THE EMERGENCE OF SPACE-TIME

The theory of general relativity forever changed our way of viewing the Universe around us: space and time are part of a dynamic, four-dimensional entity called space-time. However, in relativistic theory, space-time is classically defined, and it is known that at a fundamental level, nature is quantum. Therefore, space-time must emerge from fundamental constituents at the quantum scale. Understanding what these constituents are and how space-time emerges is one of the most fundamental and far-reaching tasks in the history of physics and human knowledge. In this article, we will examine three approaches to this formidable challenge: two from the perspective of string theory and one from the perspective of loop quantum gravity. We will also explain a fascinating relationship between the two theories that could lead to a unification of our two main approaches to quantum gravity.

Welcome to the study of the fundamental nature of space-time!

The emergence of classical space-time in string theory

In a first approach, we will address the emergence of classical space-time from general relativity in string theory. In string theory, the notion of space-time is linked to an object called a "world sheet," which describes the motion of strings in space-time. This motion describes a two-dimensional sheet:

Surface (worldsheet) formed by the motion of an open string (left)

and a closed string (right).

The motion of the strings forming the "worldsheet" is described by the so-called "sigma models." These models have the peculiarity that the g coupling that determines the strength of the interaction is not constant but depends on the gravitational field. The study of these models tells us that they are only consistent if they possess a special symmetry called "Weyl symmetry." This symmetry only manifests itself if the fields are invariant under changes in coordinates, that is, if they satisfy the equations of general relativity . This fact allows us to glimpse the emergence of classical space-time in string theory:

Strings can only reside on worldsheets whose fields satisfy the equations of general relativity. This stunning result led physicist David Tong to write:

"That tiny string is seriously high-maintenance: its requirements are so stringent that they govern the way the whole universe moves"

"That little string requires a lot of maintenance: its requirements are so strict that they govern the way the entire Universe moves."

The emergence of quantum space-time in string theory

The quantum of gravitational interaction is the graviton. Next, we'll analyze what happens to the action of the world sheet when an interaction with a graviton occurs. The action associated with the string in the world sheet is given by:

The graviton is described by a traceless symmetric tensor S. To analyze this process at the quantum level, we will use the Feynman path integral, whose value depends on the factor eiS(x). If we perturb this metric by including an interaction with a graviton, we have:

In field theory, the emergence of classical fields occurs through "coherent states," that is, through a superposition of a large number of excited states. This fact gives us a glimpse into the emergence of space-time at the quantum level: it appears to emerge from the entanglement of many quantum states .

If we consider a very large number n of states then the interaction is given by:

But this amplitude is exactly what we would obtain if we had modified our original metric in the form:

This means that, as far as the path integral is concerned, there is no difference between a curved space-time and a flat Minkowski space-time with coherent graviton excitation. This tells us that the curvature of space-time is produced by a coherent state of gravitons.

Emergence of space-time and holography

Before starting this section we must review two key concepts about the two basic objects of string theory: strings and D-branes.

1st) A Dp-brane is a p-dimensional object with open strings attached to its surface. This object separates the total 10 dimensions of spacetime into two groups: p dimensions tangent to the Dp-brane and 10-p dimensions transverse (perpendicular) to the Dp-brane.

2) Open strings have electric charges at their ends. Since these endpoints are attached to the D-branes, the branes carry electric charge (in reality, they carry an equivalent field called the Kalb-Ramond field). Similar to how the electric field has U(1) symmetry, N strings attached to the D-brane produce an SU(N) symmetry.

We are now in a position to explain the emergence of space-time in the context of holography. As is well known, superstring theory is based on a 10-dimensional space-time. Imagine that we have a very large number N of 3D branes stacked in the same region of space-time. The coupling g describes the strength of the interaction between strings. We will see what happens as we vary this parameter from a very small value to a very large value.

