Friday, July 29, 2016

Correlations and Entanglement swapping


For this week I want to do a pedagogical presentation of entanglement swapping, but I got busy with other things and tomorrow I'll go on vacation for a week in San Francisco, Lake Tahoe, and Yosemite National Park. Normally I would stay late to write the post, but this involves a lot of LaTeX and so I have to postpone this post. Sorry for the delay, I'll be back in a week.

Saturday, July 23, 2016

A question to George Musser

and

Algebra of coordinates vs. quantum mechanics number system


As I was preparing to start writing the weekly blog post I noticed a spike in readership from Lubos' blog and this seemed very odd: usually those happen after I write something and Lubos counters it, not before. 

So it turned out to be a guest post by George Musser where he touched on a thorny issue: nonlocality. Now here is what he stated:

Lubos defines nonlocality as a violation of relativistic causality - an ability to signal at spacelike separation [...] In our present understanding of physics, this is impossible [...] At times, physicists and popularizers of physics have been guilty of leaving the impression that quantum correlations are nonlocal in this sense, and Lubos is right to take them to task (for instance here, here, and here).


So here is my question to Mr. Musser: where exactly I left that impression in my post?

For the record, if I did leave this particular impression it was not my intention and in that case mea culpa: I accept  that I wrote a bad post. However my gut feeling is that Mr. Musser did not took the time to understand what I was saying.
Puzzled by this allegation I start reading the comments and (as I expected) it went downhill:

Thanks, George, for the remarks. You were telling me that you agreed about the key points but I think that your blog post makes it spectacularly clear that you misunderstand these issues just like all others whom I have criticized concerning this topic over the years.

Dear George, can we please stop this exchange that can't lead anywhere? By now, you have repeated 100% of the idiocies that are commonly said about these issues. You haven't omitted a single one. I've erased last traces of doubts on whether you are a 100% anti-quantum zealot. You surely are one.


But enough is enough of nonlocality and Lubos, Let's come back to the topic of the week.

My interest in noncommutative geometry started from a side problem, the study of Connes' toy model

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

when \(n=2\). This is an interesting problem in itself unrelated to the spectral triple. One way to understand this is to decouple \(C^{\infty}(M) \) from \(M_2 (C)\) and treat \(M_2 (C)\) as a number system for quantum mechanics. But can it be done? 

Here is the motivation. The algebraic structure of quantum mechanics can be derived in the framework of category theory because of an universal property linking products with the tensor product. As such any physical principles we impose on the tensor product induces mathematical constraints on the algebras involved. The physical principle in question is the invariance of the laws of nature under composition. This is a natural principle because the laws of nature do not change by adding additional degrees of freedom. From this one derives the Jordan algebra of observables, the Lie algebra of generators, and a compatibility condition which yields in the end Noether's theorem.

Now on the Lie algebra part one can use Cartan's beautiful theory of classification of Lie algebras and obtain the four infinite series along with the five exceptional cases. So what happens to this classification if one imposes the additional compatibility condition?

It turns out that there is an exceptional cases of interest. This correspond to \(SO(2,4)\), and we may have found an nonphysical case because this is isomorphic with \(SU(2,2)\) which violates positivity. But can this be cured?

Positivity is an additional distinct property/axiom of quantum mechanics, so there is at least a hope it can be done. In a generalized sense we can restore positivity when we consider a constraint case in the BRST formalism. However, something is lost and something is gained. What we gain is a new number system for quantum mechanics: \(M_2 (C)\), but what we lost is the Hilbert space which needs to be replaced by a Hilbert module. Physically this means that to any experimental question we ask nature we do not attach a probability like in ordinary quantum mechanics, but we attach a 4-vector current probability density respecting a continuity equation. The resulting theory contains Dirac's theory of the electron and is intimately related to Hodge decomposition.

So we did not gain anything physical in the end, but \(A = C^{\infty}(M) \otimes M_n (C)\) sits at the intersection of Connes' theory of the spectral triple with the theory of the number systems for quantum mechanics and with generalizations of the concept of norm and Hilbert spaces. It was the investigation of this toy model which made me put the effort to understand noncommutative geometry. Can the algebra of the Standard Model in the noncommutative geometry formalism be understood as a number system for quantum mechanics? The answer is no. To qualify to be a number system for quantum mechanics requires the invariance of the formalism under system composition. Only complex quantum mechanics respects this. The physical explanation is that two fermions cannot be considered another fermion for example.

Saturday, July 16, 2016

What is Noncommutative Geometry?


