Mass creation in graphene

Physicists tell me that the way that particles get mass is something to do with extra space dimensions getting curled up, giving mass to particles that would be massless without the extra dimensions. I don't really understand this, but it sounds neat.

Following the white hole in a kitchen sink and Hawking radiation from a black hole analogue, today's hot topic in the arxiv blog is a low energy system where massless particles acquire mass from the curling up of an extra dimension!

Apparently the charge carriers in graphene are very well-modelled by equations that treat them as massless (perhaps one can think of them floating in a sea of other electrons?). Some smart guy discovered that if you turn graphene into a nanotube (curling up one of the two dimensions), the equations get modified so that the electrons act as if they have acquired some mass.

Maybe some day I'll understand how that happens...

Mass generation in graphene (original paper)

Man, that is one interesting article.  Definitely heady stuff.  I definitely can't grasp what the practical applications/implications are just from reading the article.  As for understanding it, I got lost in the math.  I just don't know enough about physics and math at that level (I stopped when I got to Analytical Geometry).

There's so much to know, but at what expense to time?  I defaulted to more practical applications.  I play the guitar and piano.  I'm a network geek for AT&T.  I married Debbie, my wife, 20+ years ago.  We became foster parents, and raised 50-60 kids.  We adopted three of them, all when they were very small (Michael, now 24, Jonathan, now 19, and Emily, now 16).  There's so much more that we do along with all of that.  Music was my thing, so that's where I focused my thoughts, energy, and study.  Well, that and computers.

However, I'm thankful for guys like those that were involved with the paper.  They're the ones unseen by society as a whole.  They dream the really BIG dreams; Understanding things like the cosmos, mass, dark matter, multi-dimensionality, etc.  These big dreamers push open the envelope in every generation.

A Guardian blog post about a recent physics prize only makes it clear how very difficult the topic is. I have some understanding of gauge theories, spontaneous symmetry breaking, and still I've never got to that "aha!" moment that gives me an understanding of how a universe that would have nothing but massless particles moving at the speed of light turns into one that has lots of massive particles as well that move at lower speeds and whose energy varies with their speed.

I happen to own a copy of Dirac's own book on quantum mechanics, where he talks about the relativistic equation for the electron that he introduced. It has some weird features, one of which is that the velocity operator for it only has eigenstates with speed c, the speed of light. So electrons really only move at the speed of light. However, it was explained that there is this thing called "Zitterbewegung" (if I recall correctly) which meant that electrons keep changing their direction, so that the average speed is always less than the speed of light. I don't know how this relates to the modern view, but given that this was in the early 1930s, ideas have developed a fair bit. Quantum field theory has changed a very great deal.

[There is a wikipedia article on Zitterbewegung, which talks about the electron oscillating around its median position at the speed of light with frequency (about 1.6 × 10²¹Hz), which sounds a bit like a small curled up dimension, but may or may not be related]

They lost me at graphene can act as a laboratory for studying exotic relativistic physics.

But from what I can understand, energy is created as dimensions appear?  Can this defy the second law?  Does lessening dimensions (big bang went from 11 to 3 from what I hear) lower the energy of a system?

The first thing is that in this example (like the white hole and Hawking radiation ones), what you have is a system that can be modelled with similar mathematics to another system. As a consequence, the mathematics predicts that certain things that happen in one system has an analogue that happens in the other system. It's like an analogue computer, where electronic components model systems in the real world that have similar equations describing their behaviour.

The thing that is of principal interest here is the mass, rather than just the energy. In relativistic quantum mechanics without mass, everything has to move at the speed of light. In graphene, the charge carriers act as if they are massless, and have to move at a speed of about 1000 km/s (a fixed speed for the material). This is like photons in empty space, which are obliged to always move at the same speed. Apparently, the mathematics of the two systems are similar, with the speed of light replaced by 1000 km/s (and 3 dimensions replaced by 2)

But what happens when you roll the graphene up into a tube?  The motion of the charge carriers can have a component that goes round tube rather than along it. Imagine a corkscrew motion along the tube as being the addition of a component along the tube and another component that goes in a circle around the tube.

But this second component is a cyclic thing. If it was classical physics, it would be like an orbit, but the scale of a nanotube, one would expect the charge carrier to have its position smeared all around the tube like an electron orbiting a nucleus does.

