Clever integration tricks

Today I want to talk about a clever integration trick I learned from Achim Kempf at DICE2014. Any mathematical physicist learns clever integration tricks and one of my personal favorite is how to compute:

\(I = \int_{-\infty}^{+ \infty} e^{-x^2} dx \) 

because there is no elementary primitive function since the integral comes from the Gaussian (normal) distribution. However one can still compute the integral above quite easily by going to a 2-dimensional plane and considering the \( y \) integral as well: \(\int_{-\infty}^{+ \infty} e^{-y^2} dy \) which is \( I\) again .

So 

\(I ^2 = \int_{-\infty}^{+ \infty}\int_{-\infty}^{+ \infty} e^{-x^2} e^{-y^2} dx dy  = \int_{-\infty}^{+ \infty}\int_{-\infty}^{+ \infty} e^{-(x^2 + y^2)} dx dy \)

and the trick is to change this to polar coordinates:

\( x^2 + y^2 = r^2\) and \( dx dy = rdr d\theta\)

integration by \( \theta\) is a trivial \( 2 \pi\), and the additional \( r \) allows you to find the primitive and integrate from \(0 \) to \( \infty\).

But how about not having to find a primitive at all? Then one can try Achim's formula:

\( \int_{\infty}^{\infty} f(x) dx = 2 \pi f(-i\partial_x ) \delta (x) |_{x=0}\)

It's a bit scary looking

Happy Halloween!

but let's first prove it:

\( 2 \pi f(-i \partial_x) \delta(x) |_{x = 0} = f(-i \partial_x)  \int_{-\infty}^{+\infty} e^{ixy} dy |_{x = 0}\)

due to a representation of \( \delta (x)\):

\( \delta (x) = \frac{1}{2 \pi} \int_{-\infty}^{+\infty} e^{ixy} dy \)

Moving \( f(-i \partial_x) \) inside the integral makes this \( \int_{-\infty}^{+\infty} f(y) e^{ixy} dy |_{x = 0} \). Why? Expand \( f \) in Taylor series and apply the powers of \( -i \partial_x \) on \( e^{ixy} \) resulting into the powers of \( y \). Then recombine the Taylor series terms in \(f(y) \). Finally compute this for \( x = 0 \) which kills the exponential term and you are left only with \( \int_{-\infty}^{+\infty} f(y) dy\) and the formula is proved.

So now let's see this formula in action. Let's compute this:\( \int_{-\infty }^{+\infty} \frac{sin x}{x} dx\):

\( \int_{-\infty }^{+\infty} \frac{sin x}{x} dx = 2 \pi sin(-i \partial_x) \frac{1}{-i\partial_x} \delta(x) |_{x = 0} = \)

\( = \frac{2 \pi}{-i} \frac{1}{2i} (e^{\partial_x} - e^{-\partial_x}) (\theta(x) + \epsilon) |_{x = 0}\)

Now we can use Taylor to prove that \( e^{a \partial_x} f(x) = f(x+a) \) and from this the integral becomes:

\( = \pi (\theta(x+1) - \theta(x-1) +c - c)|_{x=0} = \pi (1 - 0 + 0) = \pi\)

So what is really going on in this formula? If we start with another representation for the Dirac delta:

\( \delta(x) = \lim_{\sigma \rightarrow 0^{+}} \frac{1}{\sqrt{\pi \sigma}} e^{-\frac{x^2}{\sigma}}\)

then:

\(\int_{-\infty}^{+\infty} f(x) dx = \lim_{\sigma \rightarrow 0^{+}} 2 \sqrt{\frac{\pi}{\sigma}} e^{\frac{{\partial_x}^2}{\sigma}} f(x) |_{x=0}\)

The exponential term is a Gaussian blurring which flattens \( f(x) \), and is in fact a heat kernel because a heat equation is actually a convolution with a Gaussian. Also the limit sigma goes to zero or equivalently one over square root of sigma goes to infinity would physically correspond to the temperature going to zero. 

