Thursday, August 27, 2015

Is China's turmoil the next Lehman Brothers?

I am taking a break from physics topics to discuss a hot topic of the day: the extreme turmoil on the stock markets due to trouble in China. I am not sure how the system works in other countries, but in US your retirement funds, your nest egg, is usually tied to the stock market fortunes as the largest investors are mutual funds. Every person have his or her tolerance to fluctuations and loss but major evaporation of retirement funds can be very nerve racking. Since I got burned in the past and learned from my mistakes, I want to share some common sense advice you will not easily find otherwise. 

The first observation is that unless you are a serious investor willing to commit time and resources into researching individual stocks it makes little difference overall on which mutual fund you select. Each mutual fund is already diversified and by watching how the DOW does you have a pretty good idea how your investments are doing. For the usual ups and downs of the market you do not need to worry at all, you will have an average return of about 8%, but what to do when there are violent price movements like the recent ones?

One of the most imbecile advice typically found on CNN by their so-called experts like Richard Quest is that you should not panic and ride out the storm. Is there a topic he is not qualified in? But when the losses exceed 8, 9, 10% this shows that major events are happening and it is not the time to stick your head in the sand and pretend it is business as usual. It is true that overreacting is bad and usually the ordinary investor crowd bets incorrectly. So you need expert advice to understand what is going on, but where do you get such a thing? One possible source is o reach out to your financial adviser, but in general this is not a good idea because those guys make money when you buy stocks and they do not offer unbiased advice. They usually give you the same cookie cutter nonsense of investing for the long term. So you have to realize that you cannot get good financial advice during an unfolding crisis and you should follow the money instead. 

If you watch the DOW value, get in the habit to see watch the transaction volume as well. The idea is that large financial institutions do have very serious experts who make decent predictions, and more important, those advice is actually followed. And when large mutual funds decide to make a large move, the volume tells the story. So the advice is simple:
  • when DOW drops significantly but the volume is close to the average, don't panic and do nothing.
  • when DOW drops significantly and the volume is two to three times the average it is time to sell. The large players decided that the outlook is negative. 
  • when DOW increases significantly but the volume is close to the average, do nothing.
  • when DOW increases significantly and the volume is two to three times the average it is time to buy. The large players decided that the outlook is positive. 
Sure, you will miss the first big drop, and the first large increase, but in the long term this does not matter. This advice is for the investors who don't have the stomach for large drops in value. If your tolerance for risk is higher or you have the luxury of a really long term investment period measured in many decades then you should not care about the stock market antics in the first place.   

Now on the current China turmoil. 

Is this a short storm, or the harbinger of a larger trouble like the Lehman Brothers collapse? Honestly, I don't know, and I think nobody knows either. China is now the second largest economy and any turmoil there has large implications. Also the current trouble is the result of irresponsible advice from Chinese leadership to ordinary citizens to buy stocks in a get rich quick scheme. This created a bubble which now burst. But what will the effect be in the psychology of ordinary Chinese citizens? Would this create unbearable social pressure which will result in the change of the political system? Would this impact the real Chinese economy? Nobody knows because this is a first in the modern history of China. When Lehman collapsed, for a few months there was no economic pain. Everyone was reassured that "the economic fundamentals are strong" and this was just a blimp on the radar. But something funny happened: the usual economic activity simply stopped like a switch was turned off because lending between banks froze solid due to sudden lack of confidence. It took a Keynes style influx of money to get the economic engine restarted.

Thursday, August 20, 2015

Where does the Hilbert space come from?

Continuing the discussion from last time, today we can put some of the pieces of the puzzle together. It is helpful to switch the discussion direction from classical to quantum and start for a moment from the quantum mechanics side to see where we want to arrive.

Why do we use a Hilbert space in quantum mechanics? This is a big topic and we cannot cover it in one (or even several posts). Right away we will restrict ourselves to the finite dimensional degrees of freedom case, thus excluding the field theory considerations and avoiding the issues raised by Haag's theorem, or the breakdown of the Stone-von Neumann uniqueness theorem. We will also skip the treatment of unbounded operators which require the theory of rigged Hilbert spaces and we will stick with boring but well behaved bounded operators. 