Scenario for very small g

When the coupling is very small we have two completely decoupled physical systems: the set of stacked D3-branes and a set of closed strings (the gravitational field) that propagate freely through the 10 dimensions of space-time:

As we saw at the beginning of this section, a set of N matching D-branes produces a system with SU(N) symmetry. Therefore, the first system consists of a supersymmetric gauge theory with SU(N) symmetry, and the second system consists of a closed string theory called type IIA superstring theory.

This scenario translates into a set of N 3D branes in a flat 10-dimensional spacetime. The distribution of these dimensions is as follows:

Let's imagine an observer in a spaceship in this spacetime. The ship heads toward point P and measures the size of the concentric circles around this point. As we approach P, the length of these circles progressively decreases until it reaches zero just as it reaches point P:

Scenario for very large g

As we increase the coupling value and reach a value where g>>1, gravitational effects become significant, and the D3-branes begin to interact. These branes carry mass, energy, and the equivalent of an electric field:

the so-called Kalb-Ramond field. Since the energy is still low compared to the natural energy of strings (1/ls, where ls is the string length), we can use the low-energy solutions of superstring theory.

The solution has a metric of the following form:

This solution represents a space-time with the following geometry:

Now the geometry around P has changed radically. Point P has been displaced to an infinite distance relative to any other point in space-time!

In this geometry our exploration ship will detect that as it approaches point P the size of the concentric circles decreases and asymptotically approaches a fixed value R. This value is the radius of the throat of infinite length.

The new distribution of the space-time dimensions is as follows:

The throat, along with the hyperbolic spacetime that asymptotically "sticks" to the flat spacetime, represents an AdS 5 geometry. At the end of the infinite throat, we have a 5-dimensional sphere. Therefore, the global spacetime at the bottom of the throat has AdS 5 x S 5 symmetry.

The key point is that a redistribution of the original spatial dimensions has occurred. Initially, we have a set of D3-branes occupying a space of 3 dimensions plus 1 time dimension, leaving us with a total of 6 transverse dimensions to the branes. As the coupling increases, five of the transverse dimensions transform into a 5-sphere, and the 3 brane dimensions plus the time dimension form the "throat" of AdS space. But then, what happened to the missing transverse dimension? The answer is that this dimension has also become part of the AdS space of the throat. If we look closely, the throat has a new dimension: the r dimension. We can say that, in some way, this dimension is emergent. Schematically, the redistribution would be as follows:

Redistribution of the original dimensions by increasing the coupling: the 3+1 dimensions of the 3-branes plus a transversal r dimension are transformed into an AdS5 space, the other 5 dimensions transversal to the 3-branes form a 5-sphere.

This whole process implies the following: at low coupling we have two disconnected systems: an SU(N) gauge theory and a type IIA superstring theory. At high coupling we have an AdS5xS5 geometry. The conclusion we can draw from this scenario is that the gauge theory and the AdS5xS5 geometry are actually the same theory! This is the famous AdS/CFT duality: a conformal theory formulated in N dimensions is equivalent to an AdS geometry formulated in N+1 dimensions.

The AdS/CFT duality has similarities with the physics of a holographic information system: all the information in a 3D image can be stored in a 2D holographic image. In this way, the CFT system stores all the information about the AdS geometry, and we say that AdS spacetime arises holographically .

The emergence of space-time in Quantum Loop Theory

One of the basic principles of Loop Quantum Theory (LQG) is that all quantities that describe geometry (lengths, areas, etc.) are quantized. This is because space-time at the fundamental level is discrete due to the existence of a minimum size: the Planck length. Therefore, if we take any volume of space, for example, a tetrahedron, all measurements we can make on it will be quantized. To describe the tetrahedron, we must choose the most appropriate independent variables. In LQG, the link between quantum mechanics and geometry originates in the simple vector product of two vectors:

In this way in LQG the faces of the tetrahedron are described as the surface determined by 1/2 of the vector product of the 2 sides that form the triangle.

