Thursday, April 17, 2014

Homotopy and Homology


Let us continue the discussion about homotopy and introduce some definitions. Let π0(X) be the set of paths in X and let π0(X,p) be the set of paths containing the point p. Remembering from last time, we want to make closed loops and consider a unit circle which parameterizes a loop. Then for each point p of X we can introduce a loop space “Ωp X” which is the space of maps from the unit circle to X and for which p is the starting point of the loop.

With those preliminaries, we can now introduce a fancy abstract redefinition of the first homotopy group (the fundamental group) as follows:

π1(X, p) = π0p X, p)

The advantage of this abstraction is that it naturally generalizes to higher levels:

πk+1(X, p) = πkp X, p)

So for example π2 represents the first homotopy group of the loop space meaning we cover X with a surface and we try to continuously deform it to a loop. In higher dimensions, this becomes a nightmare of visualization and we need a way to compute those groups algebraically (we need a mechanical method relatively free of visual intuition). And why do we want to compute those groups in the first place? The point is that spaces with different groups are distinct.

As a group, π1 is non-abelian in general, and this points the way towards a key simplification: let’s construct an abelian (commutative) group out of π1. This will be the first homology group, but we are not yet ready to discuss it. Instead let’s try to understand what abelianization means. It means that we don’t care about the point p and we replace the closed loops with cut and stitching operations, the same way a tailor is making clothes. 

For example take an inner tube which is equivalent with a doughnut, or torus.  If we cut the inner tube along its two circles we obtain a flat surface. Equivalently, if we stitch (glue, zip) a rectangular piece of cloth along the opposite sides matching the stitching direction, we obtain a torus. As a nontrivial example of a torus, in early computer games, because of memory limitation, sometimes an object exits through the right edge of the screen and reappears from the left (the same behavior at top and bottom).

Probably the best way to understand cut and stitch is to try to understand the technique of the proof of a problem mathematicians solved around 1900: what is the complete classification of two dimensional compact surfaces?

Two dimensional surfaces are easiest to visualize and it turns out that orientability and Euler’s characteristic are the only ingredients needed for a complete classification. For the record, the answer to the classification problem is that the surface is characterized by the numbers of “holes” (or genus) and the numbers of “cross caps”.

Euler’s characteristic was discussed two posts ago, and non-orientability is easy to understand using Mobius band.





Now it is (relatively) straightforward to solve the classification theorem for 2 dimensional compact surfaces. All we have to do is to cut the surface until we obtain small flat pieces, remembering the stitching back instructions. Then we need to find some equivalent way of writing stitching instructions in a “standard normalized way”. In the end we tally all “standard normalized ways” and obtain the proof.

For a beautiful presentation of the proof, please see those lectures by Norm Wildberger:


and


If you are new to algebraic topology, I highly encourage you to invest the time and energy to watching and understanding those lectures.


The technique of the theorem’s proof will naturally introduce us to a way of thinking about homology. Homology is related to the boundary of a space. The link with physics is that integration on the boundary can tell us essential things about the bulk (does Gauss theorem ring a bell?) Moreover when we jump from homology to cohomology, we’ll see that differential forms can also reveal information about space. This is the realm of the beautiful Hodge and de Rham theories which are essential to a modern understanding of physics. 

Saturday, April 12, 2014

Homotopy and Aridne’s thread


Last time we introduced Euler’s characteristic and noted that it is an intrinsic geometric invariant. We can now take the geometry story in two different directions and we will follow it both in due time.

Let us start with a game: try to cover the Earth’s surface in triangles (which can be as small as we need to follow the contour precisely). Also when we add a new triangle, color the surface green (substitute here you favorite color) not to loose count of faces, edges, and vertices. We can cover the Earth all the way around until only a last triangle is left to be colored. Up to that point, F+V-S=1 (because when we add a new vertex and we color the new triangle the number does not change), and when we completely color the surface of the Earth, at the very last step: F+V-S=2. Now imagine increasing the gravitational attraction which smoothes the surface to a perfect sphere. Nothing changes in F+V-S however. This is the key idea of homotopy: one can continuously deform a shape in such a way that no tears occur. Then the initial and final shapes are equivalent and retain some core characteristics. For example one can deform a coffee mug into a doughnut and this is called homeomorphism. Mathematically a homeomorphism is a continuous map with a continuous inverse. It is important to note that the continuous property for the inverse map is essential otherwise a line segment can be equivalent with a circle, and this contradicts the no tearing property.