For a bounded operator T on a Hilbert space it is easy to prove that \(||T^{\dagger} T|| = {||T||}^2\) as follows:

\({||T \Phi||}^2 = \langle T \Phi, T \Phi\rangle = \langle T^{\dagger} T \Phi, \Phi\rangle \leq ||T^{\dagger} T\Phi || ||\Phi|| \leq ||T^{\dagger} T || {||\Phi||}^2 \)
\({||T||}^2 \leq ||T^{\dagger} T || \)
and since 
\(||T^{\dagger} T || \leq ||T^{\dagger}|| ||T|| = {||T||}^2\)
we have:
\({||T||}^2 \leq ||T^{\dagger} T || \leq {||T||}^2\)

An algebra of bounded operators on a Hilbert space is the prototypical example of a C* algebra. A remarkable fact is the correspondence between states and representations of C* algebra given by the GNS construction. Here a representation is a linear map from the elements of the C* algebra to bounded operators on some Hilbert space.

From categorical considerations one can obtain a C* algebra without the norm axioms. To distinguish math from physics one needs to be able to make experimental predictions and this is where the states enter the picture. A state on a C* algebra gives rise to a representation of the algebra as bounded linear operators on some Hilbert space and this is how Hilbert spaces are introduced. The key ingredient for this to work is the C* norm condition: \(||T^{\dagger} T|| = {||T||}^2\). However, this norm is unique and is given by the spectral radius - an algebraic concept! So there is hope we can arrive at quantum mechanics using only algebraic methods. Now we will show how.

Coming back to the quantization discussion from the prior post, what we need to achieve is a prescription which constructs operators on a Hilbert space from functions on the phase space (also known as the cotangent bundle). Even better we should be able to start from either a Kahler, symplectic, or Poisson manifold.

We can start with the simplest case where we replace the position \(q\) and the momenta \(p\) with the operators: \(x\) and \(\frac{h}{i}\frac{\partial}{\partial x}\) in any observable \(f(p, q)\) provided \(f\) contains no products \(pq\) because the position and momenta operators in the Hilbert space do not commute and the order of the operators is ambiguous.

The next level of sophistication is Weyl quantization procedure and the details can be found here. (Please excuse me for skipping typesetting it in LaTeX.) Weyl quantization tends to preserve well symmetry properties, but a better quantization prescription is Berezin quantization which work on all Kahler manifolds when positivity is guaranteed by the Kodaira embedding theorem.

Erich Kahler

A Kahler manifold is a truly outstanding mathematical object where three concepts meet:

  • a metric structure
  • a symplectic structure
  • a complex structure
and any two define the third one. The main example is the complex projective space (endowed with the Fubini-Study metric) which is essential for quantum mechanics. It is very enlightening to see how it all works out in quantum mechanics and I'll attempt to show it below.

In classical and quantum mechanics there are two products, one symmetric \(\sigma\) and one anti-symmetric \(\alpha\) corresponding to observables and generators as follows:

Observables: \(\sigma\) = regular function multiplication on phase space OR Jordan product
Generators: \(\alpha\) = Poisson bracket OR commutator

There is also a 1-to-1 map \(J\) between observable and generators called dynamic correspondence where \(J^2 = 0\) for classical mechanics and \(J^2 = -1\) for quantum mechanics. This map corresponds to Noether's theorem.

Composing two physical systems 1 and 2 gives rise to the following fundamental composition relationship:

\(\sigma_{12} = \sigma_1 \otimes \sigma_2 + J^2 \frac{\hbar^2}{4}\alpha_1 \otimes \alpha_2\)
\(\alpha_{12} = \sigma_1 \otimes \alpha_2 + \alpha_1 \otimes \sigma_2\)

and so the symmetry and anti-symmetry of the products is preserved.

Now we want to deform the Poisson bracket and regular function multiplication of classical mechanics which respects the composition with \(J^2 = 0\) into two products which respect \(J^2 = -1\). We can do this term by term in powers of \(\hbar\) preserving associativity at each step. This is the essence of deformation quantization.

Without ado, here is the solution given by Moyal sine and cosine brackets in terms of the Poisson bracket \(\{ , \}\) in the simplest case of a flat space:

\(\alpha = \frac{2}{\hbar} sin (\frac{\hbar}{2} \{ , \})\)
\(\sigma = cos (\frac{\hbar}{2} \{ , \})\)

The star product is then \(\star= \sigma+ J\frac{\hbar}{2} \alpha\) and we arrived at quantum mechanics in phase space.

First a note: I demanded earlier to preserve associativity at each power of \(\hbar\). This is a physical requirement to be able to compose experiments sequentially and not care where we draw the boundaries between them. But this has a very interesting consequence: we have freedom of pick how we carry out the quantization at each power of \(\hbar\) step and this makes the subject of quantization non-trivial. In particular it turns out that the equivalence classes of star products on symplectic manifolds are in 1-to-1 correspondence with the second de Rham cohomology \(H^2_{dR} (M)\)!

