Wednesday, December 17, 2014

Coalgebras


Last time we introduced the coproduct which is the essential ingredient of a co-algebra. How can we understand it? If we think of the product as a machine which eats two numbers and generates another one we understand the coproduct as the same machine working in reverse. A Xerox machine can be understood as a coproduct, but a coproduct can be understood not only a a cloning machine but as an action which breaks up an elements into sub-elements. For example a complex number can be decomposed into a real and an imaginary part and each of those are nothing but other kinds of complex numbers.

One funny example comes from shuffling cards: cutting a deck of cards in two is the coproduct, while putting it back together in all possible ways is the product. Renormalization techniques in quantum field theory generates coproducts. Here is a partial list of well studied mathematical examples. The coproduct is usually expressed with the symbol \( \Delta \) and the product is represented by the symbol \(m\).

The first (and most trivial) example come from group theory. Consider finite linear combinations of group elements:

\( kG = \{ \sum_{i=1}^n \alpha_i g_i | \alpha_i \in k, g_i \in G\}\)

\(\Delta g = g\otimes g\)

This is nothing but a basic cloning operation. A bit more complex example comes from polynomial rings:

\( \Delta (x^n) = \sum_{i=0}^n (n~choose~i) x^i \otimes x^j \)
\(m(x^i \otimes x^j) = x^{i+j}\)

Much fancier example come from the cohomology ring of a Lie group, or the universal enveloping algebra of a Lie algebra which gives rise to the so-called quantum groups which have major physical applications.

For now the question is: can we generate a coproduct given a product, and a product given a coproduct? The answer is rather surprising. The answer is yes in both cases for finite dimensional cases, but in general one can only generate a product given a coproduct.

Then can we have a mathematical structure which has both a product and a coproduct? If such a structure exists, it is called a bi-algebra and this respects a compatibility relation where tau is transposition of the terms in the tensor product. 



Let's take the group example. Start from the upper left corner with \( g_1 \otimes g_2\) and move it horizontally:

\(g_1 \otimes g_2 \rightarrow g_1 g_2 \rightarrow g_1 g_2 \otimes g_1 g_2\)

Then take it down, across and up and see you get the same thing meaning the diagram commutes:

\(g_1 \otimes g_2 \rightarrow g_1\otimes g_1 \otimes g_2 \otimes g_2 \rightarrow g_1\otimes g_2 \otimes g_1 \otimes g_2 \rightarrow g_1 g_2 \otimes g_1 g_2\)

Usually this kind of commutative diagram are fancy ways of expressing mathematical identities. For the polynomial ring the commutativity of the diagram means that this holds:

(m+n choose k) = sum over i, j with  i+j = k of (m choose i) (n choose j)

Also Hopf algebras are special kinds of bialgebras and no wonder they have major applications in combinatorics.

Next time we'll talk about Hopf algebras. Please stay tuned.

Wednesday, December 10, 2014

Fun with k-Algebras


Continuing from last time, suppose we have a bilinear map \( f\) from \(V \times W\) to \(L\) where V, W, and L are vector spaces. Then there is a universal property function \(\Phi \) from \(V \times W\)  to \(V \otimes W \) and there is a unique linear map \( g\) from \(V \otimes W\) to \(L\) such that the diagram below commutes:

               \(\Phi\)
\( V \times W \)-------> \(V \otimes W\)
    \                        |
       \                     |
           \                 |
       \(f\)    \              | \( g \)
                  \          |
                     \       |
                      _|   \/
                           \( L \)

The proof is trivial: "f" is used to define a function from the free vector space \( F (V \times W) \) to \( L \) and then we make a descent by modding by the usual equivalence relation of the tensor product to define the map \( g \).

This all looks a bit pedantic, but the point is that any multiplication rule in an algebra \( A \) is a bilinear map from \( A \times A \) to \( A\) and we can now put it in tensor formalism.

In particular consider the algebra \( A \) of matrices over a field \( k \). Matrix multiplication is associative, and we also have a unit of the algebra: the diagonal matrix with the where the elements are the identity of \( k \).  This is a prototypical example of what is called a k-algebra.

