Expanded Space (Part One)
Quanta of space
Our space is determined by gravitational fields. To put it simply, material objects and/or enormous energies form gravitational fields, the arena in which Galaxies, stars, planets live and where the physical Laws of our World begin to work. Both galaxies and laws are observed in macroscopic sizes, and to describe them we use the usual Archimedean metrics. The remaining fields “live” on gravitational fields: electromagnetic, weak and strong, with all their architecture. In fact, “our” world is a gravitational field, and all other matter and other interactions are vibrations of different parts of this field.
In the microcosm, on “Planckian” scales, space begins to manifest its quantum essence. The idea of space quanta is most fully described by the theory of loop quantum gravity (LQG), successfully developed by C. Rovelli and Co.0[1]. Drawing an analogy between the electromagnetic and gravitational fields, he writes “…the key difference between photons (electromagnetic field quanta) and…gravity quanta is that photons exist in space, while gravity quanta represent space itself. Photons are characterized by the place “where they are.” The quanta of space have no place where they can be, since they themselves are a place.”
Rovelli himself treats the quantum of space with caution; for example, he cites a fuzzy object as a quantum without any detail (left, lower part of Figure 1). Instead, an analogue of field lines is introduced in the form of spin networks with nodes on quanta and defining quantum operators in Hilbert space from areas and volumes through Ashketar variables [2]the geometry of discrete quantum space and its gravitational curvatures are constructed*.
The main idea of C. Rovelli and Co.0 is simple and comes down to two sequential things, first the quanta of space, spin networks and spin foam are introduced, then in this new formalism, with the help of special variables, the Einstein equations and the Schrödinger equations are determined, with the hope that now general relativity and quantum physics will live together.
1. Discreteness of space-time, spin network and spin foam: In the theory of loop quantum gravity, the geometry of space is quantized, so operators such as area and volume acquire discrete spectra of values in terms of Ashketar variables. For example, 8πγlp2 is the minimum area, and the minimum volume will be the quantity
The figure gives an idea of the quantum structure of space, the spin network, and the discrete spectrum of areas and volumes and about the spin numbers j(1/2.3/2.5/2…) associated with the edges of the spin network.
The spin network (right side of the figure) is a graph of nodes and edges, where the edges are labeled with spin numbers corresponding to representations of the SU(2) group, and the nodes are connected by these edges and are characterized by volumes v1 (tetrahedron V1V2AB), V2(tetrahedron V2V1AC),….**. In fact, the quantization of space in the PKG is an ordinary triangulation, although in fact spin networks describe the quantum states of the geometry of space. To do this, the concept of a loop is introduced – a closed contour covering the area under consideration. The concept of a loop is key in this theory. Despite the simplicity of the definition and intuitive geometric interpretation, it carries deep internal content and is the main construct. In this sense, it is similar to the string from the theory of the same name, but the loop does not require a continuous, or indeed any background in the form of a supporting space. In PKG, loops clinging to each other themselves form the fabric of space, simultaneously representing a local basis.
That is why, in PKG, space quanta themselves cease to play a special role, the whole theory begins to focus on loops, on their combinations, on the geometry and topology of loop space. Let us repeat that the spin numbers of a spin network have both geometric and physical meaning. Geometrically, they define discrete values of areas and volumes in quantized space. Physically, they represent quantum states of elementary units of space and play a key role in describing quantum interactions and information in the quantum theory of gravity. In Figure 2, the chain mail rings mesh, forming the iron fabric of the chain mail; also in the PKG, the loops, representing a closed path of successive edges, intersect at the vertices of the spin network, forming the fabric of space. In both the first and second cases, mathematicians will see Baire sets in them. This means that despite any tricks and loopholes, the loops themselves will not allow us to construct either a measure, or compact sets, or functional integrals, and therefore, no meaningful theory. The founders of the theory cleverly avoided these questions by introducing both areas and volumes through direct products of Planck lengths, respectively, with the formation of two and three dimensions:
lp X lp = lp2 (S) And lp X lp X lp = lp3 (V),
using the Archimedean metric, thereby leaving open the questions of its introduction and problems with dimensions. How are these problems actually solved? Partly with the help of spin foam, partly with the help of density in the variables by Ashketar, partly with concepts introduced from the equations of Einstein and Schrödinger, but more on that later**. In the meantime, all creators of PKG, introducing spin networks, “keep” in mind both the left and central parts of Figure 1. So, the quantum state of a region-space is described by a set of loops surrounding it, or simply a spin network. The denser the spin network in the region, the greater the curvature of space will be recorded by Ashketar’s variables – just like in Einstein’s theory.