Second, we can see where the inner product is coming from. From the Moyal sine bracket we extract a symplectic form \(\omega^{IJ}\) and construct it's inverse \(\Omega_{IJ}\). So we have one of the three structures of a Kahler manifold: the symplectic structure. But we also have the complex structure as well because we have \(J^2 = -1\). It can be shown that \(J\) is actually a tensor or rank (1,1): \(J = J^{I }_{  J}\) and from this we get a metric tensor \(g_{IJ}\):

\(g_{IJ} = \Omega_{IK} J^{K}_{  J}\)

The complex inner product is defined now by: \(g+ \sqrt{-1}\Omega\):

\(\langle X, Y \rangle = X^{T} g Y + i X^T \Omega Y \)

where X and Y are column vectors : \(q_1, q_2,...q_n, p_1, p_2, ..., p_n\)

Time evolution preserves \(J\) and \(\Omega\), meaning they preserve the metric structure by preserving a normalization constraint:

\( \langle g \rangle - 1=  X^I g_{IJ} X^J - 1 = 0\)

The constraint Hamiltonian motion which preserves the metric structure is nothing but the Schrodinger equation is disguise!

I do not want to create the impression that this is all as simple as this. I only discussed the flat \(R^{2n}\) case above. There are many subtle and hard problems, as well as open questions. As an example, there are Poisson manifolds which do not admit a Kahler structure, but all Poisson manifolds are quantizable. How would the quantization of such a system look like? Perhaps there are no bounded operators in this case, I don't know.

Next time I'll present a concrete calculation of a standard problem in the phase space formalism of quantum mechanics. This will challenge the ontic interpretation of quantum mechanics.

Friday, August 14, 2015

Cotangent bundles, symplectic, and Poisson manifolds

For today's post I want to explore some interesting facts about classical mechanics. But why should we care about classical mechanics if nature is quantum at core? One misconception I used to have was that we should care only about the emergence of classical physics out of quantum mechanics. I was not alone in this misconception, the late Sidney Coleman supported this point of view in his outstanding talk: Quantum Mechanics in your face. But as it turns out rigorous deformation quantization starting from a Poisson manifold can help arrive at the standard formalism of quantum mechanics. 

And to achieve that we need to understand the mathematical structures involved in classical mechanics: cotangent bundles, symplectic manifolds, Poisson manifolds. Because I did not write the needed mathematical posts for symplectic geometry I will keep the discussion at high level.

The natural starting point for classical mechanics is a configuration space Q of generalized coordinates and Hamiltonian mechanics happen in the cotangent bundle \(T^* Q\): to each point of Q we attach the dual of the tangent space. The transformation between the velocities in the tangent bundle and the momenta in the cotangent bundle is made by the Lagrangian L using a Legendre transform:

\(p_i = \frac{\partial L}{\partial v_i}\)

The Hamiltonian is obtained from the Lagrangian using a Legendre transformation:

\(H(q^i, p_j) = p_i v^j - L(q^i, v^j)\)

While the Lagrangian formulation is best suited for field theory and its relativistic formulation, the Hamiltonian formalism is more natural for classical and quantum mechanics. One reason to use the cotangent bundle over the tangent bundle is the existence of a canonical "God given" 1-form:

\(\theta = \sum_{i} p_i dq^i\)

The hierarchy of spaces used in classical mechanics is:

cotangent bundle \(\subset\) simplectic manifold \(\subset\) Poisson manifold

To go from the cotangent bundle to a symplectic manifold we use the canonical 1-form \(\theta\):

\(\omega = d \theta\)

A manifold with a closed nondegenerate differential 2-form \(\omega\) is called a symplectic manifold. Not all symplectic manifolds are cotangent bundles, but due to Darboux theorem a symplectic manifold is locally a cotangent bundle. In a symplectic manifold a polarization is a Lagrangian foliation (in the tangent bundle the foliation is the map \((q, p) \rightarrow q\); picture a foliation as a slicing of the space). Existence of polarizations is essential for proving the equivalence of a symplectic manifold with a cotangent bundle when true. Moreover polarizations are key ingredients in geometric quantization. The details are much more complex but this is the 10,000 feet view.

The generalization from symplectic to Poisson manifolds happen when \(\omega\) can be degenerate. Non-degeneracy demands that the dimension of a symplectic manifold to be even, but Poisson manifolds can have odd dimensions. So how does the Poisson bracket look on a Poisson manifold?