Noncommutative geometry is not well known and is even less understood by the physics community. Part of the problem is its abstract advanced mathematics which requires a sizable effort to learn, and part of the problem is the lack of down to earth explanations of its basic ideas. I think it was Yang (the Yang from Yang-Mills) who said something like: there are two kinds of mathematical books: the ones you cannot read past the first page, and the ones you cannot read past the first sentence.

Now let's talk quantum mechanics. One thing everyone agrees with is that in quantum mechanics one encounters both discrete and continuous spectra. Another thing which almost everyone agrees with is that the notion of trajectory for elementary particles does not exist. So is there a unified body of mathematics where continuous and discrete naturally coexist, and where the notion of trajectory is not used? 

Riemannian geometry is based on the concept of metric and line element but can those ideas be somehow generalized? The starting point in understanding noncommutative geometry is to consider the concept of a real variable x. Classically this is usually expressed as a function from a subset of R into R: \(f:X\rightarrow R\). So what is wrong with this? The problem is the coexistence of the discrete with the continuum: if x has the cardinality of the continuum then the multiplicity of a discrete variable would also have the cardinality of the continuum and then we would have problems to define measure theory.

However the problem is not encountered in quantum mechanics formalism and the quantum analog of a real variable is a selfadjoint operator in a Hilbert space (there is no ambiguity about which Hilbert space because all infinite dimensional separable Hilbert spaces are isomorphic). What makes the coexistence of the continuum with the discrete possible in the quantum mechanics case is the noncommutativity of operators. Guided by this analogy the task is to extend the usual geometric concepts by using the following prescription:

-identify possible quantum mechanics inspired analogies of the usual mathematical concepts
-verify that in the commutative case they are equivalent with the usual definitions
-introduce noncommutativity and see what we obtain

The next steps comes from the Gelfand-Naimark correspondence: compact Hausdorff spaces correspond to unital C ∗ -algebras:

any commutative C*-algebra (a Banach *-algebra with \(||a||^2 = ||a^* a||\)) is isomorphic with the algebra of continuous functions vanishing at infinity on some topological space.

Connes' theory of spectral triples (A, H, D) extends Gelfand duality beyond topology into differential, homological, and spin aspects:

Riemannian Geometry <==>commutative spectral triple ==> noncommutative spectral triple ==> noncommutative geometry.

Then the following dictionary is obtained:

Commutative
Noncommutative
measure space
von Neumann algebra
locally compact space
C- algebra
complex variable
operator on a Hilbert space
real variable
sefadjoint operator 
range of a function
spectrum of an operator
integral
trace


and the list continues with many more advanced concepts like de Rham cohomology, Chern Weil theory, index theorems, etc.

So now let's revisit something which I introduced two posts ago: the concept of distance in noncommutative geometry:

\(d(x,y) = Sup \{|f(x)-f(y)|; || [D, f] || \leq 1\}\)

This definition looks strange so let me sketch how one arrives at it in the noncommutative framework. The problem with the Riemannian definition of distance:

\(d(x,y) = Inf \{\int_{\gamma} ds |\gamma~is ~a~path~between~x~and~y\}\)

is the usage of the path concept. If we have a space made out of continuous and a discrete pieces then there is no path possible which link them. So we need something which reduces to the Riemannian definition in the commutative case but which does not use the notion of trajectory.

Enter the Monge-Kantorovich optimal transport theory.

Suppose you own coal mines and factories and want to transport the coal from the mines to your factories. Transport cost money and you want to optimize the total transport price. Then one clever mathematician (Kantorovich) comes with a proposal to you: outsource the shipping problem to him and he will charge you only a loading and an unloading price. Moreover, he proves to you that in his proposal the loading/unloading prices will be lower or equal than the minimum transportation cost you face when you do the shipping yourself. The minimization problem for you became a maximization problem for Kantorovich.

Now to connect this to the distance definition, make your price be proportional with the line element. Under suitable conditions (which are fulfilled by the metric tensor) one can apply a minmax principle and you define the distance as the Kantorovich dual. Additional mathematical manipulations of the Kantorovich dual formula in case of the metric yield the Sup definition from above.  However this definition has a big advantage: it is defined regardless of the notion of a path and works in the noncomutative case as well. The key point now is that we can define the notion of distance, neighborhood and topology in cases containing discrete spaces where you only get trivial topologies by using Riemannian (commutative) geometry. 

