# Quantum Yang-Mills theory in
de Sitter ambient space formalism

M.V. Takook, J.P. Gazeau  _APC, UMR 7164_
_Université de Paris_
_75205 Paris, France_<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We present the quantum Yang-Mills theory in the four-dimensional de Sitter
ambient space formalism. In accordance with the SU$(3)$ gauge symmetry the
interaction Lagrangian is formulated in terms of interacting color charged
fields in curved space-time. The gauge-invariant field equations are obtained
in an independent coordinate description, and their corresponding color
conserved currents are computed. Faddeev-Popov ghost fields are shown to be
equivalent to their Minkowski counterparts. We obtain that the free ghost
fields are massless minimally coupled scalar fields. The problems of the
vacuum state, namely the breaking of de Sitter invariance, and the appearance
of infrared divergence in its quantization procedure, are discussed. The
existence of an axiomatic quantum Yang-Mills theory within the framework of
the Krein space quantization is examined. The infrared divergence
regularization of the interaction between the gauge vector fields and the
ghost fields is studied in the one-loop approximation. Two different
regularization methods are discussed: cut-off regularization and Krein space
regularization. A mass term for the gauge vector fields is obtained, which may
explain the mass gap and the color confinement problems at the quantum level
in de Sitter background. The large curvature limit at the early universe or
inflationary epoch is considered.

Proposed PACS numbers: 04.62.+v, 98.80.Cq, 12.10.Dm

###### Contents

  1. 1 Introduction
  2. 2 Notations
  3. 3 Lagrangian density
  4. 4 Quantization problems
  5. 5 Quantum Yang-Mills theory
    1. 5.1 Standard approach
    2. 5.2 Krein approach
  6. 6 Mass gap and confinement
  7. 7 Conclusions
  8. A
  9. B

##  1\. Introduction

Quantum field theory in de Sitter (dS) space-time has been evolving as an
exceedingly vital subject in the past decade. Historically the dS space-time,
as a curved space-time manifold with maximum symmetry, was introduced as a
solution of Einstein’s equations with a positive cosmological constant,
“maximum” meaning that it has the same degree of symmetry as the flat
Minkowski space solution [1]. The interest in the dS space has been
tremendously increasing when it turned out that it could play a central role
in the inflationary cosmological paradigm [2]. Over the past three decades, a
non-zero cosmological constant has been proposed to explain the luminosity of
the farthest supernovae [3, 4]. Therefore the dS space-time could play a
further important role in the modeling of the large-scale universe.

All of these developments make it more imperative than ever to seek a
formulation of dS quantum field theory (QFT) with the same level of
completeness and rigor as its Minkowskian counterpart. A unique Poincaré
invariant vacuum state can be fixed in Minkowski space by imposing the
positive energy condition. In a curved space-time like de Sitter, however, a
global time-like Killing vector field does not exist, and therefore the
positive energy condition cannot be imposed without any ambiguity. Symmetry
alone is not sufficient in the determination of a unique vacuum state.
Nevertheless, in dS space, symmetry considerations allow one to identify the
vacuum with a two-parameter ambiguity, say $|\alpha,\beta\rangle$,
corresponding to a family of distinct dS invariant vacuum states (see [5] and
references therein). Only the one-parameter family $|\alpha,0\rangle$, is
invariant under the disconnected de Sitter group, i.e. O$(1,4)$ [5]. By
imposing the condition that in the null curvature limit, the Wightman’s two-
point function becomes the same as the Minkowskian Wightman’s two-point
function, the other parameter $(\alpha)$, can be fixed as well. This vacuum
state, $|0,0\rangle\equiv|\Omega\rangle$, is called Bunch-Davies or Euclidean
vacuum state. However, this vacuum state is not suitable when one deals with
the massless minimally coupled (mmc) scalar fields, which causes problems for
quantization of the Yang-Mills theory and the linear gravity.

In [6, 7] Bros et al. have presented a QFT for a massive scalar field in dS
space-time that closely mimics the QFT in Minkowski space-time. They have
introduced a new version of the so-called Fourier-Bros transformation on the
dS hyperboloid, which allows us to completely characterize the Hilbert space
of the “one-particle” states in terms of corresponding irreducible unitary
representations of the dS group. In this construction, the analyticity
properties of the two-point functions play a significant role in the
interaction field quantization. Then the Bros construction was generalized to
various quantum free fields with non-zero spin in dS space in a series of
papers (see [8] and references therein).

On the other hand, quantum Yang-Mills theory can be considered one of the most
mathematically challenging topics in QFT [9]. An axiomatic quantum Yang-Mills
theory, with a mass gap and color confinement, does not yet exist111It is one
of the seven Millennium Prize Problems in mathematics that were stated by the
Clay Mathematics Institute in May $24,2000$ [9].. Recently, a dS Yang-Mills
theory was studied in an intrinsic coordinate system [10]. In the present
paper, the Yang-Mills theory is considered within the $4$-dimensional dS
ambient space formalism. We describe the interaction between the color-charged
spinor field (“quark”) and the massless color-charge vector fields (“gluons”)
in a coordinate-independent way. We obtain the gauge-invariant Lagrangian, the
field equations, and the gauge fixing terms in ambient space formalism. We
then show that the free ghost fields are the mmc scalar fields in dS space-
time.

The famous infrared (IR) divergence of the mmc scalar field and the different
ways of dealing with it were addressed by many authors over the last $30$
years. We will briefly recall the crucial results. It was commonly thought
that this IR divergence is just a gauge artifact [11]. The appearance of this
IR divergence in the Faddeev-Popov ghost fields of dS perturbative gravity in
the covariant gauge has also been noticed (for instance, see [12]). The effect
of an IR cut-off in the momentum space, which regularizes this divergence,
breaks the dS invariance. About this problem, we will here insist on the
necessity of using the Krein space quantization to preserve the dS invariance,
as was shown in the covariant quantization of the mmc scalar fields in dS
space [13, 14, 15].

The Krein approach was implemented for the massless fields with spin $1,3/2$,
and $2$ with the construction of relevant indecomposable representations [16,
17, 18]. The content of the present paper also rests upon the results of [19,
20, 21, 22, 23] which were based on the axiomatic quantum field theory. The
Krein space quantization may be used to construct an axiomatic dS quantum
Yang-Mills theory on the Krein-Fock vacuum state.

The interaction of the ghost field and the gauge vector field at the QFT level
is discussed in the one-loop approximation. We show that, due to the
regularization of the IR divergence, a mass term appears for gluons fields,
which may explain the mass gap and color confinement problems. The
regularization of the IR divergence is considered along two different paths:
A) a cut-off regularization method [5], and B) the Krein space regularization
method [24, 25, 26, 27, 28]. The first one breaks the dS invariance while the
second one preserves it.

At this point, it is necessary to recall that the concept of (proper) mass is
defined within the context of the Poincaré group symmetry, and has to be
carefully revisited within the context of the dS group symmetry. In the
present paper, a field is called “massive” when it propagates inside the
light-cone and corresponds to a massive Poincaré field representation in the
null curvature limit. We call a field “massless” if it propagates on the dS
light-cone and corresponds to a massless Poincaré field representation at
$H=0$. The concept of light-cone propagation can be precisely defined in dS
ambient space formalism [7].

The organization of this paper is as follows. In Section 2 we fix the
notations, and we make explicit the field equations for Dirac and massless
vector fields, the Lagrangian density, the conserved current, and finally, the
massless vector two-point function. In Section 3 we obtain the non-abelian
gauge-invariant Lagrangian. Then the invariant field equations and their
corresponding color conserved currents are computed. By obtaining the Faddeev-
Popov ghost fields, it is shown that these fields are mmc scalar fields. In
Section 4 the problems of the vacuum state, the breaking of the Sitter
invariance, and the appearance of infrared divergence in the quantization
procedure are reviewed. Section 5 is devoted to the quantum Yang-Mills theory,
and two different approaches are presented. The mass gap and the color
confinement problems of the Yang-Mills theory are discussed in Section 6
through the consideration of the one-loop effective action corresponding to
the gluon-ghost interaction. Finally, concluding remarks and a brief outlook
are given in Section 7.

