### The Universe in a Helium Droplet (The International Series of Monographs on Physics, 117)

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**The Universe in a Helium Droplet**

They are as follows. Canonical quantum gravity [16, 17, 43, 44, 32, 99, , 6, 54, ]. Manifestly covariant quantization [, 33, 94, 74, 7, , 21, ]. Euclidean quantum gravity [68, 90]. R-squared gravity []. Supergravity [64, ]. String and brane theory [, 98, 10]. Non-commutative geometry [26, 75]. Among these 8 approaches, string theory is peculiar because it is not field- theoretic, spacetime points being replaced by extended structures such as strings. A second set of approaches relies instead upon different mathematical structures with a more substantial but not complete departure from con- ventional pictures, i.

Twistor theory [, ].

Asymptotic quantization [67, 5]. Lattice formulation [, 22]. Loop space representation [, , , , ]. Simplicial quantum gravity [72, 1, , 2] and null-strut calculus []. Condensed-matter view: the universe in a helium droplet []. Affine quantum gravity [].

## Институт теоретической физики им. Л.Д. Ландау

After such a concise list of a broad range of ideas, we hereafter focus on the presentation of some very basic properties which underlie whatever treatment of classical and quantum gravity, and are therefore of interest for the general reader rather than just the specialist. He or she should revert to the above list only after having gone through the material in sections 2—7. This is done in the subsection below. The latter feature gives rise to the light-cone struc- ture of spacetime, with vectors being divided into timelike, null or spacelike depending on whether g X, X is negative, vanishing or positive, respectively.

The classical laws of nature are written in tensor language, and gravity is the curvature of spacetime. With hindsight, following DeWitt [39], one can say that general relativity was actually the first example of a non-Abelian gauge theory about 38 years before Yang—Mills theory []. Note that the spacetime manifold is actually an equivalence class of pairs M, g , where two metrics are viewed as equivalent if one can be obtained from the other through the action of the diffeomorphism group Diff M. The metric is an additional geometric structure that does not necessarily solve any field equation.

It tells us that, on such scales, the world can be described by a Hilbert space structure, or suitable gener- alizations. Even in the relatively simple case of the hydrogen atom, the ap- propriate Hilbert space is infinite-dimensional, but finite-dimensional Hilbert spaces play a role as well.

The operators satisfying the canonical commuta- tion relations 9 cannot be both bounded [57], whereas it would be nice to have quantization rules not involving unbounded operators and domain problems. For this purpose, one can consider the strongly con- tinuous one-parameter unitary groups having position and momentum as their infinitesimal generators. The Weyl approach is very elegant and far-sighted, with several modern applications [57], but still has to do with a more rigorous way of doing canonical quanti- zation, which is not suitable for an inclusion of relativity.

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This point of view has enormous potentialities in the quantization of field theories, since it preserves manifest covariance and the full symmetry group, being derived from a Lagrangian. There exist indeed different interpretations of quantum mechanics, e. Copenhagen [], hid- den variables [15], many worlds [60, 35]. In general relativity, timelike geodesics correspond to the trajectories of freely moving observers, while null geodesics describe the trajectories of zero-rest- mass particles section 8.

Moreover, a spacetime M, g is said to be singularity-free if all timelike and null geodesics can be extended to ar- bitrary values of their affine parameter. At a spacetime singularity in general relativity, all laws of classical physics would break down, because one would witness very pathological events such as the sudden disappearance of freely moving observers, and one would be completely unable to predict what came out of the singularity. In the sixties, Penrose [] proved first an important theorem on the occurrence of singularities in gravitational collapse e.

Subsequent work by Hawking [79, 80, 81, 82, 83], Geroch [66], Ellis and Hawking [84, 52], Hawking and Penrose [86] proved that spacetime singularities are generic properties of general relativity, pro- vided that physically realistic energy conditions hold. Very little analytic use of the Einstein equations is made, whereas the key role emerges of topological and global methods in general relativity. On the side of singularity theory in classical cosmology, explicit mention should be made of the work in Ref.