Spin foams are 4-dimensional structures that describe the evolution of 3-dimensional spin networks over time. Spin foam connects the initial and final states of the spin network cells through two-dimensional surfaces, “forming” a system of prism-pyramidal simplexes, and is used to describe the dynamics of the quantum gravitational field (middle part of Figure 3).
Spin foam is an analogue of 4-dimensional space-time in Einstein's theory. Having approximated continuous space using a spin network, an analogue of the space-time interval from General Relativity is implemented in the PTC using a spin foam, and quantum fluctuations that oscillate the quantum net. An important issue in the concept of spin foam is the issue of time. The first equations of the theory, which had not yet been fully developed, obtained by the founders Wheeler and Devitt shocked everyone; they did not contain a variable indicating time. But the solutions to these equations turned out to be our loops. For a long time, the loops did not want to give birth to anything worthwhile (as discussed above), until edges and vertices were introduced. The circle closed, time became an outcast in theory and they tried to push him out the door. One of the earlier posts talked about how and what time is replaced with, but we won’t dwell on that here. Let us only note that Rovelli reluctantly introduced quantum time into the spin foam, masking it with probabilistic characteristics: the randomness of the occurrence of fluctuations and Feyman summation along all possible paths between the initial and final state of the spin network, which ultimately led to the history of spin networks and packages. The question of history is the most dubious in the whole theory. This is due to the fact that the fundamental restrictions given in the table are suggested to us by nature itself, and they should be used as the main postulates in the theory.
Time, as we see, is not included in this table.
1. Loop quantum gravity, or loop theory, combines general relativity with quantum mechanics; it does not introduce any hypotheses other than those contained in these theories themselves, writing them in a form compatible with itself. For this, PTG uses special Ashketar variables – analogs of the metrics and Christophel-Schwartz symbols in Ricci tensors, as well as quantum postulates in the form of constraints:
1. Gaussian constraints arise from symmetries of the inner gauge group (usually SU(2)) in the three-dimensional formulation of gravity. They ensure local invariance of the theory under gauge transformations, that is, they ensure the conservation of internal gauge charges. The meaning of this invariance is that in quantum fields it becomes possible to determine the carriers of interactions, in other words, bosons: photons, gluons, etc.
2. Diffeomorphic constraints ensure the invariance of the theory with respect to space-time transformations – translations and rotations. The meaning of this invariance is that the physical laws are the same in any region of our space.
3. The Hamiltonian constraint, written in terms of quantized variables by Ashketar, contains information about matter, its interaction with gravity, and in fact is an analogue of the Einstein equation.
Note that quantum constraints transform into classical constraints in the limit of large quantum numbers, restoring Einstein’s equations in the macrocosm, and classical geometry is extracted through the eigenvalues of area and volume operators. How is continuity and smoothness realized in the large dimensions of space itself? This is where quantum fluctuations come to the rescue. Packed in spin-foam bags, a rapid sequence of births and disappearances of tetrahedral simplexes “sweeps up” the volumes in the bag, which in large numbers forms the representation of a continuous monolithic three-dimensional space. Just like if you place a coin on its edge and spin it with a click, no one from afar will be able to distinguish whether it is a real ball or a rapidly rotating disk.
The most important advantages of the theory at the moment are the elimination of singularities appearing in Einstein's classical equations in Planck regions and the introduction of space quanta.
As a benefit for space quanta, we will try to take a fresh look at Fermat’s principle and explain it using spin networks. Fermat's Principle (or Least Time Principle) is a fundamental principle in optics and wave theory that states that light travels between two points along a path that requires the least amount of time (why light is so smart remains a mystery). This means that of all the possible paths that a light beam can take, it chooses the one that provides the shortest travel time.
