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Third' Quantization Gauge Theory with Minkowski Metric, Non-Zero

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Third' Quantization Gauge Theory with Minkowski Metric, Non-Zero A ‘Third’ Quantization Constructed for Gauge Theory of Gravity Maysam Yousefian∗ and Mehrdad Farhoudi† Department of Physics, Shahid Beheshti University G.C., Evin, Tehran 19839, Iran (Dated: September 5, 2021) In general, a global and unique vacuum state cannot be constructed for a curved space. As a remedy, we introduce a curved space background geometry with a Minkowski metric tensor and locally non- zero curvature and torsion. Based on this geometry, we propose a ‘third’/vacuum quantization model as a consequence of Unruh effect. Accordingly, we introduce a ‘third’ quantization scalar field as a general coordinate transformation of spacetime for the second quantization fields. Then we show that in the classical limit, the ‘third’ quantization fields appear as Riemannian manifolds with an emergent metric on which the second quantization fields are located. This way, the standard model of field theory turns out as an effective theory. Moreover, using the proposed ‘third’ quantization fields, we build a U(1) × SU(4) Yang-Mills gauge theory for gravity. According to this gravitational model, we indicate that an analytical solution of the presented gravitational model, for the ‘third’ quantum field particle trajectory (such as a star), corresponds to the trajectory of a test particle in the Mannheim-Kazanas space. Furthermore, by using non-perturbative methods and lattice gauge theory results, we render a solution for the potential of the constructed model that can explain the galaxy rotation curves and gravitational lensing without any need to dark matter. We also address the cosmic microwave background phenomenon and the expansion of the universe. PACS numbers: 11.15. − q; 04.60. − m; 03.70. + k; 95.35. + d Keywords: Quantum Gravity; Rindler Vacuum; Dark Matter; Galaxy Rotation Curves; Gravitational Lens- ing I. INTRODUCTION There is a fundamental difference between standard gauge theories such as for quantum electrodynamics General relativity is an elegant and interesting theory (QED) and those proposed for gravity. Consider a fer- of gravity that is invariant under diffeomorphism – see, mionic test particle with mass m and charge e, in a 1 α e.g., Refs. [1–3] and references therein. Physicists have Riemann-Cartan manifold with connection Γ µν , sub- α been interested in expressing it (and, in general, gravita- jected to an external electromagnetic gauge field A . Its tion) as a gauge theory. At the early stages of gauge the- autoparallel equation is ory, in Refs. [4, 5], a gauge theory based model of gravity e x¨α +Γα x˙ µx˙ ν = (A ,α Aα )x ˙ ν , (1) was proposed. Thereafter, by introducing a gauge theory µν m ν − ,ν based on the Poincar´egroup for gravity [6, 7], other types of models have been suggested but their gauge trans- where the Greek letters run from zero to three. Adding a formation method differs compared to the standard ap- constant value to the electromagnetic gauge field will al- proach [8]. Nowadays, there are many examples of gauge ter the particle’s velocity but, due to Eq. (1), will keep its gravitational models built upon gauge gravitation the- acceleration invariant. Based on QED, a fermion cannot ory (GGT). Most of these models are developed by using be accelerated by a quantum of the second quantization differential geometry approach – see Refs. [4–8] and ref- gauge connection field. A statistical set of quanta will be needed.2 In contrast with the case for gravity, adding a arXiv:2104.05768v2 [hep-th] 5 Sep 2021 erences therein. On the other hand, in Ref. [9], a gauge theory of grav- ity called gauge theory gravity (GTG) has been construc- ted in the language of geometric algebra [10]. There is 1 The physical consequences of the existence of torsion have been an important difference between GGT and GTG. The considered in the literature. For example, the compatibility background space is Riemannian for the former, and of torsion with the equivalence principle has been discussed in Minkowski for the latter. The fact that there is not a Ref. [12], and the existence of stability in theories including tor- unique vacuum state in a curved background space [11] sion has been investigated in Refs. [13–15]. Torsion and discus- is a problem for GGT based models but not for GTG. sions about the unitarity, the existence of ghost and some is- sues related to quantum field theory have also been discussed in The GTG is the first point of inspiration for the work Refs. [16, 17]. However, considering these types of issues for the presented in this article. The second point has to do proposed model in this study will be investigated in a separate with the second quantization in quantum field theory. work. 2 In Refs. [18, 19], it has been shown that an inertial state is equi- valent to an accelerated state on which a statistical set of the second quantization creation operators is applied. (Of course, conversely in this work, we will show that an accelerated state ∗ M Yousefi[email protected] is equivalent to an inertial state on which a statistical set of the † [email protected] second quantization creation operators is applied.) 