The prototypical is the free pivoted rigid body rotation. The Hamiltonian is:

\(H(x) = \frac{1}{2}(\frac{x^2}{I_x} + \frac{y^2}{I_y} + \frac{z^2}{I_z})\)

the Poisson bracket is:

\(\{F,G\}(x) =-x \cdot (\nabla F \times \nabla G)\)

which makes the Hamiltonian equations of motion:

\(\frac{d x}{d t} = x\times \nabla H(x)\)

The most general Poisson bracket can be proven to be:

\(\{F, G\} = \sum_{i,j} \{x_i , x_j\} \frac{\partial F}{\partial x_i} \frac{\partial G}{\partial x_j}\)

and again by Darboux theorem for any Poisson manifold there are local coordinates for which the Poisson bracket takes the usual form for the even subset of coordinates for which the rank is locally constant. In other words, a Poisson manifold can be decomposed into a product of a symplectic manifold and a Poisson manifold of rank zero.

But how do we get the crazy Poisson bracket from above? The answer is symplectic reduction. I did not built the mathematical pre-requisites of moment maps to explain in an intuitive way how this works, and this will have to wait for future posts. In the meantime I can state some conclusions.

First, category arguments can reconstruct both classical and quantum mechanics. In the classical case those arguments offer no help in proving the existence and non-degeneracy of \(\omega\). All we get from category arguments are Poisson manifolds! So it is imperative to prove that one can use deformation quantization on any Poisson manifold to arrive at quantum mechanics. Fortunatelly such a proof exists and was found by Maxim Kontsevich in 1997: A remarkable fact is that the proof uses string theory insights!!!

Second, the existence of time without invariance under composition is not enough to reconstruct either classical or quantum mechanics. The best counterexample comes from soliton theory and the Korteweg-de Vries equation. A KdV equation comes from a Poisson manifold without a symplectic analog, and moreover it is a bi-Hamiltonian system. KdV admits not one, but two Poisson brackets compatible with each other!

Third, not all quantum systems come from quantization of classical systems.

Next time I'll start exploring topics of quantization. Please stay tuned.

Friday, August 7, 2015

From time to quantum mechanics

Last time I presented the case for canonical time evolution stemming from the non-commutativity of operator algebras, and today I'll start talking about the reverse implication: obtaining quantum mechanics from the existence of time. 

First, let me expand on the brief statement from the prior post that the transition from classical to quantum mechanics is NOT as simple as replacing the Poisson bracket with the commutator. We need to put this in rigorous mathematical formalism and this is known as the Dirac problem:

"Does  there exists two matrices P and Q and a correspondence \(\phi\) which, to every polynomial g in the classical variables p and q, associates a matrix \(\phi (g)\) in such a manner that:
(i) \(\phi (p) = P\) and \(\phi (q) = Q\)
(ii) to the unit function \(1 : (p, q) \in R^2 \rightarrow 1\), \(\phi\) associates the unit matrix I
(iii) \(\phi\) is linear
(iv) \((i/\hbar) [\phi (f), \phi(g)] = \phi(-\{f,g\})\) for every polynomials f and g in p and q
(v) the matrices P and Q form an irreducible system, i.e. the only matrices A satisfying [X,P] = 0 = [X,Q] are of the form \(X = \lambda I\) where \(\lambda \in C\) and I is the unit matrix?"

It turns out that the Dirac problem has no solution and the proof is actually not very complicated. What are really complicated are the solutions to the quantization problem, and different approaches reject different assumptions in Dirac's problem. 

What is important for our purposes is that in classical mechanics one encounters the Poisson bracket and the quantum mechanics we have the commutator. Moreover both of them obey Leibniz identity:

\(\{H, fg\} = \{H, f\} g + f \{H, g\}\)
\([H, AB] = [H,A] B + A [H, B]\)

In the case of the Poisson bracket this is a trivial consequence of the partial differential operators, while in the case of the commutator this is a simple algebraic identity:

\([H, AB] = HAB-ABH = HAB-AHB + AHB -ABH = [H,A] B + A [H, B]\)

We can understand the commutator and the Poisson bracket as a product "\(\alpha\)":

\(A \alpha B = [A, B]\)
\(f \alpha g = \{f,g\}\)

and this is related to time evolution. If we call T a time translation operator and o any algebraic product used in physics, the invariance of the laws of Nature under time evolution implies the following commutative diagram:

T(A) o T(B) = T(A o B) which shows that [T, o] = 0 (T after o is the same as o after T) or that time translation preserves algebraic relations.