Can we formalize the associativity and the unit using the tensor product language? Indeed we can and here is the formal definition:

A k-algebra is a k-vector space \( A \) which has a linear map \(m : A\otimes A \rightarrow A\) called the multiplication and a unit \(u: k \rightarrow A \) such that the following diagrams commute:
                     \( m \otimes 1\)
\( A \otimes A \otimes A \) ----------> \(A \otimes A\)
              |                             |
              |                             |
  \( 1 \otimes m\) |                             |  \( m \)
              |                             |
              |                             |
             \/                            \/
          \(A \otimes A\)   ---------->    \(A \)
                           \( m\)

and

                          \(A \otimes A\)
                    _                    _
                      |         |         |
\( u\otimes 1\)       /              |                \ \( 1\otimes u\)
              /                 |                    \
\( k \otimes A \)                      | \(m \)               \(A \otimes k\)
              \                 |                     /
                  \              |                 /
                       _|      \/          |_
                               \( A \)

Please excuse the sloppiness of the diagrams, it is a real pain to draw them.

So what are those commuting diagrams really saying? 

The first one states that:

\( m(m (a\otimes b) \otimes c) = m(a \otimes m(b \otimes c)) \)

In other words: associativity of the multiplication: (a b) c = a (b c)

The second one defines the algebra unit:

\( u(1_k ) a = a u(1_k )\)

which means that \( u (1_k) = 1_A \)

So why do we torture ourselves with this fancy pictorial way of representing trivial properties of algebra? Because now we can do a very powerful thing: reverse the direction of all the arrows. What do get when we do that? We get a brand new concept: the coproduct. Stay tuned next time to explore the wondeful properties of this new mathematical concept.

Wednesday, December 3, 2014

A fresh look at the tensor product

(the very first lesson in category theory)


Recently I was reviewing Hopf algebras and their applications in physics. This is a very interesting and straightforward topic on par with linear algebra which students learn in first year in college, but unfortunately not well known in the physics community. Starting with this post I will present a gentle introduction and motivation and we'll get all the way to the application in renormalization theory for quantum field theory.

The place to start is to understand what a tensor product really is. In physics one encounters tensors every step of the way and the usual drill is about covariant and contravariant tensors, but this is not what tensors are about. 

We want to start with two vector spaces V and W over the real numbers and attempt to combine them, The easiest way to do that is to have the cartesian product VxW which are the pairs of elements (v, w) each of them in their vector space. If those spaces are finite dimensional, say of dimensions m and n, what is the dimension of VxW? The dimension is m+n but we want to combine them in a tighter way such that the resulting object dimension is m*n not m+n. How can we get from the Cartesian product to the tensor product?

The mathematical answer is a bit dry so let's simply state it. We start with a free vector space over out field or real numbers F(V) and this is nothing but formal sums of elements in V such as:

\(\alpha_1 v_1 + \alpha_2 v_2 + \cdots \alpha_n v_n\)

with \(\alpha_i \in R\) and \(v_i\) in V.

Then of course we can consider F(VxW) and now let's ask what the dimension of this object is? Its dimension equals the number of elements in VxW which is infinite and so we constructed a big monstrosity. We want the dimension of the tensor product to be m*n so to get from \( F(V \times W)\) to \(V\otimes W\) we want to cut down the dimension of \( F(V \times W)\) by using appropriate equivalence relations which capture the usual behavior of tensor products.

To recap, we started with \(V \times W\) but this is too small. We expand it to \(F(V\times W)\) but this is too big, and now we'll cut it down to "Goldilocks" size by equivalence relations.

What are the properties of \(v\otimes w\)? Not too many:

  • \(\lambda (v\otimes w) = (\lambda v)\otimes w\)
  • \(\lambda (v\otimes w) = v\otimes (\lambda w)\)
  • \(v_1\otimes w + v_2\otimes w = (v_1 + v_2)\otimes w\)
  • \(v\otimes w_1 + v\otimes w_2 = v\otimes (w_1 + w_2 )\)

Then \(F(V\times W) \) modulo the equivalence relationship above is the one and only tensor product: \(V\otimes W\) with dimension m*n.

So what? What is the big deal? Stay tuned for next time when this humble tensor product will transform the way we look at products in general. 

