Friday, August 29, 2014

What if electromagnetism were a SU(2) Yang-Mills gauge theory?


Let us ignore for a moment the Standard Model, the weak force, and the spontaneous symmetry breaking and try to imagine how would the world look like if electromagnetism would not be present and would have been replaced by a SU(2) Yang-Mills theory.

To get a good grip on this, let's do a very quick review of Lie groups and algebras. A Lie group is a continuous group, and is so named after Sophus Lie, a Norwegian mathematician (no, he was not Chinese).  Like any group, a Lie group has a unit element, and because of the continuity, we can define a tangent space for this element. This tangent space is a Lie algebra. A Lie group can be recovered from the Lie algebra by exponentiation, and the elements of the Lie algebras are called generators.

For SU(2), there are three generators:

\( F_i = \frac{1}{2} \sigma_i\)

where \( \sigma_i\) are the Pauli matrices:

\( F_1 = \frac{1}{2}\left( \begin{array}{cc} 0 &1 \\ 1 &0 \end{array}\right) \), \( F_2 = \frac{1}{2}\left( \begin{array}{cc} 0 &-i \\ i &0 \end{array}\right) \), \( F_3 = \frac{1}{2}\left( \begin{array}{cc} 1 &0 \\ 0 & -1 \end{array}\right) \)

[side note - I think I was a bit overly ambitious to present the derivations of the equations in this post and I will instead restrict to just stating the results]

The field tensor \( F_{\mu \nu}\) has the usual definition in terms of the "electric" and "magnetic" fields \( E \) and \( B \):

\( F_{\mu \nu} = \left( \begin{array}{cccc} 0 & E_1 & E_2 & E_3 \\ -E_1 & 0 & -B_3 & B_2 \\ -E_2 & B_3 & 0 & -B_1 \\ -E_3 & -B_2 & B_1 & 0 \end{array} \right) \)

where:

\( F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu -i q (A_\mu A_\nu - A_\nu A_\mu ) \)

Now the essential thing is that instead of \( A_\mu = (\phi, A_x , A_y , A_z ) \) meaning the electric potential and the magnetic vector field, the four components are no longer scalars, but linear combinations of the generators. And the generators can be naively interpreted as rotations in a 3D space. Naively because for each SO(3) rotation there are two SU(2) elements, what mathematicians call a "double cover". A better physical interpretation comes from quantum mechanics where two linear combination of \(F_1 \) and \(F_2 \) are interpreted as "raising" \( I_+ \) and "lowering" \( I_- \) operators. To get to physics, in the original isospin Yang-Mills paper where SU(2) was applied to protons and neutrons, \( I_+ \) corresponded to a transformation of a neutron into a proton, while \( I_- \) corresponded to the reversed operation. In other words, the quanta of SU(2) interaction carries (an isospin) charge. For ordinary electromagnetism, the photon is not electrically charged, but non-abelian gauge interactions are no longer charge neutral.

Let us work out how \( E_x\) would look like :

\( E_x = E_1 = F_{01} = \partial_0 A_1 - \partial_1 A_0 -i q (A_0 A_1 - A_1 A_0 )  = \frac{\partial A_x}{\partial t} - \frac{\partial \phi}{\partial x}\)

the same way as in standard electromagnetism.

Let us also work out how \( B_x\) would look like :

\( B_x = B_1 = F_{43} = \partial_4 A_3 - \partial_3 A_4 -i q ( A_4 A_3 - A_3 A_4 ) \)

\( B_x =  \frac{\partial A_y}{\partial z} - \frac{\partial A_z}{\partial y} - q A_x \)

Here we pick up an additional term due to the non-commutativity. On top of all this, at each space-time point \( A's \) are no longer scalars, but they are vectors in an internal space which can carry SU(2) charges.

Beside the internal space motion, for space-time motion, the Lorentz force law is the same as in the electromagnetic case, but the inhomogeneous Maxwell's equation:

\( \partial^\mu F_{\mu \nu} = j_\nu \)

generalizes to:

\( \partial^\mu F_{\mu \nu} - iq [A^\mu , F_{\mu \nu}]= j_\nu \)

The (Dirac) current itself generalizes from:

\( j_\nu = q \psi^\dagger \gamma_\nu \psi \)

to 3 currents corresponding to the 3 generators \( F^k \) of SU(2):

\( {(j_\nu )}^k = q \psi^\dagger \gamma_\nu F^k \psi \)

So overall, the Yang-Mills theory it is quite more complicated due to the non-commutativity of the gauge group. But one thing should be clear: the magnetic field is just a very naive simplistic picture of what is going on and this mental picture only works for electromagnetism because U(1) is a commutative Lie group. The real physical objects are the "vector potentials" \( A_\mu \). Then the Bohm-Aharonov effect where measurable changes are produced by changes in vector potential while the the net magnetic field is zero is no longer counter-intuitive. The real explanation of this effect is geometrical. 

