, M., Hassan, A.K., Abbas, Z.H., Abbas, G., Hussien, A., Baker, T.:  Cooperative power-domain NOMA systems: An overview.  Sensors (Basel) 22(24) (December 2022) 9652
  * [3] Hou, T., Liu, Y., Song, Z., Sun, X., Chen, Y., Hanzo, L.:  Reconfigurable intelligent surface aided NOMA networks.  IEEE J. Sel. Areas Commun. 38(11) (November 2020) 2575–2588
  * [4] Huang, J., Wang, C.X., Sun, Y., Feng, R., Huang, J., Guo, B., Zhong, Z., Cui, T.J.:  Reconfigurable intelligent surfaces: Channel characterization and modeling.  Proc. IEEE Inst. Electr. Electron. Eng. 110(9) (September 2022) 1290–1311
  * [5] Kizilirmak, R.C.:  Non-Orthogonal multiple access (NOMA) for 5G networks.  In: Towards 5G Wireless Networks - A Physical Layer Perspective.  InTech (December 2016)
  * [6] Kumbhani, B., Kshetrimayum, R.S.:  MIMO wireless communications over generalized fading channels.  CRC Press, London, England (July 2017)
  * [7] Moozakis, C., Kerner, S.M.:  What is wireless (communications)? everything you need to know.  https://www.techtarget.com/searchmobilecomputing/definition/wireless (January 2023) Accessed: 2023-4-7.
  * [8] Sklar, B.:  Rayleigh fading channels in mobile digital communication systems .i. characterization.  IEEE Commun. Mag. 35(7) (July 1997) 90–100
  * [9] Tepedelenlioglu, C., Abdi, A., Giannakis, G.B.:  The ricean K factor: Estimation and performance analysis.  IEEE Trans. Wirel. Commun. 24(5) (May 2003) 799–810
  * [10] Xie, Z., Yi, W., Wu, X., Liu, Y., Nallanathan, A.:  STAR-RIS aided NOMA in multicell networks: A general analytical framework with gamma distributed channel modeling.  IEEE Trans. Commun. 70(8) (August 2022) 5629–5644
  * [11] Yacoub, M.D.:  The $\alpha$-$\mu$ distribution: A physical fading model for the stacy distribution.  IEEE Trans. Veh. Technol. 56(1) (January 2007) 27–34
  * [12] Zhang, D., Mumtaz, S., Huq, K.:  Chapter 2 - siso to mmwave massive mimo.  In Mumtaz, S., Rodriguez, J., Dai, L., eds.: mmWave Massive MIMO.  Academic Press (2017) 19–38