However something does look fishy in the formula. How can the integral of a function which includes the values over the entire domain be identical with a a formula containing only the value of \( f\) in only one point \( x = 0\) ? It does not! This is because \( e^{\frac{\partial_{x}^{2}}{\sigma}}\) acts nonlocally because \( e^{\frac{\partial_{x}^{2}}{\sigma}}f(x) \) is a convolution! 

More can be said, but it is a pain to typeset the equations and the interested reader should read http://arxiv.org/abs/1404.0747. Enjoy.

­Should Gravity be Quantized?

Merging quantum mechanics with general relativity is the hardest problem of modern physics. In naive quantum field theory, treating gravity quantum mechanically involves adding smaller and smaller distance contributions to perturbation theory but this corresponds to higher and higher energy scales and adding enough energy will eventually lead to creating a black hole and the overall computation ends up predicting infinities. String theory, loop quantum gravity, and non-commutative geometry have different ways to deal with those infinities, but there are also approaches which challenge the need to treat gravity using quantum theory. Those approaches are a minority view in the physics community and I side with the majority point of view because I know it is mathematically impossible to construct a self-consistent theory of quantum and non-quantum mechanics. But wouldn't be nice to be able to put those ideas to an experimental test?

Here is where a nice talk at DICE2014 by Dvir Kafri came in.  The talk was based on http://arxiv.org/abs/1311.4558 and http://arxiv.org/abs/1404.3214 .  The best way to explain is probably to present it from the end, and here is the proposed experiment (from http://arxiv.org/pdf/1311.4558.pdf ).

Penrose advanced the idea of the gravitational collapse of the wavefunction and Diosi refined this in the best available model so far. Rather than looking at decoherence of objects due to gravity, Dvir instead asks the following question: can two masses which only interact gravitationally become entangled? Direct superposition experiments are out of the question, but how about measuring some sort of residual quantum noise required to screen the entanglement from occurring in the first place? Sure, since the gravitational coupling is so weak, the noise needed to do this is really tiny, but what if we cool the experiment close to absolute zero? One experiment is not enough because at 10 micro Kelvins you expect one thermal phonon to be emitted every 10 seconds and the desired effect produces a phonon every 3000 seconds, but massively replicating the experiment in parallel might work to extract the signal (replicate 10,000,000 times! – OK this is a bit in the realm of science fiction for now but maybe future technological advances will drop the price of such an experiment to something manageable).

Dvir motivates the experiment by modeling how two distant objects can communicate by individually interacting with an intermediary object.

Here is a slide picture from Dvir’s presentation (I thank Dvir for providing me with a copy of his presentation)

Please note that position and momenta are non-commuting operators. So you apply A first, followed by B, followed by –A and the by –B. The intermediary F (a harmonic oscillator) is unchanged by this procedure, but gains a geometric phase proportional to \( A \otimes B \). In other words this is what happens:

If you break this process into n infinitesimal steps and repeat n times, by a corollary of Baker-Campbell-Hausdorff formula you get:

\( {(t/n)}^{n} = exp (-it [H_A + H_B + A\otimes B]) \)

This picture is a simple model for how two objects can become entangled, To prevent that entanglement (but still allow communication between A and B), we add a “screen” S  which captures the coupling with the environment

By the monogamy of entanglement, this can only decrease the entanglement between A and B.

Since the environment is learning about A and B through F, Dvir invokes what he calls the “Observer Effect”: a measurement of observable \( O \) necessarily adds uncertainty to an non-commuting observable \( O^{’} \). In this case, the process of screening entanglement means that all observables not commuting with A and B become noisier.

Here is an experimental setup that is analogous to the first experiment: S is a weak measurement and the purpose is to see the noise generation, which is model-independent in that the equations of motion are the same.

If a certain inequality is violated (relating the strenght \( \eta \) of the \( A \otimes B \) interaction to the noise added to the system), then the communication channel between the Alice-Bob system transmits quantum information. Analogously, if we can verify that \( \eta \) is only due to gravity (that is why there is a superconductor shield between the oscillators coupled by gravitational attraction), by observing the noise and checking the inequality we can conclude that gravity can convey quantum information. Pretty neat.

PS: I thank Dvir for providing clarifying edits to this post.