Yang-Mills is quite an interesting model and its original intention proved to be not in agreement with reality, but physicists kept studying it and it turned out that the SU(2) gauge theory does describe a physical interaction, that of the weak force responsible for particle decays but the story is a bit more complicated due to symmetry breaking. The generalization to SU(3) is straightforward, one simply change the group generators, but then brand new physics arises in the form of asymptotic freedom which explains why we do not see free quarks in nature.

Yang-Mills theory was only accepted by everyone after the proof of renormalizability was obtained in the 70s showing that a quantum field theory based on Yang-Mills does produce sensible finite predictions and all the infinities can be cured in a mathematical consistent way.

Friday, August 22, 2014

Yang's Matrix Trick


It is time to come back to the math series. Today I want to talk about a remarkable similarity spotted by Chen-Ning Yang, the same Yang from Yang-Mills' theory.



Yang-Mills gauge theory is a generalization of electromagnetism when the gauge group is non-abelian. 

Maxwell's equations can be written as:

\( F_{\alpha \beta}= \partial_\alpha A_\beta - \partial_\beta A_\alpha \)

where \( F\) is the electromagnetic tensor and \( A \) is the electromagnetic four-potential.

From Maxwell to Yang-Mills, the generalization is simply by adding the commutator of the potentials:

\( F_{\alpha \beta}= \partial_\alpha A_\beta - \partial_\beta A_\alpha + A_\alpha A_\beta - A_\beta A_\alpha \)

Now here is the magic: if you recall from a prior post the Riemann curvature tensor is:

\( R^{\delta}_{\alpha \beta \gamma} = \partial_\alpha \Gamma^{\delta}_{\beta \gamma} - \partial_\beta \Gamma^{\delta}_{\alpha \gamma} + \Gamma^{\delta}_{\alpha \mu} \Gamma^{\mu}_{\beta \gamma} - \Gamma^{\delta}_{\beta \mu} \Gamma^{\mu}_{\alpha \gamma} \)

we have the following identification which makes the Yang-Mills equation identical with the Riemann curvature:

\( A_\alpha = \Gamma^{\delta}_{\alpha \gamma}\)
\( F_{\alpha \beta} = R^{\delta}_{\alpha \beta \gamma } \)

This hints at a deeply geometrical interpretation of the gauge theory because both the Riemann curvature and Yang-Mills equations are nothing but Cartan's structural equations in disguise: 

\( F = d A + A \wedge A\)

There are 4 fundamental forces in our universe: gravity (SL(2,C)), electromagnetism (U(1) gauge theory), weak force (SU(2) gauge theory), and strong force (SU(3) gauge theory) and all four can be expressed in the form above proving that in nature curvature = force. This is easiest to understand in general relativity, but even there there is a very surprising fact requiring a big conceptual leap: even empty space can curve.

Next time we'll slowly start exploring gauge theory in depth starting with Maxwell's equations. Then all those abstract equations will become much more intuitive.

Wednesday, August 13, 2014

Quantum vs. Classical Mechanics

The search for a distinguishing principle


This is the last post discussing http://arxiv.org/abs/1407.7610 before I'll resume my prior math series.

After boiling down the essentials of quantum and classical mechanics and extracting the common algebraic structure, the question becomes "what is quantum"?


The standard answer from Dirac is that in quantum mechanics we add amplitudes, not probabilities. Even earlier, Schrodinger identified superposition. More modern takes on this starting from Hardy is that pure states are linked by continuous transformations. 

A pure state is a state which cannot be decomposed into a sum of other states. Because state spaces are convex spaces, this means that pure states reside on the boundary of the state space. In classical physics pure states form a discrete set while in the quantum world pure states form a continuous surface. What does this mean? It means that a measurement in classical physics reveals an intrinsic property of the system, but in quantum mechanics even pure states can collapse from one into another. 

But is this intuitive? Can we really claim that we understand the distinction between the classical and the quantum world? No, No, No.

Because quantum and classical physics are completely separated domains, first one cannot explain one in terms of the other, and second, there is no outside bird's eye view to introduce the concepts needed to explain them. 

