On the breakdown of space-time via constraint quantization of d\geq2 General Relativity
aa r X i v : . [ g r- q c ] F e b On the breakdown of space-time via constraintquantization of d ≥ General Relativity
Farrukh A. Chishtie
Department of Applied Mathematics, The University of Western Ontario, London,ON N6A 3K7, CanadaE-mail: [email protected]
February 2021
Abstract.
Based on the canonical quantization of d ≥ d ≥ Keywords: General Relativity, Quantum Gravity, Space-time
1. Introduction
The quantization of General Relativity (GR) leading to a full and coherent theory ofgravitation at quantum mechanical scales is an ongoing and active area of theoreticalphysics research. Quantum Gravity (QG), a theory which extends GR to such scales isbeing developed by a diversity of ways including quantum field theoretical approaches.In this regard, Dirac laid the foundation for quantizing gravity via his constraint n the breakdown of space-time of d ≥ General Relativity d ≥
2) itself. Inthe following section, we present a summary of results derived in [9-13], whereby, basedon these findings we propose that the loss of manifest covariance using the constraintquantization approach is due to its application on the full action. To support our claims,we present a counterexample regarding the successful quantization of non-Abelian gaugeYang Mills theories, where covariance is recovered in the path integral [13]. We thendeduce that this breakdown is actually a non-perturbative property of d -dimensionalGR theory ( d ≥
2) itself from the recovery of covariance and renormalizability fromlinearized versions (which have different constraint structures for 1EH and HGR actions[14,15]). This is recent work focusing on 1EH and second order (2EH) actions usingbackground and Lagrange multiplier fields and the path integral approach [18-25]. Thesefindings are consistent with existing work on space-time breakdown in GR by Penrose[26,27], whereby this breakdown in GR happens in the strong limit of the gravitationalfield of a black hole, and also with Effective Field Theory (EFT) results which hold atone loop order in the low-energy limit of the theory [28, 29]. Moreover, we find it toalso be consistent with the recent resolution of the Black Hole Information paradox byPage [30] and Marloff and Maxfield [31], which also indicates breakdown in the space-time fabric as an essential ingredient of their explanation. We also note that space-timeis an emergent feature, however there is loss of covariance in the high-energy (strongfield limit) of the theory, and that this property does not require thermodynamics as isrequired by entropic gravity [32].
2. Canonical quantization of d ≥ dimensional GR via canonical constraintformalism The d -dimensional 1EH action is defined as: L EHd = h µν (cid:18) G λµν,λ + 1 d − G λλµ G σσν − G λσν G σλν (cid:19) (1)where h µν = √− gg µν , g µν is the metric, G λµν,λ = Γ λµν − ( δ λµ Γ σσν + δ λν Γ σσµ ) and Γ λµν isthe affine connection. This choice of this action for QG is based on the property that it n the breakdown of space-time of d ≥ General Relativity d > Z [ J ] = Z D Φ D Π Dλ a Dκ a det { φ a , χ b } det / { θ a , θ b } δ ( χ b ) × exp i Z dx (cid:18) Π ∂∂t Φ − H c (Φ , Π) − λ a φ a (Φ , Π) − κ a θ a (Φ , Π) + J Φ (cid:19) (2)Here Φ are the canonical fields, Π are the conjugate momenta, H c is the canonicalHamiltonian, χ b is the gauge condition associated with the first-class constriants φ a , λ a and κ a are lagrange multipliers, while θ a and θ b are second-class constraints. Chishtieand Mckeon [9] found that these second class constraints were non-covariant, while thequantity p i q i − H c also displayed this breakdown in spacetime, rendering the path integralintractable and rendered any feasible calculation impossible. These non-standard ghostfields were also found in the case of scalar tensor theory by Chishtie and McKeon [10].We note that this same breakdown occurs in the d -dimensional action for theHamiltionan based GR (HGR) [3,4] which is: L Hd = √− gg αβ (cid:16) Γ µαν Γ νβµ − Γ ναβ Γ µνµ (cid:17) = √− gB αβγµνρ g αβ,γ g µν,ρ (3)where B αβγµνρ = g αβ g γρ g µν − g αµ g βν g γρ + 2 g αρ g βν g γµ − g αβ g γµ g νρ .For this action, it was found that the constraints are first class [11] for d > d = 2 dimensions [12] for which the following path integralby Fadeev [17] applies: Z [ J ] = Z D Φ D Π Dλ a det { φ a , χ b } δ ( χ b ) (4) × exp i Z dx Φ ∂ Π ∂t − H c (Φ , Π) − λ a φ a (Φ , Π) + J Φ ! Here, the gauge condition as well as the constraints are obtained to be non-covariant, and this in turn renders the path integral non-covariant as well. Interestingly,the canonical structure for the d = 2 1EH action was found to contain only first classconstraints, unlike for the d > d = 2 case, there is a loss of manifest covariance as for 1EHaction, hence this property applies for d ≥ n the breakdown of space-time of d ≥ General Relativity A aµ L Y M = − F aµν F aµν (5)with the covariant derivative, D abµ = ∂ µ δ ab + f apb A pµ and [ D µ , D ν ] ab = f apb F pµν , it is foundthat the constraints are first class. Upon application of the path integral for first classsystems, the following covariant path integral is obtained: Z [ J ] = Z DA aµ det( − ∂ µ D abµ ) exp i Z dx (cid:20) − F aµν F aµν − α ( ∂ · A a ) + J aµ A aµ (cid:21) (6)With this counter example, we want to illustrate that the constraint quantization doeswork out for covariant gauge theories such as Yang-Mills theory. Hence, the breakdownof covariance in the case of d ≥
3. Quantization of the linearized GR action
As shown in the previous section, there is loss of covariance in the full GR actions withthe application of the Dirac constraint formalism, however this is recovered in the weaklimit of these actions. Linearized GR actions are derived by expanding the field metrictensor around the flat metric as follows, h µν = η µν + κφ µν , (7)where h µν = h νµ , η µν is the flat metric ( η µν = diag (+ + + ... − ) and φ µν is theperturbation around this metric.In [14] and [15], the constraints derived for linear 1EH action and HGR action areshown to be different from those derived from the full theory action, and the resultingpath integral is shown to be covariant as a result.Recovery of covariance in the weaklimit of the metric tensor is an important finding, whereby this symmetry is restoredand field path integral quantization is indeed then made possible. Recently, McKeonand collaborators et al. have used background field theory approach in the weak limitof the gravitational field to render the 1EH and 2EH actions renormalizable, finite andrestricted it to one-loop order using background and lagrange multplier fields [18-25].These are also consistent with EFT results whereby quantum corrections are possible at1-loop order in the low-energy limit of 4-dimensional GR, however there is no indicationof breakdown in covariance in this approach, except that renormalizability beyond one-loop in this formalism is not possible [28,29]. n the breakdown of space-time of d ≥ General Relativity
4. Conclusions
In this letter, we propose three key findings based on the canonical structures and theloss of manifest covariance when quantization is conducted for various formulationsof d -dimensional GR. We firstly propose that this is a non-perturbative property ofthe theory itself (valid for ( d ≥ n the breakdown of space-time of d ≥ General Relativity
5. Acknowledgments
I thank Gerry McKeon for the useful comments, various discussions on this work andfruitful collaborations over the years.
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