Quasinormal Modes and all that

I think those posts gave a pretty good and mostly comprehensible idea of how you make progress on a research question. However, both of those problems, “how does stuff work in de sitter space” were exploratory questions. Certainly not trivial or well trodden, but are not complete ideas. The real goal is to have a real theory behind how stuff interacts in de Sitter space. Didn’t we already do that, you ask? No, not quite, we just did one part of that.

I’m currently working through a recent paper that starts to do that called “Quasinormal Quantizations in de Sitter Spactime.” Wheras usually in physics “normal” modes are those regular particle states tat separate from each other, ‘quasinormal’ modes are particles that start to die off at large distances, which is intuitively what we would expect based on our previous discussion of space expanding ‘too quickly.’

However, where as my calculations were more on the intuitive side (“If there were a particle, it would do this…”) this paper is actually trying to come up with a consistent quantum theory in de sitter spacetime. That is harder to do, especially with these decaying quasinormal modes.

This paper is claiming that they can do this, at least with some mysterious “R-norm” of their invention, which is pretty mysterious as regular old norms are really what we’re all after.  They claim that these quasinormal modes make a basis of states with respect to this R-norm. But what does that really mean? To carry all of this stuff out and consistently quantize a theory takes all sorts of tools, especially representation theory. It’s not an easy task, and when you’re not getting the answers you’re looking for, you have to start being clever.

These are the questions I’m trying to answer. Definitely much more elaborate than the stuff I was previously talking about, but hopefully from those posts you got a feel for what physics research is about and how we tackle problems.

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How does stuff scatter in de Sitter space?

How does stuff scatter in de Sitter space? Its the next most important question to ask. Even though we have seen that we don’t have a great notion of a particle “coming in” and a particle “going out” we can still try our best to answer this question.

In physics, “scattering” problems, intuitively, refer to a class of problems where some particles come in from a long distance away where they cannot affect each other, approach each other where they get pretty close, interact strongly for some short period of time, then leave again, going their separate ways. Scattering problems are of great importance in quantum field theory, as most physical experiments are exactly that: scatter problems. In the LHC, particles are rammed into each other at very high speeds, interact, then leave. Of course, the particles that leave are not the same particles that enter. In the LHC we have no way to see ‘how’ the particles are interacting (and indeed, quantum mechanically, this is a very poorly defined notion anyway). All we can see is what goes out and what goes in.

So particles “going in” and “coming out” are not particularly well defined in de Sitter space, but that shouldn’t stop us cold in our tracks.

Just like in the last post, when I had to set up a calculation in de Sitter space based off of what I knew about flat space, here too I tried to come up with some analog of the “LSZ reduction formula” for de Sitter space. What is the LSZ reduction formula? Well, in quantum field theory, you don’t really start out with a notion for particles. You start out with a notion for fields, as I talked about in an early post. You can calculate these “correlation functions” that tell you what the field in one point of space has to do with the field in another point in space. The LSZ reduction formula tells you how to use these fundamentally important correlation functions to calculate scattering probabilities.

And indeed, that’s what I did. First I used what I knew (the derivation of the LSZ reduction formula) and some very slight guiding physical intuition to start finding an analog of it in de Sitter space.

Actually, the first time I got it wrong. My adviser could tell that my answers were wrong because they admitted oscillatory solutions for late times, when solutions should have been exponentially decreasing as we already discussed on this blog. Sort of interesting how people who do this stuff for a long time can use their hawk eyes to quickly tell if an equation is right or wrong…

What happens to wave packets in de Sitter space?

 

Ok, so I told you about wave packets and how they expand and why they’re important. So now what happens to wave packets in de Sitter space? This was a question that I made for myself in order to get a better grip on what de Sitter space is all about. Do momentum states separate nicely like they do in flat space?

The answer, simply, is no, they don’t.

As space starts expanding exponentially quickly, points once close to each other soon grow incredibly far apart. Whereas in a non expanding universe particles are able to move farther and farther apart, in a rapidly expanding universe they get ‘stuck’ and ‘freeze,’ so the momentum of the particle becomes less and less important compared to the expansion of the universe.

Intuitively, what this means is that particles aren’t able to really `separate’ from each other, they just get stuck. There isn’t really a notion of particles moving at infinite times.

So how does one calculate something like that?