For the first part, imagine a world of triangles trying to grasp the concept of a circle. This is basically what various quantum interpretations actually attempt to do: explain the weirdness of the quantum world in terms of classical concepts: a futile approach. Each interpretation has intuitive parts, but also craziness baked in. 

For the second part, to intuitively grasp the distinction between quantum and classical physics, you need to extract yourself from this quantum universe and explain both quantum and classical physics in terms of the laws of a meta-universe where both quantum and classical mechanics are valid. No such thing exists.

A comparison with special theory of relativity is helpful here. To really understand Lorenz transformations, one first needs to free himself/herself from the concept of aether. One does not attempt to understand the constant speed of light using notions of unbounded speeds in a Galilean framework. Just consider how silly a theory of relativity "interpretation" would be along those lines:

Light appears to have a constant propagation speed because there is a "Lorenzian potential" which acts contextually in a particular reference frame measurement. However, in reality light does not have a constant propagation speed.

Now if this is silly, why is Bohmian interpretation not silly? 

To really understand quantum mechanics weirdness we need to let go of our classical prejudices. Relativity gave up the concept of aether. Nature is quantum mechanical, no ifs, ends, and buts. Isn't time for quantum mechanics to give up attempts to search for a natural distinguishing principle? It is a futile attempt.

Sunday, August 10, 2014

It is what can generate a bit (part 1)

and

It takes two to tango (part 2)


Sorry for the delay, I was on vacation and although I brought my laptop with me, I forgot the charger and I could not use it. I got internet access through my cell phone, but typing on a tiny screen is not suitable for generating a blog post.

So with a bit of delay, today I present part two: it takes two to tango.


Let me start by listing the axioms used in other approaches to derive quantum mechanics:

  1. individual systems are Jordan algebras, 
  2. composites are locally tomographic,
  3. at least one system has the structure of a qubit

Dakic and Brukner: "Quantum Theory and Beyond: Is Entanglement Special?":
  1. (Information capacity) an elementary system has the information carrying capacity of at most one bit. All systems of the same information carrying capacity are equivalent; 
  2. (Locality) the state of a composite system is completely determined by local measurements on its subsystems and their correlations
  3. (Reversibility) between any two pure states there exists a reversible transformation; 
  4. (Continuity) between any two pure states there exists a continuous reversible transformation.
Lluis Masanes and Markus Muller: "A derivation of quantum theory from physical requirements": 
  1. in systems that carry one bit of information, each state is characterized by a finite set of outcome probabilities; 
  2. the state of a composite system is characterized by the statistics of measurements on the individual components;
  3. all systems that effectively carry the same amount of information have equivalent state spaces; 
  4. any pure state of a system can be reversibly transformed into any other; 
  5. in systems that carry one bit of information, all mathematically well-defined measurements are allowed by the theory.

Chiribella, D’Ariano, and Perinotti: "Informational derivation of Quantum Theory":
  1. Causality: the probability of a measurement outcome at a certain time does not depend on the choice of measurements that will be performed later. 
  2. Perfect distinguishability: if a state is not completely mixed (i.e. if it cannot be obtained as a mixture from any other state), then there exists at least one state that can be perfectly distinguished from it, 
  3. Ideal compression: every source of information can be encoded in a suitable physical system in a lossless and maximally efficient fashion. Here lossless means that the information can be decoded without errors and maximally efficient means that every state of the encoding system represents a state in the information source, 
  4. Local distinguishability: if two states of a composite system are different, then we can distinguish between them from the statistics of local measurements on the component systems, 
  5. Pure conditioning: if a pure state of system AB undergoes an atomic measurement on system A, then each outcome of the measurement induces a pure state on system B. (Here atomic measurement means a measurement that cannot be obtained as a coarse-graining of another measurement).
  1. Probabilities, 
  2. Simplicity (K is determined by a function of N and for each given N, K takes the minimum value consistent with the axioms), 
  3. Subspaces, 
  4. Composite systems rules ( \( N_{A⊗B} = N_A N_B \) and \( K_{A⊗B} = K_A K_B \) ), 
  5. Continuity (there exists a continuous reversible transformation on a system between any two pure states of that system)
Sure, there are other axiomatization approaches which do not use composition, but why is system composition appearing so often? The answer is in quantum correlations which by Bell's theorem cannot be causally explained. 