Mathemathics is about exploring the infinitely rich and connected landscape of math. Noncommutative geometry is a quantum mechanics inspired mathematical paradigm of exploring the landscape of the algebra-geometry duality. Connes took this paradigm further and applied it in physics resulting in an alternative formulation of the Standard Model. There he first made an incorrect prediction of the Higgs boson mass. If the prediction were true, this would have given him a big physics credibility boost. So it is fair to say that the incorrect prediction caused a loss of physics credibility. Later on he had a second look at the model and saw that he overlooked something which brought agreement between theory and experiment. The overlooked element also makes predictions of new physics. Can this prediction be trusted? I would say not yet. Why? because we know the Standard Model is only an approximate description of nature and we expect supersymmetry to be discovered. So the bet of new physics hinges on the validity of the Standard Model itself-a risky strategy. On the other hand, writing off noncommutative geometry as a mathematical fantasy without physics merits is arrogant. If you want to make new contributions in physics, does it make any sense to use the state-of-the-art mathematics from 100 years ago and ignore recent advances in math?

Friday, July 8, 2016

Geometrization of the Standard Model


Now we can explain Cones' framework to the Standard Model coupled with (unquantized) general relativity. This requires a bit of a mathematical preliminary: short exact sequences:

\(0\rightarrow A \rightarrow B \rightarrow C \rightarrow 0\)

where the image of each map is the kernel of the next one. The beginning zero means that the A to B map is injective, while the last zero  means that the B to C map is surjective.

Short exact sequences come in several flavors, but we have two cases of interest:

\(0\rightarrow fiber \rightarrow totalspace \rightarrow base \rightarrow 0\)

and 

\(0\rightarrow N \rightarrow M \rightarrow M/N \rightarrow 0\)

For the Standard Model coupled with gravity we have the gauge group \(G=U(1)\times SU(2) \times SU(3)\) and the diffeomorphism group \(Diff(M)\) of the manifold \(M\). \(Diff(M)\) acts on \(G\) by permutations and the full group of symmetries \(U\) of the Standard Model and the Hilbert-Einstein action is the semidirect product:

\(U = G\rtimes Diff(M)\)

Now we introduce a toy model: a \(U(n)\) gauge theory coupled with general relativity. Suppose we have a coordinate algebra \(A\) on a non-commutative space in the following way:

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

This means that on the ordinary space-time manifold we attach at each point an \(n \times n\) complex matrix, which acts as internal degrees of freedom for the generalized notion of coordinate on this (now a) noncommutative space. We are now investigating the automorphisms of the algebra \(A\): \(Aut(A)\). One trivial subgrup of \(Aut(A)\) are the so-called inner automorphisms: \(Inn(A)\) which is constructed by sandwiching an element of \(A\) with an invertible element \(u\) like this:

\(\alpha(x) = u x u^{-1}\)

Moreover this is always a normal subgroup and we have the following short exact sequence:


\(0\rightarrow Inn(A) \rightarrow Aut(A) \rightarrow Aut(A)/Inn(A) \rightarrow 0\)

which is identical with (drum roll please...)

\(0\rightarrow U(n) \rightarrow U(n)\rtimes Diff(M) \rightarrow Diff(M) \rightarrow 0\)



So now the only thing we need to do is find an appropriate algebra \(A\) such that we get the Standard Model gauge group (subject to the constraint that there is a simple action functional identical with SM+GR action when applied to the noncommutative space). This was easier said than done, and it took Connes and collaborators (Marcolli, Chamseddine) several iterations and many years (I think over 10 years) until they fully recovered the Standard model in its entirety (including 3 generations of mass and the correct value of the Higgs boson mass). 

For the Standard Model The algebra \(A\) is a beast; if I recall correctly it has 96 dimensions. One simple earlier iteration was:

\(A=C\oplus H\oplus M_3 (C)\)

As a side note, Emile Grgin's quantionic quantum mechanics is the same as Connes' toy model when \(A = M_2(C)\).

To uncover the algebra able to reproduce all the known Standard Model experimental facts, a deep dive into K-theory and spectral triples (a spectral triple is a generalization of a \(Spin^C\) manifold) was required. As a mathematical curiosity, the non-commutative space in the Standard Model case turned out to be two copies of the manifold M, meaning a product of a continuous manifold with a discrete space. It is interesting to see the physical mapping of the Standard Model in the new language of noncommutative gometry. For example the Higgs boson links the two copies of the manifold and the two copies are a Highs boson Compton length apart, a picture reminiscent of the branes in string theory.

There were attempts to apply Connes' geometrization ideas to string theory, but they were not successful. More interesting facts about the Standard Model were uncovered by Kraimer and Connes: a particular renormalization technique was proven to be the marriage of two well known mathematical areas: Hopf algebras and Birkhoff decomposition (which has direct application in solving solitonic equations)


Friday, July 1, 2016

A new definition of distance


I started this noncomutative series with the puzzle: what are quantum correlations higher than what we would normally expect? However I want to advocate that the real puzzle is the concept of distance or separation because the notion of distance plays no role whatsoever in the quantum mechanics reconstruction project. And if everything is quantum mechanical, where does the concept of distance comes from?