The amazing Graphene

Continuing the interesting talks series from DICE2014, I was blown away by a talk by Alfredo Iorio: “What after CERN?”. Physics is an experimental science and the lack of experiments forces theoreticians to construct alternative models which most likely have nothing to do with how nature really is.

In high energy physics the experiments are extremely expensive and the price tag for a new accelerator is in the billions. Why do people need larger accelerators? Because to probe smaller and smaller regions of space you need larger and larger energies. Accelerators circulate a beam of particles in a circle to gain the required energy, and the faster they go (closer and closer to the speed of light), the heavier the particle become and they need larger and larger circle radius. To probe at the scales of the string theory for example, one needs an accelerator the size of the galaxy. So is there an alternative to this?

It turns out that there are theoretical and experimental efforts of outstanding value circumventing this brute force approach and Iorio’s research belongs into this rare breed in physics.

In the past I was blogging at FQXi about an experiment done by Bill Unruh with a laboratory waterfall which was able in principle to simulate a black hole and its Hawking radiation. However even more amazing things can be achieved with Graphene

So what is so special about this material? There are two key properties which makes it extremely interesting.

First, the hexagonal structure requires two translations to reach any atom.

Given an origin, any atom can be specified first by a linear combination of two vectors: \( (a_1 , a_2 )\): \( x_a = n_1 a_1 + n_2 a_2\) where \( n_1 , n_2\) are positive or negative integers, followed by a second translation using the vectors \( s_1 , s_2 , s_3\).

Second, the band structure in graphene is very special: the conductivity and valence band touch in exactly one point (called the Dirac point) making the structure a semi-metal:

Graphene Band

When the excitation energy is small (~ 10 \( \mu \) eV), the quasi-particle excitations respect Dirac’s equation. Two of the 4 spinor components come from the Lattice A vs. Lattice B, and the other two come from the up and down bands touching at the Dirac point.

By its very geometrical structure, graphene is an ideal simulator of spin ½ particles.

Now the hard work begins. How can we use this to obtain answers about quantum field theory in curved space-time? First we can start easy and consider defects in the hexagonal pattern. A defect changes the Euler number and introduces curvature. This is tractable mathematically for all simple defects using a bit of advanced geometry, but you don’t get very far except in the description of the phenomena in terms of critical charges and magnetic fluxes.

But if you can manufacture surfaces of negative curvature:

called Beltrami spheres then the real fireworks begins. Under the right conditions you can simulate the Unruh effect (http://arxiv.org/pdf/1308.0265.pdf ): an observer in A sees the quantum vacuum in the frame B as a condensate of A-particles. To observe this the tip of a Scanning Tunneling Microscope is moved across the graphene surface and probes the graphene quasi-particles.

More amazing things are possible like: Rindler, de Sitter, BTZ black hole, Hawking radiation.

Of course there are drawbacks/roadblocks too: the defects in manufacturing which might spoil those effects. It is unclear how accurate are manufacturing techniques at this time. Also I don’t know if the impurities effects are properly computed. Much more serious I am skeptical of the ability to maintain the hexagonal pattern while creating the Beltrami funnel. And if this is not maintained, in turn it will affect the band structure which can ruin the validity quasi-particle model of Dirac’s equation.

I brought my concerns to Alfredo and his response put my mind to ease. To avoid playing telephone, with his permission I am sharing his answer here:

“- So, you are perfectly correct when you doubt that the Beltrami shape can be done all with hexagons. In fact, this is not possible, not because of technical inabilities of manufacturers, but because of the Euler theorem of topology.

- How do we cope with that? Although at the location of the defects the Dirac structure is modified, the hexagonal structure resists in all the other places. When the number of atoms N is big enough, one can safely assume that the overall hexagonal structure dominates (even when the defects start growing, as they do with N, all they do is to distribute curvature more evenly over the surface).