Now back to "intrinsic" geometrical games: suppose you are an ant on a doughnut. How can you uncover the type of shape you are on? One way is to play the triangle game from before, but what if the space is infinite, or very large? (remember that Euler’s characteristic can change at the very last step). Fortunately help comes from ancient mythology in the form of Ariadne’s thread.



but with a twist: after you randomly journey through the surface of your space, make sure to always return to the starting point. Then try to reel the thread back on both ends at the same time. If your space has no intrinsic holes, and is simply connected, you will succeed to get all your thread back. However, if you go around a hole N times, your thread will become stuck. This may look like trivial observations, but it has extremely deep connections with electromagnetism and high energy physics as I will show over time. 

What we need to realize is that this thread game has secretly a hidden mathematical structure: a group. A group has an associative operation, an identity element, and each operation has an inverse. The group elements in our case are the trips, and the associative operation is concatenating two trips one after another. When we do that, we tie the end of the thread for the first trip to the beginning of the thread of the second trip, and try to reel in the combined thread. The inverse operation of a trip is a reversed trip, and the identity element is the “stand still” no trip.  What we just introduced is the first homotopy group (or the fundamental group). Child’s play? For the first homotopy group yes, but things will quickly get out of hand. Next time we’ll generalize Ariadne’s thread game, observe that is no longer intuitive, and seek a simplification which will lead us to the wonderful craft of tailoring clothes or as mathematicians like to call it in fancy abstract and impenetrable obscure language: homology. Who knew that fashion design and making clothes is deeply relevant to understanding Nikola Tesla’s alternative current generators?

Friday, April 4, 2014

Physics and Geometry


What exactly is cohomology?


More than a year ago I started this blog as an experiment, wondering if I would have enough interesting things to say. I think I ran out of simple interesting topics, and it is time to take the level up a notch. After all, how many interesting posts about strange correlations and coloring games in quantum mechanics can one do? Fortunately the quantum mechanics (and physics in general) is huge, but we need to understand its geometric language. There are three large mathematical mountains blocking our path to understanding physics the modern way: homotopy, homology, and cohomology theory. I will attempt to guide you through this terrain and explain it in as much intuitive terms as I can. Once we conquer this area, a vast and interesting landscape waits for us. Then we should be able to explain electromagnetism, Yang-Mills gauge theory, and the Standard Model. My end goal is to make the following statement self-evident: fermions are sections and bosons are connections in a vector bundle.

For the first couple of weeks I will start with simple geometrical topics, and after I will attend an upcoming physics conference, I will stop the geometry posts and present impressions from the conference. Then I will resume the quest to explain the beautiful area of differential and algebraic geometry.

Today’s topic will be: Euler’s characteristic


After the discovery of Euclidean geometry, nothing happened in geometry for a long time, until merchants sailing the oceans needed accurate maps. This forced introduce the idea of coordinates and distance. Oddly enough, ancient Greeks taught geometrically using constructions with ruler and compass, but they lacked the idea of coordinates. Once coordinates are introduced, a discrete mathematical structure is superimposed on a continuous domain, and essential information about geometry is encapsulated in algebraic notions. Geometry is like playing the violin, while algebra is like playing the piano. Each piano key makes a distinct repeatable sound and is easier to play complex songs because you don’t have to worry about accurately recreating the notes. In the same way, algebraic methods are much more powerful than simply visualizing potentially highly complex geometrical spaces.   

So let us start with a deep and apparent trivial observation. Draw a triangle on a piece of paper. Pick a point outside the triangle and connect it with the closest side of the triangle. You just drew two additional lines, created a new vertex, and a new face. If you repeat the process, the number of FACES+VERTEX-SIDES (F+V-S) stays the same. For the original triangle F+V-S = 1+3-3 = 1

Suppose you place the additional point inside of a triangle: then you add 2 new faces, 1 vertex, and 3 sides and F+V-S continues to stay 1 regardless of where you add the new point, inside or outside of the triangle. 