What you have is the energy of the charge carrier being broken into two parts, one along the tube, and one going round and round the tube. To someone looking at the tube as being one-dimensional this second component is rest energy, which we call mass. This energy will be quantised, i.e. only allowed to take certain discrete values (this is because the orbit around the tube is a finite length, so a wave around it has to have a wavelength that is a fraction of this length. Quantum mechanics suggests implies that the energy stored is related to the wavelength, just like the law for a photon E = h_bar nu).

With part of its energy being in this form, the charge carrier is allowed to move along the nanotube at different speeds (imagine it going in a corkscrew at 1000 km/s, so that its speed along the tube is less than 1000 km/s). If the charge carrier has more energy, presumably with the "rest energy" (around the tube) being fixed, this has to go into increasing the speed along the tube. As the energy became enormous, the speed along the tube would approach, but never reach 1000 m/s (think of a right angled triangle of the velocities, the first around the tube, the second along the tube and their sum, 1000 m/s, going in a corkscrew. Very like in special relativity as an object with mass approaches the speed of light! The reason is simply geometrical.

This is highly reminiscent of the concept that other spatial dimensions beyond the three we know can allow particles in our uncurled up 3 dimensional world to have mass (due to their oscillations in the curled up dimensions in a similar way to this). That is the main point of the first post.

[It is also reminiscent of Shroedinger's observation that in Dirac's relativistic equation for the electron, it moved at the speed of light but in an oscillatory way, so that the median speed could be finite, and this probably is also related]

I hope this has slightly improved other people's understanding of this (very tough) topic, as it has mine.

For an interesting read on some other fascinating features of the graphene quantum system, including a prediction of an observable analogue of Zitterbegung (haven't heard of this being confirmed yet), I found an article from New Scientist from a few years back. Well worth reading.

Does my lengthy post #5 make sense to anyone here who has taken a course in quantum mechanics? The ideas were based on basic examples such a plane wave and some sort of bound oscillatory (eg a particle in a potential well is quite similar mathematically to one in a circular domain. It's also related to solving equations for some sort of vibrating circle). There is probably scope for a a separation of variables, so the equation for the particle is the product of an equation for motion on a 1-dimensional circle and an equation for motion in a 1-dimensional straight line. 

I sort of understand it, although I have no quantum physics background whatsoever... but isn't the stationary energy, instead of mass, heat?  And if this stationary energy really is mass, that means that its actually energy that creates distortions in space-time fabric?  But dues that also apply for moving energy i.e. kinetic energy?  If something heavy enough is moving fast enough, can that kinetic energy also distort the fabric, rather than it just being affected by the mass of the object?

@pawn_slayer666, in answer to your 2nd and 3rd questions, yes and yes. Energy distorts the fabric of space time. Einstein's equations for general relativity relate the curvature of space-time to the distribution and flow of energy (including mass) and momentum.

Relating to the first question, I should clarify that in the nanotube, the motion around the tube is merely an analog of of the mechanism that creates mass (like the natural fixed speed of 1000 km/s or so is an analog of the speed of light). In the real world, with its 3 large scale space dimensions and 7 more curled up dimensions (according to M-theory), I understand that it is a similar sort of "motion around these curled up dimensions" that gives rise to mass (and allows things to move at less than the speed of light in the three large spatial dimensions). [Compare with a quote from the non-technical wikipedia article on M-theory: "... "strings" vibrate in multiple dimensions, and depending on how they vibrate, they might be seen in 3-dimensional space as matter, light, or gravity. It is the vibration of the string which determines whether it appears to be matter or energy, and every form of matter or energy is the result of the vibration of strings."

But as for heat, this is perhaps the most mass-like of other forms of energy. Why? Well if you heat up an object, it still sits in the same place but it appears heavier because it has acquired some more energy. i.e. it would weigh more on a scales and have more inertia if you tried to accelerate. How much heavier? Well, work out an example and you may see why we never notice this effect!

Prize question (trophy for first correct answer)

If a 1 kg object has a specific heat capacity of 1000 Joule/kg-Kelvin how much heavier does it get if you increase its temperature by 1K?

(hint - what equation do you use to convert between mass and energy?)

