According to the highly redshift observation of the Supernova Ia [1, 2], galaxy clusters [3, 4], and cosmic microwave background radiation [5], the current universe is expanding in an accelerating way.
Then our current universe may be described by the de Sitter space-time... Therefore, the construction of the quantum field theory in de Sitter space
is very important for better understanding of the evolution of the early and current universe. The
rigorous mathematical construction of quantum field theory in de Sitter space-time, based on the
unitary irreducible representations of the de Sitter group and the analyticity of the complexified de
Sitter space-time, was previously presented in [7].
(Submitted on 2 Apr 2015 (
v1), last revised 20 Jul 2016 (this version, v2))
Although many
programs aiming to construct a consistent quantum gravity theory, such as string theory [3],
loop quantum gravity [4], causal sets [5], dynamical triangulations [6] and asymptotic safety
[7] have been developed, none can claim to have attained yet a reasonable level of maturity
enabling computations in physically relevant situations. In this context, all one can do is to
treat gravity as a low energy effective field theory [8]. This is the rationale behind perturbative
quantum gravity around fixed backgrounds.
The first step toward this goal is to construct a consistent linear quantum theory of
gravity. Aside from the usual conceptual difficulties encountered in any attempt to quantize
gravity, this program has met reasonable success when applied around Minkowski spacetime
without raising significant technical difficulties. However, as soon as one tries to construct
linear quantum gravity around a non trivial background, one faces more serious technical and
conceptual difficulties. Due to its maximal symmetry and to its relevance to cosmology, one
of the most natural backgrounds one is interested in is the de Sitter (dS) spacetime, and no wonder that a huge literature was devoted to the study of QFT and perturbative quantum
gravity on de Sitter space (see the excellent review [9] and references therein)...
Finally we want to underline the fact that physics in de Sitter spacetime is far from being a clear and familiar domain of investigation and that the construction of acceptable physical observables in dS is a delicate matter. For instance the usual observables of Minkowskian QFT, like scattering amplitudes, are ill defined on dS due to the existence of horizons.
(Submitted on 14 Jan 2009)
... the study of de Sitter space-time offers a specific interest because of the regularization opportunity afforded by the curvature parameter as a “natural” cutoff for infrared or other divergences. On the other hand, some of our most familiar concepts like time, energy, momentum, etc, disappear. They really require a new conceptual approach in de Sitterian relativity. However, it should be stressed that the current estimate on the cosmological constant does not allow any palpable experimental effect on the level of high energy physics experiments, unless (see [14]) we deal with theories involving assumptions of infinitesimal masses like photon or graviton masses.
(Submitted on 27 Jan 2010)
The linear quantum gravity in de Sitter space was studied thoroughly by Iliopoulos
et al. [9, 10]. They have shown that the pathological large-distance behavior (infra-red divergence)
of the graviton propagator on a de Sitter background does not manifest itself in the
quadratic part of the effective action in the one-loop approximation [9, 10]. This means that
the pathological behaviour of the graviton propagator may be gauge dependent and so should
not appear in an effective way as a physical quantity. The linear gravity (the traceless rank-2
“massless” tensor field) on de Sitter ambient space formalism was built rudimentary from the
minimally coupled scalar field [11]. It has been shown that the application of Krein space
quantization to the minimally coupled scalar field in de Sitter space has resulted in removal
of infrared and ultraviolet divergences and henceforth naturally maintained the principle of
causality [11, 12, 13]. Construction of linear gravity in de Sitter universe through Krein space
quantization will {be} present{ed} in this paper, pursuing a similar path {to} the earlier works
[13, 14]. A notable consequence of this construction is absence of divergence in Green function
at large distances, resulting in removal of infra-red divergence [13, 14]. This method was applied
in various area of QFT and/or QED where all resulted in natural renormalization of the
solutions [15, 16, 17, 18, 19].
