The "Loop Quantum Gravity" (LQG), designed by Carlo Rovelli and Lee Smolin in 1990, based on the concept of "spin networks", designed in the early '70s by Roger Penrose, is one of the most fascinating and elegant theories of gravity.

In physics a spin network is a type of diagram which can be used to represent states and interactions between particles and fields in quantum mechanics. From a mathematical perspective, the diagrams are a concise way to represent multilinear functions and functions between representations of matrix groups. The diagrammatic notation often simplifies calculation because simple diagrams may be used to represent complicated functions. Roger Penrose is credited with the invention of spin networks in 1971, although similar diagrammatic techniques existed before that time.

Spin networks have been applied to the theory of quantum gravity by Carlo Rovelli, Lee Smolin, Jorge Pullin and others.

A spin network, as described by Penrose in 1971, is a kind of diagram in which each line segment represents the world line of a "unit" (either an elementary particle or a compound system of particles). Three line segments join at each vertex. A vertex may be interpreted as an event in which either a single unit splits into two or two units collide and join into a single unit. Diagrams whose line segments are all joined at vertices are called closed spin networks. Time may be viewed as going in one direction, such as from the bottom to the top of the diagram, but for closed spin networks the direction of time is irrelevant to calculations.

Each line segment is labeled with an integer called a spin number. A unit with spin number n is called an n-unit and has angular momentum nh/2л. For bosons, such as photons and gluons, n is an even number. For fermions, such as electrons and quarks, n is odd.

In loop quantum gravity (LQG), a spin network represents a "quantum state" of the gravitational field on a 3-dimensional hypersurface.

The loop quantum gravity is also known by terms of gravity loop quantum geometry and quantum canonical general relativity. It has been proposed as a quantum theory of spacetime which attempts to unify seemingly incompatible theories of quantum mechanics and general relativity. This theory is part of a family of theories called canonical quantum gravity. It is a quantum theory of gravity in which the real space in which happen all the other physical phenomena is quantized.

The LQG retains the basic aspects of general relativity, such as invariance to coordinate transformations, and at the same time, using the quantization of space and time at the Planck scale, feature of quantum mechanics. In this sense it combines general relativity and quantum mechanics. Critics of the LQG often refer to the fact that the theory doesen't predict the existence of extra dimensions of space-time, or supersymmetry. The response of the LQG proponents is that at present, despite repeated experimental researches, there is no experimental evidence or other dimensions or supersymmetric particles, so the additional dimensions of space and time, both supersymmetry must be considered speculative hypotheses found to run faster than they do on Earth.

Abstract from “on LQG” by Carlo Rovelli.

