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

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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 transformation 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. [19], from curved space is highly non-unique [11], defining a global the quantum field theory point of view, we first show vacuum state for it is out of reach. However, given that the concept of vacuum in the Minkowski spacetime is well-defined, if one can somehow replace the background 3 Most tensors become simple in this system, and it has the ability manifold with it, then it might open a way to bypass the to reflect important physical aspects of the spacetime however, problem of vacuum definition there. it does not alter reality. 3

L iω(ξ τ) R iω(ξ+τ) that the Unruh effect can lead to a kind of ‘third’ quant- +â †ωe − θ(t x)+â+†ωe θ(x + t) − − ization as vacuum quantization. For this issue, we rep- L iω(ξ+τ) +â † e θ( x t) , (11) resent a Rindler vacuum as a coordinate transformation +ω − − of a Minkowski vacuum. This task will be performed i where θ(x) is the Heaviside step function. The corres- by defining a ‘third’ quantization operator in terms of a ponding annihilation field,ϕ ˆ =ϕ ˆ , and the Hamiltonian partition function of the second quantization fields, which a c† is given by acts on the unique Minkowski vacuum constructed in the previous section. Based on this procedure, we obtain a ∞ ˆ R R R R L L L L general coordinate transformation of the Minkowski va- H = ω aˆ+†ωaˆ+ω +â †ωaˆ ω +â+†ωaˆ+ω +â †ωaˆ ω , cuum by applying a ‘third’ quantization scalar field to − − − − ω=0 the Minkowski vacuum. We interpret this procedure as a X (12) ‘third’ (or, vacuum) quantization. This approach defines where the zero-point energy has been omitted. Moreover, a way for representing gravitational connections in terms the second quantization independent operators for the of a partition function of the second quantization fields. Minkowski vacuum (say, ˆb ω and its Hermitian adjoint) Hopefully this should pave a way towards a quantum have also been defined [18]± in terms of the second quant- process for gravitation. ization operators of the Rindler frame with a uniform In this regard, it is also well-known that in an accel- proper acceleration, erated reference frame, in the right and left sides of the ˆ πω/α R Rindler wedges with a uniform proper acceleration (say, b ω 1 1 e− aˆ+ω − = − . α) as ˆ 2πω/α πω/α L b+† ω! √1 e− e− 1 ! aˆ+†ω! 1 αξ 1 αξ − − R : (x, t) = (α− e cosh ατ, α− e sinh ατ) (13) 1 αξ 1 αξ (7) Now, we intend to find out the inverse relation of (8), ( L : (x, t) = ( α− e cosh ατ, α− e sinh ατ), − − i.e. getting the Rindler vacuum with a uniform proper the Minkowski vacuum appears as a thermal bath, acceleration from the Minkowski vacuum. Accordingly, where (ξ, τ) and (x, t) are respectively the Rindler and similar to the technique used in Ref. [19] in obtaining Minkowski spacetime coordinates. Indeed, in Ref. [19], relation (8), we indicate that the Rindler vacuum can via the Bogolyubov transformation [20], the Minkowski be achieved in terms of the Minkowski-Fock space in the vacuum has been shown to be equivalent to a gas of the form of Rindler quanta in a thermal equilibrium. Accordingly, in two dimensions, the relation between 0 >= Φˆ † 0 >, (14) | R,α α| M the Minkowski vacuum and the Rindler vacuum with a uniform proper acceleration has been represented as [19] with

0 >= Ψˆ 0 >, (8) ∞ M α R,α ˆ πω/α ˆ ˆ | | Φα† exp iW ∗ e− b+† ωb† ω (15) ≡ − − − ~ ˆ " ω=0 # where (in the natural unit c =1= ) operator Ψα, in X the Rindler space, is and the corresponding Hamiltonian

ˆ ∞ πω/α R L R L Ψα exp iW + e− aˆ+†ωaˆ+†ω +â †ωaˆ †ω . ∞ ≡ − − ˆ ˆ ˆ ˆ ˆ " ω=0 # H = ω b+† ωb+ω + b† ωb ω , (16) − − X (9) ω=0 X Here, the term eiW is the vacuum persistence amplitude with [19] where again the zero-point energy has been omitted. To prove the claim of relation (14), it would be sufficient to ( 2π/α)H 2 Im(W ) = ln[tr e − ] = (πα/6)δ(0), show that the act of any of the corresponding annihilation (10) operators on this Rindler vacuum vanishes. For this task Re(W ) α=0 = 0 and Re(W ) α=0 = . | | 6 ∞ and without loss of generality, for instance, we perform R the procedure for the annihilation operatora ˆR , wherein, The argument of the Dirac delta function is energy,a ˆ+†ω, +ω L R L by transformation (13), we have aˆ+†ω,â †ω anda ˆ †ω are respectively the right and left parts − − of left (+) and right ( ) moving creation operators (with ˆ ˆ πω/α − 4 b ω + b+† ωe− energy ω) of a massless scalar field ϕˆ =ϕ â +ϕ ˆc. The R ˆ aˆ+ω 0R,α >= − Φα† 0M > . (17) corresponding creation field, in the Rindler frame, is | √1 e 2πω/α | − − dω R iω(ξ τ) ϕˆ (ξ, τ) aˆ † e θ(x t) Then, by employing definition (15), it is straightforward c √4πω ω − ≡ − − to show that relation (17) vanishes. ´ h On the other hand, in Ref. [19], it has been indicated that the Minkowski- and Rindler-Fock spaces are unit- 4 For simplicity, we have chosen a massless scalar field. arily inequivalent. We furthermore prove that every two 4