This vector product defines four vectors (L0, L1, L2, L3) perpendicular to the four faces. These vectors contain all the information about the tetrahedron's geometry. For example, the surface area of the faces is given by the magnitude of the vector L, and the tetrahedron's volume is given by:

Furthermore, due to rotational symmetry the sum of the four vectors is zero:

Due to this rotational symmetry, the values of the four vectors form a group: the SU(2) group. This is the group that determines spin, i.e., angular momentum!

This is the link we were looking for between geometry (spacetime) and quantum mechanics: since the L vectors describing the tetrahedron represent angular momentum values, and these values must be quantized in LQG, we can use the quantized values associated with quantum spin. These values are given by the representations of the SU(2) group. Therefore, the quantized values of the area are given by:

Where is a constant with units of area. Therefore, for each value of angular momentum j, there corresponds a value of a surface S. This is how spacetime "emerges" in LQG.

Towards a relationship between String Theory and Quantum Loop Theory

The previous sections based on string theory (ST) highlight a very important shortcoming: the theory initially starts from a pre-existing space-time in which the strings that generate the worldsheet move. This is why it is said that string theory is not independent of the environment. The definitive quantum theory of gravity should be able to explain how space-time emerges from fundamental quantum components.

Loop quantum theory (LQG) is able to solve this shortcoming because it does not start from an initial space-time, but rather, space-time emerges from a pre-geometric structure defined by the theory. However, LQG has another fundamental shortcoming: it is not clear how the classical space-time of general relativity emerges from the pre-geometric structure of the theory. Therefore, some physicists have proposed a possible link between the two theories: LQG would provide the pre-geometric structure of space-time, while CT represents the effective theory that describes the dynamics of that pre-geometric structure and gives rise to the space-time we observe. We will briefly explain this possible and fascinating relationship below.

The values of j allowed by LQG are represented in a graph of the type:

Given a surface S defined on an LQG graph, we can calculate the total area as follows: we divide the total area into a very large number I of area units. The area of these units will be:

Where ji is the SU(2) representation assigned to the i-nth cell of the surface.

The key point relevant to our discussion is that once a surface is defined in an LQG graph, it can be expressed as a sum of "fundamental units of area." This sum is performed by counting all the bridges in the graph that traverse the entire surface we have defined. The fundamental unit of area is expressed in terms of the angular momentum j, which is a state of the SU(2) group, that is, a spin value:

Representation of a single bridge of an LQG graph: the bridge of the graph "punctures" the surface and defines a unit area. The area is thus expressed in terms of the angular momentum j of the SU(2) representation. (The minimum possible value of area is obtained by replacing j by 1/2, which is the minimum value of j; see the first expression.)

Let's now look at how area is defined in string theory. In string theory, the action of the worldsheet is given by the so-called Nambu-Goto action:

Where the square root of hAB is precisely the area of the worldsheet. To derive the dynamics, we must calculate the minimum extreme of the action (principle of least action).

Next, we ask the key question: assuming there is no pre-existing spacetime and therefore we cannot measure lengths or areas, how could we define the object hAB in the Nambu-Goto action? It turns out that LQG has the answer to this crucial question. Suppose we have a large number of strings moving in a pre-geometry without any structure, without any defined metric. The worldsheets of each of the strings will intersect each other a large number of times. Following the principles of LQG, we can assign to each of these intersection points an area unit whose value is given by the angular momentum j of the string that intersects the worldsheet. In this way, the total area of the worldsheet will be the expected value (the statistical mean) of all these area units:

Since strings carry mass and energy, for a very large number of them, one expects the tendency to avoid string overlaps—to minimize the number of collisions between strings. This amounts to minimizing the expectation value of the area operator. But performing this operation amounts precisely to minimizing the area and the Nambu-Goto action. We therefore conclude that the Nambu-Goto action arises from the need to minimize the expectation value of the LQG area operator. This is the surprising relationship between string theory and loop quantum gravity!

This relationship could lead to the unification of both theories, which would be a huge step towards the final theory of quantum gravity.

Sources:

Connecting Loop Quantum Gravity and String Theory via Quantum Geometry , Spacetime in String Theory