In the infinitesimal case in natural units (ignoring the usual factors of h bar and such): \(T = I + \epsilon H \alpha\) 

Substitution in T(A) o T(B) = T(A o B) yields:

\(((I + \epsilon H \alpha) A)  o ((I + \epsilon H \alpha) B) = ((I + \epsilon H \alpha )(A o B))\)

which to first order in \(\epsilon\) is:

\(H \alpha (A o B) = (H \alpha A) o B + A o (H \alpha B) \)

which is known as Leibniz identity. (A trivial observation is that when \(o = \alpha\), the Leibniz identity becomes the Jacobi identity and this gives rise to a Lie algebra.)

Now the heavy mathematical lifting follows using category theory arguments (invariance under composition = universality of the theory) to completely recover the algebraic structure of quantum and classical mechanics. Then using geometric or deformation quantization (to avoid the lack of the solutions for Dirac's problem) one obtains the usual Hilbert space formalism for quantum mechanics. Therefore the Poisson bracket and commutator are the only mathematical realizations of Leibniz identity for theories of nature obeying invariance under composition.

The starting point of quantum mechanics reconstruction using category theory arguments is Leibniz identity and this follows from infinitesimal time translations of the commutative diagram from above.

Moreover, in the usual Hilbert space formulation Leibniz identity corresponds to unitarity. This cuts both ways and violation of unitarity implies violation of Leibniz identity. And so now we have a big mathematical problem:

collapse postulate -> unitarity violation -> violation of Leibniz identity -> no Hilbert space formalism for quantum mechanics!!! 

It makes no more sense to talk about Hilbert space, operators, etc and this is clearly impossible. Something must give. Is it that there is no real collapse (MWI)? Or do we need to add contextual protection (Bohmian, QBism)? Or maybe there is an extension of quantum mechanics (GRW)?

How can this be solved? The so-called measurement problem just got much more serious than a simple problem of philosophical interpretation. There is good news however: the same categorical arguments which highlighted the problem in the first place, point the way to a most natural solution: unitary dynamical generation of superselection rules similar with spontaneous symmetry breaking. To be continued...

Friday, July 31, 2015

It's about time

After discussing some well known results in quantum mechanics, I will start presenting my approach for solving the measurement problem which was my talk at the Vaxjo conference. This will take several posts, and today I start the discussion by presenting some ideas about time.

Time is essential in Hamiltonian mechanics which is the natural way to transition from classical to quantum mechanics. Usually one is introduced to the subject by the naive statement that we replace the Poisson bracket with the commutator. The story is much more subtle (and interesting) than that and today I want to explore only some parts of those issues with the remaining advanced topics to be discussed at a later time.

It turns out that there is a circularity problem: time is derived from quantum mechanics and quantum mechanics is derived from time. Let me state upfront that I do not yet know how to solve this very interesting and hard problem.

Today I'll present the claim that time has a quantum mechanical origin. The main proponents of this are Alain Connes and Carlo Rovelli in their Thermal Time Hypothesis. I have talked about this last year, but now I want to go in depth in the mathematical reasoning. As a pre-requisite the reader should understand the concept of short exact sequences and I have a review of this topic here.

I am not attempting to justify the canonical gravity point of view and oppose it to string theory, but I want to present the mathematical reasons of why time may have a non-commutative (quantum mechanical) origin. The mathematical underpinnings are von Neumann algebras and the Tomita-Takesaki theory.

The von Neumann algebras are the generalization of measure theory: they reduce to the study of measureable spaces when we restrict to the commutative case.

Suppose \(M\) is a von Neumann algebra in a Hilbert space H, \(a\) is an element of \(M\), and we have an operator S defined by:

\(S a \psi = a^* \psi\)

Then S admits a polar decomposition:

\(S = J \Delta^{1/2}\)

with \(\Delta\) a positive self-adjoined operator and J anti-unitary.

The Tomita-Takesaki theory proves that:

\(JMJ = M^{'}\): M has the same size as its commutant
\(\Delta^{-it} M \Delta^{it} = M\): there is an one-parameter group of automorphism  \(\sigma^\phi_t\) which gives the time flow in the Heisenberg picture.

So far so good, but stronger statements can be made. If we impose the KMS condition, then the automorphism is unique. Still, it depends on the choice of the state \(\phi\). The really powerful result is that in the exact sequence:

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

where \(Inn(M)\) is the normal subgroup of inner automorphisms: \(T\rightarrow aTa^*\), the automorphism \(\sigma^\phi_t\) does not depend upon the choice of state \(\phi\) and hence it is a canonical time evolution.