Rindler’s vacua with different uniform proper accelera- respectively.5 In this way, we are able to define a general tions are also orthogonal to each other. For this purpose, coordinate transformation of the Minkowski vacuum us- via relations (14), (15) and (16), we achieve ing these scalar ‘third’ quantization operators in terms of a statistical set of the second quantization fields, which ′ ˆ ′ ˆ < 0R,α 0R,α >=< 0M Φα Φα† 0M >= can apply the desired acceleration to a particle of the ′ ∗| ′ ∗ | | ′ iRe(W W ) Im(W +W ) π(1/α +1/α)H e − − tr e− . (18) second quantization fields. Indeed, this transformation of coordinates deforms spacetime for the second quant- h i Then, by using the simple relations ization fields. Since the act of operator (24) on the Minkowski vacuum produce a scalar field, such a field 4 ( 2π/α)H ∞ 1 does not cause any torsion or curvature. Of course, it is tr e − = (19) 1 e 2πω/α clear that the relations of the ‘third’ quantization fields ω=0 − Y − with each other are similar to the relations of the second and quantization fields with each other. However, in the next section, while introducing a constant h (analogous with 2πω/α 2πω/α′ the Dirac constant in the second quantization fields), we (1 e− )(1 e− ) − πω/(1/α−+1/α′) < 1, (20) indicate that the states obtained by acting (24) on the 1 e− " p #α=α′ Minkowski vacuum, in the classical limit h 0, play − 6 the role of a Riemannian manifold on which the→ second and relations (10), relation (18) reads quantization fields are located. To generalize the presented formulation to four dimen- < 0 ′ 0 >= δ ′ . (21) R,α | R,α α α sional spaces, one can simply use the procedure of the Moreover, due to relation (10) and definition (15), it is quantum field theory given in Ref. [18]. In this regard, consider spaces with coordinates (x, x ,t) and (ξ, x , τ) clear that as ⊥ ⊥ ′ ∗ ′ ∗ iRe(W W ) Im(W +W ) < 0M Φˆ † ′ Φˆ α 0M >= e − − =0. (22) 1 αξ 1 αξ α R : (x, x ,t)=(α− e cosh ατ, x , α− e sinh ατ) | | ⊥ ⊥ 1 αξ 1 αξ L : (x, x ,t)=( α− e cosh ατ, x , α− e sinh ατ), Therefore, the Minkowski vacuum and the Rind- ( ⊥ − ⊥ − ler vacua with different uniform proper accelerations, (25) not only are perpendicular to each other, but each of where ξ is a spatial dimension in the direction of proper the latter ones (as ‘third’ quantization states) can also acceleration and the components of x denote the other be obtained via the defined operator (consisted of a stat- ⊥ two spatial dimensions. Hence, relations (11), (13) and istical distribution function of the second quantization (15) will respectively change to operators), which acts on the Minkowski vacuum. Thus, these vacua form a set of orthogonal bases for their cor- R † † ϕˆc(ξ, τ, x ) dω aˆ ωk⊥ v ωk⊥ θ(x t) responding Fock space with operators Φˆ † and Φˆ α as the ⊥ ≡ − − − α L ´ R † † h † † creation and annihilation operators with relation +â ωk⊥ v ωk⊥ θ(t x)+â+ωk⊥ v+ωk⊥ θ(x + t) − − − L +â † k v† k θ( x t) , (26) ˆ ˆ ′ +ω ⊥ +ω ⊥ < 0M [Φα, Φα† ′ ] 0M >= δαα . (23) − − | | i Accordingly, we have established a kind of ‘third’ (or, R πω/α ˆ aˆ+ω, k⊥ 1 e− b ω, k⊥ vacuum) quantization procedure. − = 1 − − L −2πω/α πω/α aˆ † k √1 e e 1 ˆb† As every second quantization operator is usually in- +ω,+ ⊥ ! − − ! +ω,+k⊥ ! dexed with a momentum in a certain direction, these (27) ˆ ˆ ‘third’ quantization operators Φα† and Φα have also been and indexed with a uniform proper acceleration in its corresponding direction. Analogously, a vacuum state, in which ∞ πω/α ˆ ˆ† ˆ† Φα† exp iW ∗ e− b+ω,+k⊥b ω, k⊥ . (28) each point has a different acceleration, can be obtained ≡ − − − − ω=0 via the act of an operator on the Minkowski vacuum. In P Here two dimensions, such a Hermitian operator can also be 1 ˆ 2 k formed from the Fourier transformation of operators Φα† sinh(πω/α) ⊥ αξ i(k⊥.x⊥ ωτ) k | | v ω ⊥ 4π4α K(iω/α)( α e ) e ± and Φˆ α as ± ≡ h i (29) ˆ ∞ ˆ i(αx qt)/c2 ˆ i(αx qt)/c2 with K (µ) as the modified Bessel function, and k as Θ(x, t)= Φα† e − + Φαe− − , (ν) ⊥ α= the corresponding wave vector [18]. −∞ h i X (24) where α/c2 and q/c2, in this presented ‘third’ quantization, are analogous with the wave number and the 5 2 3 angular frequency in the second quantization fields, Obviously, the usual unit of q is (lenght) /(time) . 5

