Geometric Optics with Lie Groups

Donald Barnharth, from Optical Software introduced me with
another very interesting application of Lie groups in geometric optics.

This should not be a surprise because geometric optics as well as classical mechanics can be expressed in terms of a variational principle. In classical mechanics we have the minimal action and in geometric optics we have Fermat's principle, which states that the trajectory of light rays minimize the optical path length defined as

where n is the refraction index, ds is the differential arc length and C is the path. The whole machinery of classical mechanics can be translated to geometric optics with the corresponding Hamiltonian formulation as the phase space formulation of geometric optics.

The key idea of Hamiltonian mechanics is that the trajectory of the particles can be seen as active continuous symplectic transformations. For example, one has the following time-evolution operator



where H is the Hamiltonian (independent of time), with the Lie operator defined in terms of the Poisson brackets as



and


The same formalism can be applied to geometric optics with some particular adjustments. The time-evolution operator is replaced by the transformation that propagates the optical phase-state in the space. For practical applications this transformation is factorized in a perturbative-like product expansion that reminds the Fer expansion as



where, the first factor represents the para-axial approximation while the rest represent the corresponding higher order corrections.

References

1. V. Lakshminarayanan, Ajoy Ghatak, and K. Thyagarajan, Lagrangian Optics
   
2. Alex J. Dragt, A Lie connection between Hamiltonian and
   Lagrangian optics
   
3. Kurt B. Wolf, Geometric Optics on Phase Space, Springer, 2004
   
   

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Relativistic Many-body dynamics

An intuitive generalization of the explicitly covariant single-body relativistic dynamical equation to multiple multiple interacting particles is difficult. The no-go theorem of Currie Jordan and Sudarshan is an example of the difficulties to device such explicitly covariant version of relativistic interacting particles. I do not know if a satisfactory elegant solution was found.

A book that explains this problem is

FROM CLASSICAL TO
QUANTUM MECHANICS
Giampiero Esposito, Giuseppe Marmo

Also see
Form of relativistic dynamics with world lines

The Lorentz-Dirac Force

There is a naive general idea that we already understand classical mechanics very well, but I do not think that is the case. There are many fundamental questions in classical mechanics without a satisfactory answer. One of them is the description of the trajectory of a charged particle in the presence of an electromagnetic field. Yes, we have the Lorentz force, but it does not take into account the effect of the radiation of the charged particle in the acceleration process. The Lorentz-Dirac force is an attempt to include the effect of the radiation but unfortunately, the solutions of this equation are pathological. Most books only mention this situation but Baylis's book in electrodynamics devotes a complete chapter this subject.
Electrodynamics: a modern geometric approach
By William Eric Baylis (chapter 12)

Returning from IMUC

I am finally writing again after my return from the magnificent International Mathematica User Conference 2009, where I learned about the features of the future Mathematica 8 and met many people. I am going to devote a series of entries about IMUC 2009, but some general observations and things that I learned are

• Mathematica is growing and improving at an increasingly faster rate.
• The new Mathematica notebook will have nearly the same capacity found in LaTeX for creating static documents. However, is the dynamic capability what makes it revolutionary and much better than LaTeX.
• The Mathematica kernel is eventually going to support efficient tensor computation. These routines are going to be implemented in separated modules, so that Tensorial (the tensor package I develop with David Park and Jean-Francois Gouyet) will benefit from these new features. Moreover, now I have more ideas on how to independently improve Tensorial.
• The rendering of 3D images with CUDA is truly amazing and I am waiting to play with it when it gets implemented in the kernel. Now I also know that my next laptop has to have an NVidia graphics card.
• I am also waiting for the Mathematica plugin for web browsers, so that I can directly publish mathematica notebooks on the web without the need to export any html.
  

Hyperdeterminants

The concept of hyperdeterminant is as new to me as a few days and is introducing me into a fascinating new branch of algebra that I did not even suspect. I am very familiar with tensors and how they can be seen as multidimensional arrays that generalize matrices, but the concept of hypermatrices goes further.

The hyperdeterminant was invented (discovered) by the famous mathematician Cayle, who gave us many things including the Cayle transform, useful to approximate the exponential of anti-Hermitian matrices, and the amazing Cayle-Hamilton theorem of linear algebra.

There is active research today and I even found a blog
hyperdeterminant.wordpress.com

More recently I found that Thomas Wolf and Sergey Tsarev are implemening serious computer work in order to find hyperdeterminants of higher order
Hyperdeterminants as integrable discrete systems.