Well, the first step, and I think the hardest step, is setting up a calculation. Physics is a network of equations and interpretations between these equations. If you’re curious about some physical process and want to investigate it, you have to use what you know to follow the right steps and set up the right formulas to calculate.

The second part entails using all of your acquired calculation skills from taking classes and doing homework to  rewrite your formula in a more useful form that admits a simple interpretation. Usually this requires using some justified approximations. Most physical equations are just good approximations that get all of the essence of the physics without carrying a lot of extra baggage.

This is especially true in quantum mechanics / quantum field theory. Exact solutions are almost always hopeless, but luckily there’s a very large toolkit of approximation schemes that are a pain to learn but vital to learn.

 

 

What happens to wave packets?

So I’ve told you what de Sitter space is, and that I’m interested in studying it, but what am I interested in studying? Well, to start things off, we want to know what happens to particles moving in de Sitter space. When we’re talking about particles, we’re talking about quantum field theory (see previous entry).

What happens to particles in a non-expanding universe? Well, because of the Heisenberg uncertainty principle, its position and its momentum cannot be perfectly determined, but certainly it can start roughly at some position with roughly some momentum (as well as some uncertainty in both).  That particle then starts moving in the direction of it’s momentum. Because both it’s position and its momentum are uncertain, the region over where this particle might be also starts expanding. However, there is something interesting when you look at the equations: a particle that starts off with a lot of position uncertainty eventually has less position uncertainty than a particle that starts off with very little position uncertainty.

Let me explain: the Heisenberg uncertainty principle is

\sigma _{x}\sigma _{p}\geq {\frac {\hbar }{2}}~~

that the uncertainty in the position multiplied by the uncertainty in the momentum is greater than Planck’s (reduced) constant over two. This means that a particle with a very uncertain momentum can have a very well defined position, and a particle with a very uncertain position can have a very well defined momentum.

A particle that starts out with very little uncertainty in its position must have a lot of uncertainty in its momentum, which actually makes it spread out faster than a particle that starts out with a very uncertain position but a very well defined momentum!

This might make sense intuitively, and can be pictured as follows:wave packets

The particle that starts out with more momentum uncertainty spreads out quicker than the one that starts out with a very little.

However, this fact is backed up by a mathematical calculation which I am not showing you here.

Why do we care about this?  Well think about this: a particle with one exact momentum value must have infinite position uncertainty. That is: if you know for 100% certainty what the momentum is, you have absolutely no idea where it is. However, it is very important, quantum mechanically speaking, that states with different momenta do not overlap. This is the same thing as saying that a state with a definite momentum will never be measured to have a different momentum. But how can this be if definite momentum states overlap with the entire universe?

Well, what if instead of a definite momentum we mearly had an almost definite momentum with almost no uncertainty? Well, then two states with different momenta would separate out and eventually have no overlap, so they are indeed different states.

This may seem like word juggling the way I am explaining it but I can assure you that it’s not, and this property of momentum states is very important. We can define perfect momentum states by thinking about merely very good momentum states and then pretend that they are perfect.

But what happens in an expanding universe? This took longer to explain than I thought so I will tell you next post…

New Year New Stuff

Hello! It’s a new year and there’s new stuff to talk about. I figured that to kick this blog off I’d take you through some calculations I did for my research. I would like to give you the reader some idea of what carrying out a typical physics calculation entails. This will take me a few posts so stick with me.

I will begin by telling you about de Sitter space. When I say “space” I mean that it is a space just like how our universe is a “space.” What does it mean from one space to be different from another “space” in physics? Well, certainly there’s simple stuff to start off. For example, a sphere is a different sort of space from a donut in the sense that one cannot be deformed into the other. However, in physics we’re usually thinking more concretely than that. Different spaces have different notions of distance. For example, you could imagine a universe where all the distances stay the same for all time. This is not the case of our universe, as our universe is expanding. Thus the distance between two points is increasing with time. Perhaps “space time” is a better term to use than just “space” in this context, but as everyone else calls it “de Sitter space” and not “de Sitter space time” we will too.

What is de Sitter space? Imagine our 3D world as some sort of 3D sphere. I don’t mean a sphere like a ball with an interior. When physicists or mathematicians use the word “sphere,” they mean the surface such a ball. So the surface of a ball that you’re used to is a 2D sphere, as the surface of something 3D is 2D. It is impossible to really picture a 3D sphere as it is the boundary of a “4D ball.” There is no way to walk out of a 3D sphere just as there is no way to walk off the surface of our earth (with is a 2D sphere). Our universe very well may be, on the largest scale, a 3D sphere like this, although we’re not sure yet. If it were such a 3D sphere, it would be a very very big one.