IV. RELATION BETWEEN THE ‘THIRD’ AND where the emerged metric is SECOND QUANTIZATION FIELDS i µν 1 1 p /m √ gg b− j − ij i j 2 Consider a statistical set of particles of the second − ≡ p /m (ρ0b/m)δ + p p /m − − quantization fields, which is specified in the form of a (35) partition function similar to (15) as a scalar ‘third’ quant- and p = ▽S0. This result reveals that a perturbation ization field with the creation and annihilation operators in a scalar ‘third’ quantization field (e.g., field Θ), as a (ˆb†, ˆb) and the action second quantization field, is located on the emergent of a Riemannian manifold.10 The maximum possible speed ∞ ˆ ˆ ˆ ˆ (i.e., the usual c) of such second quantization field turned S b+† ωb† ω+ b+ωb ω . (30) 2 ∝ − − out to be c ρ0b/m, as the realization of interaction ω=0 ≡ X properties of a scalar ‘third’ quantization field. This is- If an extra particle of the second quantization fields is sue is similar to the sound subject,11 in which the speed added to this set, action (30) will slightly alter by δS. of sound is the realization of the interaction properties of Given the similarity of exp ( iS/h) with operator (15), the second quantization fields [30]. Indeed, the properties the relation between a particle− of the second quantization (such as temperature and mass) of the second quantiza- fields and a scalar ‘third’ quantization field can be com- tion fields constitute the properties of sound propagation parable to the relation between δS and the pilot wave environment. exp( iS/h) in the ‘pilot wave theory’6 [21]. To clarify Also, it is known that the properties of elastic envir- some− aspects of this comparison, let us consider a well- onments and their sound waves (as perturbation of the known example in the acoustic black hole topic as follows. second quantization fields) are the realization of the prop- Analogous with the Gross-Pitaevskii equation [22, 23], erties of the second quantization fields in a scale of energy. we envisage that the ground state of a quantum system Accordingly, while considering the last two sections, ana- of identical bosons is described as7 logous with the sound waves and through the presented ‘third’ quantization point of view, we envisage that the h2▽2 ih∂ + b Θ 2 Θ(x, t)=0, (31) standard model of particle physics and its parameters t 2m − | | would emerge from the properties of the ‘third’ quantiz- 12 where Θ is a scalar ‘third’ quantization boson field, ation fields in the corresponding scale of energy. m is the mass of field Θ and b is representative of the self-interaction power of Θ with the usual unit (lenght)2/(time)3. Afterwards, let us purposely set Θ V. MODELING ‘THIRD’ QUANTIZATION ≡ FIELDS √ρ exp( iS/h) and perturb it as ρ ρ0 + ερ1 and − 8 → S S0 + εS1. Hence, by substituting these perturba- tions→ into Eq. (31), while using the ‘pilot wave theory’ In Sec. III, we have introduced a ‘third’ quantization and working in the classical limit h 0 (i.e., neglecting scalar field, which due to the scalar nature of it does not the quantum potential term), in the→ first approximation, cause any curvature and/or torsion. Now in this sec- we achieve the corresponding Hamilton-Jacobi and the tion, we intend to make a gravitational model for the continuity equations, namely ‘third’ quantization fields in such a way that it would have the following properties. First, we require that it p. ▽ S1 possesses the ‘Minkowski-Cartan’ geometry presented in ∂ S bρ = 0 (32) t 1 − m − 1 Sec. II. Second, due to the success of the Weyl grav- and ity as a renormalizable gravity theory [37] in explaining solar [38], galactic [39, 40], extra-galactic and cluster ∂tρ1 ▽. (ρ0 ▽ S1 + ρ1p) /m =0. (33) scales [41] gravitational phenomena, we want to have a − structure and solutions similar to the Weyl-Cartan the- 9 Then, by substituting Eq. (32) into Eq. (33), it gives ory. µν We also want the model to indicate similarities between ∂µ √ gg ∂ν S1 =0, (34) − a rotating black hole or a rotating star with fermionic