At first sight the problem seems pointless. Distance is something that we measure with a meter stick, meaning that between any two points we count the minimum numbers of meter sticks needed to connect the two points:

\(d(x,y) = Inf \{\int_{\gamma} ds |\gamma~is ~a~path~between~x~and~y\}\)

The study of the generalization of measure theory on pathological spaces led to a very different definition which is equivalent with the usual one in the standard case, but works under any circumstance:

\(d(x,y) = Sup \{|f(x)-f(y)|; || [D, f] || \leq 1\}\)

where \(f\) is a scalar value function subject to the constraint  that it does not vary too rapidly as controlled by the operator norm of the commutator \([D, f]\), and where \(D\) is the Dirac operator.

Moreover, there is a relationship between the Dirac operator and the infinitesimal line element \(ds\):

\(ds = 1/D\)

and the homotopy class of \(D\) represents the K-homology fundamental class of the space under consideration.

Alain Connes


I won't explain the technical details as I did not present the building blocks required, but coming back to physics, this new definition means that the notion of distance is spectral, and therefore it is fundamentally quantum mechanical in nature. As such the Tsirelson bound puzzle evaporates as everything (correlations from norm and distance from spectral information) comes from quantum mechanics and therefore quantum correlations do not cry out for an explanation as Bell put it.

There is much much more to noncommutative geometry both on the mathematical and the physical side. Noncommutative geometry is expressed in what is called a "spectral triple":

\((A, H, D)\)

where \(A\) is the algebra of coordinates on the geometric space, H is the Hilbert space of A and of the line element \(ds = D^{-1}\), and D is the Dirac operator. 

Dirac operator..., Hilbert space..., this is all physics. Is there a relationship between the noncommutative geometry and the Standard Model? Indeed it is and it is very deep. 

The symmetry group G of the Standard Model action together with the Einstein-Hilbert action is the semi-direct product of the group of gauge transformations with the diffeomorphism group. 

At this point Alain Connes asked two questions:
  1. Is there a space X such that Diff(X)  = G?
  2. Is there a simple action functional identical with SM+GR action when applied to X? 

For "normal" spaces the answer is negative, but recall that noncommutative geometry is about "pathological" spaces, and the answer in this case is positive. Connes calls this "Clothes for the SM beggar". I'll talk about this next time. Please stay tuned.

Friday, June 24, 2016

Noncommutative Geometry


Before jumping into the topic for today, let me say a few words about Brexit: UK put a show of monstrous selfishness and hypocrisy: after colonizing half the globe, now they complain about immigrants?

Back to physics (and math). Last time I stated that geometry requires a generalization, so what does this all mean? There are many ways one can approach this, but let's do it in historical fashion and start with the duality:

Geometry - Algebra

It is informative to remember how ancient Greeks did geometry. For them everything (including the proofs) were a geometric construction with straightedge and compass and they had no concept of coordinates.


It was not until 1637 that geometry and algebra were married by Descartes with what we now call a Cartesian coordinate system. Subsequently mathematicians started realizing that geometry and algebra are nothing but distinct languages describing the very same thing. The first geometry theorem unknown to ancient Greeks was discovered in 1899 and the proof was done by purely algebraic arguments.

The power of algebra is higher than that of geometry because it is easier to formalize abstractions in algebra. In algebra one easily encounters noncommutativity and one example is operator non-commutativity in quantum mechanics. But if algebra is dual to geometry, what kind of geometric spaces would correspond to a non-commutative algebra? What does it mean that "the algebra of coordinates is non-commutative"?

The simplest example is that of a torus. Recall the old fashion arcade games where your character exits through the right side of the screen and re-enters though the left side? Similarly if you move past the top edge you re-emerge at the bottom. Topologically this is a torus. Now suppose you move in straight line in such a way that the ratio of your horizontal and vertical speeds is an irrational number. Slice the torus with trajectory lines respecting this ratio. What you get is a pathological foliation because all "measurable functions" are almost everywhere constant and there are no non-constant continuous functions. 

Other pathological examples are: the space of Penrose tilings, deformations of Poisson manifolds, quantum groups, moduli spaces, etc.

From the mathematical side, one can explain away all those pathological cases one by one, but this is missing the forest because of the trees. The duality from above now becomes:

Quotient spaces - Non Commutative Algebra

where we replace the commutative algebra of constant functions along the classes of an equivalence relation by the noncommutative convolution algebra of the equivalence relation.