In 1915 Einstein realized that gravity also had to be described by a ﬁeld theory in order to be consistent with special relativity. As long as we stay within the classical regime, rather than the quantum one, the gravitational ﬁeld deﬁnes a 4D continuum. We can therefore still think of the ﬁeld as a sort of spacetime, albeit one that bends, oscillates and obeys ﬁeld equations. However, once we bring quantum mechanics into the picture this continuum breaks down. Quantum ﬁelds have a granular structure – the electromagnetic ﬁeld, for example, consists of photons – and they undergo probabilistic ﬂuctuations. It is difﬁcult to think of space as a granular and ﬂuctuating object. We can, of course, still call it “space”, or “quantum space”. But it is really a quantum ﬁeld in a world where there are only ﬁelds over ﬁelds, and no remnant of background space. John Wheeler of Princeton University suggested that spacetime must have a foam like structure at very small scales and, along with Bryce DeWitt now at Texas University, he introduced the idea of a “wavefunction over geometries”. This is a function that expresses the probability of having one spacetime geometry rather than another, in the same way that the Schrödinger wave function expresses the probability that a quantum particle is either here or there. This wave function over geometries obeys a very complicated equation that is now called the Wheeler–DeWitt equation, which is a sort of Schrödinger equation for the gravitational ﬁeld itself. These ideas were brilliant and inspiring, but it was more than two decades before they become concrete. The turn around came suddenly at the end of the 1980s, when a well deﬁned mathematical theory that described quantum spacetime began to form. The key input that made the theory work was an old idea from particle physics: the natural variables for describing a Yang–Mills ﬁeld theory are precisely Faraday’s “lines of force”. A Faraday line can be viewed as an elementary quantum excitation of the ﬁeld, and in the absence of charges, these lines must close on themselves to form loops. Loop quantum gravity is the mathematical description of the quantum gravitational ﬁeld in terms of these loops. That is the loops are quantum excitations of the Faraday lines of force of the gravitational ﬁeld. In low energy approximations of the theory, these loops appear as gravitons – the fundamental particles that carry the gravitational force. In LQG the loops themselves are not in space because there is no space. The loops are space because they are the quantum excitations of the gravitational ﬁeld, which is the physical space. It therefore makes no sense to think of a loop being displaced by a small amount in space. There is only sense in the relative location of a loop with respect to other loops, and the location of a loop with respect to the surrounding space is only determined by the other loops it intersects. A state of space is therefore described by a net of intersecting loops. There is no location of the net, but only location on the net itself; there are no loops on space, only loops on loops. Loops interact with particles in the same way as a photon interacts with an electron, except that the two are not in space like photons and electrons are. Elementary grains of space are represented by the nodes on a “spin network”. The lines joining the nodes, or adjacent grains of space, are called links. Spins on the links (integer or half- integer numbers) are the quantum numbers that determine the area of the elementary surfaces separating adjacent grains of space. The quantum numbers of the nodes determine the volume of the grains. The spins and the way they come together at the nodes can take on any integer or half-integer value, and are governed by the same algebra as angular momentum in quantum mechanics. The idea that there cannot be arbitrary small spatial regions can be understood from simple considerations of quantum mechanics and classical general relativity. The uncertainty principle states that in order to observe a small region of spacetime we need to concentrate a large amount of energy and momentum. However, general relativity implies that if we concentrate too much energy and momentum in a small region, that region will collapse into a black hole and disappears. Putting in the numbers, we ﬁnd that the minimum size of such a region is of the order of the Planck length – about 1.6× 10–35m. Loop gravity had begun to make this intuition concrete, and a picture of quantum space in terms of nets of loops was emerging. But at the time we did not really under stand what that meant. But space is more than just a collection of volume elements. There is also the key fact that some elements are near to others. A “link” of the net – the portion of loop between two nodes – indicates precisely the quanta of space that are adjacent to one another. Two adjacent elements of space are separated by a surface, and the area of this surface turns out to be quantized as well. In fact, it soon became clear that nodes carry quantum numbers of volume elements and links carry quantum numbers of area elements. Each node in a spin network determines a cell, or an elementary grain of space. Nodes are represented by small black spheres and the links as black lines, while cells are separated by elementary surfaces. Each surface corresponds to one link, and the structure builds up a 3D space. When the surfaces are pulled away we can see that the sequence of links form a loop. These are the “loops” of loop quantum gravity. Finally, for the moment there has not been any direct experimental test of the theory. A theoretical construction must remain humble until its predictions have been directly and unambiguously tested. This is true for strings as well as for loops. Nature does not always share our tastes about a beautiful theory. Maxwell’s theory became credible when radio waves were observed.

The MT ("Marius Theory") was created by Marius when he still did not know the LQG. The striking similarity of the concepts brought Marius to deal with this theory. The outcome was that the seemingly "surreal" conclusions of MT (substantial identity between space, energy, mass and matter: "S.E.M.M. logic”, absence of time, pushing gravity, standing aether and more ...) all things were not so fanciful.

Where MT makes substantial difference respect to LQG is in the quantization of a normal 3d Euclidean space, rather than the 3d hypersurface of LQG which, according to Marius, would need set free from temporal dimension standing that "time arrow" is irrelevant for the closure of loops. Of course this does not allows to conciliate GR and QM but, according to Marius, the spin networks, such as the quantization of 3d space, represent only the nearest approximation of the underlying physical reality, that is an a-temporal matrix made up by standing electromagnetic waves.

In the "compact" form of MT : DU = nhv, the quantization term “n” represents the number of times that the path between two orbits in a gravitational field of a point mass can be divided into Planck’s sizes" : 10 ^ -35 m. The thing that generates energy is electromagnetism represented by the second term of equality: nhv, with v frequency of the radiation. The application of this formula allowed Marius to calculate, on a path more or less likely, the frequencies necessary to move some solar system planets from the tidal Sun’s orbit since to the actual one and, with his large surprise, to discover that these frequencies have the sizes of normal high-energy gamma rays. The quantization term n, which is the fundamental parameter of the formula, contains all the variables involved : the energy required to compact the, at the beginning, fluid and incandescent mass of the planet in a spherical form starting out from Roche’s orbit of the Sun, the gain of angular momentum during removal since to parking at a certain distance from the sun with a certain angular speed. All this thanks, Marius believes, to having seen the planet mass point and, therefore, able to make work all the quantum space really crossed by the “distributed” mass along the path, regardless by the time it takes.

Stefano Gusman

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