6 In Ref. [21], this theory has been used to illustrate the relation between the first and zeroth quantization or the classical state. 10 In Refs. [25–28], instead of employing the Gross-Pitaevskii equa- 7 This equation is a non-relativistic equation with respect to a tion, other kind of equations have been used. However, their maximum speed related to the ‘third’ quantization fields (say C). final results are similar to those obtained in this section. In another research [24], we are working on the parameters h and 11 In Ref. [29], we have examined similarities between the elastic C via physical constants. waves and the second quantization fields. 8 Obviously ρ = |Θ|2, where it can analogously be interpreted as 12 It is accepted that those constants that play a fundamental role the corresponding probability density, and S is as the corres- in the standard model of particle physics (such as the fine struc- ponding action with the unit of h. ture constant) vary with the spatial and temporal changes of 9 The i and j indices are assumed to run from 1 to 3. gravitational fields [31–36]. 6

ab elementary particles. Indeed, there is a not-so-new hy- where igAµ /(hC) is a ‘third’ quantization connection pothesis that black holes and elementary particles are gauge field with its related SU(4) group symmetry in- comparable. In the older viewpoints, e.g. Refs. [42–44], dices a and b, ieAµ/(hC) is a ‘third’ quantization U(1) efforts were made to show that elementary particles are vector gauge field, g and e are coupling constants (i.e., some kind of black holes. However more recently, altern- SU(4) and U(1) charges, respectively). Besides, ative views have emerged [45–47] that hypothesize black ab ab ab ab C holes behave like elementary particles via the double copy Dµ = δ ∂µ i(gAµ + eδ Aµ)/h (40) − theory. Finally, we intend to have a kind of symmetry µ similar to the one introduced in Ref. [48], proposing that is the gauge covariant derivative and γ ’s are the Dirac small scales in physics can simulate large scales. matrices. Indeed, this Yang-Mills theory is a principal ab C C In this regard, first there is a correspondence U(1) SU(4) bundle with igAµ /(h ) and ieAµ/(h ) as the connections× on it, ψa as a section of an associated between rotating black holes and fermionic elementary µ ab particles [42–47], and second, the rotation wave of mi- spinor bundle and γ Dµ as the induced Dirac operator of the induced covariant derivative on the associated crostructures of a granular medium has a fermionic be- a havior [49–54]. Accordingly, while assuming that stars bundle. Now, if we make a map between the section ψ of the spinor bundle in this Yang-Mills theory and the sec- are microstructures of cosmic structure as a granular me- a dium, we consider stars similar to fermionic fields.13 In tion e µ of the vector bundle in the ‘Minkowski-Cartan’ this way, the similarity between a fermionic fundamental geometry, we can state that the presented Yang-Mills the- particle and a star can be guaranteed. On the other ory also has the ‘Minkowski-Cartan’ geometry. hand, as stars constitute a statistical set of particles of In this regard, we employ an auxiliary constant fermi- the second quantization fields, it is plausible to assume onic Dirac spinor (say, ι) and its conjugate, ¯ι, with the those as ‘third’ quantization fermionic fields. In addi- conditions tion, since we have chosen a gauge theory for the model, ¯ι ι =4 and ¯ιγ ι =0. (41) we presume that a gauge connection (as a gravitational α field) can change the speed and movement of stars. Accordingly, if we consider At this stage, given that we want a model similar to a a the Weyl gravity, by considering the structural similarity e µ ¯ιγµψ /ψ and eaν ψ¯aγν ι/ψ (42) between the (general) Yang-Mills theory and the Weyl- ≡ ≡ Cartan gravity [55–58], we choose a Lagrangian akin to respectively as a complex tetrad field and its conjugate, the Yang-Mills theory based on symmetry group U(1) then, by using (13), (37), (38) and the Fierz identit- SU(4) as × ies [59], those will lead to

C ¯ µ a C a b a a a YM = ψa (i hγ Dµ b m δb ) ψ (e e )= η and ψ¯ ι ¯ι ψ = ψ¯ ψ . (43) L − µ aν µν a a 1 1 (36) ℜ G abGµν F F µν . 14 − 4 µν ab − 4 µν Hence, the corresponding torsion field turns out to be In this Lagrangian, ψa is a spinor fermionic ‘third’ quant- T a D a eb D a eb . (44) µν ≡ ν b µ − µ b ν ization field and ψ¯a is its conjugate (both as the field of stars, analogous with the quark-like particles) with com- Thus, we have stablished a map between the section ψa of pleteness relation the spinor bundle in the presented Yang-Mills theory and the section ea of the vector bundle in the ‘Minkowski- a ¯ 2 I µ ψ ψa/ψ = /4, (37) Cartan’ geometry. which lead to Before we continue, let us highlight that Lagrangian (36) is similar to that mentioned in Refs. [55–57] for the a 2 ψ¯aγµγν ψ /ψ = ηµν , (38) Weyl-Cartan theory. For this purpose, we make a map between the fields of Lagrangian (36) and the fields of ¯ a ab where ψ = ψaψ . Also, Gµν and Fµν represent the the Weyl-Cartan theory as curvature or strength field tensors, namely p g ab ab ab ab ab g a cb a cb i CAµ ω µ, Gµν =∂µAν ∂ν Aµ + i C Aν cAµ Aµ cAν , h → − h − (39) e (45) i A K , Fµν = ∂µAν ∂ν Aµ, C µ µ − h →