But that’s not all: de Sitter space is expanding. Well, actually, first it contracts, then it expands. To be specific: the radius of this 3D sphere ‘starts out’ as infinite, then rapidly contracts to some finite radius, then starts expanding again. This expansion is roughly exponential, i.e., very fast.

de Sitter space is not a good model for our universe, as our universe is not expanding exponentially. It did expand exponentially during ‘inflation’ in the moments after the big bang, and it may expand exponentially in the distant future due to dark matter (the jury is still out on that one) but it is not now.

However, this space has very many nice properties and is used to do a lot of very interesting theoretical physics, so that’s what we’re studying for now.

How to read

kid_reading_boring-e1459526032134I thought I would say a few words on how to read physics and math texts, which is always at least half the work of doing research. The first step is to make sure you aren’t ever spinning your wheels when you read. If you get stuck on some point, or get confused, don’t waste your time. Read something else, come back tomorrow, or ask some one to clear up some points for you. It’s never good to just waste 3 hours pulling your hair out. (1 hour is acceptable.) Going along with that, always make sure you are awake. Sometimes you’re in a mood where you can read for 8 hours straight, and other times you can’t concentrate at all.

As for actually reading, make sure you understand the calculations being performed as well as possible. It should go without saying that you should be following around with a notebook of your own, doing the calculations discussed. It is far too easy to think you understand something. In physics research, it is always the most important to be able to calculate, not just be good at reading books. This is the hardest part of learning, to learn something well enough that you can reproduce it when you want to in the way you want to.

Reading is often an extremely slow process. A page every half an hour is pretty fast pace. A page every hour is also good. Often times I’ll have a moment of realization that I’ve been reading one page for an hour and wonder how much time I’ve spent reading all the physics I’ve ever read in my entire life.

What’s incredible about the process is that after all that time you really do learn something, even if it takes you 2 months. At any given moment it feels like you aren’t learning that much, but when you look back you realize how far you’ve come.

 

What does it actually mean to do theoretical physics research?

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It’s difficult to get across, even to students of physics. The first thing I’d like to say is that people have the impression that all physicists are out there searching for a “grand unified theory of everything,” or “quantum gravity” or something. This isn’t even close to true. For starters, most physicists are experimentalists, working on things far removed from particle physics. Maybe they do condensed matter physics, nuclear physics, quantum optics, or something completely different. These branches of physics will probably not be extremely affected by the discovery of a grand unified theory. However, each of these fields have their own theoretical physicists. Not all theoretical physicists are working on things like string theory. They are found in all branches of physics. You may wonder, what is it that they are doing? Aren’t their respective fields “done”? Not at all. Even when we have a theory of basic physics, how these laws interact in larger phenomena can still be hard to determine. Very often, effects are discovered in the laboratory and then only later theoretically explained!

The number of physicists working on quantum gravity is about 1%. What is it that they are doing, exactly? Aren’t they all trying to find a grand unified theory? Well, not exactly. Theoreticians spend a lot of time on theories that in no way describe our world, and usually never will. They study universes that have very different properties from our own, or particles that definitely do not exist. The point of studying these things, firstly, is that they are interesting! They have very interesting properties and motivate a lot of interesting new physics. Sometimes, in these made up universes, we can find properties that look a lot like our own, even if they have other properties that do not match up. But by studying what sorts of theories have these properties, we can try to reproduce these effects elsewhere.

Theoretical physicists also study certain things an extreme amount, hoping that they shed light on far off branches of physics. The Black Hole Information Paradox is a great example that I am studying. Basically, stuff can fall into a black hole and never be seen again. Stephen Hawking showed that black holes can radiate off their mass in “hawking radiation” and eventually disappear. So what happens to that information that went into the black hole? Does it just disappear? Physicists have many ideas for what happens, although there is still a lot of work to be done, with new interesting papers coming out all the time.

Why are they studying these black holes? Many are studying them because, perhaps, by studying this black hole we can learn more about the interaction of gravity and quantum field theory. We assume that both are manifestations of one theory. If we truly understand what is happening here, we might hopefully be able to unite quantum mechanics and gravity into one coherent framework.