My math took a different path (computer science) so most of the math part of all that is totally greek to me.  But I'm visual-spacially oriented, and I can visualize it quite easily from that description.  And it makes sense intuitively for the graphene tube (transforming a 2-space object to 3-space) to project to 3-space being affected by curling on additional dimensions.

It would be interesting to consider what would happen if the tube (cylinder) were replaced by a sphere.  For lack of better terminology, would the charge carriers acquire an additional freedom of motion?  And what might the resulting relationship between energy and speed be in a doubly curled system?  Perhaps quanta at the peaks and/or troughs of an interference pattern dependent on wavelength?

Edit: and how about a toroid?

Well done, RainbowRising! Trophy winging its way to your collection.

@DaveShack.  You bring up a very interesting issue, of how several curled up dimensions are combined. While with large scale dimensions (which we assume are effectively infinite) there is only one way to combine them, there are many ways to combine curled up dimensions. A large part of this issue is topology.

I would say in the graphene nanotube analogy, the one dimension of length corresponds to the 3 dimensions of space in our world, and the one circular dimension corresponds to the 7 curled up dimensions in string theory. The main features of the analogy are that there are both the familiar large scale dimensions and also dimensions curled up on a tiny scale that cannot be directly observed, and that waves propagate the same way in the tiny dimensions except that, obviously, they come back to where they started.

If you had a sphere, you would have no large scale spatial dimensions, but two curled up spatial dimensions. But another way to combine two curled up dimensions is to form the product of two circles and get a torus, which is rather simpler. People have studied the modes of vibration of theoretical toruses (simply 2-dimensional Fourier series, I think) and 2-spheres (rather more complicated because of the positive curvature, whereas toruses are "flat"). If you imagine a bell made of metal in the shape of either (suspended by a thin wire, to isolate it), the question would be about how it could ring (which is about standing waves in the object). To confuse things, a torus in our 3-D world is geometrically distorted compared to the product of 2 circles, as the inside of the torus is smaller than the outside. This would distort or change the solutions somewhat.

We have reached the borders of my current knowledge here, as I am not sure how the 7 curled up dimensions in M-theory are combined. Someone needs to do some deep delving!

[P.S. when I look this stuff up, I am not entirely clear even how many dimensions are curled up, never mind how they are curled up. Very difficult stuff!]

I read Michio Kaku's book Hyperspace a few months ago.  It said something about during the big bang when it started with 10 dimensions but 3 got big, and 7 shrunk to really small.  I may or may not have said this before on this forum, but how does a dimension "expand" or "shrink".  Supposing we were looking at line-land, with one hidden dimension, we would see all of space as a thin rectangle, being 3-D observers.  The flatland universe would have begun as a small circle or rectangle, but why would it suddenly elongate like that?  Same for our universe, why are there 3 big dimensions with 7 curled up?  All else aside, wouldn't it make more sense if 5 got big and 5 got small?

Could it be that there really is no "size" of a dimension?  All 10 are there, we just biologically adapted to live in 3?  A flatlander could not exist, because a biological pathway would cut the body in 2.  3 is the fewest needed dimensions to live in, that we can have a metabolic pathway.  Maybe we would expend unnecessary energy if we were to reside as higher dimensional beings?  And if everything else were 3 dimensional, we could not sustain life.  Imaging eating 2-dimensional food.  No energy gained.  So is it possible we are merely not adapted to live in or experience higher dimensions?

Being the "bleeding edge" of science, I'm not sure even an expert could give a definitive answer. There is increasing understanding of the "what", not so much of the "why". One relevant fact is that the universe is agreed to have been ludicrously small at the stage when the 3 large dimensions appeared. It might be possible to say that all the dimensions were curled up, and 3 of them unravelled? As for whether it would make more sense if 5 were big and 5 small, I don't see any reason for this. Would you prefer 2 time dimensions and 2 space ones instead of our 1+3 as well? Curled up dimensions are radically different, and I don't see an easy reason why there should be the same number. I understand there is no doubt about the number of large dimensions - this number has a very strong effect on many things.