13 In Ref. [29], we have shown that the spin wave of microstructures 14 of an elastic medium behaves similar to a fermion wave. Hence, Given the U(1) gauge symmetry in the presented model, relation the rotation of a microstructure corresponds to the spin of a (44) under the U(1) gauge transformation can also be written as fermion particle. Here, in comparison, we have considered that a −1 a a b T µν = ψ ¯ι (Dν bγµ − Dµ bγν ) ψ . the spin of stars and planets correspond to the spin of fermionic particles. 7

ab where ω µ is the spin connection as the Lorentz gauge and group, Kµ is a Weyl vector as the dilatation gauge. Thus, a b a b ν c the curvature tensors of the corresponding connections ( Dν bDµ c Dµ bDν c γ ψ /ψ − can be defined as C 2 (52) m ˆαγβ ˆ a g = S Sαγβγµψ /ψ, Ω ab i G ab, 2h2 µν ≡ hC µν (46) e where SˆαγβSˆ =3h2I/2 has been used. By multiplying Ωµν i CFµν ∂µKν ∂ν Kµ. αγβ ≡ h → − ¯ι into Eqs. (51) and (52), while using definitions (42) and a Accordingly, let us consider the Lagrangian S µν , we will have C C C µν a m µν a a b a b m a = S T + S S Dν be µ Dµ be ν = S µν (53) LYM 3 a µν h2 a µν − h2 (47) h2C2 h2C2 + Ω abΩµν + Ω Ωµν , and 4g2 µν ab 4e2 µν 3(mC)2 µν αµν a α a a b a b cν a where S = ψψ¯ γ Sˆ ι, S = ¯ιγ Sˆ ψ /ψ, Dν bDµ c Dµ bDν c e = e µ. (54) a a α µν αµν − 8 ˆ h σµν = i [γµ,γν ] /2 and Sαµν γα, σµν /8 is the corresponding spin tensor operator.≡ This{ Lagrangian} resembles Now, due to relations (45) and (46), Eqs. (53) and (54) the one mentioned in Refs. [55–57] for the Weyl-Cartan are comparable to the Einstein-Cartan equations.17 theory.15 Now, by using relation (37) and the Dirac equa- Moreover, it is obvious that Eq. (54) corresponds to tion obtained via the variation of Lagrangian (36) with the Einstein equation with constant curvature (see e.g., a respect to ψ¯a and ψ , and making some manipulations, Ref. [17]), and hence some solutions of Eq. (54) for one obtains four-velocity of fermionic fields can correspond to the µ a b ¯ ¯ µ a bI Schwarzschild-de Sitter solution [60] for four-velocity of γ Dµ bψ ψa = ψaγ Dµ bψ /4 (48) test particles. Also, as Lagrangian (36) and Eqs. (53) and and (54) are U(1) gauge invariance, the proposed Yang- a Mills theory results a U(1) transformation of tetrad e µ b a ¯ µ a ¯ µ b ψ Dµ bψaγ = Dµ bψa γ ψ I/4. (49) (in analogue with a conformal transformation in the Weyl-Cartan gravity) that is also the solution of Eq. Then, by employing relations (37), (41), (48), (49), and (54). Thus, in addition to the Schwarzschild-de Sit- the Fierz identities, one can show that (36) is the same ter solution, the Mannheim-Kazanas solution [61] is also as (47). valid.18 With these explanations, the similarity between In what follows, we show that the equations obtained the Mannheim-Kazanas solution and the phenomeno- from the variation of Lagrangian (36) are also related logical potential between quarks (as represented fields to the ones obtained from the Einstein-Cartan theory. based on the (general) Yang-Mills theory) is plausible, To perform this task, first by factoring out γµ from the as pointed in Ref. [61]. obtained Dirac equation, we will have16 The difference between the presented Yang-Mills the- mC ory and the Weyl-Cartan theory is that the former D a ψb = i γ ψa. (50) µ b − 4h µ possesses the ‘Minkowski-Cartan’ geometry whose base space is the Minkowski space and its fiber space is a gen- Then, by constituting the wedge product of γµ and Dµ eral manifold with U(1) SU(4) symmetry. Whereas, the into Eq. (50), and making some manipulation, one gets latter has the Weyl-Cartan× geometry whose base space is mC a general manifold with torsion and the Weyl connection, ( D a γ D a γ ) ψb/ψ = σ ψa/ψ ν b µ − µ b ν 2h νµ and its fiber space is the Minkowski space. Topologically, (51) mC mC = γ , σ γαψa/ψ = Sˆ γαψa/ψ 8h { α νµ} h2 ανµ

