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# Quantum Tunneling in QFT and GR

Eternal inflation and bubble collisions.

Quantum Tunneling on QFT and GR
. This report was submited as my Part III essay in the University of Cambridge (2020). Working under the supervision of Prof. Fernando Quevedo, I did a literature review on the generation of bubbles in a quantum field through tunneling. For this, the main goal was to replicate the results from the 1980 paper by Coleman and de Luccia.

The essence of this project was to study the implications of quantum tunneling on the universe we live in. To give the full picture on the change of physical mechanics on the last century, I start the project by explaining why quantum mechanics broke all the intuition we had from classical mechanics.

In physics, a fixed point is the value for which the system remains in equilibrium, and it is given where the potential has vanishing slope. However, the stability of these fixed points depends on how the system reacts to small perturbations. For instance, every minimum of the potential will be stable, while all the maxima will be unstable. Moreover, to go from one minimum to another one, the system requires having enough energy to climb up the potential barrier separating them both.

However, with the development of quantum mechanics, this picture was shown to be incomplete. The process mentioned above (Quantum tunneling) implies that a state can travel through a potential barrier which is more energetic than the system itself. This is something totally forbidden in classical mechanics, and the probability of this happening depends on the shape of the potential barrier.

Therefore, in classical mechanics every minimum is stable, while in quantum mechanics only the global minimum is. As working with complex potentials is very difficult, there are nice approximations to find the results.Thus, I started by using the WKB approximation on N-dimensional quantum mechanics. As quantum field theory (QFT) can be thought as an infinite-dimensional quantum mechanical system, I just had to use the results for N=∞. To go thorugh these calculations, I closely followed the paper by Copeland et al. Fig. 1: Generic potential

Once I had an approximation for the probability of tunneling happening in a quantum field, I studied the implications of this process on a potential as the one in Fig. 1. For instance, we need to understand how tunneling affects the configuration space of the field. For this, I followed the original paper by Coleman, where he introduces the Bounce and works with a field initially at rest in a local minimum. After some time, tunneling to the global minimum takes place, leading to the creation of an expanding bubble with different energy inside. To work out the properties of this bubble, it is used the thin-wall aproximation.

One of the most interesting things of the universe is that there exists an absolute energy. Even though in classical and quantum mechanics we just care about differences of energy, the universe seems to know when a system has energy in it or not. This is because a field resting at a positive minimum of the potential acts as a positive cosmological constant, leading to the accelerated expansion of the universe. Depending on the sign of the vacuum the field is resting at, we can have deSitter, Minkowski or Anti-deSitter spaces.

As I wanted my results to describe the universe we live in, I studied the probability of dS to dS tunneling. Thus, the universe is originally expanding at high speeds because of the field being in the false vacuum, but after some time a bubble of true vacuum forms. As we can tell in Fig. 1, the true vacuum has less energy than the false one. This will make the space inside the bubble to expand slower than the space outside the bubble. The whole description of the geometry of the whole space can be found in my essay. Moreover, it was found that General Relativity sets upper bounds on the possible radius the bubble can take.

In the same way we can have one bubble being nucleated, there is a probability of generating two or more bubbles. As one might expect, if this bubbles are close enough, they will collide with each other, leaving imprints on the primordial universe.

Therefore, in the last section of my essay I estimated the number of collisions we shall expect on a universe like ours. For this, I started by defining a metric on the whole space, and then calculating the available space where a second bubble can form. As we have a good approximation of the tunneling probability density, I just had to substitute into the result the data that we have from our universe. From this, I got an expression for the number of expected collisions our universe should have gone through so far. This number depends on the speed of expansion both inside and outside the bubble.

However, we do not have any evidence of being inside a bubble, so we cannot have any estimation of the number of collisions. But we can do the opposite; by observing anisotropies on our nightsky, we might be able to find evidence for the nucleation of bubbles, and so get an approximation of the shape of the potential that is leading the expansion of our universe.

In conclusion, in this project I learned about how to study the effects of a quantum field when gravity is considered. Normally, it would just lead to the accelerated expansion of the universe, as showed in my project at Nottingham. However, in a quantum system there can be wild things such as quantum tunneling. This will also influence the universe once gravity is included, creating bubbles of space which expand at a different rate. Thus, this essay is good starting point to study how to parametrize non-homogeneous expansions, which can be further used if it is ever found an anisotropy on the expansion of our Universe.

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