So what is the big deal? The big deal is that if you start with a triangle and add a point in space above the original plane then F+V-S changes. For a tetrahedron this number is now 2 because when you add a point in space, you create 3 new faces instead of 2 and F+V-S is able to tell distinguish between plane figures and geometrical figures in space.

But can't we simply just see that a geometric objects is in space and not in a plane? Sure, but this works because we embed the object into the ordinary 3D space. If we are talking about crazy complex 26-dimesional objects for example we lack the intuitive embedding and we need a way to use intrinsic object characteristics to be able to say something meaningful.


The key point is that F+V-S which is called Euler’s characteristic is an intrinsic geometrical invariant. To be continued…

Friday, March 28, 2014

Is Time travel Possible? (part 2 of 2)

Here is the conclusion of my interview for a popular science magazine: Science Illustrated in Denmark.













1:
Could you clarify why you find it highly unlikely that time travel is allowed by the laws of nature? Why do you believe, that the merger of general relativity and quantum physics leads to a theory (of everything) that will not allow time travel / closed timelike curves?

2:
When you build your own wormhole, you can only go back in time to the very moment a time loop was created. But couldn’t you – in theory - use an existing wormhole (if you have the technology to open a microscopic wormhole wide open) to go back further?

3:
Delayed choice quantum eraser experiments and the non-locality of quantum mechanics seems to indicate that the quantum world is “above” space and time. So, could we live in a world of self-consistently evolving quantum spacetime fields, which would work as a banana-peel-solution to the paradoxes?


1. Could you clarify why you find it highly unlikely that time travel is allowed by the laws of nature? Why do you believe, that the merger of general relativity and quantum physics leads to a theory (of everything) that will not allow time travel / closed timelike curves?

The answer is a bit complex and I’ll start with a detour. Quantum mechanics teaches us that position and velocity cannot be measured simultaneously with arbitrary precision. This goes under the name: Heisenberg uncertainty principle. Special theory of relativity shows that there is a maximum speed limit in the universe and nothing can go faster than the speed of light. Combining quantum mechanics with special relativity results in something completely new: creation and annihilation of particles (hence anti-particles). Why is this so? Suppose you try to pinpoint the location of a particle with arbitrary precision by putting it into a box and squeezing the box on all sides. By Heisenberg uncertainly principle, when the box is squished to nothing, because the position is known exactly, the velocity uncertainty goes to infinity and will be possible to have velocities higher than the speed of light. Since nothing goes faster than the speed of light, the higher velocity is only apparent because new particles are generated and we detect another particle instead of the original one. The theory for combining quantum mechanics with special theory of relativity is called quantum field theory and is the most successful theory of nature we have so far (the current measurement and prediction accuracy is better than a part in a billion). In quantum field theory, the lowest possible energy state is called the vacuum. Vacuum is not the absence of things, and it is a very violent place where virtual particles and antiparticles get created and eventually destroyed. Quantum field theory obeys a fundamental principle of physics called unitarity which means that information cannot be created or destroyed. A time loop violates unitarity because it can create new information out of nothing. When general relativity meets quantum mechanics, by time machine solutions, or by the simpler example of a black hole, unitarity is violated and information is no longer conserved. One may recall the debate between Leonard Susskind and Stephen Hawking on the black hole information paradox. The same thing is at play here and by its construction the best candidate for unification between general relativity and quantum mechanics, string theory, preserves information conservation and rejects time loops. If information conservation is violated, quantum field theory predicts that the universe is heating up and we simply don’t see this happening. Hawking also proposed a quantum field theory mechanism to prevent time loops. The moment general relativity is about to create a causal time loop, the virtual particles in the vacuum start traveling around the loop draining the energy out of it. The end result is that the time loop collapses. Since we lack the precise unified theory, Hawking’s computation is only speculative at this time.

2. When you build your own wormhole, you can only go back in time to the very moment a time loop was created. But couldn’t you – in theory - use an existing wormhole (if you have the technology to open a microscopic wormhole wide open) to go back further?