Friday, November 28, 2014

The Quantum Cheshire Cat

Before I begin let me reveal the answer for the last post. If no skipping is allowed, the best strategy can achieve 50%. However when not answering is an option, the maximum win rate now jumps to 75%!!!. Here is the original source of the puzzle: http://www.relisoft.com/Science/hats.html where the problem was put in terms of hats. The full answer is here: http://www.relisoft.com/Science/hats3.html and please open it only after fully giving up trying to solve the problem yourself.

Now back to today's topic. Schrodinger's cat is sooo... last century. Meet the Cheshire Cat from Alice in Wonderland:

‘All right,’ said the Cat; and this time it vanished quite slowly, beginning with the end of the tail, and ending with the grin, which remained some time after the rest of it had gone.
‘Well! I’ve often seen a cat without a grin,’ thought Alice, ’but a grin without a cat! It’s the most curious thing I ever saw in my life!’

Can this be even possible? Well,... quantum mechanics is stranger than common sense and indeed it is "possible". To understand how this works, please read this clearly written paper: http://arxiv.org/pdf/1202.0631v2.pdf by the heavyweights: Yakir Aharonov, Sandu Popescu, Daniel Rohrlich, and Paul Skrzypczyk. 




But how can we detect part of the wavefunction? The answer is weak measurements. However, this requires many repeated measurements to extract the information. Fine, but is this real? Indeed it is, and it has been observed in an actual experiment: http://www.nature.com/ncomms/2014/140729/ncomms5492/pdf/ncomms5492.pdf

From the quantum mechanics point of view, this is all relatively trivial, but from the general public impact it has a certain "sex appeal" and this is where journals like Nature thrive. Despite the large impact factor, the intrinsic science content in such journals is rather below mediocre which prove that packaging and good marketing sells. It is important to generate excitement about science in the society at large, but if you are not careful this can easily starts the slippery slope of tabloidization.

As a case in point, the quantum Cheshire cat. Stranger things occur in an interferometer like the one above when weak measurements are involved. By adding a second interefometer in the top arm of the larger interferometer, Lev Vaidman showed that in certain cases the particle (electron, cat, etc) circulating inside the inner interferometer has no connecting paths with the outside circuit, which in effect means that it forms a causal loop. And this too is revealed by weak measurement. Unfortunately I do not have a paper reference for this, but I consider this effect more interesting than the quantum Cheshire cat. The reason the quantum Cheshire cat is now hyped by Nature and other science outlets (which are not talking about Vaidman's more interesting case) has to do with the popularity of the story of Alice in Wonderland and not with its intrinsic scientific value. 

Friday, November 21, 2014

Variations on Romanian Whist Game


Now I want to go back to quantum mechanics for a bit and what better way than by playing card games. To this day I enjoy a type of Whist game, called Romanian Whist and one variant on this is when on the one games you get a card and you place it on your forehead without looking at it. Each player can see all other player's card values, but not his own. 

So let us now imagine the following "game": there are three players, and each is dealt a card. The cards could be either red or black as below.


Each player places his/her card on his forehead and is able to see the other two player's card color but not his own. Then each player tries to guess the color of their card. The guesses are done simultaneously and independently. The game is won if everyone guesses correctly. What is the best strategy which maximizes the chances of winning, and what is the maximum probability of success?

So what does this have to do with quantum mechanics? If you recall Bell's theorem, quantum mechanics is all about correlations which cannot be explained by shared randomness. But what if some measurements are allowed to be discarded? (This is called the detection loophole). Can we achieve higher correlations?

So now in the game above let's change the rules a bit: each player is allowed to enter a guess or abstain. The game is won if at least one person makes a correct guess and there are no incorrect guesses. What is the maximum possible chance of winning the game under the detection loophole variant?

Let us summarize the questions:
  1. What is the best strategy and what is the maximum chance of winning the game when everyone is forced to take a guess?
  2. What is the best strategy and what is the maximum chance of winning the game when you are allowed the freedom to answer or not?
I got this problem from the internet and I will not reveal the source this time because they also have the solution. Do not try to search for it because I restated it on purpose to prevent spoiling the fun of solving it, but I will give full credits next time. You will be surprised to find out just how much better the odds become when skipping an answer is allowed.