More important, for the spin 1/2 case, Bell produced an exact hidden variable model which obtains all quantum mechanics predictions for one particleThis means that considerations of only one particles (systems) are not enough to distinguish between classical and quantum mechanics! Hence the need to consider COMPOSITE systems. 

Now what http://arxiv.org/abs/1407.7610 shows is that composition considerations are extremely powerful because they constrain the algebraic properties. The best analogy is that of a fractal:


with the key difference that invariance under tensor composition implies that the self-similarity pattern is IN PLACE. This is how system composition constraints the dynamic. 

Composition demands either:
  • Quantum mechanics (elliptic composability)
  • Classical mechanics (parabolic composability)
  • Split-complex quantum mechanics (hyperbolic composability)
It is what can generate a bit kills the hyperbolic case.

Then how can we separate the quantum from the classical case? Some quantum mechanics reconstruction proposals (starting with Hardy's) talk about "a continuous reversible transformation" between any two pure states. Next time I'll address this issue and its relationship with quantum mechanics interpretations. 

Saturday, August 2, 2014

It is what can generate a bit (part 1)

and

It takes two to tango (part 2)


So now the cat is out of the bag. The paper deriving quantum mechanics from physical principles is now public and on the archive http://arxiv.org/abs/1407.7610 and I'll wait for about a month to collect feedback before submitting it for publication. Please don't be afraid to ask any questions, no matter how silly they may seem. Also, if you like the paper and/or this blog please vote for the paper on Scirate: https://scirate.com/?range=3 In the meantime I am going to explain what is going on under the mathematical cover, and what is the physical intuition.

The paper is based on two pillars: information and composite systems. Today I'll cover the information side. John Wheeler asked: "to describe how information is fundamental to the physics", how it comes from bit. In his book: "Our Mathematical Universe" Max Tegmark proposed this idea that reality is nothing but mathematics. While this got a less than cordial reception in the academic and philosophical circles, something along the lines of "not even wrong but we want to be polite because we like his grant money from FQXi", I think on one hand the cold welcome is not deserved but the on the other hand the idea is not fully baked. Let me explain.

The bad reception was due to the distinction between object and language used to describe the object. This is certainly a very serious objection, but consider this: if the laws of nature are relational, objective existence independent of everything is certainly an illusion. Sure, there may be different (mathematical) languages expressing the same thing, but ultimately reality is made only out of relative relationships. To the extent that mathematics is about relationships, and reality is about relationships, reality is mathematical, and I am a Platonist just like Tegmark. The "unreasonable" effectiveness of math explained Wigner. 



On the other hand Tegmark's idea is not fully baked because it is only an entertaining interpretation devoid of consequences, a parlor trick good to sell a book. "Show me the money", show what is the consequence of this idea! There are none because we do not need to know how we are like mathematical theorems, but the other way around: what is the distinction between the abstract world of math and concrete reality? Sticks and stones may break my bones, but when is the last time you saw on TV that "Person X was injured by Pythagoras' theorem"?

So here is the first principle I want to discuss: ``it is what can generate a bit''. In other words, positivity. There are three composability solutions in the QM reconstruction paper: elliptic, parabolic, and hyperbolic. We know that elliptic composition corresponds to quantum mechanics, and parabolic composition leads to classical mechanics, but what is this hyperbolic composition. What if this is something physical as well? But before I packed my bags to go collect my Nobel prize on predicting a brand new physics more Earth shattering than the Higgs' boson, I wanted to understand why this may be unphysical (quantum and classical mechanics are ubiquitous and we can expect the same thing on the hyperbolic case as well if it were physical). 

The answer was that in the hyperbolic case one cannot define an "objective reality" for which one can make predictions which can be tested against experiments. Why? Because in this case you cannot eliminate negative probabilities. Objective physical reality demands to be able to generate information. If you cannot do that, you are not an "it", but just another abstract mathematical relationship of no ontological value.

Ontology = ability to generate information

There are many papers attempting to distinguish classical from quantum information. Similarly there are many attempts to derive Born's rule. This is all misguided because of counterexamples. Born's rule and quantum information is not as universal as people think. They only form a particular very important flavor in the quantum garden. 

Figuring out the complete classification of quantum information is a big open problem and a prerequisite first step. Finding a natural physical principle distinguishing quantum from classical mechanics is an impossible task just like finding an explanation for why there is a maximum speed limit in the universe. A maximum speed limit cannot have a dynamic explanation, because it has a kinematic origin. The elliptic composability class cannot have a parabolic explanation either. Sure, there are equivalent descriptions, like: existence of quantum superposition, existence of continuous transformations between pure states, but are those formulations really intuitive? Different quantum mechanics interpretations can provide intuitive explanations for this only at the expanse of sweeping the dirt under the rug for other things. Different quantum interpretations means different dirt and different rugs. 