Yes in theory, no in practice. A wormhole has a delicate part, its “neck” which by general relativity will collapse very quickly unless it is kept open by negative energy. Think of negative energy as a credit card: you spend what you don’t have, and you have to pay it back eventually with interest (this is also because of Heisenberg uncertainty principle, but this time not for position and velocity but for energy and time). To keep a wormhole open for a decent amount of time, you need to keep feeding it negative energy and every time the interest compounds. Some extremely advanced alien civilization has to maintain the wormhole open for us to be able to see the extinction of the dinosaurs (like rolling the balance from one credit card to another credit card with a higher credit limit). And for that long amount of time, the energy required could easily exceed the entire energy of our galaxy.

3: Delayed choice quantum eraser experiments and the non-locality of quantum mechanics seems to indicate that the quantum world is “above” space and time. So, could we live in a world of self-consistently evolving quantum spacetime fields, which would work as a banana-peel-solution to the paradoxes?

This is correct. And this was shown in a precise mathematical way by David Deutsch using quantum mechanics. There may not be there any “banana peel” but it will feel like it.  However, this only solves the grandfather paradox. The lack of information conservation problem still remains and this is against quantum mechanics.

You can’t have the cake and eat it too:  you can’t have a quantum mechanics solution of the grandfather paradox while rejecting quantum mechanics because of lack of information conservation.

I think is safe to say that we all love “Back to the future” movies. I’d love to have a flux capacitor installed in my car and as a physicist I am saddened by the realization that time travel is almost surely impossible. However, physicists pursue time travel questions because they test the limit of our current understanding and the quest can provide hints of how to uncover the ultimate “theory of everything”.

Friday, March 21, 2014

Is Time Travel Possible? (part 1 of 2)


I will take a couple of weeks break from quantum mechanics to talk about the possibility of time travel. As it happens, I was asked a few questions for an article in the popular science magazine Science Illustrated in Denmark. The instructions were to keep the answers short, but I could add additional info to be used as seen fit by the editor. Also the explanation level should avoid being technical. This generated an interesting exchange which I will show in this and next post. Enjoy.

1:
Is it – in theory – possible to travel back in time? Does nature allow such time travel?

2:
If so, will it ever become possible to construct a time machine capable of transporting human beings back in time?

3:
And if this might be the case, how is paradoxes like the grandfather paradox prevented?

1, Is it – in theory – possible to travel back in time? Does nature allow such time travel? The answer to the possibility of traveling back in time is not yet known, but time travel is highly unlikely. Einstein’s general relativity theory – a very successful physical theory at large scale - allows many solutions which exhibit time travel but general relativity is at odds with quantum mechanics –a very successful theory at small scale - and so far there is no known physical theory which consistently combines them. There are several proposals being considered, like for example string theory, but only when such a theory will be validated by experiments we could have a definite answer to the possibility or impossibility of time travel.

Background info:
It is important to understand how time travel solutions occur in general relativity and what it means. Einstein’s general relativity equations are local laws and they do not forbid global behavior like traveling back in time. Since space and time are not rigid, they can be twisted and stretched by the presence of mass and if you continue doing it in certain ways, you can eventually manage to turn time back on itself. This is not unlike how one can turn a car all the way around in an empty parking lot. Then all sorts of paradoxes can occur and to understand them physicists studied for example how billiard ball games can be played in the presence of a time machine. To avoid paradoxes, the billiard ball may collide with its younger self in a self-consistent manner, but here is the catch: consider replacing the billiard ball with an egg. When it collides with its younger self it will go “splat” and create a paradox. The only way to prevent the paradox is for the egg not to break, and this means that: global consistency conditions required to avoid paradoxes imply non-physical local behavior and this does not agree with our current knowledge of nature.

2:
If so, will it ever become possible to construct a time machine capable of transporting human beings back in time?

The answer is a double no. First, assuming that time travel is actually permitted by nature, you can only go back in time to the very moment a time loop was created. In other words, nobody will be able to go back in time to witness the extinction of the dinosaurs, or the invention of the light bulb. Second, the energy required to bend space-time on itself is of galactic magnitude and you need to harvest the energy of an entire galaxy to bend space-time on itself. A more practical approach is by creation of a wormhole, but this requires negative energy and while negative energy is a real possibility, when you generate negative energy you have to pay it back with interest. To create a macroscopic wormhole large enough for a human to pass through you need yet again an immense source of energy.

