False Vacua I: The End of the Universe As We Know it

A few weeks ago, I wrote the first script of what I hoped would become a pop-sci podcast. Since this podcast, if it ever happens, will not happen for a rather long time, I figured it’d be fun to make it a series of blog posts. Many apologies if the post seems too simplistic or condescending….

The subject of false vacua is fascinating in its own right, but also nice because it brings together many different areas of theoretical physics. The discussion encapsulates many current areas of beautiful physical research such as quantum field theory, gravity, cosmology, supersymmetry and string theory. There is also a science-fiction like quality about the subject; After all, we’ll be talking about the possibility that at any second a bubble might form out of nothing and eat you. Hopefully, after these podcasts/blogposts the viewer/reader will appreciate that nowadays science fiction is somewhat superfluous; some of the craziest ideas are found in modern physics.

The fusion of Einstein’s special relativity and quantum mechanics under the heading of “Quantum Field Theories,” which describes things that are both very small and very fast.

How do we test quantum field theories in the laboratory? Well, we take very small things like subatomic particles: electrons, protons, etc., and we accelerate them to close to the speed of light and collide them into each other. Huge particle accelerators are built around the world to accomplish exactly this. Quantum field theories (QFT’s) will tell us what to expect from a collision; Experiementalists can measure what’s happened and see if the theory is correct.

This will not be a discussion about QFT’s. But we will quickly list a few things about them. There are many different possible, consistent QFT’s. Every QFT is different: each describes different types of particles that have different charges and interactions between the particles. Mathematically, we can easily write down a quantum field theory that corresponds to particles that may not even exist in nature!

For example, for the rest of this podcast/blogpost I will work with a set of quantum field theories with only one type of particle, which I’ll call the \phi particle. Now, no \phi particle has ever been observed in nature, but that doesn’t mean I can’t write down such a QFT. This QFT would tell me how the \phi particle would scatter and interact with other \phi particles, if \phi particles existed. Now, in real life things are much more complicated: there are electrons, muons, quarks, protons, a veritable zoo of particles! Such a zoo makes a quantum field theory a little bit more complicated. So, to make things simpler, we’ll just pretend there’s one type of particle in the universe, the \phi‘s. This is just a crutch; it is an interesting toy that we can play with. In the first blogpost we’ll derive some physical qualities of these \phi models, and I’ll later discuss what generalizes to our actual, more complicated universe.

Ok, so there it is, we live in a world with one particle, the \phi. Ok, well how does our theory tell us about the \phi? Well, it turns out that a QFT is specified by a function, the “potential energy,” denoted here, V(\phi). If you handed a picture of V(\phi) to a physics grad student he or she, in a night without sleep, could work out all of the details about how a \phi particle scatters and interacts with other \phi particles.

Let’s take a theory that has a particularly simple picture of V(\phi). Now by their nature, these podcasts/blogposts will be relatively imprecise. We’re going to have to begin talking in pictures, and somewhat metaphorically.

Let’s think of the potential energy picture as a valley in between two infinitely tall hills. If I place a ball at the bottom of the valley, it will stay there and not move. The ball sitting at the bottom of the valley should be though of as “the vacuum”. It corresponds, in the quantum field theory, to there being “nothing”. Since everything in this fictional universe is \phi particles, it corresponds to there being no \phi particles around. What, in this pictorial version of the theory, corresponds to one \phi particle? Well, that corresponds to a ball that oscillates, goes back and forth on the two hillsides. Adding more $\phi$ particles, roughly corresponds to having the ball go further and further up in the hillside.

Now what if I had taken a different V(\phi)? What if I took the old one and turned it upside down?

Well from our previous experience, the vacuum state, the situation where there is no particles, corresponds to a place where I can put a ball and where it will stay put forever. You might say: place the ball at the very top of the hill. Surely, if you were very careful, the ball would not move; it would stay there forever and that would be the vacuum state. You’d almost be right. Let’s say I carefully balance a ball at the top of the hill, and then let go. Ok, it’s good for now. However, any small movement, any small perturbation, no matter how infintesimal will cause the ball to roll down the hill one way or the other. In fact, tiny unmeasureable quantum fluctuations will serve to nudge the ball off the top. The ball is horribly unstable perched up at the top of the hill. Correspondingly, this theory has an unstable vacuum. The ball will roll down the hill forever. Physicists can calculate what this means: energetic \phi particles will be created out of thin air. The rolling of the ball corresponds to creation of these energetic particles. Things will get hotter and more energetic with no end! Surely these are not very nice or interesting theories to work with; they clearly do not describe our universe very well!