In special theory of relativity one first needs to give up the attempt to understand the maximum speed and give up concepts of aether. In quantum mechanics one first needs to give up the attempt to explain quantum superposition. Quantum superposition, quantum correlations, elliptic composability class are primitive concepts. Causal explanations of quantum correlations, are not only silly, but impossible due to Bell's theorem.

Tuesday, July 29, 2014

Quantum Mechanics Reconstruction 


I want to announce the paper: http://arxiv.org/abs/1407.7610:  

Quantum Mechanics reconstruction from invariance of the laws of nature under tensor composition


Quantum and classical mechanics are derived from 4 physical principles:
  • the laws of nature are invariant under time evolution, 
  • the laws of nature are invariant under tensor composition, 
  • the laws of nature are relational, 
  • positivity (the ability to define a physical state). 

Quantum mechanics is singled out by a fifth experimentally justified postulate: nature violates Bell's inequalities.

I will put the Standard Model math explanation series on hold for a bit and in subsequent posts I'll explain this result. ALL of quantum mechanics formalism follows from those 4+1 physical principles in a rigorous, constructive, step by step argument. Both the Hilbert space and the state space realizations are derived. 

The axioms are minimal: 

- Composition (tensorial, categorical) arguments are needed because there are classical physics models for quantum mechanics for a single particle. Correlations between systems are the essential quantum characteristic.
- Information theoretical arguments (positivity) are used because composition arguments produce a third unphysical solution. However there is no "it from bit", but: "it is what can generate a bit".


Friday, July 25, 2014

Cartan structural equations


Elie Cartan was a French mathematician who made fundamental contributions in differential geometry and group theory.


Today I want to continue the discussion regarding curvature and talk about Cartan's modern approach to differential geometry. Cartan died in 1951, three years before Yang and Mills wrote their seminal paper on gauge theory and he did not see how his ideas are used by Nature in all the four fundamental interactions: gravity, electromagnetism, weak force, and strong force.

Last time I presented the Christoffel symbols and I noted that they depend on the coordinate system used. Riemann uncovered an intrinsic geometric object called Riemann curvature tensor which expressed in terms of the Cristoffel symbols is:

\( R^{\delta}_{\alpha \beta \gamma} = \partial_\alpha \Gamma^{\delta}_{\beta \gamma} - \partial_\beta \Gamma^{\delta}_{\alpha \gamma} + \Gamma^{\delta}_{\alpha \mu} \Gamma^{\mu}_{\beta \gamma} - \Gamma^{\delta}_{\beta \mu} \Gamma^{\mu}_{\alpha \gamma} \)

Now recall the exterior product: \( a \wedge b = a\otimes b -b \otimes a\) which is skew-symmetric: \( a\wedge b = - b \wedge a\) 


\( \omega^\delta_\gamma = \Gamma^{\delta}_{\beta \gamma} dx^\beta \)

and compute \( d \omega^\delta_\gamma \) and \( \omega^\delta_\mu \wedge \omega^\mu_\gamma \) and compare the sum with the curvature equation in terms of \( \Gamma \) we get Cartan's local structural equation:

\( \Omega^\delta_\gamma = d \omega^\delta_\gamma + \omega^\delta_\mu \wedge \omega^\mu_\gamma \)

where \( \Omega^\delta_\gamma = \frac{1}{2} R^{\delta}_{\alpha \beta \gamma} d x^\alpha \wedge d x^\beta \)

In matrix form: \( A = \omega^\delta_\gamma \) is the local connection, and \( F = \Omega^\delta_\gamma \) is the local curvature.  

Cartan's local structural equation is:

\( F = d A + A \wedge A\)

which differentiated yields Bianchi's local identity:

\( d F = F \wedge A - A \wedge F \)

Now those two equations are a thing of maximal physical beauty. Sure, they look alien and vaguely interesting in this abstract mathematical form which Cartan derived, but inside them hides Maxwell's equations, Yang-Mills gauge theory, and Einstein's general relativity. In the next posts it will be our job to extract the physics from them.  

For example if \( A \) is the electromagnetic 4-potential because the electromagnetic gauge theory is abelian, \( A \wedge A = 0 \) and the two equations from above are nothing but Maxwell's equations as we will show later. Please stay tuned.