# Are there levels of quantum gravity

## Loop quantum gravity

Loop quantum gravity (LQG), sometimes also called loop quantum gravity (also called loop quantum gravity) in German, is a theory of quantum gravity. The aim is to unite the concepts of quantum theory with those of general relativity theory (GTR). In other words, an attempt is made to quantize the gravitational field. The related gravitational field quantum is called in the quantum field theories (QFT) graviton. In the LQG one rather distances itself from the concept of the graviton and describes the quanta of the theory as*Loops* (German loops).

### granular spacetime

In the GTR, space-time is a deformable, four-dimensional, continuous structure, a**Space-time continuum**that is curved by forms of energy such as mass. The classical theories do not provide for principles such as fuzziness or discrete physical quantities (quanta). The spacetime appears sufficiently smooth, continuous and shows at most 'holes' in the form of curvature singularities (the black holes are an example of this). However, quantum theory and in particular the Planck scale derived from it with the ART implies that with very strong gravitational fields on very small length scales a**Grain of spacetime** must appear. The length scale, where the grain occurs, is in the area of the Planck length, i.e. to only 10^{-33} cm. This is the smallest unit of length that physicists know. Granulation requires a theory that goes beyond the GTR and takes into account the quantum character of spacetime: a **Quantum gravity**.

### Successes of quantized field theories

For a long time there was only one serious candidate for this theory, the string theories. This was mainly motivated by the fact that these theories contained a particle with spin 2 (tensor boson). This was precisely a desired property for the graviton. Therefore the string theories were suddenly attractive for a quantized theory of gravity. The mathematical apparatus of quantum field theories proved to be very successful in describing elementary particles, such as quantum electrodynamics, quantum chromodynamics, electroweak theory, large unified theories, in general, which prove gauge theories. This conception revealed a unified view in the form of the standard model of particle physics. The theorists also tried to apply this apparatus of quantum field theories to the strings.

### Strings & Branes

It turned out that thread-like strings can be generalized morphologically to form two-dimensional structures, the **Branes**. However, the string theories require the existence of extra (spatial) dimensions (six or seven *Further* Room dimensions in addition to the known three). They also require supersymmetry (SUSY) validity to be consistent with the physical principle of causality. The consequence of the validity of SUSY is that the particles of the Standard Model known to this day have so-called super partners. The particle zoo is thus significantly enlarged, and many new SUSY particles must then exist. **But:** So far none of these SUSY particles has been observed experimentally, nor have any other spatial dimensions besides the known three.

### The beginnings of the loop theory of gravity

This motivated the search for alternative concepts that pursue completely different approaches than the string theories; Concepts that do not necessarily require extra dimensions and SUSY. This is exactly what loop quantum gravity is, a theory whose beginnings go back to 1986. Worked in said year *Amitaba Sen* and *Abhay Ashtekar* on a new formulation of the ART with the help of new mathematical variables that one **Connections** (engl. *connections*) is called. The Christoffel symbol of the GTR (associated with Einstein's field equations and the covariant derivative), which clearly replaces Newton's gravitational force in a geometric description of gravitation, is, mathematically speaking, a connection (connection). In the *Ashtekar formulation* of the ART, the field variable is no longer the metric, but a so-called one **self-dual context** (which differs from the Christoffel symbol). Since then this has also been called *Ashtekar variables*.

In the same year, 1986, reformulated *Ted Jacobson* and *Lee Smolin* the Wheeler-DeWitt equation in these new variables. They also found a class of exact solutions to these equations: the **Wilson loops** (German Wilson loops). In order to understand what kind of mathematical objects these are, one has to go back further - unfortunately it is now conceptually quite complicated because an explanation without mathematics is not appropriate. The mathematically less interested reader may skip the following section.

### LQG in the Language of Mathematics: Holonomies and Wilson Loops

With **Holonomy** is also called the *Parallel propagator* (Holonomy operator), i.e. the operator that implements a parallel transport (a transformation, compare also Lie derivation) of a relationship, a vector or a tensor along a closed curve on a spacetime (mathematically a *Riemannian manifold*) accomplished. Holonomy satisfies certain mathematical properties, above all it forms a mathematical group that*Holonomy group*.

Now has a special meaning **track** of the holonomy operator (sum of the diagonal elements if the operator is represented as a matrix). The trace, based on a certain path through space-time, is used to determine parallel-transported physical quantities when a particle propagates along the path through space-time.

Let us take an electron moving through space-time as an example. Using the parallel propagator, one can investigate how the observable spin changes. The curvature of spacetime rotates the spin vector as it moves along the closed curve. If you now calculate the trace of the associated parallel propagator, you have found a measure of how the observable spin has changed. The requirements for the curves are that they are 'smooth' and that they do not overlap.

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