##  2\. Notations

The dS space-time can be identified with the $4$-dimensional hyperboloid
embedded in the $5$-dimensional Minkowski space-time as:

(2.1) $X_{H}=\left\\{x^{\alpha}\equiv x\in{\rm IR}^{5}|\;\;x\cdot
x=\eta_{\alpha\beta}x^{\alpha}x^{\beta}=-H^{-2}\right\\}\,,\;\;\alpha,\beta=0,1,2,3,4\,,$

with $\eta_{\alpha\beta}=$diag$(1,-1,-1,-1,-1)$ and $H$ is like Hubble’s
constant parameter. The dS metric element is:

(2.2)
$\mathrm{d}s^{2}=\left.\eta_{\alpha\beta}\mathrm{d}x^{\alpha}\mathrm{d}x^{\beta}\right|_{x\cdot
x=-H^{-2}}=g_{\mu\nu}^{dS}\mathrm{d}X^{\mu}\mathrm{d}X^{\nu}\,,\;\;\mu=0,1,2,3\,,$

where the $X^{\mu}$’s form a set of $4$-space-time intrinsic coordinates on
the dS hyperboloid, and the $x^{\alpha}$’s are the ambient space coordinates.
Let us introduce the global coordinates system in terms of the intrinsic
coordinates system $(X^{\mu})=(t,\chi,\theta,\varphi)$, $t\in{\rm IR}$,
$0\leq\chi,\theta\leq\pi$, $0\leq\varphi<2\pi$ as:

(2.3) $\left\\{\begin{array}[]{clcr}x^{0}&=H^{-1}\sinh Ht\\\
x^{1}&=H^{-1}\cosh Ht\sin\chi\cos\theta\\\ x^{2}&=H^{-1}\cosh
Ht\sin\chi\sin\theta\cos\phi\\\ x^{3}&=H^{-1}\cosh
Ht\sin\chi\sin\theta\sin\phi\\\ x^{4}&=H^{-1}\cosh
Ht\cos\chi\,,\end{array}\right.$

which are appropriate to take the zero curvature limit. The analyticity
properties of QFT in dS space are described with the introduction of the
complexified de Sitter spacetime $X_{H}^{(c)}$:

$X_{H}^{(c)}=\left\\{z=x+\mathrm{i}y\in\mathbb{C}^{5};\;\;\eta_{\alpha\beta}z^{\alpha}z^{\beta}=(z^{0})^{2}-\vec{z}.\vec{z}-(z^{4})^{2}=-H^{-2}\right\\}$
(2.4) $=\left\\{(x,y)\in{\rm IR}^{5}\times{\rm
IR}^{5};\;\;x^{2}-y^{2}=-H^{-2},\;x.y=0\right\\}\,.$

The forward and backward tubes in $\mathbb{C}^{5}$ are defined as
$T^{\pm}={\rm IR}^{5}+\mathrm{i}V^{\pm}$. The domain $V^{+}$(resp. $V^{-})$
stems from the causal structure on $X_{H}$:

(2.5) $V^{\pm}=\left\\{x\in{\rm
IR}^{5};\;\;x^{0}\gtrless\sqrt{\parallel\vec{x}\parallel^{2}+(x^{4})^{2}}\right\\}\,.$

Their respective intersections with $X_{H}^{(c)}$ are denoted by:

(2.6) $\mathcal{T}^{\pm}=T^{\pm}\cap X_{H}^{(c)}\,,$

and will be called forward and backward tubes of the complex dS space,
respectively. Finally the “tuboid” above $X_{H}^{(c)}\times X_{H}^{(c)}$ is
defined by [7]:

(2.7)
$\mathcal{T}_{12}=\\{(z,z^{\prime});\;\;z\in\mathcal{T}^{+},z^{\prime}\in\mathcal{T}^{-}\\}\,.$

In the ambient space formalism, the tangential derivative on the dS
hyperboloid reads as
$\partial_{\beta}^{\top}=\theta_{\alpha\beta}\partial^{\alpha}=\partial_{\beta}+H^{2}x_{\beta}\,x\cdot\partial\,,$
where $\theta_{\alpha\beta}=\eta_{\alpha\beta}+H^{2}x_{\alpha}x_{\beta}$ is
the transverse projector. The transverse-covariant derivative acting on a
tensor field of rank-$l$ is defined by:

(2.8)
$\nabla^{\top}_{\beta}T_{\alpha_{1}\cdots\alpha_{l}}\equiv\partial^{\top}_{\beta}T_{\alpha_{1}\cdots\alpha_{l}}-H^{2}\sum_{n=1}^{l}x_{\alpha_{n}}T_{\alpha_{1}\cdot\alpha_{n-1}\beta\alpha_{n+1}\cdots\alpha_{l}}\,.$

The second-order Casimir operator $Q_{0}$ of the dS group SO${}_{0}(1,4)$ in
its representation acting on scalar fields is written as:

(2.9)
$Q_{0}=-H^{-2}\partial^{\top}\cdot\partial^{\top}=-H^{-2}g^{dS}_{\mu\nu}\nabla^{\mu}\nabla^{\nu}=-H^{-2}\Box_{H}\,,$

where $\Box_{H}$ is the Laplace-Beltrami operator on the dS space-time.

By expressing the Casimir eigenvalue equation in terms of the infinitesimal
generators of the group representation and proceeding with a dS-Dirac field
factorisation of the quadratic terms, one obtains the first order equation
[19, 29]:

(2.10)
$\left(-\mathrm{i}H\not{x}\gamma\cdot\partial^{\top}+\mathrm{i}2H+H\nu\right)\psi(x)=0,\;\;\nu\in{\rm
IR},\;\;\not{x}=x\cdot\gamma\,,$

where $\nu$ is the parameter of the principal series representation, as was
introduced in [19]. This equation can be derived from the least action
principle applied to the following spinor field action:

(2.11)
$S[\psi,\bar{\psi}]=\int\mathrm{d}\sigma(x)\mathcal{L}(\psi,\bar{\psi})=\int\mathrm{d}\sigma(x)\bar{\psi}\gamma^{4}\left(-\mathrm{i}H\not{x}\gamma\cdot{\partial}^{\top}+\mathrm{i}2H+H\nu\right)\psi(x)\,,$

where $\bar{\psi}=\psi^{\dagger}\gamma^{0}\gamma^{4}$ and
$\mathrm{d}\sigma(x)$ is the $4$-dimensional dS-invariant volume element. The
latter is expressed in the ambient space formalism as [7]:

(2.12) $d\sigma(x)=\left.\frac{dx^{0}\wedge dx^{1}\wedge dx^{2}\wedge
dx^{3}\wedge dx^{4}}{d(x\cdot
x+H^{-2})}\right|_{X_{H}}=d\sigma(x^{\prime})\,.$

Here we need five $\gamma^{\alpha}$ matrices instead of the usual four ones in
the Minkowski space-time. They generate the Clifford algebra associated with
the five dimensional Minkowskian metric as:

$\\{\gamma^{\alpha},\gamma^{\beta}\\}=\gamma^{\alpha}\gamma^{\beta}+\gamma^{\beta}\gamma^{\alpha}=2\eta^{\alpha\beta}\;\;\;,\;\;\;\gamma^{\alpha{\dagger}}=\gamma^{0}\gamma^{\alpha}\gamma^{0}\,.$

By using the Noether’s theorem and the invariance of the field equation (2.10)
under the global $U(1)$ transformation
($\psi^{\prime}=e^{-\mathrm{i}\Lambda}\psi,\,\Lambda=$ constant), the
conserved current can be obtained [30]:

$J^{\alpha}(x)=\frac{\delta\mathcal{L}(\psi,\bar{\psi})}{\delta\partial^{\top}_{\alpha}\psi}\frac{\delta\psi}{\delta\Lambda}=-H\bar{\psi}(x)\gamma^{4}\not{x}\gamma^{\alpha}\psi(x)\,.$

It is easily verified that this current satisfies the conditions [19, 30]:

(2.13) $\partial^{\top}\cdot J(x)=0\,,\;\;\;\;x\cdot J(x)=0\,.$

For the massless vector field (spin $=1$), one also starts from the Casimir
operator and its eigenvalue, uses the infinitesimal generators and some
algebraic relations, to finally obtain the following field equations [20, 16]:

(2.14) $\left(Q_{0}-2\right)K_{\alpha}(x)+2x_{\alpha}\partial^{\top}\cdot
K(x)+H^{-2}\partial^{\top}_{\alpha}\partial^{\top}\cdot K=0\,.$

This equation is invariant under the gauge transformation:

(2.15) $K_{\alpha}\longrightarrow
K^{\prime}_{\alpha}=K_{\alpha}+H^{-2}\partial^{\top}_{\alpha}\Lambda(x)\,.$

Here $\Lambda(x)$ is an arbitrary differentiable scalar field. The condition
of transversality $x\cdot K(x)=0$ restricts the five-vector field to live on
the dS hyperboloid and guarantees that it should be viewed as a vector-valued
homogeneous function of the ${\rm IR}^{5}$-variables $x^{\alpha}$ with some
arbitrarily chosen degree $\lambda$ [29]:

(2.16) $x^{\alpha}\frac{\partial}{\partial
x^{\alpha}}K_{\beta}(x)=x\cdot\partial\,K_{\beta}(x)=\lambda K_{\beta}(x)\,.$

The massless vector field equation can be derived from the following action
integral:

(2.17)
$S[K]=-\frac{1}{4}\int\mathrm{d}\sigma(x)\left(\nabla^{\top}_{\alpha}K_{\beta}-\nabla^{\top}_{\beta}K_{\alpha}\right)\left(\nabla^{\top\alpha}K^{\beta}-\nabla^{\top\beta}K^{\alpha}\right).$

Like for its flat space counterpart, the gauge fixing is accomplished by
adding to (2.14) a gauge fixing term with parameter $\zeta$ [16]:

(2.18) $\left(Q_{0}-2\right)K_{\alpha}(x)+2x_{\alpha}\partial^{\top}\cdot
K(x)+\zeta H^{-2}\partial^{\top}_{\alpha}\partial^{\top}\cdot K=0\,.$

With the simple choice $\zeta=\frac{3}{2}$, the vector two-point function
reads [16]:

