GEOMETRY, GRAVITY AND GAUGE SYMMETRIES

In the previous article, we saw how practically everything we know about particle physics and fundamental forces revolves around the fundamental concept of gauge symmetry. The three fundamental forces included in the Standard Model of particle physics appear to locally restore the SU(3), SU(2), and U(1) gauge symmetries, which consequently gives rise to the strong force, the weak force, and electromagnetism, respectively. However, what about gravity? Does gauge symmetry have anything to do with gravity? As we know, we still don't have a quantum theory of gravity, so if there were a connection between gauge symmetry and gravity, perhaps we could derive a way to quantize gravity. In this article, we will see how this relationship leads us to a profound connection between the geometry of spacetime and the geometry of the internal spaces where the gauge fields reside.

Special relativity and symmetry

Special relativity implies that all non-inertial observers must measure the same conserved physical quantities. These conserved quantities are a consequence of a fundamental external (space-time) symmetry: Poincaré symmetry. This symmetry includes translational invariance and Lorentz symmetry. The importance of this symmetry group is captured in the definition of a fundamental particle: "A fundamental particle is defined as an irreducible representation of the Poincaré group." This basically means that a fundamental particle is an entity that must remain invariant under translations and Lorentz transformations.

General relativity, geometry and gauge symmetries

The key question we want to answer in this article is the following: Can gravity be formulated as a gauge theory like the other three fundamental forces? An important difference that we must keep in mind is that the gauge symmetries of the Standard Model are internal symmetries while the external symmetries of the Standard Model are internal symmetries.

of gravity are symmetries of space-time itself and are therefore external symmetries. Can we find any relationship between these internal mathematical spaces and the external physical spaces of space-time?

As we saw in the previous article, when a local gauge symmetry was created, a new term appeared in the Lagrangian of the field in question. To correct this new term and restore the gauge symmetry, we must use the covariant derivative, for example, in the following way:

In general relativity, tensors are used to ensure that coordinate systems remain valid for any observer. However, it is well known that in differential geometry, partial derivatives do not transform tensorically and are therefore useless for comparing two different reference systems:

If we compare the previous expression with the definition of a tensor, we see that we have an extra term (the term on the right). To make the transformation covariant (equivalent for all reference frames), we would need to use a derivative that would cancel this extra term, that is, a derivative defined as:

In this way we could define a covariant derivative of the form:

And thus manage to cancel the extra term that we obtained in the "usual" partial derivative.

It is easy to see that the process we just described is very similar to the one we followed in the previous article to transform a global gauge symmetry into a local gauge symmetry. In fact, transforming a global symmetry into a local one is similar to going from special to general relativity: special relativity implies that fields must remain invariant under translations and Lorentz transformations. These transformations imply global spacetime symmetries since they occur with the same magnitude at all points in spacetime. If we make these symmetries local, we make the translations change with different magnitudes depending on spacetime; that is, variations in position (velocities) are now allowed in translations depending on space or time.

¡ Exactly what you'd expect in accelerating reference frames! We're beginning to glimpse the link between gauge symmetries and gravity!

The final term of the covariant derivative we saw in the last expression (the corrector term) is linked to a term called the connection. The connection plays a central role in general relativity because it describes the curvature of spacetime. This is the link we were looking for between gravity and gauge symmetries! We now see a clear similarity between gauge symmetries and the covariance principle of general relativity: for all observers to measure the same locally invariant quantities, a new term representing a new force (gravity) must appear, and the effect of this new term is to curve spacetime.