Background info: The reason enormous amounts of energy are required is that gravity is the weakest force in our universe. Mass is equivalent with energy (E=mc2) and you are required to have a large amounts of mass (corresponding to even larger amounts of energy) to bend space and time. Titanic was hard to turn by a small rudder. Turning space-time all the way to itself is extremely hard with weak gravity. And how do we know gravity is a very weak force? After all it does not look that way when we fall for example. Consider a magnet on a refrigerator. The small magnetic attraction between the magnet and the sheet of metal can easily overcome the gravitational pull of the entire planet.

3:
And if this might be the case, how is paradoxes like the grandfather paradox prevented?

There are only two paradoxes generated by time travel: the grandfather paradox and creation of information from nothing. For the grandfather paradox there are two solutions: “the banana peel type solution” and the splitting the world into multiverses in quantum mechanics. Here is how they work.

In the grandfather paradox, you go back in time and you try to kill your grandfather thus preventing your own birth. But suppose at the key moment of the murder when you want to shoot your grandfather you step on a providential banana peel, slip and miss. And no matter what you try, there is always something which goes wrong and nature always conspires against you. Another solution is the multiverse idea in quantum mechanics. The moment you shoot your grandfather, the universe splits in two identical copies, one in which you fire the gun, and one in which you don’t fire the gun (call them universe A and universe B). The bullet from universe A jumps into universe B and kills the grandfather in universe B. Since that was not your grandfather in universe A there is no contradiction in universe A. In universe B, a bullet out of nowhere kills the grandfather which prevents your birth in universe B. But because you did not fire the gun in the first place there, universe B is also free of paradoxes.

For the creation of information out of nothing, the paradox goes as follows: As a young person you meet an old person who hands you the blueprints of how to construct a time machine. You work your entire life building it and as an old person you take the blueprints with you, hop in the time machine, and go back in time handing the blueprints to your younger self. So far there is no contradiction, but who wrote the blueprints in the first place? This paradox has no known solution.

Background info: As farfetched as it sounds, the splitting universe solution is actually correct and is based on real science. Splitting the universe in quantum mechanics is one of the several interpretations of the theory and since other interpretations are possible it can be taken as a narrative which can help visualize complex mathematical computations. The banana peel arguments may seem to contradict free will but here is a simple counterargument (I think originally given by Novikov http://en.wikipedia.org/wiki/Novikov_self-consistency_principle ): it is my free will to walk on the ceiling but the laws of physics prevent it.

Saturday, March 15, 2014

Quantum mechanics in your face

Again


Two weeks ago I discussed the GHZ-M argument and I listed the exceptional talk by late Sidney Coleman. Today I want to revisit the strange nature of quantum mechanics and show how it violates common sense and classical intuition.  

This time I will present an argument introduced by late Asher Peres in his classic book: “Quantum Theory: Concepts and Methods” which uses the discrete measurement outcomes for spin.

Now I assume everyone is familiar with the idea that spin is quantized and takes only discrete values when measured. But what does this mean and why this is counterintuitive? Spin measurement can be done by a Stern-Gerlach experiment



silver atoms evaporate from an oven, pass through a velocity selector, then go through an inhomogenous magnet before hitting a detector. Classical physics predicts that the magnet causes the precession of the atoms and a vertical deflection in a continuous range from +μ to –μ (here I skipped the details of the derivation but please take my word for it).

But what is the experimental result? Only two deflection results are obtained and so “spin is quantized”.

But what is so special about it? What is special is that we can rotate the orientation of the magnet and still obtain only two outcomes because of the rotational symmetry. And this can generate a classical contradiction. Here is how:

Pick there orientations of the measurement direction, e1, e2, e3, 120 degrees from each other. By symmetry, e1+e2+e3=0 (here we add them as vectors). Now assume that the atoms have an intrinsic magnetic moment μ along a certain direction. In general the experimental outcome is computed classically to be the scalar product of μ with the particular measurement direction: e1, e2, e3.

Summing the outcomes we get: μ 1+ μ 2+ μ 3 = μ.(e1+e2+e3)  =  0 because e1+e2+e3=0 However, this means we are adding three numbers of the form +1 or -1 (the actual experimental results) to obtain zero and this is a mathematical impossibility.