So, in the theories we will work with from now on, our V(\phi) will always have at least one place where you can place a ball and have it remain there stably. These correspond to points of V(\phi) which are lower than all of the ones around them. Not surprisingly, these points, which correspond to vacua, are called “minima”.

Let’s look at another potential, corresponding to a different theory of \phi particles.

\"Vaccum A\" is on the left, \"vacuum B\" on the right...

This will be our favorite potential from now on! For this picture, innocent it may seem, contains all the information a physicist would need to know to tell you when our universe is going to end! (duh duh duh!)

In the picture, there are two possible places I could place a ball, point A (the left minimum) and point B (the right minimum). You’ll notice that point A is higher in potential energy than point B, this will play an important role in things to come.

Actually, I’ve been lying to you. If I just gave this picture to our disheveled looking grad student and asked him/her to calculate the mass of the $\phi$ particle, or how two \phi particles scatter off each other, he/she would be very confused. He’d stay up all night (sorry for the straying from gender neutrality!), pounding his head against the wall wondering how his life went so very, very wrong. He’d finally come back at the end of the long night and ask: “Which vacuum are you talking about?” And he’d have a good question. The physics of the quantum theory actually depends on which vacuum we use. From this picture, we know it’s either vacuum A or vacuum B, but we can’t tell just by looking at the picture which one it is. If we lived in A, \phi particles around us would correspond to little oscillations about A. While if we lived in B, particles would correspond to oscillations around B. These oscillations could be very different in nature. For example, \phi particles in the A vacuum could be very heavy and also very strongly attracted to one another, while \phi particles in the B vaccum might be very light, and relatively weakly attractive. The universe around us, made entirely of \phi particles, would depend very much on whether or not our vacuum was A or B.

How does the universe choose which vacuum we’re in? Well, this depends very much on the physics of the early universe. After the big bang, everything was extremely hot and energetic. Let’s represent the vacuum by a red ball. In this hot soup, the vacuum was not yet stable and it too, was bouncing around. Slowly, the universe cooled, and when it was done, the vacuum ball either was on the left side, corresponding to vacuum A, or on the right, corresponding to B. Without knowing the full details of the early universe we cannot say how the vacuum came to be what it is now. Let’s assume that we are ignorant of this and that which one we ended up in was largely a matter of random chance. From now on, let’s assume that we’re in the vacuum that is at higher potential energy, vacuum A.

Now, if we’re in vaccum A, we know a few things. Let’s assume that in this crazy \phi only universe, there are \phi people. We would know that the physics of vacuum A make it possible for there to be life (we’d infer this from the fact that we, the \phi people existed). The \phi particles have exactly the right mass, and exactly the right strength of the interactions between them for them to make little \phi people. Life surely requires a very delicate balance that is very dependent on the physics of the \phi particles.

Chances are, that if we’d ended up in the B vacuum after the universe had cooled, there would be no life around! That’s a scary proposition! Man, I’m sure glad, as a \phi person, that we ended up in the A vacuum.

But what if we’re not out of danger yet?! I told you before that phi particles correspond to oscillations of the vacuum. What if we oscillated so far, that the ball was able to get over the hill and into the B vacuum? This means that we’d have to take some \phi particles, and make them extremely fast and then collide them; only this could precipitate such a disasterous event. The height of the hill would tell us just how energetic we’d have to make the particles. Fortunately, for most realistic theories, the hill is too high, our accelerators simply do not have enough energy to move us from the A vacuum to the B vacuum. Good news!

So far, so good. We \phi people are existing. We’re happy living in our $\phi$ A world. We’re sure glad that we didn’t end up in a \phi B world, we wouldn’t exist. They don’t have \phi movies, \phi music, \phi beautiful women in the B universe. There are no long, romantic walks on the beach in the \phi B universe. There’s probably not even a beach. All there is, probably is a lot of empty space. It’s a pretty depressing universe.

But, we’re ok. Living in our A universe. Whew.

Not so fast! In fact, we should remember that our theory is a quantum field theory and that strange things can happen with quantum mechanics. If our world didn’t involve quantum mechanics, then our picture of balls and hills would be the whole story, but it’s not. In fact, there’s a finite probability that at any given time, the ball can go right through the hill, as if it weren’t even there! This phenomenon is called “quantum tunneling.” You should picture our vacuum ball somehow managing to dig a hole through the hill. Then, when it gets to the other side of the hill, it rolls down to become a B vacuum. The dreaded, lifeless, humorless B vacuum. The B vacuum that is as fun as doing taxes.