17 However, these two equations are more comprehensive than the Einstein-Cartan equations. This comprehensiveness is due to the 15 The diﬀerence between Lagrangian (47) and the corresponding fact that these two equations also include the Weyl connection. one mentioned in Ref. [56] is that in Lagrangian (47) the spin In addition, the connection in the Einstein-Cartan gravity is a replaced by the curvature of the special conformal transformation spin connection, whereas in these equations, the connection is vector and the square of the spin (as the matter part) has been a gauge connection of the SU(4) symmetry group. Thus, the added to it. In addition, the symmetric group of the gravitational solutions of the Einstein-Cartan gravity equations in this partic- connection in Ref. [56] is SO(1, 3) while, in Lagrangian (47), it ular example can be regarded as a special case of the solutions is U(1) × SU(4). of these two equations. 16 Note that, every solution of Eq. (50) is also a solution of the ob- 18 In Ref. [62], it has been shown that the Schwarzschild-de Sitter tained Dirac equation, whereas any solution of the Dirac equation and the Mannheim-Kazanas solutions are related via a conformal is not necessarily a solution of Eq. (50). transformation. 8

the total space of the presented Yang-Mills theory cor- should be relative to the speed C), one possible method responds to the total space in the Weyl-Cartan gravity, is to display the potential as a Fourier transform of the where its fiber and base spaces have been displaced. propagator of the connection field in the real time form- It is known that one of the major advantages of the alism [67, 68]. Another method, while using the Wilson (general) Yang-Mills theory over the Weyl-Cartan theory loop [69, 70] in the lattice field theory, is to display the and other gravitational theories is that, in general, it is potential as a well-tested theory and its phenomenological behaviors ∂ W (r, t) are well-known, see e.g., Refs. [63, 64]. Indeed, the result V (r) = lim t , (57) t of calculations in this theory corresponds to the result of −→∞ W (r, t) observations with good accuracy [65, 66]. In this respect, in the next section, we use the methods of the (general) wherein the Wilson loop is defined as Yang-Mills theory to obtain the effective potential func- µ ab tion for the fields of the presented Yang-Mills theory to W (r, t) exp dx Aµ , (58) ≡ P ˛ explain the trajectory of stars in the gravitational fields of galaxies and the gravitational lensing. with as the path-ordering operator. AsP a few examples (that can also be applied to the presented theory as a Yang-Mills one), in addition to VI. GALAXY ROTATION CURVES AND some interesting anisotropic solutions [67], the real part GRAVITATIONAL LENSING of some isotropic potential functions of the non-Abelian Yang-Mills theory (that has been obtained through the In this section, we first address the gravitational poten- calculations of the methods mentioned in Refs. [67–73]) tial obtained from the Yang-Mills theory presented in the are previous section. Due to Eqs. (53) and (54), the traject- α µr σ µr ory of a fermion (as a star) in the presented Yang-Mills (V (r)) = C e− + (1 e− ), (59) theory can correspond to the trajectory of a test particle ℜ − r µ − in the Weyl-Cartan theory of gravity. That is, for the mentioned analytical solution, the trajectory of a fermion α µr 2σ µr µr (V (r)) = C e− + (1 e− ) σre− (60) in the former theory corresponds to the trajectory of a ℜ − r µ − − test particle in the Mannheim-Kazanas space. In this regard, the static symmetrical metric of the Mannheim- and Kazanas solution (as an analytical solution of the Weyl 2 2 α 2σ/µ µr 2σ/µ conformal gravity) for a point particle is (V (r)) = C − e− , (61) ℜ − r − r 2 2 1 2 2 2 2 2 ds = f(r)dt f − (r)dr r (dθ + sin θ dφ ) (55) − − with temperature dependence parameters C, α, µ and σ. Besides, the parameter µ has been known as the Debye with the gravitational potential g as 00 mass, which is the result of the Debye screening phe- β nomenon [71]. The Debye mass, at a temperature below f(r)= V + γr λr2. (56) 0 − r − a certain temperature (or at a radius less than a certain radius), is equal to zero, in which case the potential func- Here V0, β, γ and λ are some constants of integration. It tion of (59), (60) and (61) becomes the so-called Cornell has been shown that such a potential can explain the (or funnel) potential (as one of the most popular poten- gravitational phenomena from solar system to cluster tial models) scale without using the concept of dark matter [37–41]. However, in a Yang-Mills theory, the screening and con- α′ 19 V (r)= C′ + σ′r. (62) finement effects are important aspects of the theory. In − r what follows, we address the deviation of the effective potential function from the analytical