Although negative norm states appear in our method, by imposing the following conditions
they are effectively removed and the unitarity of theory is preserved:
i) The first condition is the ”reality condition” in which the negative norm states do not
appear in the external legs of the Feynmann diagram. This condition guarantees that the
negative norm states only appear in the internal legs and in the disconnected parts of the
diagram.
ii) The second condition is that the S matrix elements must be renormalized in the following
form:
Sif ≡ probability amplitude = <physical states, in|physical states, out>/ <0, in|0, out>.
This condition eliminates the negative norm states in the disconnected parts.
In previous methods the choice of vacuum state directly affected the expectation values of
energy momentum tensor. In the present method, however, vacuum expectation values are
independent of the choice of modes. Although the expectation value of the energy momentum
tensor for physical states are dependent on the choice of modes, the expectation value of vacuum
states remains uniquely the same.
(Submitted on 28 Aug 2012 (
v1), last revised 8 Jun 2016 (this version, v3))
Delivering the Higgs boson ...
The Higgs field ΦH is thought to exist everywhere in the universe. The interaction of this fundamental field with vector and spinor fields results in the appearance of mass for some fields through the Higgs mechanism and the spontaneous symmetry breaking. Whereas the interactions between the various fields are usually described by using the gauge formalism, the interaction between Higgs and spinor fields is expressed in terms of a Yukawa type potential, which does not pertain to the gauge theory framework. Establishing a firm theory, based on the gauge principle, for the interaction between Higgs and spinor fields is still an important challenge.
In this paper, we explain how the massless minimally coupled (mmc) scalar field in de Sitter (dS) space could play a similar role to the special one Higgs field plays in the standard model. One essential step is to show that the mmc scalar field can be written in terms of a massless conformally invariant (mcc) scalar field and an arbitrary constant five-vector Aα in dS ambient space formalism. Then we prove that a non-zero term in the two-point function appears everywhere in the dS universe and that, at short relative distance, we have the Hadamard behavior, which guarantees the observation of the field quanta or “Higgs particle”.
We know that gauge invariance has two handicaps: (1) appearance of infrared divergence and (2) breakdown of background space-time symmetry. Fixing the gauge and introducing a Gupta-Bleuler triplet formalism remove these problems. These difficulties are also present in the quantization of the mmc scalar field in dS space (see [1] and references therein). Here, we propose a new transformation for the mmc scalar field in dS ambient space formalism, and we reformulate the interaction between scalar and spinor fields. We show that the Yukawa interaction can be reconstructed with an appropriate fixing of the parameters in the null curvature limit.
We first recall the abelian gauge theory in dS ambient space formalism. Then we show that for a specific class of abelian gauge parameters our local transformation becomes equivalent to the latter. Finally, the one-loop correction of the scalar-spinor interaction is presented...
The de Sitter ambient space formalism has permitted to define a unique Bunch-Davies vacuum state for a quantum field theory that includes mmc scalar field and linear quantum gravity. The infrared divergence is non-existent in either the quantization of the scalar field Φmmc , or the linear quantum gravity. In this construction the two-point functions are all analytic. Moreover, it is clear that the mmc field can be viewed as similar to the Higgs field. By using this formalism one can also define the interaction between scalar and spinor fields ... Now we have all the necessary building blocks for the construction of a unitary super-gravity in dS universe [20] and unified theory of all interactions in de Sitter space. This will be considered in a forthcoming paper.
(Submitted on 23 Dec 2016 (
v1), last revised 13 Mar 2017 (this version, v5))
.... and a super-gauge field offspring ?
The following principles are used as the axioms in our construction {of a quantum field theory in 4-dimensional de Sitter (dS) space-time}:
• (A) As it is indicated by observation, our universe can be well approximated by the de Sitter space-time with its symmetrical group SO(1, 4).
• (B) The quantum fields actually are fundamental objects and their corresponding free field operators must be transformed by the unitary irreducible representation of the de Sitter group.
• (C) The interaction between these fields are governed by the gauge principle (gauge theory).
• (D) The conformal symmetry is preserved in the early universe.