The argument can be criticized because we are reasoning counterfactually and there is no experiment possible to measure all three simultaneously. In fact the three measurements are mutually incompatible. The contradiction still stands if (as Asher Peres put it) a measurement is a passive acquisition of knowledge. The strange world of quantum mechanics where objective reality does not exist before measurement is forced upon us by the humble discrete measurement outcome.

Friday, March 7, 2014

The Detection Loophole in Bell Test Experiments


Caroline Thompson's Chaotic Ball*


Continuing the discussion on Bell inequalities, supporters of local realism challenged the experimental verification by means of several experimental loopholes. One of these loopholes, the detection loophole, is very interesting because it can precisely reproduce quantum mechanics predictions in an EPR-B experiment.

The key idea is that spin measurements can have three outcomes: +1, -1, and no detection. In 1970, Philip Pearle found such an example and computed a minimum no-detection limit of 14% required to reproduce the minus cosine correlation, but the math there is cumbersome and it was not explored further. (What this shows is that detector efficiency needs to exceed 86% to close the detection loophole). I will not discuss Pearl’s model, but I will show instead an intuitive (but inexact) model found by Caroline Thompson in 1996 (http://xxx.lanl.gov/abs/quant-ph/9611037 ).

Let us start with the minus cosine correlation between spin measurement in EPR-B. In The EPR-B experiment, a source of two electrons initially in spin zero state emits two electrons in opposite directions and their spin is measured on two directions A and B making an angle α between them. In any experimental model (quantum mechanics, classical mechanics, and local hidden variable models) there are three fixed correlation values: -1 for α = 0, 0 for α = 90 degrees, +1 for α = 180 degrees. Quantum mechanics formalism and experiments show that the measurement outcome correlation is –cos α. But what is so special about this? When α = 0 by conservation of spin, if we measure the left electron on a direction and obtain an outcome, we naturally expect that if we measure the right electron on the same direction we obtain the opposite outcome.

However, the catch is in the slope of the correlation: since the differential with respect to α of –cos α is +sin α, the tangent to the correlation curve at α = 0 is zero. Let’s think of this for a minute of what it means: if we have a slight deviation in the two measurement angles, the correlation stays the same. In quantum mechanics this is true, because a Bell state is a superposition of two wavefunctions. So what? What this has to do with anything? In the classical case, the electron has a definite direction of spin independent of measurement and the correlation curve has a constant slope of 1 (the three fixed points are connected by a straight line). In quantum mechanics, the correlation curve slope at α = 0 is zero because there is a compensation effect in measurement outcomes due to superposition.

Now back to the detection loophole: can we imagine a simple classical system where not all measurements generate an experimental outcome, and still the correlation curve at α = 0 is zero? Late CarolineThompson came with such a simple system, and is as follows:

Consider a uniformed colored ball which spins randomly around its center. Pick two opposite points and write N and S on the ball (for North and South Pole). Let the ball spin chaotically and look at the ball from two different directions A and B and at certain time intervals. The two experimentalists write down what they see: N, S, or nothing.  If  the observers are close to the ball, due to the reduction in the field of vision, there are some bands on the ball which nobody can see and they generate the “nothing outcome”.



To compute the correlation, the experimentalists have to discard the cases where nothing was detected by one or both observers and the surprise is that the correlation exhibits a flat correlation curve at 0 and 180 degrees.

As I stated earlier the model is not exact, but it raises the question of the detector efficiency in experimental tests and questions the validity of the observed correlation as an argument against local realism because the observed correlations can be an artifact of incomplete detection. In general to create realistic models of the EPR-B experiments using the detection loophole (and exact models do exist) one needs to have unfair sampling depending on the angle between the measurement direction and the intrinsic spin direction. Since the undetected outcomes are by their very nature hidden from the experimentalist, who is to say that Nature obeys fair sampling? After all we want to describe Nature as is, and not to force our preconceptions of fair sampling on the experiments.


Do not expect however to prove quantum mechanics wrong by a few clever hidden variable models exhibiting the EPR-B correlation using the detection loophole. No experiment to date contradicted quantum mechanics predictions. Also in a few years it is expected that loophole free experiments confirming Bell theorem will become feasible.  

* I thank Richard Gill of making me aware of Caroline Thompson's work