(2.19) $\mathcal{W}_{\alpha\alpha^{\prime}}(x,x^{\prime})=\langle\Omega\mid
K_{\alpha}(x)K_{\alpha^{\prime}}(x^{\prime})\mid\Omega\rangle=D_{\alpha\alpha^{\prime}}(x,x^{\prime},\partial_{x})\mathcal{W}_{mcc}(x,x^{\prime})\,,$

where $D_{\alpha\alpha^{\prime}}(x,x^{\prime},\partial_{x})$ is the following
bi-tensor differential operator,

(2.20)
$D_{\alpha\alpha^{\prime}}(x,x^{\prime},\partial_{x})=\theta_{\alpha}\cdot\theta^{\prime}_{\alpha^{\prime}}+H^{-2}\partial^{\top}_{\alpha}\left[\partial^{\top}\cdot\theta^{\prime}_{\alpha^{\prime}}+H^{2}x\cdot\theta^{\prime}_{\alpha^{\prime}}\right].$

The symbol $\mathcal{W}_{mcc}(x,x^{\prime})$ stands for the Wightman two-point
function of the massless conformally coupled scalar field [7, 31, 32]:

(2.21)
$\mathcal{W}_{\mathrm{mcc}}(x,x^{\prime})=\frac{-H^{2}}{8\pi^{2}}\left[P\frac{1}{1-\mathcal{Z}(x,x^{\prime})}-\mathrm{i}\pi\epsilon(x^{0},x^{\prime
0})\delta(1-\mathcal{Z}(x,x^{\prime}))\right]\,.$

Here the symbol $P$ means the principal part and

$\epsilon(x^{0}-x^{\prime 0})=\left\\{\begin{array}[]{clcr}1&x^{0}>x^{\prime
0}\\\ 0&x^{0}=x^{\prime 0}\\\ -1&x^{0}<x^{\prime 0}\\\ \end{array}\right..$

The function $\mathcal{Z}$ is given by:

(2.22) $\mathcal{Z}(x,x^{\prime})=-H^{2}x\cdot
x^{\prime}=1+\frac{H^{2}}{2}(x-x^{\prime})^{2}\equiv\cosh
H\sigma(x,x^{\prime})\,,$

and $\sigma(x-x^{\prime})$ is the geodesic distance between two points $x$ and
$x^{\prime}$ on the dS hyperboloid.

##  3\. Lagrangian density

In this section, a direct generalization of the abelian gauge theory expressed
in terms of the dS ambient space [30] to the non-abelian gauge theory is
presented. In this formalism, the gauge theory construction is utterly similar
to its Minkowski counterpart [33]. The dS-Dirac spinor field equation (2.10)
is not invariant under the following non-abelian gauge transformation:

(3.1) $\psi^{\prime}(x)=e^{-\mathrm{i}\Lambda^{a}(x)t^{a}}\psi(x)\equiv
U\left(\Lambda(x)\right)\psi(x)\,.$

The $\Lambda^{a}$’s are local group parameters and the $t^{a}$’s are
generators of the $\mathrm{SU}(3)$ group. They satisfy the following
commutation relation:

(3.2)
$[t_{a},t_{b}]=\mathrm{i}C_{ab}^{\;\;\;c}t_{c},\;\;a,b,c=1,2,\cdots,8\,,$

where $C_{ab}^{\;\;\;c}$’s are the real structure constants of
$\mathfrak{su}(3)$.

Like the flat Minkowskian case, local gauge symmetry with its space-time-
dependent transformation can generate the gauge interaction in the dS
universe. For obtaining a gauge-invariant Lagrangian for spinor field, it is
necessary to replace the transverse-covariant derivative
$\nabla^{\top}_{\beta}$ with the gauge-transverse-covariant derivative
$D_{\beta}^{K}$ which is defined by:

(3.3)
$D_{\beta}^{K}\equiv\nabla^{\top}_{\beta}-\mathrm{i}gK_{\beta}^{a}t_{a}\,\,,\;\;x\cdot
K^{a}=0\,,$

where $g$ is a real coupling constant. The gauge fields $K_{\beta}^{a}(x)$ are
associated with the group generators $t^{a}$. In order to have a simple gauge
transformation for the gauge-covariant derivative of the spinor field as:

(3.4)
$D_{\beta}^{K}\psi(x)\longrightarrow\left[D_{\beta}^{K}\psi(x)\right]^{\prime}=e^{-\mathrm{i}\Lambda^{a}(x)t^{a}}D_{\beta}^{K}\psi(x)\equiv
D_{\beta}^{K^{\prime}}\psi^{\prime}(x)\,,\quad
D_{\beta}^{K^{\prime}}=UD_{\beta}^{K}U^{-1}\,,$

the connection fields or vector gauge potentials $K_{\beta}^{a}$ must be
transformed in the following form:

(3.5)
${K^{\prime}}_{\beta}^{a}t_{a}=U(\Lambda)K_{\beta}^{b}t_{b}U^{-1}(\Lambda)+\frac{\mathrm{i}}{g}U\left(\partial^{\top}_{\beta}U^{-1}\right)\,.$

Then the dS-Dirac gauge invariant equation reads as:

(3.6)
$\left(-\mathrm{i}H\not{x}\gamma.D^{K}+2H\mathrm{i}+H\nu\right)\psi(x)=\left(-\mathrm{i}H\not{x}\gamma\cdot{\partial}^{\top}-Hg\not{x}\gamma\cdot
K^{a}t_{a}+2H\mathrm{i}+H\nu\right)\psi(x)=0\,.$

By using the curvature $\mathcal{C}$ of the Lie algebra of the gauge group
$\mathrm{SU}(3)$, one obtains the field equations for the vector fields
$K_{\beta}^{a}$. The gauge group curvature is:

$\mathcal{C}\left(D^{K}_{\alpha},D^{K}_{\beta}\right)=\frac{\mathrm{i}}{g}\left[D^{K}_{\alpha},D^{K}_{\beta}\right]=F_{\alpha\beta}^{\;\;\;\;a}t_{a}\equiv\,{\bf
F}_{\alpha\beta}\,,\;\;\;{\bf F}_{\alpha\beta}^{\prime}=U{\bf
F}_{\alpha\beta}U^{-1}\,,$

where

(3.7)
$F_{\alpha\beta}^{\;\;\;\;a}=\nabla^{\top}_{\alpha}K_{\beta}^{\;\;a}-\nabla^{\top}_{\beta}K_{\alpha}^{\;\;a}+gC_{bc}^{\;\;\;\;a}K_{\alpha}^{\;\;b}K_{\beta}^{\;\;c}\,,$

together with the transversality properties
$x^{\alpha}F_{\alpha\beta}^{\;\;\;\;a}=0=x^{\beta}F_{\alpha\beta}^{\;\;\;\;a}$.
The $\mathrm{SU}(3)$ gauge invariant action or Lagrangian in the dS background
for the gauge field $K_{\alpha}^{\;\;a}$ is:

$S[K^{a}]=-\frac{1}{2}\int\mathrm{d}\sigma(x)\;\mathrm{Tr}\left({\bf
F}_{\alpha\beta}{\bf
F}^{\alpha\beta}\right)=-\frac{1}{2}\int\mathrm{d}\sigma(x)F_{\alpha\beta}^{\;\;\;\;a}F^{\alpha\beta
b}\,\mathrm{Tr}\left(t_{a}t_{b}\right)\,,$

where summing over the repeated indices is used. The normalization of the
structure constants is usually fixed by requiring that, in the fundamental
representation, the corresponding matrices of the generators $t_{a}$ are
normalised such as [34]
$\mathrm{Tr}\left(t_{a}t_{b}\right)=\frac{1}{2}\delta_{ab}$. Then the action
becomes

$S[K^{a}]=-\frac{1}{4}\int\mathrm{d}\sigma(x)\left[\left(\nabla^{\top}_{\alpha}K_{\beta}^{\;\;a}-\nabla^{\top}_{\beta}K_{\alpha}^{\;\;a}\right)\left(\nabla^{\top\alpha}K^{\beta
b}-\nabla^{\top\beta}K^{\alpha b}\right)\delta_{ab}+O({K}^{3})\right]\,,$

where non-linear terms in the field equation appear in $O(K^{3})$ and describe
the interaction between the gauge potentials $K_{\beta}^{a}$. One can see that
the gauge potential or connection $K_{\beta}^{a}$ in the linear approximation
is a massless vector field with action integral (2.17).