solution (56) and/or However, in the other areas, the potential changes lin- the Cornell potential (see (62)). early with temperature [67–73]. Nevertheless, with very In the literature, for obtaining a suitable potential small amounts of the Debye mass, the shape of their po- for the (general) Yang-Mills theory, the non-perturbative tential functions is in the form of (56). methods of Yang-Mills theory (such as lattice field The imaginary part of the potential function of (59), theory) have been employed. Actually, in the non- (60) and (61) has been attributed to the Landau damping relativistic limit (note that in our case, such a limit phenomenon [74], which–while assuming that the above potentials are valid in the presented Yang-Mills theory– its qualitative consequences analogously cause formation or destruction of stars in our case. However, in general, 19 Given the similarity between the presented Yang-Mills theory the value of the imaginary part of the potential is less and the Weyl-Cartan one, such effects are expected for the Weyl- than the real part of it. In addition, at short distances, Cartan theory as well. its value is close to zero and increases with increasing 9 distance [67–73]. In comparison to the presented the- hadron consists of a number of valence quarks and a large ory, such a result indicates that no star formation would number of seaquarks that are float in the viscous sap of occur near the center of a galaxy, and it should take gluons [63, 64]. In a proton as a hadron, most of the intra- place farther away from the center of galaxy. Actually, proton pressure is generated by the field of gluons (as it has been observed that the H II regions (wherein star connection fields), and sea quarks have a much smaller formation takes place) are in the arms of spiral galax- role in generating intra-proton pressure [66]. Given the ies or around irregular galaxies [75, 76]. It is noticeable similarity between the presented Yang-Mills theory and that these results are not limited to the SU(3) symmetry the (general) Yang-Mills theory, by qualitatively compar- group [77–79]. ing a galaxy and its stars with a hadron and sea quarks Let us now examine a specific case in almost more de- within it, such a property of intra-proton pressure would tails. As mentioned above, the potential of the presented be similar to property of dark matter, which contains the Yang-Mills fields behaves like the analytical solution (56) largest share of mass in a galaxy and causes the gravity and/or the Cornell potential in a small radius. In con- rotation curves. nection with the quantitative explanation of the gravitational lensing phenomenon on a larger scale than clusters (although this potential is somewhat successful [80–83]), VII. CONCLUSIONS AND OUTLOOK in Ref. [40], it has been shown that its calculated value is slightly higher than its observed value. This result means We have introduced a new geometry with non-zero that as distance increases, the potential function deviates local curvature and torsion with a Minkowski metric from the analytical solution (56) and/or the Cornell po- tensor. This is named ‘Minkowski-Cartan’ geometry. tential. Indeed, it can be considered that, with increasing The tangent bundle over the Riemannian-Cartan man- distance (and hence with increasing potential), the effect ifold M has been replaced with an M bundle over the of vacuum polarization appears and the potential beha- Minkowski space. In this way, the total space has not vior deviates from the analytical solution (56) and/or the changed but the spin connection is not necessarily an- Cornell potential. Such an effect of vacuum polarization tisymmetric and the connection becomes a contorsion would become apparent when the vacuum non-Abelian tensor. Since the base space is Minkowski, then a unique permittivity value is different from one [67, 70, 84–86]. and well-defined vacuum can be constructed. Therefore, for the effective potential in the momentum We have shown that the quantum of gravitational con- space, one makes [67, 70, 84–86] nection fields cannot be realized via second quantization. V (k) Due to this issue, we have defined a field in such a way V (k) V˜ (k)= (63) → ε(k,T ) that its quantum is equivalent to a statistical set or a partition function of second quantization fields’ quanta. This where T is temperature, k is momentum, V (k) is the ana- result is then used to show that the Unruh effect leads to lytical solution of potential, and the non-Abelian permit- a ‘third’ quantization as