What I had in mind was the hypertederminant of at least 5x5x5 cubic matrices, but now I known this is an exceedingly difficult problem. Now my question is if there is one way to find a good approximation.

MMM,.... I just, learned about other alternative definitions of hyperdeterminants, so the possibilities are still open.

Inverted Retina

If you did not know, we have our retinas inverted in the sense that the light that enters our eyes has to pass through many layers of nerves, blood vessels and all the wiring before reaching the photo detectors themselves. Even more troublesome, is the fact that it seems that the images are completely distorted and blurred when they arrive the photo-detectors. However we surely know we can see very well. Why did we evolve this feature? and how do we really see? Some of the answers can be found in this absolutely amazing paper

V. D. Svet and A. M. Khazen, About the formation of an image in the inverted retina of the eye, Biophysics, Volume 54, Number 2 / April, 2009, pp 193-203

The key point of this article is that we process the images in blocks. This means that we collect a sequence of images, which we process together in order to improve the signal/noise ratio.

There is a recent independent article here, suggesting that we actually see images in discrete sequences, which is completely consistent with Svet's theory.

I met Dr Svet in Windsor, Canada, where he gave a few seminars about topics concerning sonar imaging and detection. I expect to write more entries in my blog about other topics of his research.

What really matters is to challenge our brains

Our brains thrive when we learn new challenging things. It is not enough to practice what we already know well
Effect of challenging our brains in our brains

Rodolfo Sanchez Ph.D.

I was very glad to hear again from my old Bolivian friend Rodolfo Sanchez, who earned his Ph.D. degree in physics in Germany and now is working as a postdoctoral researcher at the GSI institute

Rodolfo Sanchez PhD

This world in small because it happens that he is now collaborating with Dr Gordon Drake, who is professor at Windsor, Canada.

Tree Fractal

I was a little bit surprised to find one of my Mathematica fractals as part of a collection of other fractals, as you can see in the October 17, 2004 entry

http://nylander.wordpress.co/2004/10/

Calculation of the unitary part of the Bures measure for N-level quantum systems

My paper about the Bures measure was published

Calculation of the unitary part of the Bures measure for N-level quantum systems

The Bures measure can be written as the product of two factors. One corresponding to the populations and the other corresponding to the unitary transformation. Many people arrived to formulas for the Bures volume and the volume of the unitary part of the Bures measure, but in this paper we give an explicit expression of the measure itself in terms of even balls.

Much more about measures is going to come the public very soon!

Fisher Information In Natural Selection

I introduced myself with the concept of Fisher information in my readings about quantum estimation and the Cramer-Rao bound. It was when I read the book

B. Roy Frieden, Physics from Fisher Information: A Unification,

which showed me that the Fisher information may even play an important role in the foundations of physics itself. Unexpectedly, I run into the Fisher information again when I was reading about Jeffreys' prior for Bayesian estimation. Now, I came across with another interesting paper about the role of Fisher information in Natural selection

Frank, S. A. 2009. Natural selection maximizes Fisher
information. Journal of Evolutionary Biology 22:231–244

Can we use this in order to refine genetic/evolutionary optimization algorithms?

Quantum estimation

Qantum estimation/metrology is a research field closely tied with quantum information theory. Everything was born with the Heisenberg uncertainty principle, when we realized that in general measurements can be non-commutative in contrast with the classical world where everything is commutative i.e. the order of the measurements is irrelevant. The consequence of the non-commutativity is that two given observables may be incompatible for simultaneous arbitrarily precise measurements.

The next step was done by Helstrom in the 70's who introduced the Symmetric Logarithmic Derivative (SLD) operator to obtain the Quantum Fisher information. It was later in the 80's when Wootters came with the refined and elegant concept of quantum statistical distinguishability, which was tied with the work of Helstrom by Braunstein and Caves (1994) in their famous paper Statistical distance and the geometry of quantum states. This paper is remarkable because it also established the connection with the completely independent work of Uhlmann about the Bures metric, which was born in the generalization of the Berry phase for mixed states.

More recently, we witnessed many important developments in the quantum single parameter estimation with one unexpected twist. Now we know that quantum mechanics and in particular quantum entanglement have the potential to defeat their classical estimation methods in the quest for higher precision. For example, you can see my older post about NOON states and quantum interferometry.

However, this new knowledge cannot be applied in the real world until we learn how to generate a considerable number of entangled states in a reliable way.

You can also see my wikipedia article about the Bures metric

Learning from being wrong

Being wrong seems to be something to avoid, but it may be the price to pay if we want to improve or optimize something. This is the general philosophy behind Genetic/Evolutionary algorithms and even statistical sampling.