The gauge invariant Lagrangian density for the vector and spinor fields in the
simplest form can be written as:

(3.8)
$\mathcal{L}(K,\psi)=-\frac{1}{4}F_{\alpha\beta}^{\;\;\;\;a}F^{\alpha\beta
a}+H\bar{\psi}\gamma^{4}\left(-\mathrm{i}\not{x}\gamma\cdot
D^{K}+2\mathrm{i}+\nu\right)\psi\,.$

From the Euler-Lagrange equations one obtains [35]:

(3.9) $\nabla^{\top}_{\alpha}F^{\alpha\beta}_{\;\;\;\;a}=gF^{\beta\gamma
b}C_{ba}^{\;\;\;c}K_{c\gamma}+gH\bar{\psi}(x)\gamma^{4}\not{x}\gamma^{\beta}t_{a}\psi(x)\,,$
(3.10) $\left(-\mathrm{i}H\not{x}\gamma\cdot{\partial}^{\top}+2H\mathrm{i}+\nu
H\right)\psi(x)=gH\not{x}\gamma\cdot K^{a}(x)t_{a}\psi(x)\,.$

In this case the conserved current reads as:

(3.11)
$\mathcal{J}_{\alpha}^{a}=J_{\alpha}^{a(K)}+J_{\alpha}^{a(\psi)}=-gF_{\alpha\beta
b}C^{ba}_{\;\;\;c}K^{c\beta}-gH\bar{\psi}_{i}(x)\gamma^{4}\not{x}\gamma_{\alpha}(t^{a})_{ij}\psi_{j}(x)\,,$

where $i,j=1,2,3$ are the colour indexes of spinor fields. By using the
equation (2.13), one verifies that this current satisfies the following
conditions:

(3.12)
$x\cdot\mathcal{J}^{a}=0\,,\;\nabla^{\top}\cdot\mathcal{J}^{a}=0=\partial^{\top}\cdot\mathcal{J}^{a}\,.$

By comparing this current with the abelian case (2.13), we see that the term
$J_{\alpha}^{a(K)}$ is due to the color vector fields in dS space-time.

The two conditions of transversality, $x\cdot K=0$, and divergencelessness,
$\nabla^{\top}\cdot K=0$, are combined in one mathematical relation,
$\partial^{\top}\cdot K=0$. It is called the _generalized Lorenz gauge_ ,
which will be considered in this paper. For the infinitesimal gauge
transformation, the equation (3.5) becomes:

(3.13)
$(K^{\Lambda})_{\alpha}^{a}=K_{\alpha}^{a}+\frac{1}{g}\partial_{\alpha}^{\top}\Lambda^{a}+C_{cb}^{\;\;a}K^{b}_{\alpha}\Lambda^{c}\,,$

and the generalized Lorenz gauge is rewritten as:

(3.14) $\chi^{a}=0=\partial^{\top}\cdot(K^{\Lambda})^{a}=\partial^{\top}\cdot
K^{a}-\frac{H^{2}}{g}Q_{0}\Lambda^{a}+C_{cb}^{\;\;a}\partial^{\top}\cdot
K^{b}\Lambda^{c}\;.$

This equation fixes the gauge parameter through the equation:

(3.15) $\left(\delta^{ac}H^{2}Q_{0}-gC_{cb}^{\;\;a}\partial^{\top}\cdot
K^{b}\right)\Lambda^{c}=0\,.$

Like its Minkowskian counterpart, the Faddeev-Popov ghost procedure can be
implemented by using the expression:

$\det
M_{ab}=\det\left|\frac{\delta\chi^{a}(y)}{\delta\Lambda^{b}(x)}\right|=\det\left(\delta^{ab}H^{2}Q_{0}-gC_{cb}^{\;\;a}\partial^{\top}\cdot
K^{c}\right)\,.$

Let us introduce the ghost field $\Phi$ to absorb this factor in the
Lagrangian density. The Lagrangian density of the ghost field for the
generalized Lorenz gauge is obtained in the same way as its Minkowskian
counterpart. By using the equation (3.14) and the path integral formulation,
the Lagrangian density of the ghost field in the generalized Lorenz gauge is
found to be:

(3.16)
$\mathcal{L}(\Phi,\Phi^{\dagger})={\Phi^{a}}^{\dagger}\left[\delta^{ac}H^{2}Q_{0}-gC_{cb}^{\;\;a}\partial^{\top}\cdot
K^{b}\right]\Phi^{c}(x)\,.$

It is important to note that the field equation for the free ghost fields is
the same as for the mmc scalar field. Within the dS context, the scalar free
ghost fields transform analogously to the mmc scalar field [14] under the
action of an indecomposable representation of the dS group. Concerning the
gauge group indices $a$ for $\Phi^{a}$, they transform under the adjoint
representation analogously to the gluon fields. The many “particle” states are
constructed from the Fermi-Dirac statistics resembling its Minkowskian
counterpart.

The gauge fixing classical Lagrangian density for the interaction between
colour charged particles reads as:

(3.17)
$\begin{split}\mathcal{L}_{c}(\psi,\bar{\psi},K^{a},\Phi,\Phi^{\dagger})&=H\bar{\psi}\gamma^{4}\left(-\mathrm{i}\not{x}\gamma\cdot
D^{K}+2\mathrm{i}+\nu\right)\psi\\\
&-\frac{1}{4}F_{\alpha\beta}^{\;\;\;\;a}F^{\alpha\beta
a}-\frac{\zeta}{2}\partial\cdot K^{\;a}\partial\cdot
K^{\;a}+{\Phi^{a}}^{\dagger}\left[\delta^{ab}H^{2}Q_{0}-gC_{cb}^{\;\;a}\partial^{\top}\cdot
K^{c}\right]\Phi^{b}\,,\end{split}$

where $\zeta$ is a gauge fixing parameter. The spinor fields correspond to the
quarks with three colors, and the vector fields correspond to the gluons with
eight combinations of the three colors. Eight colors ghost fields are also
presented. In the quantization of the ghost field appears an IR divergence in
the propagator, and its regularization, thanks to a cut-off (see [36]), breaks
the dS invariance. The mmc scalar field is invariant under the global
transformation ($\phi$ \+ const.). On the quantum level this feature is
equivalent to a gauge theory [14] and also to the two-dimensional QFT [37].

##  4\. Quantization problems

For the quantization of the Yang-mills theory, i.e. quantization of the
Lagrangian density (3.17), first, the quantization of the free fields are
considered, and then, by using the perturbation theory, the quantization of
the interaction part is presented. For the free field quantization, the
propagator has to be calculated. For the interaction field, the renormalized
effective action is a significant quantity that must be obtained. There are
three types of free fields in the Yang-Mills theory: spinor fields, massless
vector fields, and mmc scalar fields. The quantization of the spinor field can
be performed without any problem. The quantization for the vector field is
discussed in the next section. In this section, we describe the quantization
of the mmc scalar fields.

In the quantization procedure for the mmc scalar field, three problems emerge
that are related to each other: (1) appearance of IR divergence, (2) breakdown
of dS background symmetry, and (3) absence of de Sitter-invariant Fock vacuum
state [5, 36]. These difficulties arise from the zero-mode problem associated
with a global transformation of the field equation ($\phi^{\prime}=\phi$ \+
const.). Similar problems exist in quantum gauge theory. An indefinite metric
quantization can be used to solve them [38]. In the case of the mmc scalar
field the so-called Krein space quantization, which was developed in [13, 14],
represents a consistent alternative to overcome these obstacles: removing the
infrared divergence and preserving the dS invariance. Concerning the third
problem, the alternative is to introduce the Krein-Fock vacuum state as a
workable remedy.

As proved by Allen [5], the covariant canonical quantization procedure with
positive norm states fails in the mmc case. The Hilbert space generated by the
positive modes, including the zero mode ($\phi_{0}$), is not de Sitter
invariant [5, 14]:

$\mathcal{H}=\left\\{\sum_{k\geq 0}\alpha_{k}\phi_{k};\;\sum_{k\geq
0}|\alpha_{k}|^{2}<\infty\right\\}\,.$

It means that the positive modes solutions are not closed under the action of
the de Sitter group generators. Nevertheless, one can obtain a fully covariant
quantum field operator by adopting a new construction, which consists in
adding all the conjugate modes to the original ones. Consequently, one now
deals with an orthogonal sum of a positive and a negative inner product space,
which is closed under an indecomposable representation of the dS group [14].

Since the spinor field and vector field are studied in ambient space
formalism, and for the unification of the formalism, the mmc scalar field is
presented in ambient space formalism, which permits us to better understand
the breaking of dS invariance. In the ambient space formalism one easily
constructs the mmc scalar field $\Phi_{\mathrm{mmc}}$ in terms of the massless
conformally coupled scalar field (mcc) $\Phi_{\mathrm{mcc}}$, and an arbitrary
constant five-vector $A=\left(A^{\alpha}\right)$ [8, 39]:

(4.1) $\Phi_{\mathrm{mmc}}(x)\equiv\left[A\cdot\partial^{\top}+2H^{2}A\cdot
x\right]\Phi_{\mathrm{mcc}}(x)\,,$

It is reminiscent of the arbitrariness in the choices of the vacuum state. The
mcc scalar field two-point function is constructed on the Bunch-Davies vacuum
state (2.21).

Using the equation (4.1), the analytic (in its domain) two-point function for
the massless minimally coupled scalar field can be written as:

(4.2)
$W_{\mathrm{mmc}}(z,z^{\prime};A)=\left[A\cdot\partial^{\top}+2H^{2}A\cdot
z\right]\left[A\cdot\partial^{\prime\top}+2H^{2}A\cdot
z^{\prime}\right]W_{\mathrm{mcc}}(z,z^{\prime})\,,$

where $z=x+\mathrm{i}y$, $z^{\prime}=x^{\prime}+\mathrm{i}y^{\prime}$ and
$W_{mcc}$ is the analytic two-point function of the mcc scalar field [7]:

$W_{mcc}(z,z^{\prime})=\frac{H^{2}}{8\pi^{2}}\frac{-1}{1-\mathcal{Z}(z,z^{\prime})}\,.$

The Wightman two-point function $\mathcal{W}_{\mathrm{mcc}}(x,x^{\prime})$,
Equation (2.21), is the boundary value of the analytic function
$W_{\mathrm{mcc}}(z,z^{\prime})$ (in the distributional sense, according to
the theorem $A.2$ in [7]).