2 constant value to the gravitational connection field will In the tetrad/frame formalism,3 for a tangent bundle, change the particle’s acceleration. Inspired by this com- the local relation between the tetradic components of parison, we envisage that a quantum of the gravitational metric (e.g., gab) and its coordinate ones are connection field cannot be a quantum of a second quant- µ ν ization field. Instead, the effect of it should be similar to gab = ea eb gµν , (2) the effect of a statistical set of, or a partition function for, where the Latin letters serve to label the tetradic bases. the quanta of the second quantization gauge connection For a Minkowski frame (i.e., gab ηab), the metricity fields. ≡ gab;µ = 0 makes the spin connection to be For a theory of quantum gravity, we require that the ωab = ωba . (3) effect of a quantum of the gravitational connection field µ − µ be equivalent to that of a statistical set, or a partition From the topological point of view, the tangent bundle function, of second quantization quanta. Before defining is over the base manifold M. If one exchanges the fiber such a field, one must remedy the problem of defining with the base space in the mentioned case, in general, a unique vacuum state. For this issue, in Sec. II, we one will have first introduce a geometry with a Minkowski background ab ba metric by making changes to the topology of a Riemann- gµν ηµν ω µ = ω µ Cartan manifold (i.e., a manifold with non-zero torsion). ≡ γ γ 6 − (gµν;α = Γ µαgγν Γ ναgµγ =0 → (Γµνα = Γνµα. In Sec. III, we define a field with the required feature − − − (4) as described above. Based on such a field, we explain Besides, the connection Γ is just the contorsion that the Unruh effect [18] can lead to what can be in- µνα tensor. In this case, the tangent space (as a Minkowski terpreted as a ‘third’ quantization. Then, we show that space with the Greek letters) is considered as the base a Rindler vacuum (as a coordinate transformation of the space, and the manifold M (with Latin letters) as the Minkowski vacuum) can be obtained by applying a ‘third’ fiber one. quantization operator (in terms of a statistical set, or a For a local infinitesimal transformation Λa on the partition function of, the second quantization fields) on b manifold M, a Minkowski vacuum. a a b Based on this approach, we also obtain a general co- δe µ =Λ be µ, (5) ordinate transformation of the Minkowski vacuum by ap- we obtain plying a ‘third’ quantization scalar field operator on the a a a b a b Minkowski vacuum. Subsequently, in Sec. IV, we indicate δω cµ Λ c,µ +Λ bω cµ ω bµΛ c. (6) that in the classical limit, the ‘third’ quantization field, ≃ − R regardless of whether it is a gravitational field or not, This transformation is a local GL(4, ) gauge transform- ation that is isomorphic to local U(1) SU(4) gauge plays the role of a Riemannian manifold on which the × second quantization fields are located. In Sec. V, we em- transformation. Such a transformation corresponds to ploy the proposed ‘third’ quantization fields to develop a diffeomorphism in the Riemann-Cartan geometry re- gauge theory of gravity. Based on this model, in Sec. VI lated to (3). We refer to the situation of case (4) as we show that some gravitational phenomena at different a ‘Minkowski-Cartan’ geometry, because the metric is scales can be explained without the need for dark mat- Minkowski but with a non-zero torsion. Such a geo- ter existence. Conclusions are drawn in the last section metry, in terms of degrees of freedom, corresponds to the where we also discuss the phenomenological implications Riemann-Cartan geometry with property (3). Accord- related to the cosmic microwave background (CMB) and ingly, this ‘Minkowski-Cartan’ geometry corresponds to ab the expansion of the universe. a principal U(1) SU(4) bundle, with ω µ as a connec- a × tion on it. e µ is a section of the associated vector bundle, and the symbol ‘;’ is the induced covariant derivative on it. II. MINKOWSKI METRIC WITH NON-ZERO CURVATURE AND TORSION III. QUANTUM INTERPRETATION OF COORDINATE TRANSFORMATION VIA A ‘THIRD’ QUANTIZATION PROCESS One of the controversial issues in curved space quantum field theory is about the definition of vacuum state. In general, as the notion of vacuum state in a Inspired by the well-known approach of Ref.
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