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A Vacuum of Quantum Gravity That Is Ether Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 24 May 2021 doi:10.20944/preprints202105.0576.v1 Article A Vacuum of Quantum Gravity That is Ether Sergey Cherkas 1,† 0000-0002-6132-3052 and Vladimir Kalashnikov 2,† 0000-0002-3435-2333 1 Institute for Nuclear Problems, Bobruiskaya 11, Minsk 220030, Belarus; [email protected] 2 Dipartimento di Ingegneria dell’Informazione, Elettronica e Telecomunicazioni, Sapienza Universitá di Roma, Via Eudossiana 18, 00189 - Roma, RM, Italia; [email protected] * Correspondence: [email protected] † These authors contributed equally to this work. Version May 19, 2021 submitted to Symmetry 1 Abstract: The fact that quantum gravity does not admit a co-variant vacuum state has far-reaching 2 consequences for all physics. It points out that space could not be empty, and we return to the notion 3 of an ether. Such a concept requires a preferred reference frame for, e.g., universe expansion and black 4 holes. Here, we intend to discuss vacuum and quantum gravity from three essential viewpoints: 5 universe expansion, black holes existence, and quantum decoherence. 6 Keywords: vacuum energy; preferred reference frame; vacuum state; quantum gravity 7 1. Introduction. 8 From the earliest times, people are thinking how empty is a "nothing," i.e., a space-time, which 9 contains no matter. Some believe that a "nothing" is unreachable, and space-time always contains 10 "something," i.e., ether [1]. If ether is like some matter, then there must be a reference frame at which 11 ether is at rest "in tote," i.e., it represents some stationary medium. After the development of the 12 quantum field theory (QFT), it was founded that the vacuum actually contains a number of virtual 13 particles-antiparticles pair which are created from vacuum and then annihilated during the time 1 14 Dt = m , where m is a particle mass. That leads to the experimentally observable effects such as 15 anomalous electron magnetic moment, Lamb shift of atomic levels [2] and proves that a vacuum is not 16 empty. However, a soup of the virtual particles-antiparticles pairs is not ether and does not prevent 17 freely motion of the test particle, because QFT vacuum state is invariant relatively to the Lorentz 18 transformations of coordinates. 19 Taking a gravity into account seems insists more on the existence of ether and preferred reference 20 frame mainly due to difficulties with gravity quantization. In particular Einstein-Aether [3] and 21 Horava-Lifshitz [4] theories have been developed (see [5,6] for phenomenological implications). 22 Another argument for the existence of the proffered system of reference is the vacuum energy 23 problem. If the zero-point energy is real, one needs to explain why it does not influence universe 24 expansion. One of the solutions is modifying the gravity theory. That may violate invariance relatively 25 to the general transformation of coordinates, e.g., as it is in the Five Vectors Theory (FVT) of gravity, 26 where local Lorentz symmetry is also violated [7]. It should also mention CPT invariance violation 27 [8], which have a number of manifestations both under the Minkowski space-time [9,10] and in the 28 presence of gravity [6,11]. 29 2. Vacuum state and QG Without knowing the quantum gravity (QG) theory, one could hardly say definitely about a vacuum state. However, even toy QG models could provide some suggestions regarding this issue. In particular, one could try to build a model in which gravity is represented only by a spatially nonuniform Submitted to Symmetry, pages 1 – 10 www.mdpi.com/journal/symmetry © 2021 by the author(s). Distributed under a Creative Commons CC BY license. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 24 May 2021 doi:10.20944/preprints202105.0576.v1 Version May 19, 2021 submitted to Symmetry 2 of 10 scale factor a(h, r). Certainly, within the GR frameworks, such a model is not self-consistent [12]. Nevertheless, there exists the (1+1)– dimensional toy model [12] including a scalar field f(t, s) and a scale factor a(t, s): Z 1 Z S = L dt = −a02 + (¶ a)2 + a2 f02 − (¶ f)2 dsdt, (1) 2 s s 30 where t is a time variable, s is a spatial variable, and prime denotes differentiation over t. Here, 31 as in GR, the fields evolve on the curved background a(t, s), which is, in turn, determined by the 32 fields (the only scalar field f(t, s) is considered, but there could be a lot of them). The relevant dL 2 0 33 ( ) ≡ = Hamiltonian and momentum constraints, written in the terms of momentums p s df0(s) a f , dL 0 34 ( ) ≡ − = pa s da0(s) a 1 H = −p2 − (¶ a)2 + a2 p2 + ¶ f)2 = 0, (2) 2 a s s P = −pa¶sa + p¶sf = 0, (3) 35 obey the constraint algebra similar to GR. The action (1) could be rewritten in the form of a string on the curved background [12] Z 2 p ab A B S = d x −g g (x)¶aX ¶bX GAB(X(x)), (4) where x = ft, sg. The metric tensor gab(x) describes the intrinsic geometry of a (1+1)-dimensional 1 A manifold and GAB(X(x)) describes the geometry of a background space. Taking X = fa, fg, and the metric tensors gmn, GAB(X) in the form of 2 2 ! ! −N + N1 N1 1 0 g = , G = 2 , N1 1 0 −a 36 where N and N1 are the lapse and shift functions, respectively, results in (1), which is written in the 37 particular gauge N = 1, N1 = 0. 