After some calculations, one obtains from (4.2) [39]:

(4.3)
$\begin{split}&W_{\mathrm{mmc}}(z,z^{\prime};A)=\frac{-H^{2}}{8\pi^{2}}\times\\\
&\frac{(\mathcal{Z}-3)\left[(H^{2}A\cdot z)^{2}+(H^{2}A\cdot
z^{\prime})^{2}+H^{4}A\cdot z\,A\cdot
z^{\prime}\,\mathcal{Z}\right]+6H^{4}A\cdot zA\cdot
z^{\prime}-(1-\mathcal{Z})H^{2}A\cdot
A}{\left[1-\mathcal{Z}(z,z^{\prime})\right]^{3}}\,.\end{split}$

This function is not dS invariant concerning the variables $z$ and
$z^{\prime}$. Instead, we have to consider the $A$-labeled family of two-point
functions (or “vacuum state”) for which the following dS invariance holds:

(4.4)
$W_{\mathrm{mmc}}(Rz,Rz^{\prime};RA)=W_{\mathrm{mmc}}(z,z^{\prime};A)\quad\mbox{for
all}\ R\in\mathrm{SO}_{0}(1,4)\,.$

This means that it is built from a one-particle state which does not transform
under the unitary irreducible representation of the dS group. $A$ is a
constant five-vector. It is a sort of polarization vector [40] which can be
chosen as one of the $5$ vectors forming an orthonormal basis of ${\rm
IR}^{5}$, the latter carrying the fundamental five-dimensional representation
of the dS group. Hence, the explicit form of the two-point function $W_{mmc}$
depends on the chosen $A$. Its construction involves the tensor product of two
representations of the dS group: $(1)$ the complementary scalar representation
related to the massless conformally coupled scalar field [8], and $(2)$ the
fundamental five-dimensional representation [40].

As a first simple example of a choice of orthonormal basis in ${\rm IR}^{5}$
(for the dS metric) , one considers the set $\\{A^{(l)}\,,\,l=0,1,2,3,4\\}$
obeying [40]:

(4.5)
$\sum_{l=0}^{4}\sum_{l^{\prime}=0}^{4}A^{(l)}_{\alpha}A^{(l^{\prime})}_{\beta}=\eta_{\alpha\beta}\,,\;\;\;A^{(l)}\cdot
A^{(l^{\prime})}=\eta^{ll^{\prime}}\,.$

With this choice and by summing the $5$ corresponding two-point functions one
obtains the constant trivial solution (for $z\neq z^{\prime}$) [39]:

(4.6)
$W_{mmc}(z,z^{\prime})=\left[\partial^{\top}\cdot\partial^{\prime\top}+2H^{2}z\cdot\partial^{\prime\top}+2H^{2}z^{\prime}\cdot\partial^{\top}+4H^{4}z\cdot
z^{\prime}\right]W_{mcc}(z,z^{\prime})=\frac{-H^{2}}{8\pi^{2}}\,.$

It is precisely the constant solution obtained by Allen [5]. In this case, we
have restored the trivial SO$(1,4)$ invariance, but then we meet the problem
of the non-existence of a Fock vacuum state [5].

A second example is provided by the elementary choice

(4.7) $A\equiv(1,0,0,0,0)\,.$

We then obtain the O$(4)$ invariant two-point function:

(4.8)
$W_{\mathrm{mmc}}(z,z^{\prime})=\frac{-H^{2}}{8\pi^{2}}\frac{(\mathcal{Z}-3)\left[(H^{2}z^{0})^{2}+(H^{2}z^{\prime
0})^{2}+H^{4}z^{0}\,z^{\prime 0}\,\mathcal{Z}\right]+6H^{4}z^{0}z^{\prime
0}-H^{2}(1-\mathcal{Z})}{\left[1-\mathcal{Z}(z,z^{\prime})\right]^{3}}\,.$

This two-point function is free of logarithmic divergence but it breaks the dS
invariance, as expected. This result was previously discussed by Allen-Folacci
in [5, 36]. As a matter of fact, the Allen-Folacci expression has the
following logarithmic divergence:

(4.9)
$\mathcal{W}^{\mathrm{AF}}_{mmc}(x,x^{\prime})\propto\left[\frac{1}{1-\mathcal{Z}}-\ln(1-\mathcal{Z})+\cdots\right]\,.$

Going back to the general expression (4.3) for $W_{\mathrm{mmc}}$, its
dominant term at large values of $\mathcal{Z}\sim(z-z^{\prime})^{2}\sim-z\cdot
z^{\prime}\longrightarrow\infty$, reads as

$-\frac{H^{4}}{8\pi^{2}}\left[\frac{A\cdot z\,A\cdot z^{\prime}}{z\cdot
z^{\prime}}+\frac{(A\cdot z)^{2}+(A\cdot z^{\prime})^{2}}{(z\cdot
z^{\prime})^{2}}\right]:=f(z,z^{\prime},A)\,.$

One observes that at large separated points, we get a $A$ dependent value,
which generally depends on the respective directions of $z$ or $z^{\prime}$
with respect to $A$. As an illuminating example, we fix
$z^{\prime}=(0,0,0,0,H^{-1})$ and we choose $z=H^{-1}(\sinh Ht,0,0,0,\cosh
Ht)$, with $t\longrightarrow\infty$, then we obtain:
$\lim_{z\to\infty}f=-\frac{H^{4}}{4\pi^{2}}(A_{4})^{2}$. Importantly, $A$
might be related to the cut-off introduced by Allen in [5]. Also, might this
behaviour explain the appearance of a mass, analogous to the Higgs mechanism?

In contrast, we assert that within the framework of Krein space quantization,
as is described below in Subsection 5.2, de Sitter invariance is preserved, as
exemplified with Equation (5.9), and we obtain a constant term for large
separated points.

##  5\. Quantum Yang-Mills theory

In QFT, the two-point function and the time-ordered product propagator are the
significant quantities that must be calculated. The first one is used for the
construction of an axiomatic quantization [41] and the second one for the
calculation of the effective action or S-matrix element in the perturbation
theory [42]. It is well known that for a quantum gauge theory, an indefinite
metric quantization must be used [38]. The covariant quantization of the
massless vector field and mmc scalar field require an indecomposable
representation of the de Sitter group. The indecomposable representations are
not unique and depend on gauge fixing parameters. The existence of an
axiomatic quantization of the quantum Yang-Mills theory is considered in two
different approaches.

In the first approach, the positive frequency solution of the field equation
is used, and it is called the standard approach. In this approach, the field
operator is defined with the help of coordinate-independent dS plane waves
(modes). The construction is based on the analyticity requirements in the
complexified pseudo-Riemannian manifold. The field equation’s positive and
negative frequency solutions are used in the second approach, based on Krein
space quantization. It is called the Krein approach in this paper.

###  5.1. Standard approach

The axiomatic QFT in dS space was constructed from an analytic two-point
function for massive and mcc scalar field by Bros et al. [6, 7]. Then it was
generalized to the spinor field [19] and massless vector field $K_{\beta}$
[16]. Here, we generalize the previous work to the massless vector field
$K_{\beta}^{a}$ in the Yang-Mills theory. The free vector field equation for
$K_{\beta}^{a}$ is the same as for the free massless vector field $K_{\beta}$
in dS space. Differences appear at the interaction level or for the source of
the fields. On the classical level for the vector field $K_{\beta}$, the
source originates in the electrically charged spinor fields, whereas in the
present case, the source of the free vector field $K_{\beta}^{a}$ stems from
the colored spinor fields and as well the vector fields $K_{\beta}^{b}$, with
$a\neq b$, i.e. $K_{\beta}^{a}$ is not a source for itself on the classical
level, which means that there is no vertex with two same-colored vector fields
$K_{\beta}^{a}$.

In the quantization procedure of the non-abelian gauge theory,
$K_{\beta}^{a}$, two types of mmc scalar fields appear. The first type stands
for the scalar and longitudinal modes of the free vector field, which is
equivalent to the massless vector field $K_{\beta}$. These scalar fields are
necessary for a covariant quantization of the free massless vector filed, and
they appear in the Gupta-Bleuler triplet construction, i.e. the free vector
field operator transforms under a specific indecomposable representation of
the dS group [16]. The other type appears as ghost fields, and they are needed
to absorb the singularity of the Feynman path integral. These two types of
auxiliary fields cannot propagate in the physical space or the external line
of the Feynman diagrams. The first one is completely decoupled from the theory
due to the conserved current, and it does not propagate in the internal line
of the Feynman diagrams. However, the second one (ghost fields) interacts with
the vector fields, which cannot be decoupled from the theory. Therefore it can
propagate in the internal line of the Feynman diagrams, and it must be handled
with precision.