This is the main mathematical result (by Connes) which needs to be applied to physics to understand what is going on. And this is what Connes and Rovelly do in their thermal time hypothesis paper. But there is more to the story. The short exact sequence from above was the starting point of Connes non-commutative geometry description of the Standard Model which led to the geometric unification of gauge theory weakly coupled with gravity in the non-commutative framework (no quantum gravity here: no background independence arguments to justify loop quantum gravity vs. string theory). In the simplest model of the noncommutative description of the \(SU(n)\) gauge group weakly coupled with non-quantized gravity, the short exact sequence from above is equivalent with:

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

where M here is a spin Riemannian manifold. 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. The diffeomorphism shuffles (acts on) the group of gauge transformations.

If we take the main lesson of quantum mechanics to be that of non-commutativity of operators, we can construct a generalization of commutative mathematics into a non-commutative domain:

measure space
von Neumann algebra
locally compact space
C- algebra
vector bundle
finite projective module
complex variable
operator on a Hilbert space
real variable
sefadjoint operator infinitesimal compact operator
range of a function
spectrum of an operator
vector field
closed de Rham current
cyclic cocycle
de Rham complex
Hochschild homology
de Rham cohomology
cyclic homology
Chern character
Chern-Connes character
Chern-Weil theory
noncommutative Chern-Weil theory
elliptic operator
spin Riemannian manifold
spectral triple
index theorem
local index formula
group, Lie algebra
Hopf algebra, quantum group
action of Hopf algebra

In the non-commutative domain (quantum mechanics) one encounters a universal definition of a time flow which has no counterpart in commutative mathematics (or classical physics). In this sense time is a mathematical necessity of quantum mechanics arising out of operator non-commutativity.

But the implication works in the other way too. Starting with the necessity of time, we can consider infinitesimal time evolution and extract the Leibniz identity. And in the categorical approach or quantum mechanics reconstruction the Leibniz identity is the main starting point.

Next time I'll expand on this and the implication for the measurement problem.

Friday, July 24, 2015

The pictorial formalism of quantum mechanics

Today I want to talk about a nice formalism of quantum mechanics developed by Samson Abramsky and Bob Coecke. The underlying mathematical formalism is that of category theory

The pictorial formalism describes systems and processes. The best analogy (with very good reason) is with computer science. There FORTRAN was one of the earlier languages of functional programming. In Fortran one writes functions which take an input, perform some transformation (does a computation) and generate an output. One can formally represent such a program with a box connected by two wires: the input and the output. From a high level perspective it does not matter what goes on inside the box. More important than the inner workings is that those functions can be executed one after another and can be combined like Lego to generate complexity.

Similarly in the pictorial formalism one encounters linear transformations of the wavefunction and those transformations can be combined for complexity. What makes it all interesting is that one can operate simultaneously on several inputs (like on two particles in a singlet state), or on parts of composite quantum systems.

The pictorial rules are extracted from the usual Hilbert space formulation to guarantee agreement with quantum mechanics standard computations. The Choy-Jamilkowsky isomorphism is baked in from the beginning in the approach. Here are some primitive concepts. A state is a process with no input and one output, and a test (measurement) is a process with an input and no output:

The combination of a state followed by a test gives you the probability. 

Other primitive notions are tensor product \(\otimes\) and composition \(\circ\):

\(f \otimes g = f~~while~~g\)
\( f \circ g = f~~after~~g\)

A fundamental relationship is this:

\((g_1 \otimes g_2) \circ (f_1 \otimes f_2) = (g_1 \circ f_1)\otimes (g_2 \circ f_2)\)

which can be proven pictorially by inspecting this diagram:

(g1 while g2) after (f1 while f2) = (g1 after f1) while (g2 after f2)

Quantum information is trivially represented in this approach. For example here is how teleportation protocol is drawn:

Alice shares with Bob a Bell pair (the bottom triangle which represent a state) and Alice performs a bipartite measurement on the qubit to teleport (the leftmost line) and one of the Bell particle (the upper triangle). Then she transfers a classical bit to Bob who can use it on his half Bell pair particle to recover Alice's original qbit. The information flow can be continuously traced as in the line on the right. Sometimes the information seem to flow from the future to the past, but the line can be deformed by pulling the ends to straighten it and restore the causal order. 