Friday, November 14, 2014

Understanding the Standard Model


Today I want to talk high level about the Standard Model and try to extract its insights. The Standard Model is basically quantum field theory and there are excellent books available on the topic. Personally I learned QED long ago from Mandl and Shaw – an excellently balanced and clear introductory textbook (I never read the second edition). If you want to really understand what is going on and not miss the forest because of the trees, an outstanding book is Zee’s "Quantum Field Theory in a nutshell” (maybe in a watermelon-the book has 518 pages). However, I do not recommend it as a first book but read it only after reading the first 10 chapters of Mandl and Shaw. For the serious practitioner earning a living computing Feynman diagrams, Peskin and Schroeder is the gold reference.

Mathematically, the basic idea is that of fiber bundles: just think of them as a common rug. However, Zee has a much better pictorial representation for physicists: consider the space-time like a giant mattress. Jump on it and you create a particle (excitation) at that coordinate. Of course you have bosons and fermions. Let's discuss the simplest case: the electron and electromagnetism. Let's forget about spin and spinors for the moment. The probability to detect the electron is given by the quantum wavefunction which is attaches a complex value at each space-time point. All expectation values are insensitive to an overall complex number phase, and by Noether theorem, invariance under this symmetry implies a law of conservation: the charge is conserved. Now let's add relativity which demands that signals cannot go faster than the speed of light. However this is at odds with the uncertainty principle and the way out is to go to second quantization which demands pair production of particles and anti-particles. If the global symmetry is changed into a local symmetry, we get to the idea of fiber bundles and local phase changes demand an adjustment in computing derivations. In other words we have what mathematicians call a connection and physicists call a potential. The potential turns out to be a vector potential \(A_\mu \) and this is the electromagnetic potential. Upon quantization, the picture now becomes that of Feynman diagrams: the vector potential which comes from a local phase mismatch between neighboring points is now a virtual photon in Feynman diagram.



So this is the basic idea. One more thing I learned in graduate school is that such diagrams were known before Feynman, but they were not computed relativistically. Feynman major contribution was to compute them relativistically which introduced about an order of magnitude overall simplification in their computation. As an apocryphal story, Feynman once attended a physics conference where someone presented the result of six month of computation using non-relativistic diagrams which Feynman managed to double check in thirty minutes using his method and found a mistake. 

Now electromagnetism is known to be invariant under gauge transformations, and the way you couple the electron to the photon (known as minimal coupling) preserves this gauge invariance. 

So far so good, electromagnetism is an interaction based on exchanging a particle (a virtual photon) and since the photon is massless, the range of the interaction is infinite. How about the other interactions? It turns out all other interactions are basically the same thing and electromagnetism generalizes into Yang-Mills gauge interactions. Here is how is done: In quantum mechanics there is a 1-to-1 correspondence between observables and generators. Observables are hermitian operators which obey a Jordan algebras, while generators are anti-Hermitean operators which obey a Lie algebra. From generators of Lie algebras one constructs by exponentiation a Lie group and in this case we are talking about \( SU(n) \). Q: how many generators are for \( SU(n) \)?  A: \(n^2 - 1\). For electromagnetism the gauge group is \( U(1) \) which has elements of the form \( e^{i \phi} \), the weak force has three generators (the Pauli matrices), and the strong force has eight generators. 

The generators correspond in second quantization to emission and absorption of one quanta of interaction (photon, W+, W-, Z, 8 gluons) and they are 4-vectors just like \( A_\mu \). The key difference between electromagnetism and Yang-Mills is that the generators do not commute. Why? Expressed them as matrices: \( A_\mu = (A_0, A_x, A_y, A_z ) \) where each \(A \)  is a \( n \times n\) matrix. Physically this means that they carry "charge". In electromagnetism there is only one electric charge, in weak interaction there are two charges (which mixes electrons with neutrinos and up quarks with down quarks), and in the strong  interaction there are three charges (red, green, and blue). Alternatively, an electron or a neutrino is the same physical particle which becomes an electron or a nutrino upon measurement just like an electron has a spin which becomes up or down when passed through a Stern-Gerlach device. Because the field lines carry charge, unlike in electromagnetism, for two charges in the \( SU(3) \) and higher hypothetical \( SU(n>3)\) cases the field lines are parallel because it is energetically more advantageous. What this meas is that quarks cannot be free because separating them adds energy to the point where two more quark-antiquark particles are formed in the middle. For the strong interaction the only possible states which are allowed in nature are the singlet state of zero color charge, all other states requiring an infinite amount of energy to be created. For \(SU(3)\) there are only two possible singlet color states corresponding to 3 or 2 quarks (proton, neutron for 3 quarks, mesons for 2 quarks). 