Let us first examine the free massless vector field. The standard approach is
based on the following massless bi-tensor two-point function:

(5.1)
$\mathcal{W}_{\beta\beta^{\prime}}^{aa^{\prime}}(x,x^{\prime})=\langle\Omega\mid
K_{\beta}^{a}(x)K_{\beta^{\prime}}^{a^{\prime}}(x^{\prime})\mid\Omega\rangle\,,$

where $|\Omega\rangle$ can be chosen as the Bunch-Davies vacuum state and
$x,x^{\prime}\in X_{H}$. These functions have to satisfy the following
requirements:

  1. a)

Indefinite sesquilinear form: For any $5$-component test function
$\big{(}f_{\beta}^{a}\big{)}\in\mathcal{D}(X_{H})$, we have an indefinite
sesquilinear form that is defined by

(5.2) $\int_{X_{H}\times
X_{H}}f^{*\beta}_{a}(x)\mathcal{W}_{\beta\beta^{\prime}}^{aa^{\prime}}(x,x^{\prime})f^{\beta^{\prime}}_{a^{\prime}}(x^{\prime})\mathrm{d}\sigma(x)\mathrm{d}\sigma(x^{\prime})\,,$

where $f^{*}$ is the complex conjugate of $f$ [7]. $\mathcal{D}(X_{H})$ is the
space of $C^{\infty}$ functions with compact support in $X_{H}$ and with
values in $\mathbb{C}^{5}$.

  2. b)

dS Covariance: The two-point function satisfies the covariance property

(5.3) $R^{-1}\mathcal{W}(Rx,Rx^{\prime})R=\mathcal{W}(x,x^{\prime})\,.$

where $R\in\mathrm{SO}_{0}(1,4)$.

  3. c)

Locality: For every space-like separated pair $(x,x^{\prime})$, i.e. $x\cdot
x^{\prime}>-H^{-2}$ or $\mathcal{Z}<1$, we have:

(5.4)
$\mathcal{W}_{\beta\beta^{\prime}}^{aa^{\prime}}(x,x^{\prime})=\mathcal{W}_{\beta^{\prime}\beta}^{a^{\prime}a}(x^{\prime},x)\,.$

  4. d)

Normal analyticity:
$\mathcal{W}_{\beta\beta^{\prime}}^{aa^{\prime}}(x,x^{\prime})$ is the
boundary value (in the sense of distributions) of an analytic two-point
function $W_{\beta\beta^{\prime}}^{aa^{\prime}}(z,z^{\prime})$. The
analyticity properties of the tensor two-point function in the tuboid
$\mathcal{T}_{12}=\\{(z,z^{\prime});\;\;z\in\mathcal{T}^{+},z^{\prime}\in\mathcal{T}^{-}\\}$
follow from the analyticity properties of the dS plane waves solutions [7].

  5. e)

Transversality: The transversality with respect to $x$ and $x^{\prime}$ is
guaranteed since the dS modes solutions are transverse by construction,

(5.5)
$x\cdot\mathcal{W}(x,x^{\prime})=0=x^{\prime}\cdot\mathcal{W}(x,x^{\prime})\,.$

The free vector field two-point function reads as:

$\mathcal{W}_{\beta\beta^{\prime}}^{aa^{\prime}}(x,x^{\prime})=\delta^{aa^{\prime}}D_{\beta\beta^{\prime}}(x,x^{\prime},\partial_{x})\mathcal{W}_{mcc}(x,x^{\prime})\,,$

where $D_{\beta\beta^{\prime}}(x,x^{\prime},\partial_{x})$ is the bi-tensor
differential operator (2.20) and $\mathcal{W}_{mcc}$ is the two-point function
of the mcc scalar field (2.21). These functions must entirely encode the
theory of the generalized free fields on dS space-time $X_{H}$ and allow us to
build the quantum field operators. Nevertheless, this approach does not work
for the Yang-Mills theory due to the appearance of ghost fields as a necessary
part of the vector fields $K^{a}_{\alpha}$. In this theory, for fixing the
gauge (3.15), the ghost field is necessary and appears in the Feynman
propagators’ internal line. Within this framework, an axiomatic quantum field
theory for the scalar field mmc does not exist and then an axiomatic quantum
Yang-Mills theory cannot be constructed.

Suppose we can ignore this difficulty and use the standard approach to
quantize Yang-Mills theory. We then face serious difficulties with this
construction, namely, a unique dS invariant vacuum state does not exist with
this theory, the regularization of the IR divergence breaks the dS invariant,
and it is not clear whether the theory is renormalizable or not at all orders
of the perturbation due to the breaking of the dS invariance.

###  5.2. Krein approach

We focus on the quantization of the mmc scalar fields since for the other
fields the generalization is straightforward. In Krein space quantization, the
decomposition of the field operator into positive and negative norm parts
reads as [14, 15]:

(5.6)
$\Phi_{\mathrm{mmc}}(x)\equiv\frac{1}{\sqrt{2}}\left[\varphi_{p}(x)+\varphi_{n}(x)\right]\,,$

where

(5.7) $\varphi_{p}(x)=\sum_{k\geq
0}a_{k}\phi_{k}(x)+h.c.\,,\;\;\varphi_{n}(x)=\sum_{k\geq
0}b_{k}\phi^{*}_{k}(x)+h.c.\,.$

The positive mode part, $\varphi_{p}(x)$, is the scalar field as was used by
Allen. The crucial departure from the standard QFT based on CCR lies in the
following commutation relations:

(5.8)
$[a_{k},a^{{\dagger}}_{k^{\prime}}]=\delta_{kk^{\prime}}\,,\;\;\;\;[b_{k},b^{{\dagger}}_{k^{\prime}}]=-\delta_{kk^{\prime}}\,.$

It is important to note that we now have a Krein-Fock vacuum state. Within
this Krein space framework, the two-point function is the imaginary part of
the two-point function of the positive mode solutions and it reads as [43]:

(5.9)
$\begin{split}\mathcal{W}_{\mathrm{mmc}}^{k}(x,x^{\prime})&=\mathrm{i}\;\mathrm{Im}\left(\mathcal{W}^{\mathrm{AF}}_{mmc}(x,x^{\prime})\right)=\langle\Omega^{k}\mid\Phi_{\mathrm{mmc}}(x)\Phi_{\mathrm{mmc}}(x^{\prime})\mid\Omega^{k}\rangle\\\
&=\frac{\mathrm{i}H^{2}}{8\pi}\epsilon(x^{0}-x^{\prime
0})[\delta(1-\mathcal{Z}(x,x^{\prime}))-\theta(\mathcal{Z}(x,x^{\prime})-1)]\,,\end{split}$

where $\theta$ is the Heaviside step function, and $\mid\Omega^{k}>$ is the
Krein-Fock vacuum state. This function is free of IR divergence, is dS
invariant, and tends to a constant quantity in the large separated points
limit. It is built on an indefinite metric space, which is not positive
definite.

The “Feynman” propagator or time-ordered product propagator in Krein space
quantization reads as:

$G_{T}^{k}(x,x^{\prime})=-\mathrm{i}\langle\Omega^{k}\mid
T\Phi_{\mathrm{mmc}}(x)\Phi_{\mathrm{mmc}}(x^{\prime})\mid\Omega^{k}\rangle$
(5.10)
$=\frac{H^{2}}{8\pi}\left[\delta(1-\mathcal{Z}(x,x^{\prime}))-\theta(\mathcal{Z}(x,x^{\prime})-1)\right]\,,$

where $T$ is the time ordering operator relative to the ambient coordinate
$x^{0}$ [7]. This function is causal and free from any IR and ultraviolet
divergence. It owns light-cone singularity. This singularity can be absorbed
in the fluctuation of the quantum metric and does not appear in the
calculation of the vacuum expectation value of the physical quantity [24].
Then, considering the quantum field theory in Krein space quantization with
quantum metric fluctuations included, it is proved that all singular behaviors
of the free scalar Green functions are removed. The new function $\mathcal{Z}$
is introduced in order to take into account the metric fluctuations:

(5.11) $g_{\mu\nu}=g_{\mu\nu}^{dS}+h_{\mu\nu},\;\;|h|<|g^{dS}|\,.$

It expands as $\mathcal{Z}=\mathcal{Z}_{0}+\mathcal{Z}_{1}+\cdots$, where
$\mathcal{Z}_{0}$ is given by the equation (2.22) and $\mathcal{Z}_{1}$ is the
first-order perturbation due to the gravitational wave $h_{\mu\nu}$. In order
to take into account the fluctuations of the quantum metric (see Appendix A)
and in analogy with the minkowskian case, the Green function (5.10) must be
replaced with the following equation [24]:

(5.12) $\langle
G_{T}^{k}(x,x^{\prime})\rangle\approx\frac{H^{2}}{8\pi}\left[\frac{1}{\sqrt{\pi(1-\langle\mathcal{Z}_{1}\rangle)^{2}}}\exp\left(-\frac{(1-\mathcal{Z}_{0})^{2}}{(1-\langle\mathcal{Z}_{1}\rangle)^{2}}\right)-\theta(\mathcal{Z}_{0}(x,x^{\prime})-1)\right]\,.$

The expectation value $\langle\mathcal{Z}_{1}\rangle\neq 1$ is related to the
effect of first order quantum metric fluctuations. It determines the density
of gravitons [44]. More precisely $\langle\mathcal{Z}_{1}\rangle$ in (5.12)
features the quantum metric fluctuations due to the quantum field operator
$h_{\mu\nu}$. In the limit $\langle\mathcal{Z}_{1}\rangle\longrightarrow 1$
the delta function is reobtained:

(5.13) $\lim_{a\longrightarrow 0}\frac{1}{\sqrt{\pi
a^{2}}}\exp\left(-\frac{x^{2}}{a^{2}}\right)=\delta(x)\,.$

This means that the light-cone fluctuations disappear in this limit.