As pointed out in Ref. Interestingly, near the singularity the spatial points essentially decouple, i.

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Moreover, the use of quali- tative Hamiltonian methods leads naturally to a billiard description of the asymptotic evolution, where the logarithms of spatial scale factors define a geodesic motion in a region of the Lobachevskii plane, interrupted by geomet- ric reflections against the walls bounding this region. Chaos follows because the Bianchi IX billiard has finite volume [27]. A self-contained derivation of the billiard picture for inhomogeneous solutions in D dimensions, with dilaton and p-form gauge fields, has been obtained in Ref.

All related phenomena can be described by an antisymmetric rank-two tensor field, and derived from a one-form, called the potential.

More- over, inertial and gravitational mass, conceptually different, are actu- ally unified as well. The physics community is now familiar with a picture relying upon four fundamental interactions: electromagnetic, weak, strong and gravitational. The large-scale structure of the universe, however, is ruled by gravity only.

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All unifications beyond Maxwell involve non-Abelian gauge groups either Yang—Mills or Diffeomorphism group. At least three extreme views have been developed along the years, i. Protons, gravitons and gluons are viewed as collective excitations of this liquid []. For this purpose, it may be useful to describe the main ideas of the Arnowitt-Deser-Misner hereafter referred to as ADM for- malism. The basic geometric data of this decomposition are as follows [53]. This yields the intrinsic geometry of the three-space. This is known once we are able to compute the spatial part of the covariant derivative of the normal n to St.

The tensor K is called extrinsic-curvature tensor, or second fundamental form of St. According to a well-established terminology, N is the lapse function, and the N i are the shift functions.

Other useful forms of the boundary term can be found in [68, ]. Hereafter, we assume that this step has already been performed. As we know, consistency of the quantum constraints is proved if one can show that their commutators lead to no new constraints [53]. For this purpose, it may be useful to recall the equal-time commutation relations of the canonical variables, i. The first two commutators are obtained by using Eq.

As we said before, following Dirac, the operator version of constraints should annihilate the wave function since the classical constraints are first-class i. Thus, by using the definition of structure constants of the general coordinate-transformation group [32], i. However, all factors appearing in this term have homogeneous linear transformation laws under the three-dimensional coordinate-transformation group. They thus remain undisturbed in position when commuted with Hj [32]. All of them, being gauge theories, need supplementary conditions, since the second functional derivative of S is not an invertible operator.

Such operators play a leading role in the one-loop expansion of the Euclidean effective action, i. Grigory E.

### International Series of Monographs on Physics

There are fundamental relations between three vast areas of physics: particle physics, cosmology and condensed matter physics. The fundamental links between the first two areas, in other words, between micro- and macro- worlds, have been well established. There is a unified system of laws governing the scales from subatomic particles to the Cosmos and this principle is widely exploited in the description of the physics of the early Universe. The information exchange from a prior to a present universe would be modeled on the template of what would be required, and of what dimensional embedding is needed to do so.

Furthermore, what is obtained should be reconciled with an additional constraint which will be put in the next page. Note that Corda [25] has modeled adiabatically-amplified zero-point fluctuations processes in order to show how the standard inflationary scenario for the early universe can provide a distinctive spectrum of relic gravitational waves. De Laurentis, and Capozziello [26] have further extended this idea to give a qualified estimate of GW from relic conditions which will be re produced here. The upper range for appears to be about Hertz.

## Григорий Ефимович Воловик

Needless to state, though, if drifted to a value of then the upper bound to Hertz. And, we suggest that Hz, if is set higher, i. We at the close refer the readers to Appendix C for crucial considerations as to the emergence of gravitational astronomy as this relates to a summary as to how to confirm the models so referenced in this paper, as to work by Corda, and the LIGO GW team which is of potentially revolutionary import as far as observational astronomy confirming these ideas so presented. The author thanks Dr.

Fangyu Li, of Chongqing University is thanked for lending his personal notes to give substance to the content of page 10 of this document.