The gauge theory of gravity

If we want to rigorously obtain a gauge theory of gravity, the most obvious way to proceed would be to carry out the same process as we did with the internal gauge fields, that is, take the global symmetry of relativity and transform it into a local symmetry to see if, by trying to restore the symmetry, a new "gauge" field appears that implies a new fundamental force. This process has been carried out taking as a basis the fundamental group of relativity: the Poincaré group P(1,3), which is the semidirect product of the T(4) translation group and the Lorentz group SO(1,3). The results obtained from this process are impressive, although with an unexpected "twist": physicists have not obtained the equations of general relativity as expected, but the equations of an equivalent theory: the Einstein-Cartan theory. The Einstein-Cartan theory produces the same results as general relativity and can be considered a generalization of Einstein's theory for torsioned spacetimes. Currently, with current technology, it is not possible to experimentally differentiate whether we live in a twisted or non-twisted Universe, since the effects are very small. Considering zero torsion, then, the theory of general relativity and the Einstein-Cartan theory are equally valid. The conclusion from this is of crucial importance: It is possible to obtain a fully viable and operational theory of gravity based on the principles of gauge symmetry.

Gauge symmetries and string theory

Although not as firmly established as the Einstein-Cartan theory, there are other, even more general gauge theories of gravity. One of these is the Poincaré gauge theory. Poincaré symmetry can be further "augmented" by including an even more general symmetry that interchanges bosons and leptons: supersymmetry.

In this way we "ascend" a little further and obtain the theory of supergravity.

Supergravity constitutes a low-energy limit of the most general and promising theory we have for unifying the four fundamental forces: superstring theory.

Gauge theories of gravity: GR=General relativity EC=Einstein's theory Cartan TG=Translation Gauge Theory PG=Poincare Gauge Theory CG=Conformal Gauge Theory AdSG=Anti-deSitter Gauge Theory SuGRA=Supergravity WG=Weyl(-Cartan) Gauge Theory MAG=Metric-Affine Gauge Theory. The theories represented with a circle (GR and EC) are experimentally verified and validated theories, the rest are candidates pending validation.

Physicists Kaluza in 1919 and Klein in 1926 succeeded in formulating the equations of general relativity in 5-dimensional space-time, achieving something extraordinary: the unification of general relativity and electromagnetism. Increasing the number of space-time dimensions from 4 to 5 allowed the effect of gauge symmetries to be included in the equations of motion: charge is the motion of a neutral particle in the 5th dimension, while electric fields arise due to the motion of particles in the new dimension. Moreover, coordinate transformations in the 5th dimension are equivalent to gauge transformations of the U(1) group. Currently, the Kaluza-Klein theory remains only a theoretical framework (albeit one with extraordinary consequences), but it serves to illustrate the intricate relationship between internal and external symmetries.

(Super)string theory is a theoretical framework described in 10 dimensions that constitutes our best candidate for unifying the four fundamental forces of nature. This theory generically incorporates gauge symmetries since branes (multidimensional extended objects) always naturally incorporate an SU(N) gauge field. This theory uses higher gauge symmetries than the Standard Model since it is based on groups such as SO(32) or E(8)xE(8) and constitutes a unifying geometric framework for both external and internal symmetries. This unifying framework not only aspires to become a quantum theory of gravity but to become a theory of all the fundamental forces and fields that exist: the so-called Theory of Everything.

Conclusions

In these last two articles, we have attempted to demonstrate what is probably one of the most fundamental and powerful aspects of the Universe we inhabit: the concept of symmetry. The power of this concept is such that it can explain all known interactions, all the forces of nature, physical quantities, the reasons for their values, and the reason for their existence. In fact, in particle physics, we know that only gauge-invariant quantities can be measured experimentally. But there's more: the symmetries we see are merely the "cinders" of greater symmetries that existed when the Universe had very high energy. It is likely that by following the "trail" of symmetry, we can approach the "fundamental symmetry" from which everything originated and explain the very nature of space-time and the fundamental forces. There seems to be a link between the external space-time symmetries we all see and recognize and the internal symmetries; this link is very reminiscent of the intricate connection between physics and mathematics.

Sources:

Spontaneous Symmetry Breaking and the Higgs Mechanism. Andrew E. Blechman. December 13, 2000, Gauge theories of gravitation , Geometrical Interpretations of Gauge Theory