The gauge group of the Standard Model is \(U(1)\times SU(2) \times SU(3) \) which means that the following particles are possible:

\(up_{r} ~~~ down_{r}\)
\(up_{g} ~~~ down_{g}\)
\(up_{b} ~~~ down_{b}\)
\(e ~~~ \nu\)

The strong force mixes the top three rows, while the weak force mixes the two columns. The particles here form what is called a "family". There are two more families identical from the point of view of gauge symmetries, but different in mass (heavier). The origin of the families is unknown and a possible explanation comes from string theory. 

Now all the particles (photon, Ws, Z. gluons, electrons, neutrinos, quarks) are massless or nearly massless (compared with the energy level at the unification scale which is the natural energy).  

Adding mass to photons, Ws, Z, gluons in the Lagrangian spoils gauge invariance, but this can be restored if it is part of another field called the Higgs field. How does this work? A zero mass particle, just like the photon has two degrees of freedom corresponding to two perpendicular modes of oscillation (two polarizations). Making a massless particle into a massive one adds a longitudinal degree of freedom which must come from some other field. If you recall the "Mexican Hat potential", a Higgs field has two modes of oscillation: radially (corresponding to the Higgs particle) and circular corresponding to a massless Goldstone boson. The Goldstone boson combines with a massless particle like W and gives rise to a massive W. This is why W and Z particles are massive and because of it particle decay is relatively slow. How do particle decay? Take a heavy quark-antiquark combination, they form a W particle in a mechanism similar with vacuum polarization is QED and then W decays into a lighter combination of electron and antineutrino. 

The Higgs mechanism works only for W+, W-, and Z, not for photons or gluons. There are two more twists in the Higgs process. First, this mechanism breaks the symmetry but the singlet boson mixes with the electromagnetism boson and results in the massive Z and the massless photon. The photon is not really the photon before symmetry breaking! Second, how do the fermions acquire mass?

Fermions interact with other fermions through the minimal coupling from above. However, they also couple with the Higgs field by "Yukawa coupling" and this is how they get mass. Fermion masses has to do with the fact that in nature the mirror symmetry is broken. Left handed particle behave differently than right handed particle in weak interaction. If you look at Dirac's equation and write it in terms of left handed and right handed components, the fermion propagation mixes them up. Without Yukawa coupling for fermions the left-handed physics would have been completely equivalent with right handed physics. What happens is that Dirac's equation is valid only in the approximation that the Higgs field does not excite radial oscillations and the mass of the fermions depend on the radial value of the Mexican Hat potential at the minima and the coupling coefficient.

Now if the Higgs Mexican hat potential would have had a different minima, or if the Yukawa coupling would have been different, then our universe would have looked completely different. Take the up and down quark masses for example. If those masses would have been the other way around, then the neutron would have been stable and the proton would have decayed into it preventing the formation of atoms. Chemistry and life would have been impossible. 

Our universe is the way it is because we are trapped in a local energy minima. What generates this minima? Can an ant walking on the surface of an apple understand the concept of the apple and how it got there? I think one of the triumphs of string theory is actually predicting the landscape of local minima despite the criticism of the lack of predictive power. This is the Copernican principle in action. We are in no way special. The Earth is one of the planets of the solar system, the Sun is one of the 100 billion stars in out galaxy, the Milky Way is one of the 100 billion known galaxies, and probably our universe is one of the  \( 10^{100}\) possible local energy minima each with very different particles then the ones in our universe. Quantum mechanics "multiverse" (from MWI) is almost sure gobbledygook, but eternal inflation, bubble universes, and the multiverse landscape are almost surely real.