Now, if you stay at small \( \lambda \) (large energy E), you see all local effects of having defects, and the lattice structure cannot be efficiently described by a smooth effective metric (essentially, since the \( \lambda \) and E we talk about here are those of the conductivity (or \( \pi \) ) electrons that live on the lattice (they don't make the lattice, that is made by other electrons, belonging to the \( \sigma \) bonds), we realize that when their wavelength is big enough, they cannot see the local structure of the lattice, just like large waves in the sea are insensitive to small rocks. Hence, for those electrons, the defects cannot play a local role, but, of course they keep playing a global, i.e., topological, role, e.g., by giving the intrinsic curvature (as well known, in 2 dimensions the Gauss-Bonnet theorem links topology and geometry: Total Curvature = 2 \( \pi \) Euler Characteristic).

- Thus, if I was good enough at explaining the previous points, you should see that the limit for big \( r \) (that is small curvature \( K = \pm 1/r^2 \)) is going in the right direction, in all respects: 1. the number of atoms N grows; 2. the energy \( E_r \sim 1/r \) (see Fig. Graphene Band) gets small, hence the \( \lambda \) involved gets big, hence 3. the continuous metric \( g_{\mu \nu} \) well describes the membrane; 4. the overall Dirac structure is modified, but not destroyed, and, the deformations are given by a ''gauge field'', that is of the fully geometric kind. Indeed, this gauge field describes deformations of the membrane, as seen by the Dirac quasi-particles. The result is a Dirac field (we are in the continuum) in a curved spacetime (i.e. covariant derivatives of the diffeo kind appear). In arXiv:1308.0265 we discuss all of this in Section 2.

- There is also an extra (lucky!) bonus in going to big \( r \), that is the reaching of some sort of horizon (more precisely, that is a conformal Killing horizon, that, for a Hawking kind of phenomenon, is more than enough). Why so? The issue here brings in the negative curvature. In that case the spacetime (the 2+1 dimensional spacetime!) is conformal (Weyl related) to a spacetimes with an horizon (Rindler, deSitter, BTZ). Something that does not happen for the positive curvature, the sphere, that in graphene is a fullerene-like structure. In fact, the latter spacetime is conformal (Weyl related) to an Anti deSitter, that, notoriously, does not have an intrinsic horizon.

Now, once you learn that, you also learn that surfaces of constant negative Gaussian curvature have to stop somewhere in space (they have boundaries). That is a theorem by Hilbert. For small \( r \) (large curvature) they stop too early to reach the would-be-horizon. For large \( r \), though, they manage to reach the horizon. Fortunately, for that to happen, \( r \) needs not be 1 km (that would not be an impossible Gedanken experiment, but still a tremendous task, and just unfeasible for a computer). The job is done by \( r = 1 \) micron! That is something that made us very happy: the task is within reach. It is still hard for the actual manufacturing of graphene, but, let me say, it turned into a problem at the border between engineering and applied physics, i.e. it is no longer a fundamental problem, like, e.g., the mentioned galaxy-size accelerator.

- We are actively working on the latter, as well. In this respect, we are lucky that these ``wonders’’ are happening (well... predicted to be happening) on a material that is, in its own right, enormously interesting for the condense matter friends, hence there is quite a lot of expertise around on how to manage a variety of cases. Nonetheless, you need someone willing to probe Hawking phenomena on graphene, while the standard cond-mat agenda is of a different kind. Insisting, though, very recently I managed to convince a composite group of condensed matter colleagues, mechanical engineers, and computer simulations wizards, to join me in this enterprise. So, now we are moving the first steps towards having a laboratory that is fully dedicated to capture fundamental physics predictions in an indirect way, i.e. on an analog system.

What we are doing right now, between Prague, Czech Republic (where I am based) and Trento, Italy (where the ``experimentalists`` are sitting), is the following:

First, we use ideal situations, i.e. computer simulations, hence we have no impurities nor substrates here. There no mention is made of any QFT in curved space model. We only tell the system that those are Carbon atoms, use QM to compute the orbitals and all the relevant quantified, perform the tight binding kind of calculations. Thus, the whole machinery here runs without knowing that we want it to behave as a kind of black hole.

What we are first trying is to obtain a clear picture of what happens to a bunch of particles, interacting via a simplified potential, e.g., a Lennard-Jones potential, constrained on the Beltrami. This will tell us a lot of things, because we know (from similar work with the sphere, that goes under the name of generalized Thomson problem, see, e.g., the nice work by Bowick, Nelson and Travesset) that defects will form more and more, and their spatial arrangements are highly non trivial.