In Genetic/Evolutionar algorithms, the key ingredient is the introduction of mutation. In most cases a mutation will be destructive, but there will be occasions when the mutation will be beneficial. We improve by discarding the destructive mutations, while keeping the beneficial ones.

It was Thomas Bäck from Leiden University who told us how Toyota promotes mutation in the production system in order to make more efficient cars.

In statistical sampling, the objective is to explore the space of a probability distribution (explore the opportunities). Here we have the Metropolis (and Metropolis-Hastings) algorithm, where one is forced to be wrong sometimes in order to explore the probability distribution.

Ben Schumacher, the author of the quantum noiseless coding theorem has something to tell about this
Ben Schumacher on "Being wrong"

Disclaimer: The only way to profit from being wrong (mutate) is to be ready to correct ourselves as fast as possible.

A scientific approach to science education

In 2006, when I was in Windsor, I had the opportunity to meet Carl Wieman, the winner of the physics Nobel price in 2001 for his experimental work in the production of the Bose-Einstein condensate. However, something that called my attention even more was his research and discoveries about science education. He showed us with experimental data how our current approach is far for being the most efficient way to teach science. Even worse, sometimes teaching science as we do it today, leads to students with even more misconceptions!

More about his research in education can be found HERE

Quantum/Classical transition

The nature of the quantum/classical transition is according to my opinion, the most important fundamental question of physics. There are many approaches and many insights that have consequences on the continuing debate on the interpretation of quantum mechanics and its mysterious features such as the collapse of the wave function. Some of the approaches and related topics, not necessarily independent from each other are

1. The Ehrenfest theorem (My favorite).
2. The Koopman- von Neumann equation for the classical wave function.
3. The Wigner function for a quantum formulation in the phase space.  Moyal brackets and Weyl quantization [1]
4. Feynman Path integrals and decoherence.
5. The classical spinor formalism. Electrodynamics: a modern geometric approach By William Eric Baylis
   
6. The limit to the Thomas-Fermi model by Elliott H. Lieb
7. The analogy of classical statistical mechanics with quantum mechanics inspired on the Wigner function by  Blokhintsev. Another researcher, A. O. Bolivar pursues this idea further but it seems that he was unaware of the previous work by Blokhintsev. 
8. Related with the previous approach is also the work by Amir Caldeira and Anthony J. Leggett, who proposed a quantum dissipation model.  
9. The very intriguing derivation of the Schrodinger equation from classical mechanics with complexified Brownian motion by Edward Nelson.  
10. The p-mechanics formalism, which exploits the representations of the Heisemberg. This method is also closely related with the Weyl quantization.
11. Geometric quantization, is another method inspired on the Weyl quantization trying to maintain a coordinate-free procedure based on differential geometry.

[1]  Zachos, C. and Fairlie, D. and Curtright, T., Quantum mechanics in phase space: an overview with selected papers, World Scientific Pub Co Inc, 2005

The creation of the universe in 6 days

And God created the universe in 6 days, but in order to give us a nice sky full of beautiful shiny stars, he decided to simulate the light as though as it was coming from thousands, millions and billions of years ago in the past. This was necessary because the speed of light is so fast that in 6 days there would be no chances for us to see anything, not even the light from our closest star, excepting our own sun. We would not be able to see the Milky Way as we see it today, not even in 6 thousand years.

However, God did his job so well that there is no way for anybody to find any hint of crack in the simulation. Unfortunately, this implies that humans will never be able to prove that the universe was created in 6 days, no matter how hard they could try.

Shannons's noisy-chanel coding theorem

This is an amazing theorem that completely defies intuition and lies at the core of classical information theory. One way to defeat the noise is to encode the original information with added redundancy. The price to pay is a reduction on the efficiency of transmission, also called transmission rate. Intuition says that as the communication error goes to zero, through redundancy coding, the transmission rate should go to zero as well. NOT TRUE!. Shannon actually found a finite bound for the transmission rate, even as the communication error goes to zero.
http://en.wikipedia.org/wiki/Noisy-channel_coding_theorem

Princeton Physics Plasma Laboratory

The director of the Princeton Physics Plasma Laboratory, Stewart Prager, gave today an overview presentation about the research carried out in the institution. They design and construct tokamaks for nuclear fusion and collaborate with other institutions and projects such as ITER.. After seeing the steady progress in this research field, I am much more optimistic about the feasibility to have one day a commercial energy generator plant based in nuclear fusion. Something that also called me the attention was the importance of Lithium as coating material for the inner surfaces of the plasma chamber.