The Krein approach is based on the massless two-point function (5.9) and it
must satisfy the following requirements (to be compared with those for the
function (5.1)) [14, 15, 45]:

  1. A)

Indefinite sesquilinear form: For any test function
$f(x)\in\mathcal{D}(X_{H})$, we have an indefinite sesquilinear form that is
defined by

(5.14) $\int_{X_{H}\times
X_{H}}f^{*}(x)\mathcal{W}_{\mathrm{mmc}}^{k}(x,x^{\prime})f(x^{\prime})\mathrm{d}\sigma(x)\mathrm{d}\sigma(x^{\prime})\,,$

where $f^{*}$ is the complex conjugate of $f$. $\mathcal{D}(X_{H})$ is the
space of $C^{\infty}$ functions with compact support in $X_{H}$ and with
values in $\mathbb{C}$.

  2. B)

dS Covariance: The two-point function satisfies the covariance property

(5.15)
$\mathcal{W}_{\mathrm{mmc}}^{k}(Rx,Rx^{\prime})=\mathcal{W}_{\mathrm{mmc}}^{k}(x,x^{\prime})\,.$

where $R\in\mathrm{SO}_{0}(1,4)$.

  3. C)

Locality: For every space-like separated pair $(x,x^{\prime})$, i.e. $x\cdot
x^{\prime}>-H^{-2}$ or $\mathcal{Z}<1$, we have:

(5.16)
$\mathcal{W}_{\mathrm{mmc}}^{k}(x,x^{\prime})=\mathcal{W}_{\mathrm{mmc}}^{k}(x^{\prime},x)\,.$

These properties are not sufficient to generalize the Krein space quantization
to the quantum theory for interacting fields such as Yang-Mills. For this
purpose, we need the analyticity properties of the vacuum expectation value of
the field operators [7, 41]:

(5.17) $\langle\Omega^{k}\mid
T\,\phi(x_{1})\phi(x_{2})\cdots\phi(x_{n})\mid\Omega^{k}\rangle\,.$

Since there is not yet rigorous mathematical proof for the analyticity
properties of the vacuum expectation value of the field operators in Krein
space quantization, we use the Krein space quantization, including quantum
metric fluctuations, as a method of quantum field regularization (for details,
see [24]). Then we impose a supplementary condition on the two-point function
in view of generalization to the quantum theory for interacting fields:

  1. D)

Light-cone fluctuations: The delta function of the internal line in the vacuum
expectation value of the time-order product of field operators (5.17), is
regularised in the following way:

(5.18)
$\delta(1-\mathcal{Z}_{0}(x,x^{\prime}))\,\mapsto\,\frac{1}{\sqrt{\pi(1-\langle\mathcal{Z}_{1}\rangle)^{2}}}\exp\left(-\frac{(1-\mathcal{Z}_{0})^{2}}{(1-\langle\mathcal{Z}_{1}\rangle)^{2}}\right).$

This condition D) supersedes the previous analyticity property d) requested
for (5.1). $\langle\mathcal{Z}_{1}\rangle$ is a complex function of the space-
time coordinates $x^{\alpha}$ [44]. We nevertheless have
$\langle\mathcal{Z}_{1}\rangle\neq 1$ due to the quantum fluctuations, and
then the replacement (5.18) results in a regular function on the light-cone.

The generalization of the approach à la Krein to spinor and vector fields is
straightforward and can be constructed by replacing the two-point function
with its imaginary parts and the condition D). Some of the advantages of this
approach can be mentioned, namely,

  * •

existence of a unique Krein-Fock vacuum state,

  * •

preservation of dS invariance,

  * •

renormalizability of the theory at all perturbation orders,

  * •

explanation of mass gap and color confinement issues in a covariant way, as is
discussed in the next section.

If we take Condition D) as a Light-cone fluctuations principle in Krein space
quantization, we get an axiomatic quantum Yang-Mills theory within the
framework of the dS ambient space formalism, and this approach can explain the
mass gap and color confinement problems in a covariant way. This principle
makes sense since it is supported by physical evidence. The gravitational
field exists, $g_{\mu\nu}=g_{\mu\nu}^{dS}+h_{\mu\nu}\,$ and at the Planck
scale the quantum linear gravity is essential, $\Delta h=\sqrt{\langle h\cdot
h\rangle-\langle h\rangle^{2}}\neq 0\,$. Fluctuations of the dS light-cone are
induced by quantum linear gravity, and then the Dirac delta function supported
by the classical light-cone must be replaced along Condition D) [24, 44].
Since this principle only changes the internal line of the Feynman diagrams,
it leads to the regularization of the theory, and then
$\langle\mathcal{Z}_{1}\rangle$ can be considered as a regularization
parameter. Through the renormalization of the theory, observable physical
results are achieved [24].

##  6\. Mass gap and confinement

The IR divergence of the massless minimally coupled scalar field resembles
features of a massless field theory in $2$-dimensional space-time [46]. The
regularization of the infrared divergence in $2$-dimensional massless field
theory results in a mass gap [47] and a confinement [48]. Motivated by these
facts, let us examine if a similar mechanism can be relevant to the Yang-Mills
theory in dS space-time. Here two approaches for this problem, which were
presented in the previous section, are discussed.

From the classical Lagrangian (3.17), there is no explanation for the mass gap
and the color confinement in the dS Yang-Mills theory. On the QFT level, the
effective Lagrangian or “quantum Lagrangian” can be established through the
loop correction to the classical Lagrangian:

(6.1)
$\mathcal{L}_{q}=\mathcal{L}_{c}+\hbar\mathcal{L}_{1}+\hbar^{2}\mathcal{L}_{2}+\cdots\,.$

In the one-loop approximation the appearance of ghost field propagators occurs
in the Feynman diagrams. By considering the gluon self-energy diagram (or
gluon mass term) the following expression appears in the effective action,
which contains two vertices (each vertice has one vector field and two ghost
fields):

(6.2)
$\cdots+\int\mathrm{d}\sigma(x)\mathrm{d}\sigma(x^{\prime})T\left[{\Phi^{a}}^{\dagger}(x)C^{abc}\partial^{\top}\cdot
K^{c}(x)\Phi^{b}(x)\right]\left[{\Phi^{a^{\prime}}}^{\dagger}(x^{\prime})C^{a^{\prime}b^{\prime}c^{\prime}}\partial^{\prime\top}\cdot
K^{c^{\prime}}(x^{\prime})\Phi^{b^{\prime}}(x^{\prime})\right]+\cdots\,.$

Having in view the finding of a mass term for the vector fields, we consider
two vector fields that are connected with the two propagators of the mmc
scalar field. Let us go through the two different approaches discussed in the
previous section.

  1. A)

Standard approach: With the dS flat coordinates used by Allen, the Green
functions are calculated by using continuous sums of the type
$\int_{\mu}^{\Lambda}\mathrm{d}kf(k)$ [5, 36], where $\mu$ is an IR cut-off or
regulator, and $\Lambda$ is an ultraviolet cut-off. This regulator excludes
the range $k\in[0,\mu]$ from the mode sum, so the zero modes’ contribution is
dropped. The introduction of the cut-off $\mu$ and $\Lambda$ in the propagator
of the mmc scalar field entails the appearance in Equation (6.2) of a mass
term of the following type for the gluon fields (see Appendix B):

$\int\mathrm{d}\sigma(x)\mathrm{d}\sigma(x^{\prime})M_{cc^{\prime}}^{\alpha\alpha^{\prime}}(\mu,\Lambda;x,x^{\prime},\partial,\partial^{\prime})K_{\alpha}^{c}(x)K_{\alpha^{\prime}}^{c^{\prime}}(x^{\prime})\,.$

The explicit form of this integral is not essential for our purpose here.
Although this mass term breaks dS invariance, it can explain the short-range
force of the strong interaction and the mass gap problem of the quantum Yang-
Mills theory in dS space.

  2. B)

Krein approach: In this approach the propagator of the mmc scalar fields is a
regular function (5.12), and by replacing it in equation (6.2) give:

$\int
d\sigma(x)\mathrm{d}\sigma(x^{\prime})M_{cc^{\prime}}^{\alpha\alpha^{\prime}}(H,\langle\mathcal{Z}_{1}\rangle;x,x^{\prime},\partial,\partial^{\prime})K_{\alpha}^{c}(x)K_{\alpha^{\prime}}^{c^{\prime}}(x^{\prime})\,.$

In this method, the dS invariance is preserved, and one can explain the short-
range force of the strong interaction and the mass gap problem. In this
expression $H$ and $\langle\mathcal{Z}_{1}\rangle$ play the role of the cut-
off parameters.