Th{e ambient space} formalism allows us to construct quantum field operator in a rigorous mathematical framework based on complexified pseudo-Riemannian manifold and the group representation theory. The field equations for various spin fields are obtained by using the second order Casimir operator and dS algebra machinery. The procedure of defining the Lagrangian from the field equations for the free fields and the gauge theory are presented in ambient space formalism. Finally the quantum field operators, the quantum states and the two-point functions are presented for various spin fields. For simplicity, we recall some interesting results of this formalism:
...
• Using the dS plane waves in ambient space formalism, the construction of quantum field theory in dS space is very similar to its counterpart in the Minkowski space-time.
• The dS space predicts the existence of a maximum length for an observable (or equivalently, an ”event horizon”) in the x-space {playing a role similar to Minkowski space-time}. The uncertainty principle results in to the existence of a minimum size in the ξ -space {similar to energy-momentum}. We know that the total volume of this parameter in Hilbert space or ξ -space, is finite. As a direct consequence, the total number of quantum states in the Hilbert space turns out to be finite physically [41]. This is the one of the most important results of this formalism.
• The massless fields with s = j = p ≥ 3 cannot propagate on the dS space in our formalism. The massless fields with spins j = 0, 1/2, 1, 3/2, 2 and 5/2 may exist in our universe.
• A novel N = 1 de Sitter superalgebra can be attained by the use of spinor field and charge conjugation in the ambient space notation [49].
• Gauge vector fields Kα(x) with j = p = 1 can be considered as the potential of the electromagnetic, weak and strong nuclear forces. The spin- 2 gauge field KMαβγ(x) in the gauge gravity model may be considered as a part of gravitational field. The case j = p = 3/2 corresponds to a vector-spinor gauge potential Ψα(x) with anti commutation relations. The consequence of this formalism is the natural appearance of super-gravity.
• The two spinor generators would not generate a closed super-algebra. Therefore these spinor generators must engage their couplings with the dS group generators and the vector-spinor gauge potential Ψα(x) with the gauge potentials KMαβγ(x). Then the gravitational field may be considered as: classical gravitational field θαβ or dS background, the spin-2 gauge potential KMαβγ(x) and the vector-spinor gauge potential Ψα(x).
• The spin-2 gauge potential KMαβγ(x) can be quantized and their quantum field operators transform according to the indecomposable representations of dS group. The physical states transform simultaneously as the unitary irreducible representations of dS and conformal groups.
• One can construct the Hilbert space and then the Fock space for the universe including the gauge potentials KMαβγ(x) and Ψα(x). The total number of quantum states for our universe turns out to be finite.
• Previously, the Krein space quantization (Hilbert space ⊕ anti-Hilbert space) has been used for the extraction of the covariant quantization of minimally coupled scalar fields that suppresses the infrared divergences but breaks the analyticity [18]. To remedy this later drawback, we have used the identity (VII.1) in ambient space formalism. This innovative formalism permits one to quantize the massless minimally coupled scalar field in terms of a massless conformally coupled scalar field. It means that the quantum field operator (VII.11) and the two-point function (VII.13) of minimally scalar field can be constructed on the Bunch-Davies vacuum state. Then the problem of infrared divergence of the linear gravity in the dS space is completely solved on the Bunch-Davies vacuum. The theory is also analytic.
• The quantum massless tensor (-spinor) field can be reformulated in terms of a polarization tensor (-spinor) and a massless conformally coupled scalar field. Its analytic two-point function can also be written in terms of the analytic two-point function of conformally coupled scalar field and a polarization tensor (-spinor).
(Submitted on 5 Mar 2014 (
v1), revised 3 Sep 2014 (this version, v3), latest version 19 Jul 2016 (
v5))
Comments
Post a Comment
Cher-ère lecteur-trice, le blogueur espère que ce billet vous a sinon interessé-e du moins interpellé-e donc, si le coeur vous en dit, osez partager avec les autres internautes comme moi vos commentaires éclairés !
Dear reader, the blogger hopes you have been interested by his post or have noticed something (ir)relevant, then if you are in the mood, do not hesitate to share with other internauts like me your enlightened opinion !