vacuum quantization. In this re- tivity is [84, 86] gard, we have indicated that the Minkowski vacuum and the Rindler vacua with different uniform proper acceler- ΠL(0,k,T ) ε(k,T )=1+ 2 , (64) ations form an orthogonal basis of a Fock space, which k are related to each other by scalar ‘third’ quantization ˆ ˆ wherein ΠL(0,k,T ), in its simplest case, is the static limit creation and annihilation operators (i.e., Φα† and Φα). of the longitudinal connection gauge self-energy, which In this way, we are able to define a general coordinate is equal to µ2 [84]. Now, if we assume V (k) being the transformation of the Minkowski vacuum via acting the ˆ ˆ Cornell potential (62) in the momentum space (as a spa- Fourier transform of Φα† and Φα on the Minkowski va- cial case of solution (56)), then the Fourier transform of cuum. Using the ‘pilot wave theory method’ and working V˜ (k) in relation (63) will be equal to (61). On the other in the classical limit (i.e., neglecting the quantum poten- hand, in this case, solution (61) is the same as the potential term), the ‘third’ quantization scalar fields play the tial used in the scalar-tensor-vector theory (MOG) [87], role of Riemannian manifold on which the second quant- wherein it has been claimed that this potential is capable ization fields are located. of explaining the gravitational phenomena [88–94]. Given that these ‘third’ quantization scalar fields lack An interesting aspect of the presented Yang-Mills the- curvature and torsion, we build an U(1) SU(4) Yang- ory is that the connection fields can form a bound state Mills model of gravity. This model corresponds× to the on their own (something like glueballs in the QCD) Weyl-Cartan gravity based on general covariance. An without any need for fermionic particles (stars). Thus, analytical solution for the ‘third’ quantum field’s particle the connection fields in the presented Yang-Mills theory (such as a star) trajectory of the model corresponds to can be a good alternative to dark matter. For more the trajectory of a test particle in the Mannheim-Kazanas explanation, we refer to an example in the subject of space and/or the Cornell potential. Solutions for a poten- QCD. As an example of a phenomenon described via tial of the model addressing large scales could be obtained the (general) Yang-Mills theory, one can refer to pro- by using loop corrections and non-perturbative, lattice tons. It has experimentally been determined that each gauge theory, results. Essentially, such solutions corres- 10 pond to modified gravity models capable of explaining The CMB fluctuations would be indicative of microstates galaxy rotation curves and gravitational lensing. and microstructures of the ‘third’ quantization fields. Based on (65), the universe can either be static for the Perspectives on Cosmology— A ‘third’ quantiza- case of a completely real TQF or dynamic (expanding or tion field is represented by a statistical set of the second collapsing) for a complexL Lagrangian, as quantization fields or a partition function of those. Ana- logously, the universe could be represented by a partition + i V function of the ‘third’ quantization fields with a Lag- LTQF −→ LTQF Im rangian (of stars and the gravitational field among those), t (66) Ψ exp dt V . say , such that | universe|∝ ˆ Im LTQF t0 t The Hubble law confirms an expanding universe at the Ψ exp i dt . (65) universe ˆ TQF ∝ − t0 L present era. Now, as mentioned in Sec. IV, the ‘third’ quantization fields play the role of a Riemannian mani- In this manner, and since the CMB radiation is a global fold for the second quantization fields, such as photons. image of the statistical universe, then this radiation Therefore, the expansion of the universe implies the ex- would also be a statistical state. Indeed, it is well-known pansion of the Riemannian manifold for photons of the that the CMB intensity in terms of frequency is equi- CMB, which leads to a decrease in the CMB temperat- valent to the chart of a black-body radiation. On the ure. other hand, the number density of CMB as a black-body radiation [95] and the number density of a gas of the Rindler quanta at thermal equilibrium [19] are quantit- atively equivalent. As mentioned in Sec. III, the Rindler ACKNOWLEDGEMENTS quanta are partition functions of the second quantization fields. Therefore, the CMB should also be a partition We thank the Research Council of Shahid Beheshti function of the second quantization fields (i.e., photons). University.

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