Basically it all boils down to a generalization of measure theory. It is well known that the proper way to generalize measure theory is by von-Neumann algebras, and this is how quantum mechanics enters the picture (although historically non-commutative geometry arose from quantum mechanics and the work to classify von-Neumann algebras).

Next time we are going to dive deeper into non-commutative geometry and we will encounter the Dirac operator.

Friday, June 17, 2016

Norm and correlations


Continuing the discussion from last time, today I want to talk about the norm of a linear operator and it's implication for the maximum correlations which can be achieved in nature: Tsirelson's bound. The very same norm definition would later on play a key role in what would unexpectedly become in the end a "geometrization" of the Standard Model coupled with (unquantized) gravity.

In a Hilbert space, the definition of the norm of a bounded linear operator is:

\(||A|| = sup (||Au||/||u||)\) for \(u\ne 0\)

The most important properties of the norm for bounded operators are the triangle inequality:

\(||A+B|| \leq ||A|| + ||B||\)



and a multiplication identity which guarantees the continuity of multiplication:

 \(||AB|| \leq ||A||  ||B||\)

(can we call this a triangle inequality for multiplication?)

On the basis of the triangle inequality, one may be tempted to explore the association of the notion of physical distance with the notion of the norm of an operator in a Hilbert space, but this is a dead end. The triangle inequality for operators is essential for quantum mechanics because it ensures the usual notions of convergence in functional analysis (most of functional analysis follows from it). The name of this blog is elliptic composability, and the "elliptic" part follows from the triangle inequality above. If one imagines a quantum mechanics where the triangle inequality is reversed, then one arrives at the unphysical hyperbolic quantum mechanics based on split-complex numbers which violates positivity which in turns prevents the usual definition of probability as a positive quantity.

There turns out however to be a deep and completely counter-intuitive relationship between the "sup" in the norm definition and the notion of physical distance, but more on this in subsequent posts-don't want to spoil the surprise, I only want to whet the (mathematical) appetite a bit.

Now back to correlations. Suppose we have four operators \(\sigma_\alpha, \sigma_\beta, \sigma_\gamma, \sigma_\delta\) such that:

\({\sigma_\alpha}^2 = {\sigma_\beta}^2 = {\sigma_\gamma}^2 = {\sigma_\delta}^2 = 1\)
and
\([\sigma_\alpha, \sigma_\beta] = [\sigma_\beta, \sigma_\gamma] = [\sigma_\gamma, \sigma_\delta] = [\sigma_\delta, \sigma_\alpha] = 0\)

If we define an operator \(C\) as follows:

\(C= \sigma_\alpha \sigma_\beta + \sigma_\beta \sigma_\gamma + \sigma_\gamma \sigma_\delta - \sigma_\delta \sigma_\alpha\)

Then it is not hard to show that:

\(C^2 = 4 + [\sigma_\alpha, \sigma_\gamma][\sigma_\beta, \sigma_\delta]\)

From the triangle inequalities we have in general that:

\(||[A, B]|| = ||AB - BA|| = ||AB + (-B)A|| \leq ||AB|| + ||-BA|| \)
\(= ||AB|| + ||BA|| \leq ||A||||B|| + ||A||||B|| = 2||A||||B||\)

and so 

\(|| [\sigma_\alpha, \sigma_\gamma]|| \leq 2 ||\sigma_\alpha|| ||\sigma_\gamma|| = 2\)
\(|| [\sigma_\beta, \sigma_\delta]|| \leq 2 ||\sigma_\beta|| ||\sigma_\delta|| = 2\)

Therefore:

\(||C^2|| \leq 4+4\)
\(||C|| \leq 2 \sqrt{2}\)

And this is Tsirelson's bound because \(C\) appears in the left hand side of the CHSH inequality.

Now one can read about textbook derivation of Tsirelson bound in many places, but the key point is that quantum correlations have their origin in the notion of operator norms in a Hilbert space. Nowhere in the derivation we have used the notion of distance or causality. Quantum correlations are a mathematical consequence of the quantum formalism which are in turn is a consequence of considerations of composition and information. Quantum mechanics is nothing but composition and information, and correlations (both quantum and classical) are nothing but considerations of composition and information as well. 

The usual way we understand classical correlations as generated by a common cause is a parochial view due to our classical intuition. Yes, a common cause can generate correlations, but correlation does not imply causation

So what does all of this have to do with the notion of  distance and that of space-time? To uncover the link we would need first to generalize the notion of a space in geometry. Crazy talk? Not when it is based on von Neumann algebras research which led to a Fields medal. Please stay tuned...