When this is clear, we want to get to a point where we tell the machine that we have N points, and she (the machine) plots the Beltrami of those points. i.e. it finds the minimum, the defects, etc. This would be the end of what we are calling: Simulation Box 1 (SB1).

When SB1 is up and running, we fix a N that is of the order of 3000, take away points interacting with Lennard Jones, and substitute them with Carbon atom, i.e. we stick in the data of Carbon, the interaction potential among them, and then let a Density Functional computation go. The latter is highly demanding, computer-time wise, but doable. With this we shall refine various details of the theory, look into the structure of the electronic local density of states (LDOS), although the \( r \) we can get with N = 3000 is still too small for any Hawking anything. That is the first half of SB2.

The work of SB1 and first half of SB2, can be done with existing machines and well tested algorithms. But we need to go further, towards a big \( r \) (the 1 micron at least... although I would be happier with 1 mm, but don't tell my experimentalist friends, they would kill me!). This is possible, but we are going into the realm of non tested algorithms, of dedicated machines (i.e. large supercomputers, etc). Nonetheless, figures of the order of N = 100K (and even whispered N = 1 million) are in the air. That would be second half of SB2, i.e. when the Hawking should be visible.

That is the road I can take with the current group of people involved. I don't give up though the idea of getting someone to actually do the real graphene thing. But this would only mean a handle of a very large number of points, to the expense of more impurities, substrates, etc. Indeed, the SB2 (the computer simulations of true Carbon interactions) would be so accurate, that myself (and, most importantly, the cond-mat community) would take those results as good (if not better, because `fully clean`) as the experiments.”

In conclusion this is an extremely exciting research direction. 

The topological structure of big data

One interesting talk at DICE2014 was a talk by Mario Rasetti on understanding the bid data of our age.

You may wonder what does this have to do with physics, but please let me explain. First when we say big data, what are we really talking about? The number of cataloged stellar object is \( 10^{21}\). Pretty big, right? But consider this: in 2013 there were 300 billions email sent, 25 billions SMS messages made money for phone companies, 500 million pictures were uploaded on Facebook. In total from those activities mankind produced in 2013 \( 10^{21}\) bytes. And the every year we produce more and more data. 

How much is \( 10^{21}\) bytes? How about 323 billion copies of War and Peace. Or 4 million copies of all of Library of Congress. In four years it is estimated that we will produce \( 10^{24}\) bytes which is larger than Avogadro's number!

Now how can we get from data to information to knowledge and then wisdom? From computer science we know to lay all this data sequentially and people considered vector spaces for this. But does this make sense? For example, if we take a social network like Twitter what we have are simplicial complexes. What Mario Rasetti proposed was to extract the topological information from those kinds of large data sets. In particular he computes the Homology groups and Betti numbers which were discussed on this blog on prior posts, and the reason is that the algorithmic complexity is polynomial in computation time.

We know that if we triangulate a manifold and omit one point we obtain different topological invariants just like puncturing a three dimensional balloon results in a two dimensional surface. Therefore in computing the Betti numbers we get fluctuations but as more and more nodes are included into computation the fluctuations stabilize. 

The link with physics and Sorkin's Causal Set theory is obvious and the same techniques can be applied there. However Rasetti did not go into this direction and instead he cited the application of the method to biology. In particular, he was able to clearly distinguish if a patient took a specific drug vs. a placebo from the analysis of the brain MRI image which looked identical to the naked eye. 

Recently I saw an article on what Facebook sees in posting patterns when we fall in love:

Now all this looks really scary. Imagine the power of information gathering and topological data mining in the hands of (bad) governments around the world. And not only governments. Big companies like Facebook are abusing the trust of their users and perform unconscionable sociological tests by manipulating advertising for example. In the biological area, human cloning is rejected because the general population understands the risks, but the understanding of big data and the ability to mine it for correlations and knowledge is badly lagging behind the current technical ability. More violation of privacy scandals will occur before the public opinion will put pressure to curb bad behavior of abusers of trust.