My poor drawings in Paint do not do justice to this very powerful method to represent the information flow. In explaining this approach I have a major weaknesses: I do not know the latest standardization of symbols. However I understand that a book on the pictorial formalism will soon become available and this will clarify the notation.

In the meantime I want to encourage the reader to look up for themselves this amazing approach. Here is an old but good reference:

In the pictorial approach complex quantum computations becomes child's play. One can even compare this method with Feynman's diagrams in terms of simplification of computation. 

I said in prior posts that I was not aware a few years ago that my research area of reconstruction of quantum mechanics from physical principles is categorical in nature. But I can now take it a step further and create a pictorial proof of my results (it looks like a tangled mess so I won't draw it here). For this I need to introduce the concept of products and coproducts. In general any product (e.g. complex number multiplication, group operation, etc.) can be understood as a machine which takes two elements and generate another element. Flip the machine around and you get the related concept of a coproduct:

Because quantum mechanics is universal, it applies just as well to single physical systems and to composite systems which are represented using the tensor product ["while", \(\otimes\)]. It is the interplay between products, coproducts and the tensor product which completely determines the algebraic structure of quantum mechanics. All I need is a basic starting point: a product which appears naturally. And this is the Leibniz identity which comes out of the fact that the laws of nature are stable and unaffected by the passage of time. In the infinitesimal case this generates the Leibniz identity which is nothing but the good old fashion product rule of differentiation. Two mathematical representations of this product are the commutator and the Poisson bracket and they correspond to the quantum mechanics Hilbert space and phase space formulations. But I will talk more about this in future posts. 


I was asked on Twitter the following excellent question:
"Has use of this pictorial/categorical formalism led to any new results? (as Feynman diagrams certainly did)"

I do not want to reveal without permission the identity of the person who asked this question, but I want to give here an extended reply which was not possible under Twitter's insane character limit.

My answer is that the pictorial formalism did not led to new results (as far as I know) as this formalism is strictly a reformulation of the Hilbert space formalism and what you can do in one you can do in the other one. The categorical approach proved its usefulness in quantum mechanics reconstruction in explaining why the results were the way they were, but this was hindsight, an "aha moment". The original motivation came from a very different direction: the attempt to find a common axiomatization for classical and quantum mechanics. 

On Feynman diagrams I am not sure what were the new results which came from it. And here is why I say this. When I was in grad school studying QFT from Mandl and Shaw, the professor did not follow the book for the first half of the semester but instead he forced us to use non-relativistic pre-Feynman diagrams, just to appreciate what Mr. Feynman actually achieved. The non-relativistic diagrams were painful to compute, and you have like 16 non-relativistic diagrams for one relativistic Feynman diagram, but you can actually do exactly what Feynman diagrams could, just with a lot more work.

Thursday, July 16, 2015

Joy Christian's program of achieving quantum correlations with Clifford algebras

In the last post I explained how the algebra of the projector operators cannot always be Boolean, otherwise the Hilbert space formalism of quantum mechanics is invalid. Today I will stay in the classical-quantum divide area and I'll talk about an invalid proposal by Joy Christian which generated a lot of debate (and acrimony). When I attended the Vaxjo conference people looking up my archive record saw that I have argued against this proposal and I was asked to explain why it is invalid.

The story begins with the EPR-B experiment and the derivation of the correlation

\(-a \cdot b\)

between Alice and Bob when Alice orients her detection device on direction a and Bob orients his on direction b. So the corelation curve is minus the cosine of the detection angles (the blue line below):

In this experiment the two spin 1/2 particles are in a singlet state:

\( |\Psi\rangle = \frac{1}{\sqrt{2}}( |up \rangle_{left} |down \rangle_{right} - |down \rangle_{left} |up \rangle_{right} )\)

and because the observables are \(a \cdot \sigma\) and \(b \cdot \sigma\) the correlation is:

\( \langle \Psi | (a \cdot \sigma)\otimes(b \cdot \sigma) | \Psi \rangle\)

So how can we compute this? We use an identity:

\( (a \cdot \sigma)(b \cdot \sigma) = -a\cdot b +i (a\times b) \sigma\)

which yields the final answer because \( \langle \Psi | \sigma| \Psi \rangle = 0\) as the mean value for both Alice and Bob are zero for any direction because we started with a total spin zero state (a singlet state).

Now Joy noticed this identity and thought that it would be nice if he could use it in a classical setting to recover the \(-a\cdot b \) correlation. There is a "little" problem: how to make the pesky \(i (a\times b) \sigma\) term disappear? 