What would such a tunneling event look like to us, sitting in our A vacuum world? Well, when the tunneling takes place, a small, microscopic bubble would nucleate somewhere in our universe. I am not using the word “bubble” metaphorically; amazingly enough the physics of this tunneling event is exactly analogous to the physics of bubbles nucleating when you boil hot water on the stove. So somewhere in our A universe, a microscopic bubble nucleates. Very quickly, the bubble begins to get bigger and bigger, it expands. After a very short time, the bubble is expanding at the speed of light! What’s inside the bubble? Well, it’s the B universe, with all its nasty B physics! Inside the bubble, there is no chance for life, joy or sex!. This bubble is expanding at the speed of light, eating the A universe as it expands and turning into the deadly B universe. It is an impending bubble of doom and it is headed straight for us, eating everything in its path!

If there is any comfort for our sordid fate it is this: we will not know that the bubble has eaten us until we are dead already. In fact, nobody will be even able to tell us that the bubble is coming: since its munching away at the speed of light, any signal that could be sent to warn us about the bubble will arrive at the exact same time that we are being eaten. We feel no pain though: the timescales on which our neurons fire and tell our brains “this hurts, I’m being eaten by a giant vacuum bubble”, are much longer than the time it would take for the bubble to eat our entire body. We go ignorantly and painlessly, but not with dignity: we have been eaten by a goddamned bubble.

If we told our disheveled grad student that we were in vacuum A, he could, again by staying up all night, calculate the probability per unit time per unit volume that a tunneling event would occur, that is, he could predict the probability of a bubble forming and eating all of us. This may sound like science fiction, but in fact, it’s a beautiful (if somewhat terrifying) effect of modern physics!

In fact, if we lived in the A vacuum and knew about it, we probably wouldn’t have to worry too much. If we knew the age of the universe to be say 14 billion years, as it is in our universe, then the probability of a bubble nucleating would probably be rather low, since it hasn’t happened in the last 14 billion years. We can probably bank on at least another few billion years or so without the bubble of doom.

I should stress this disclaimer; every time a new particle accelerator is being built, some very crazy and silly people object because they think it will create a black hole that will eat us all. The probability of something like that happening (like the probability of tunneling from a vacuum A to vacuum B in your lifetime) is astronomically low. Chances are probably better of you winning the lottery while getting struck by lightening while having a beer with George Clooney, all while you’re reading this blogpost.

By the way, I should mention, that if our vacuum, the vacuum of life and joy were actually the B vacuum, we’d have absolutely nothing to worry about. The ball cannot tunnel uphill. In fact our vacuum “wants,” to be at the lowest energy possible, and will resort to tunneling if necessary. It will not resort to tunneling to go uphill. This is directly analogous to the fact that an actual, physical ball will roll downhill if it can, but not uphill.

In the situation described before, where we live in vacuum A that is of higher energy than vacuum B, vacuum A is sometimes called the “false vacuum”. It is “false” because it will eventually decay, via ominous bubble, to the “true vacuum,” of lower energy, B.

Ok, this was all a toy universe, though–a universe made of only \phi particles. How are such theories, with false vacuua, possibly applicable to physics of this universe? More on that in posts to come…


6 responses to “False Vacua I: The End of the Universe As We Know it

  1. Wouldn’t it tunnel from B to A given Energy > V_Amin(\phi) ?

    I’m told that potentials from the Linear Harmonic Oscillator, or any other isospectral oscillator appears in disguise when dealing with homogeneous gases, since the equidistant energy levels stand for transitions occurring when integer number of particles entering or leaving the system. Is that accurate? How’d you find a good V(\phi) for a two (different) particle gas?
    Thanks. Can’t wait for the next one.

  2. Very nice essay! You’ve got a few instances of $\phi$ which didn’t render as \phi.

  3. Looking forward to the next post in the series!

  4. After pondering this for a little while, a question arises in my feeble mathematician’s mind: what happens if both vacuua have the same energy?

  5. Interesting, very interesting. Is it because we are inside the buble we will never find out if we are in a false vacuum?

    The higher energy bubble would be expanding at the speed of light? The universe expands at the speed of light doesn’t it? Or is a false vacuum also expanding at the speed of light? If that is the case would a true vacuum bubble ever be able to destroy the entire false vacuum?

  6. Pingback: Compactification in String Theory Part I: Motivation « Imaginary Potential

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