THE MYSTERIOUS NATURE OF TIME

In the previous article, we looked at four "clues" about the mysterious nature of time. In this article, we will discuss the last and perhaps most important possible approach to the true nature of time. This last "clue" leads us to a strange new view of space-time, one in which time emerges from the interaction of a large number of tiny "fundamental entities."

Entropy and statistical mechanics

Virtually every popular science book on the "arrow" of time mentions that time could have a thermodynamic origin. In thermodynamics, a large number of small molecules or atoms (for example, in a gas) interact, producing a fairly complex dynamic. However, by making a series of reasonable assumptions, the overall behavior of the whole at the macroscopic level is

It comes down to a few simple rules. The second law of thermodynamics tells us that entropy (the degree of "disorder" or degrees of freedom) always increases. This process is irreversible and produces a clear "direction of motion" toward increasing entropy. This "flow" bears a remarkable similarity to the passage of time. Could time be related to an irreversible process arising from the interaction of many elementary components? Statistical mechanics studies precisely these

processes and the simplest and most successful model is called the Ising model.

The Ising model

Consider a very large two-dimensional lattice of spins that can be in two states up (0) or down (1) and that is located within a magnetic field. In this system, two opposing "forces" intervene: on the one hand, the interaction between nearby spins causes the closest spin to have a certain tendency to orient itself in the same direction as its companion. This first force tends to order the system and

represents energy (the system tends toward the lowest-energy state, which is the one with aligned spins). The second "force" represents entropy, which tends to disorder the system. That is, there is a competition between energy and entropy, and the balance is governed by temperature. At high temperatures, there is a lot of free energy available, and therefore entropy wins: the spins are arranged randomly, and the system tends to be disordered (paramagnetic phase).

When the temperature drops below a certain critical value, the energy factor wins, and the spins spontaneously orient themselves in one direction to form a low-energy arrangement (ferromagnetic phase). This explains why certain metals, when cooled below a certain critical temperature (Curie temperature),

and they solidify and become magnets (when these solidify in the magnetic phase they maintain a certain order in which the oriented spins behave like a permanent magnet).

Dualities and the renormalization group flow

Imagine that we begin to visualize the spin lattice by zooming in, that is, increasing the display scale. If we increase the scale by a certain fixed value, the lattice will appear exactly the same but viewed at a larger scale, that is, at a lower energy (greater distances equal lower energies and vice versa). Therefore, the Lagrangian representing the energy of the system will have decreased by an amount such that it goes from being L to being L'. Mathematically, we can calculate the value of L' in terms of L and "compensate" for the energy difference by modifying the system parameters so that we have an equivalent system. By repeating this step many times, the Lagrangian appears to "shift" through the space of possible energy states; this "shift" is called the renormalization group flux.

In 1997, theoretical physicist Juan Maldacena proposed the existence of a duality that would represent one of the most important theoretical advances in quantum gravity in recent decades. This duality establishes that a conformal quantum field theory (CFT) in D dimensions without gravity is equivalent to an AdS spacetime (a Universe with negative curvature) with gravity in D+1 dimensions. In the Ising model, the properties of the system are completely defined by the partition function:

If we consider the limit where the distance between the spins tends to zero, the statistical mean of the spins behaves exactly like the values of a continuous field Ø which allows us to express the previous partition function as a path integral:

This is precisely the expression used as the partition function in conformal quantum field theory (CFT), which means that our 2-dimensional Ising model behaves like a CFT and therefore, according to the AdS/CFT duality, has a dual spacetime with 3-dimensional gravity.

The emergency of time

The AdS/CFT duality shows us the following equivalence:

This indicates that the dz dimension "emerges" from a CFT in a lower dimension. It would therefore be natural to ask whether a theory with gravity and a time dimension could emerge from a timeless CFT of the type CFT: ds2=dx2. That is, would it be physically possible to implement a duality of the following type:

Now we will perform the "miracle" of the creation of time: we take a one-dimensional "lattice" of spins described by the Ising model, that is, a one-dimensional CFT. What will the dual theory be with gravity? When applying renormalization techniques, it seems natural to identify the distance between spins as the "time sections" and therefore identify the renormalization factor as time.