38 The system (4) manifests invariance relatively to the reparametrization of the variables t, s which 39 is analog of the general transformation of coordinates in GR. Here, the (1+1)– infinite string (i.e., an 40 “immersed” space-time t, s) corresponds to the observable metric gmn, while the metric GAB has no a 41 physical meaning. The transformations of coordinates t˜ = t˜(t, s), s˜ = s˜ (t, s) reflects rather physical 42 than gauge symmetry, because one living on a string could perform some experiments allowing to 43 determine the concrete metric gmn. Considering quantization of the model (4), let us first imagine that the metric GAB is the Minkowski one, i.e., equals to diagf1, −1g. Then, fixing the gauge by taking gmn as the Minkowski metric results in the equation of motion for Xm in the form of a wave equation: 00 2 Xm − ¶ssXm = 0. (5) 44 Quantization of the wave equation in the Heisenberg picture allows defining the vacuum state [13]. 45 Then one could develop a perturbation theory expanding the metric GAB around the Minkowski one, 46 as it has been done for the string on the curved background [14]. Thus, by analogy with the model 2 47 (4), one may suggest that the vacuum state in QG has to be noninvariant relatively to the general 48 transformations of coordinates, i.e., implies some particular gauge. 1 It is not the background on which the field f(h) evolves! 2 If it exists, most probably it does not exist at all. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 24 May 2021 doi:10.20944/preprints202105.0576.v1 Version May 19, 2021 submitted to Symmetry 3 of 10 3 49 Let us emphasize the difference between GR and the conventional string theory In the string 50 theory, the reparametrization invariance is a gauge symmetry. It does not matter for calculation of the 51 gauge-invariant observables in which gauge they are calculated. In GR, an invariance relatively to the 52 general transformation of coordinates has a physical meaning because, besides the gauge-invariant 53 observables, one could also measure some quantities allowing to determine the actual system of 4 54 reference (i.e., the gauge) . 55 3. Vacuum energy problem as a criterion for finding the preferred reference frame 56 In the previous section, we argued that the vacuum state for the model (4) exists when the metric 57 gmn is the Minkowski one. Thus, the vacuum state is not re-parametrically invariant. By analogy, 58 one may think that the vacuum state, which is invariant relatively to the general transformation of 59 coordinates, does not exist in QG. 60 The more simple problem is to define the vacuum state on the fixed background. However, even 61 in this case, the exact vacuum state exists only for some particular space-times. In other cases, the 62 vacuum state has only an approximate meaning [15,16] and could be defined, for example, for a slowly 63 expanding universe. In this case, a criterion for choosing a preferred reference frame is the cosmological constant problem. This problem arises when one computes the density of vacuum energy of the quantum fields. Using the Pauli hard cutoff of the 3-momentums kmax [17,18] allows expanding the zero-point energy density into the different parts Z kmax 1 2p 2 2 2 rv = k k + a m dk ≈ 4p2a4 0 (6) 1 k4 m2k2 m4 m2a2 max + max + + + 2 4 2 1 2 ln 2 ... 16p a a 8 4kmax 4 1 kmax 64 The main part of this energy density 16p2 a4 scales as radiation. In the frame of GR, it results in an 65 extremely fast universe expansion that contradicts the observations [19]. 66 Mutual cancellation of this main part from bosonic and fermionic degrees of freedom demands 67 exact supersymmetry, which does not observed to date [20]. The remanding parts in (6) are also very 68 large, but one could suggest that there exist the sum rules for masses of bosons and fermions (the 69 condensates should be taken into account, as well) which provide a mutual compensation of these 70 terms [18,21].
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