So Joy came up with the following proposal: half the particle pairs obey:

\( (a \cdot \sigma)(b \cdot \sigma) = -a\cdot b +i (a\times b) \sigma\)

and the other half obey this:

\( (a \cdot \sigma)(b \cdot \sigma) = -a\cdot b -i (a\times b) \sigma\)

and so when averaged you get to the quantum correlation: \(-a \cdot b\)

But how can this be possible? It is all in the sign of \(\sqrt{-1}\) Joy claimed. When complex numbers are represented in a plane, the imaginary unit corresponds to the vertical axis. So for half of the particle pairs we draw the imaginary axis bottom up, and for the other half up bottom. But do we really get the cancellation? Nope because \(a\times b\) is a pseudo-vector which upon this reflection against the horizontal axis changes signs as well and the identity remains:

\( (a \cdot \sigma)(b \cdot \sigma) = -a\cdot b +i (a\times b) \sigma\)

and all of Joy's ill fated proposal is based on a "forgotten" -1 sign.

But if Joy would have presented his proposal like this, it would not have gotten very far. Instead Joy explained it all using the language of Clifford algebra which is not at all familiar to physicists. Also there was an associative faulty narrative about "topologically complete reasoning".

The main discovery of Joy was a no man's land at the intersection of math, physics, and philosophy: the mathematicians understanding Clifford algebra knew nothing of Bell, the physicists did not know how to counter Joy's philosophical narrative, and the philosophers had no clue of Clifford algebra. Add to this Joy's aggressive and patronizing defense of his proposal and you get a perfect storm of controversy.

The full story of debunking this nonsense would make for a nice soap opera. I was not the first who noticed the mathematical errors in Joy's proposal, I was the third out of four. Also I was not the first who wrote a paper about it, I was the second one out of three, but I was the first who uploaded it on the archive. There were other archive replies to Joy before me but nobody actually bothered to double check his math. The first reply by Marcin Pawlowski came very close to point out the problem but Joy's reply managed to discourage his critics into challenging his math:

"More specifically, the critics culminate their charge by declaring that, within my local realistic framework, it would be impossible to derive “a scalar in the RHS of the CHSH inequality. QED.” If this were true, then it would certainly be a genuine worry. With hindsight, however, it would have been perhaps better had I not left out as an exercise an explicit derivation of the CHSH inequality in Ref.[1]. Let me, therefore, try to rectify this pedagogical deficiency here."

And so people thought at that time that Joy is wrong, his physics and philosophical arguments were nonsense, but his math was correct and it was not a good idea to challenge him at that. But as it turned out all his math was only smoke and mirrors with more and more mathematical mistakes to cover up the prior ones, and I can write up an entire book about it.

All this controversy has hopefully came to an end with Joy resigning his FQXi membership, but he actually never accepted he was wrong and continues to this day to call his critics arguments: "strawmen arguments".

There was only one person who had more energy and spent more time than me debunking Joy's claims and this is Richard Gill

and the physics community owes him a debt of gratitude for putting this nonsense to rest. There were also two good things coming out of this challenge to Bell's theorem.

First, Sascha Vongehr came with what he called a Quantum Randi Challenge: show you beat Bell's theorem on a computer or shut up. With programming help from Cristi Stoica I came up with this simple Java Script program which runs in any web browser which anyone can use to try to disprove Bell's theorem until they really understand why it is an impossible task.

Second, James Owen Weatherall actually manage to fulfill Joy's hope to eliminate the extra term in a mathematically valid model which was not using Clifford algebras. But then would this count as a "disproof" of Bell's theorem? NO because the actual experimental outcomes are +1 and -1 and the correlations must be computed using them and not in a space of make-believe statistics. 

So even if Joy's math were valid, it would not represent in any way a "disproof" of Bell's theorem. 

Now what I found completely amazing was that after resigning his FQXi membership Joy received encouragements to continue the fight. It is not clear what fight. The fight to prove +1 = -1? Didn't I say soap opera?

Post Script: In case anyone has questions regarding any past or present mathematical, physical, or philosophical claims by Joy, feel free to ask here and I will answer. 
Post Post Script: in an online reply at Sci.Physics.Foundations, Joy claims his model it is not about the sign of sqrt(-1). Oh yes it is, by Hodge duality. And Joy knows that because he used to call the two Hodge duality for distinct Clifford algebras "Joy duality" which earned him high marks in his crackpot index for naming equations after yourself. The moderator of that blog, FreddyFizzx is in cahoots with Joy to promote Joy's ideas and suppress sane opposite points of view.