Thursday, November 6, 2014

Clothes for the Standard Model beggar


Although there were several other interesting talks at the DICE2014 conference, I'll not talk about them because they are right in my active research area and I do not want to present half-baked ideas and work in progress.

One very interesting talk was given by Fields Medal winner Alain Connes:



I won't talk about this because I do not fully understand it (and him pulling a Houdini-type disappearance after the talk like all the 100+ physicists at the conference were infected with Ebola prevented the opportunity to ask in depth questions). But I do want to present some general ideas about Connes-Chamseddine approach to the Standard Model which occurs naturally in Connes' non-commutative geometry setting.

Now when you look at the action for both general relativity and the standard model you notice the groups of invariances. In general relativity you have the group of diffeomorphims, and in the Standard Model you have the group of gauge transformations. Diffeomorphisms are easy to understand because they mean that there is no preferred system of reference in nature. One way to unify of gravity with Standard Model is to try to understand diffeomorphism as a gauge group. This is a faulty interpretation and Streater talks about it in his famous "lost causes". Now Connes' idea is the other way around: we can obtain the gauge group from diffeomorphism in a suitable generalized space. And so Connes' unification is geometrical in nature.

To get to the full Standard Model is not at all easy and is best to see how all this works on a toy model: gravity coupled with SU(n) Yang Mills theory. Start with a manifold M and consider continuous functions on it: \( C^{\infty} (M) \). Now let's add at each point an internal space described by an \( n \times n\) complex matrix representing the inner degrees of freedom of a Yang-Mills theory. Then consider the involutive algebra \( A\) of \( n \times n\) matrices of smooth functions on the manifold M:

\( A = C^{\infty} (M, M_n (C)) = C^{\infty} \otimes M_n (C)\)

Now the fireworks: the inner automorphism \( Inn (A) \) is isomorphic with the gauge group \( \mathcal{G}\) and the short exact sequence:

\( 1 \rightarrow Inn (A) \rightarrow Aut(A) \rightarrow Out(A) \rightarrow 1\)

is equivalent with:

\( 1 \rightarrow \mathcal{G} = Map(M, SU(n)) \rightarrow Diff(X) \rightarrow Diff(M) \rightarrow 1\)

And so the full group of invariance on a new space \( X = M \times M_n (C) \) is the semidirect product of the diffeomorphisms on M with the gauge group and the diffeomorphism shuffles (acts on) the group of gauge transformations.

Generalizing the space from a regular manifold to a product of the manifold with a discrete non-commutative space F: \(X = M \times F\) by pure geometrical concept of diffeomorphism in the new space generates general relativity coupled with new gauge degrees of freedom which can be understood as inner fluctuations of the metric. 

Now for the connection with non-commutative geometry: because in nature there is no absolute coordinate system, to specify a position one needs to use geometric invariants, and in particular, there is an alternative description of them using spectral information. Connes makes the point that the very definition of a meter now uses a certain laser wavelength information-a spectral concept. From non-commutative geometry Connes introduced a spectral triple (A, H, D) where A is an algebra, H is a Hilbert space, and D is Dirac's operator to have an alternative encoding of the geometric information in a diffeomorphic setting. For the Standard Model the job was to find an appropriate spectral triple which will generate the Einstein-Hilbert action of general relativity and the Standard Model action.

And so for the Standard Model beggar its clothes come as follows: the algebra A comes from the gauge group information, the Hilbert space comes from fermions and a spin manifold, and the Dirac operator comes from the Yukawa coupling matrix. In the end, while the new equivalent description does represent a simplification, the algebra \( A \) is rather complex as it involves the three generations of matter and the full theory is not as neat as the toy model.  One last key point: how can we define the notion of distance in a space which contains a discrete space? In non-commutative geometry there is a suitable generalization using the norm of an operator which works even for discrete spaces.  

Much more can be said about this approach to the Standard Model, but I only wanted to present a 10,000 feet impressionistic view of it. I only want to state one more thing: Connes-Chamseddine approach predicts new physics beyond the Standard Model and rejects the "big desert hypothesis" in order to correctly predict the Higgs mass and so the theory is falsifiable