The behaviour of the two-point function for the massless vector field in the
inflationary epoch, ($H\to$ large value), allows us to understand better the
mass gap and the confinement problems in this limit. Let us here follow the
standard approach. Let us pick up two points:
$X^{\mu}=(t,\chi,\theta,\varphi)$ and $X^{\prime\mu}=(0,0,0,0)$ in terms of
the global coordinates system (2.3). Then in the limit of large $H$, we have:
$\mathcal{Z}=-H^{2}x\cdot x^{\prime}=\cos\chi\cosh
Ht\approx\frac{1}{2}\cos\chi e^{Ht}\,$. In this limit the two-point function
of the conformally massless scalar field reads as:

(6.3)
$\mathcal{W}_{\mathrm{mcc}}(x,x^{\prime})=\frac{H^{2}}{8\pi^{2}}\frac{1}{\frac{1}{2}\cos\chi
e^{Ht}}=\frac{H^{2}}{4\pi^{2}\cos\chi}e^{-Ht}\,.$

Then the massless vector two-point function (2.19) can be written in terms of
the mcc scalar two-point function as follows ($\mathcal{Z}\neq 1$) [16]:

$\mathcal{W}_{\alpha\alpha^{\prime}}(x,x^{\prime})=\left[-\theta^{\prime}_{\alpha^{\prime}}\cdot\theta_{\alpha}\,\mathcal{Z}\frac{\mathrm{d}}{\mathrm{d}\mathcal{Z}}+H^{2}(\theta^{\prime}_{\alpha^{\prime}}\cdot
x)(\theta_{\alpha}\cdot
x^{\prime})\left(2\frac{\mathrm{d}}{\mathrm{d}\mathcal{Z}}+\mathcal{Z}\frac{\mathrm{d}^{2}}{\mathrm{d}\mathcal{Z}^{2}}\right)\right]\mathcal{W}_{\mathrm{mcc}}(x,x^{\prime})\,$
$=\frac{H^{2}}{8\pi^{2}}\left(\eta_{\alpha\alpha^{\prime}}+H^{2}x^{\prime}_{\alpha}x^{\prime}_{\alpha^{\prime}}+H^{2}x_{\alpha}x_{\alpha^{\prime}}-H^{2}x_{\alpha}x^{\prime}_{\alpha^{\prime}}\mathcal{Z}\right)\frac{\mathcal{Z}}{(1-\mathcal{Z})^{2}}$
(6.4)
$-\frac{H^{4}}{4\pi^{2}}\left(x^{\prime}_{\alpha}x_{\alpha^{\prime}}-x^{\prime}_{\alpha}x^{\prime}_{\alpha^{\prime}}\mathcal{Z}-x_{\alpha}x_{\alpha^{\prime}}\mathcal{Z}+x_{\alpha}x^{\prime}_{\alpha^{\prime}}\mathcal{Z}^{2}\right)\left(\frac{\mathcal{1}}{(1-\mathcal{Z})^{2}}+\frac{\mathcal{\mathcal{Z}}}{(1-\mathcal{Z})^{3}}\right)\,.$

At large values of $H$ the latter behaves as:

(6.5)
$\lim_{H\rightarrow\infty}\mathcal{W}_{\alpha\alpha^{\prime}}(x,x^{\prime})\Longrightarrow-\frac{H^{4}}{4\pi^{2}}x_{\alpha}x^{\prime}_{\alpha^{\prime}}\left(\frac{3\mathcal{\mathcal{Z}^{2}}}{2(1-\mathcal{Z})^{2}}+\frac{\mathcal{\mathcal{Z}^{3}}}{(1-\mathcal{Z})^{3}}\right)\approx-\frac{H^{4}}{8\pi^{2}}x_{\alpha}x^{\prime}_{\alpha^{\prime}}\,.$

Terms linear in the coordinates $x^{\alpha}$’s and ${x^{\prime}}^{\alpha}$’s
respectively appear in this limit. Such a behavior of the two-point function
may be used to explain the mass gap and the color confinement problems [9]. In
the null curvature limit ($H=0$) one can see that this behavior disappears as
well.

##  7\. Conclusions

The non-abelian gauge-invariant Lagrangian density is calculated in the dS
ambient space formalism. We obtain that the ghost fields are the mmc scalar
fields. We have examined the possibility of an axiomatic QFT for the Yang-
Mills theory within the framework of Krein space quantization. Due to the
regularization of IR divergences, a mass term is obtained for the gluon fields
in the one-loop approximation. This mass term may be used to solve the mass
gap and color confinement problems in dS Yang-Mills theory. We show that in
the large curvature limit ($H\rightarrow\infty$), early universe, or
inflationary time, a term linear in system coordinates appears in the vector
two-point function. In a forthcoming paper, we plan to consider the cluster
decomposition principle by using the dS covariance, locality, the mass gap,
and the principle D) within our Krein approach.

Acknowledgements: The authors are grateful to Edouard Brezin, Eric Huguet,
Jean Iliopoulos, and Jacques Renaud for discussions. We are grateful to the
referee for precise and valuable comments. One of the authors (MVT) would like
to thank le Collège de France and l’Université de Paris for their financial
support and hospitality.

## Appendix A

The effect of linear quantum gravity on the scalar Green’s function can be
calculated by considering the interaction between two fields. For the sake of
simplicity, the massless scalar field with minimal coupling to the
gravitational field is considered and Minkowski space time is chosen as the
background. In this case, the total classical action reads:

(A.1)
$S_{c}[g,\phi]=\int\sqrt{-g}\;\mathrm{d}^{4}x\left(R+\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi\right)\,,$

where $R$ is the scalar curvature. In the linear approximation of the
gravitational field ($g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu},\;|h|<|\eta|$) and
with the definition $\sqrt{-g}\equiv 1+f(h)$, where $f$ is a differentiable
function, we have [49]:

(A.2)
$\begin{split}S_{c}[g,\phi]&=\int\left(1+f(h)\right)\mathrm{d}^{4}x\left(\frac{1}{2}\left[(\partial_{\mu}h^{\mu\nu})(\partial_{\nu}h)-(\partial_{\mu}h^{\rho\sigma})(\partial_{\rho}h^{\mu}_{\sigma})\right.\right.\\\
&\left.\left.+\frac{1}{2}\eta^{\mu\nu}(\partial_{\mu}h^{\rho\sigma})(\partial_{\nu}h_{\rho\sigma})-\frac{1}{2}\eta^{\mu\nu}(\partial_{\mu}h)(\partial_{\nu}h)\right]+\frac{1}{2}\left(\eta^{\mu\nu}+h^{\mu\nu}\right)\partial_{\mu}\phi\partial_{\nu}\phi\right)\,.\end{split}$

Consider the term $f(h)\eta^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi$
appearing in the Lagrangian density. On the quantum metric fluctuation level
it may be written as:

(A.3) $f(\langle
h^{2}\rangle)\eta^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi\,.$

If we assume that $\langle h^{2}\rangle$ is constant, this term may be
interpreted as a counterterm for the scalar field. It modifies the scalar
Green function and the factor $f\equiv Z_{\phi}$ plays the role of a wave
function renormalization parameter. The other quantum effects show up through
the quantum effective action, which can be written as:

$S_{q}[g,\phi]=S_{c}[g,\phi]+\hbar S^{(1)}[g,\phi]+\cdots\,,$

where $S^{(1)}$ is the one-loop approximation.

## Appendix B

In this appendix a brief procedure for calculating
$M_{cc^{\prime}}^{\alpha\alpha^{\prime}}(\mu,\Lambda;x,x^{\prime},\partial,\partial^{\prime})$
is presented. We start out with the following expression,

$T\left[{\Phi^{a}}^{\dagger}(x)C^{abc}\partial^{\top}\cdot
K^{c}(x)\Phi^{b}(x)\right]\left[{\Phi^{a^{\prime}}}^{\dagger}(x^{\prime})C^{a^{\prime}b^{\prime}c^{\prime}}\partial^{\prime\top}\cdot
K^{c^{\prime}}(x^{\prime})\Phi^{b^{\prime}}(x^{\prime})\right].$

It can be divided into four terms where, for simplicity, solely one of them is
considered:

$C^{abc}C^{a^{\prime}b^{\prime}c^{\prime}}T\left[{\Phi^{a}}^{\dagger}(x)\Phi^{b}(x)\partial^{\top\alpha}K^{c}_{\alpha}(x)\right]\left[{\Phi^{a^{\prime}}}^{\dagger}(x^{\prime})\Phi^{b^{\prime}}(x^{\prime})\partial^{\prime\top\alpha^{\prime}}K^{c^{\prime}}_{\alpha^{\prime}}(x^{\prime})\right].$

Since we are looking for a mass term for the vector fields, then the vector
fields do not participate in the time order product. For scalar fields, the
time-ordered product and vacuum expectation value result in the following
expression:

$C^{abc}C^{a^{\prime}b^{\prime}c^{\prime}}\delta^{a}_{b^{\prime}}\delta^{b}_{a^{\prime}}\left[G_{T}(x,x^{\prime})\right]^{2}\partial^{\top\alpha}\partial^{\prime\top\alpha^{\prime}}K^{c}_{\alpha}(x)K^{c^{\prime}}_{\alpha^{\prime}}(x^{\prime})\,.$

It is part of the expression $M_{cc^{\prime}}^{\alpha\alpha^{\prime}}$ in
which $G_{T}(x,x^{\prime})$ is the time-ordered scalar Green function. The
latter is different from what is yielded by the standard model and by the
Krein approach.

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