When we make this identification we obtain a dual space-time defined by:

This is the metric of a 1-dimensional spacetime expanding with a certain cosmological constant! From a timeless CFT, a 1-dimensional Universe expanding in time would emerge! The evolution of this dual 2-dimensional Universe (1 spatial and 1 temporal) would be like this:

The advance of the renormalization flow means that we increasingly consider larger scales, which reproduces the effect of an expanding Universe with a certain cosmological constant (a dS Universe like ours). The renormalization factor (the number of times we have performed the operation) plays the role of time.

Furthermore, this operation has allowed us to glimpse the implementation of a duality that physicists have been pursuing for decades: the dS/CFT duality.

The constant ko has an arbitrary value that has no effect in the early stages of the renormalization group flow, however, as the flow progresses ko starts to have a non-zero value and therefore begins to have measurable effects. This is exactly the behavior of the vacuum energy in theories

quantum fields!

This brings us to a fascinating question: The cosmological constant we observe

in our Universe and which constitutes the so-called "dark energy," could it be formed by the vacuum energy of the dual CFT fixed by the ko parameter? As we will see below, the answer could be affirmative.

Our 4D Universe emerging from a dual 3D CFT

All of this so far seems like "abstract mathematics" with no connection to our real world. However, the relationship between physics and mathematics once again demonstrates its enormous power. Our universe has three spatial dimensions and one temporal dimension, and is approximately dS. Therefore, if we consider the existence of the dS/CFT duality, its "dual universe" would be a CFT defined in three dimensions. What would happen if we compared the evolution of our universe according to our standard cosmological model (the sigma CDM model) with the evolution given by an Ising model defined in three dimensions? The answer is as follows:

The similarity is striking. Of course, one can't expect an exact match since we've chosen certain parameters arbitrarily. However, the curves are very similar and offer a clear explanation for the mystery of the cosmological constant— this would be the vacuum energy of the dual CFT!

There is also other surprising evidence that indicates this could be the right path to understanding the true nature of time: every universe with a positive cosmological constant, like ours, has a horizon that depends on the observer. It has long been known that, as seen from an inertial observer, this horizon has a certain temperature. The microscopic origin of this temperature is a complete mystery, and calculations indicate that its value is:

T=1/2Πlo

In the Ising model we have considered, the critical point would be located at infinity, that is, on the visible horizon, and the value of the temperature at this point is:

T=1/2Πkclo. The temperature at the critical point kc is the same as the horizon temperature in a dS Universe! The horizon temperature in our dS Universe would be the same as the critical temperature in the dual QFT!

Conclusions

Studies based on statistical mechanics and the AdS/CFT duality indicate that time appears to arise as a result of the interaction of multiple fundamental entities (a spin network in the Ising model) that "resided" in the dual quantum theory. If this is true, the flow of time we observe in our 4D Universe would be a consequence of a "cooling" (an advance of the renormalization flow toward lower energies) of the particle-fields that reside in the dual 3D QFT. The characteristics of this 3D quantum system would explain many of the great enigmas of current physics, such as the origin of the cosmological constant or the origin of the temperature of our visible horizon. In this way, the origin of time would have a holographic explanation.

All of this leads us to a series of fascinating questions:

- What are the fundamental components from which the time dimension would arise (lepton spin networks, strings, etc.)?

- In the AdS/CFT duality, the dual CFT is considered to reside at the asymptotic edge of AdS spacetime. Does the dual CFT reside at the asymptotic edge of our Universe? How does the interaction between this CFT and our 4D Universe (the "bulk") occur?

- Is the entropy of the dual CFT critical temperature the entropy of our AdS horizon? Could this relationship help explain the mystery of black hole entropy?

Although we still have much to discover about the true nature of time, we are glimpsing new clues about its nature, and the final answer promises to forever change our conception of the Universe we inhabit.

Sources: Emergent time axis from statistical/gravity dualities