If n is the refractive index, c is the speed of light in a vacuum, v is the speed of light in a given medium, and d is the distance, then the time it takes light to travel between points A and B is minimal:
where n(s) is the refractive index along the path.
Fermat's principle, originally formulated for light, also applies to particle motion in quantum mechanics and classical mechanics. It generalizes to the broader Principle of Least Action in physics, which states that the movement of any particle occurs along a trajectory that minimizes the action – the Lagrangian integral over time. This principle applies to all physical systems and is fundamental to the description of dynamics in classical mechanics, quantum mechanics and relativity.
In the last post we described the movement of particles using the Pilot Wave. Briefly, it looks like this: a wave “runs” from one quanta of space to another at certain moments, transmitting to them encoded data about the particle from its wave packet.
Thanks to these data, a particle is born on a quantum of space, and dies when the wave leaves this quantum.
It turns out that the movement of a wave can influence the configuration of the spin network, subjecting it to a certain transformation.
Is such a transformation possible, does it not destroy the lattice structure of the spin network? The study of the properties of lattice classes in terms of the embeddability or non-embedding of certain finite lattices is a classical direction in lattice theory. The very concepts of embeddability and non-embeddability of lattices are important in order theory and algebraic structures. They help determine which structures can be integrated into each other while maintaining certain properties and operations.
A well-known classical result is Dedekind's theorem [3], according to which the lattice is modular (partially ordered and obeys the modular law) if and only if it does not contain a sublattice isomorphic to lattice N5, see Fig. 4.
Another result in this direction is given by Birkhoff's theorem [4], according to which a lattice is distributive (another law) if and only if it does not contain sublattices isomorphic to the lattices N5 and M3, see Fig. 1. We were able to prove the theorem that the lattices in Fig. 5 satisfy both of the requirements we need.
An amazing property of these lattices is their arbitrarily long extension, in contrast to the known classes of modular lattices.
Applying the proven theorem, it can be argued that Pilot waves can transform the spin network according to Fig. 5, configuring the nodes so that the natural propagation of the wave in flat space will occur along “straight” lines.
Obviously, in a curved space, geodesics will serve as analogues of straight lines, so the theorem will work in this case as well. Thus, Fermat's principle, previously accepted as an axiom, finds its natural explanation within the framework of quantum space.
Concluding the consideration, we note that some important features in the construction of the theory of PKG, which give important consequences, eluded the developers ***).
Literature:
1. CARLO ROVELLI AND FRANCESCA VIDOTTO “Covariant Loop Quantum Gravity”
2. A. Ashketar “New Variables for Classical and Quantum Gravity I” “Physical Review Letters” (volume 57, number 18, pages 2244-2247) 1986
3. R. Dedekind, ¨Uber die von drei Moduln erzeugte Dualgruppe, Math. Ann., 53, (1900),
4. G. Birkhoff, On the structure of abstract algebras, Proc. Cambridge Philos. Soc., 31 (1935).
*) Let us explain the idea with which the curvature of discrete quantum space is determined. Imagine being at the North Pole and walking south until you reach the equator. At the same time, you carry with you an arrow that points forward. When you reach the equator, you turn left without changing the direction of the arrow. It still points south, which is now on your right. Walk a little east along the equator and then turn back north, again without changing the direction of the arrow, which will now point back.
When you return to the North Pole, your route will close, forming a loop, but the arrow will no longer point in the same direction as when you started (Fig. 3). The angle at which the arrow turns when going around the loop serves as a measure of the curvature of a given space.
In PKG theory, we sum up all the loops in the region for which we want to determine the curvature, choosing a spin network (Figure 1) representing the quantum state of the geometry of this region and its evolution through the spin foam, taking into account quantum constraints. Then the curvature will finally be expressed through the derivatives of the connection by Ashketar:
Recently, the construction of loop quantum gravity is introduced as a transformation of the Einstein and Schrödinger equations to Ashtecker variables, within the framework of ordinary space-time, thereby closing questions with dimension.
**)
Recently, the construction of loop quantum gravity is introduced as a transformation of the Einstein and Schrödinger Equations to Ashketar variables, so questions with dimension have become irrelevant
***) Fermat's principle, minimal film surfaces on loops, finite element method and barycentric coordinates for calculating the effects of gravity in small areas.