One current school of thought is that there are a vast number of possible types of universes, each with different laws of physics (about 10^500 in one estimate), and possibly all of them exist. One thing that can vary is the number of big and small dimensions. 3D seems rather special, as it allows gravitationally stable systems to arise (there seem to be problems in both <3 and >3 dimensions), but it is largely speculation to say what would happen in all the other types of universe by mechanisms we are not familiar with. The strong anthropic principle says that our universe has the laws it has because it permits us to exist (to put it another way, the only universes in which beings think about these sorts of things are the ones in which they can exist). [BTW, you can have energy in a 2D universe, and Conway's game of life suggests you can have the complexity necessary for life, even if you can't have any 3D structures! Actually the number of dimensions is no real indication of complexity - the natural numbers include as much complexity as anything else we can analyse, really]

the only think I understand of it is the note

Whenever possible, I try to look at things from a cosmological darwinian viewpoint (prepare now for a large stretch of the imagination). In answer to why we have however many dimensions are present, I therefore like to think of what this has to do with gravity, because how strong or weak gravity is will clearly affect how many black holes are made. We know that gravity is weak. One particular model, which I will not necessarily say that I espouse, but that I will be put forward just for the sake of some answer, however likely or unlikely, is RS1. RS1 is concerned with some extra dimensions of course (5 total I believe), and the point is that it attempts to address the hierarchy problem. Therefore, I can answer that we may have such and such certain dimensions because these dimensions contribute to the strength of gravity, which ought to be selected for or against in some way by cosmological darwinism. Notice that this is not unique of RS1, I don't think, which I think is good for me so that I am not too limited. In the end, to me it all boils down to cosmological darwinism.

I am not smart enough to answer your question about why there is only one time dimension, but perhaps it is somewhat related.

Concerning what type of life is permitted in other possible universes (which I think exists, but not necessarily 10^500, which is a very "stringy" thing to say), I would agree that the possiblity of 2D life looks more likely to occur than someone like Hawking would think. I think it would be perhaps very intricate, but I think it is possible for 2D creatures to eat, for example. Put the topologists to work on this, or something. Concerning higher than three dimensional life, I think it sounds less likely perhaps, but I do not really know. All I know is that atoms wouldn't exactly be stable in higher dimensions, in terms of the orbits of their electrons, and this is very fundamental, so I can only imagine that life could have a very hard time. Yet I suffer from a lack of imagination.

I am familiar with (and have been convinced by) the arguments about the number of space dimensions. If orbits and atoms could not be stable, a universe like ours would certainly be impossible. I cannot eliminate the possibility that a universe without atoms and planetary systems might still permit a complex self-replicating information structure of a different type to develop (this I see as the fundamental property of life). A lot of imagination would be needed to explore the possibilities of one of a universe with 2 large space dimensions. An early attempt was made by the imaginative Edwin Abbot in 1884.

So then, what exactly was your question?

I espouse Schrodinger's definition of life, a "negative entropy pump", in whatever form. This allows a lot of flexibility.

I am not sure why you assume I had a question. I merely try to explore the truth, in the hope of seeing more clearly.

OK. I thought you had posted a note saying that you wanted your question answered here, to the link you supplied.

You may have been thinking of my attempt at an enlightening example in post #9 (solved and trophy awarded )

Anyhow, heading off on one the themes that often proves interesting in these discussions, you know I've been fond of the Conway game of life for a long time, not just because I was impressed by its creator in my years at Cambridge. After I gave up maths research, I wrote software for a couple of the early microcomputers, and one of my early programs was a version of the game of life for a Sinclair ZX spectrum. It ran a 64 x 44 array at about 1 frame a second, if I recall on a 3.75 Mhz 8 bit processor with 48k of RAM . I think I chose to make the space toroidal (i.e. the top of the screen wrapped around to the bottom, and the left to the right) which of course made it a little different to the usual version.

After 26 years of increase in computational power (and some way better programmers than me), things have moved on a little (lol). It's only recently that I came across the wonderful open source life program called Golly. If there was a pop chart for best written programs, I would place my vote for this beauty. It can handle arrays of cells of stupendous size through extraordinarily efficient indirect methods, and also run at exponentially increasing speed (no kidding) using one of its modes. This allows you to see patterns that develop over 10^50 generations (or more), for example. It reminded me of the technological singularity in this respect. There are lots of enlightening built in examples, and it's trivial to experiment with more. The two basic features that are essential are zooming the view in and out, and increasing and decreasing the speed of the simulation (both over a stupendous range).

Get a copy of Golly and play with it! Time very well worth spending.