Saturday, June 21, 2014

Siméon Denis Poisson: a birthday

Siméon Denis Poisson was born at the beginning of Summer 1781, on June 21st (like today), to a French royal soldier. He would die in 1840, at age of 58. He was a top French mathematician, geometer, and physicist of his era.

Both Lagrange and Laplace were his advisers. Liouville, Dirichlet, and Carnot (of the thermodynamic cycle fame) were among his students.

Revolutionary changes in France have influenced him in many professional ways. You may read it elsewhere.

But I want to mention his work. It's remarkable how many insights have been named after him.

First, there are some things he couldn't boast about. Some physicists are excessively conservative. Poisson was clearly an example. He would be the last leading opponent of the wave theory of light. As a hardcore reactionary, he would promote the corpuscular theory of light and the opinion that light simply could not interfere. Poisson was proven wrong by Augustine-Jean Fresnel.

During Poisson's last desperate attempts to prove that Fresnel was an idiot, Poisson discovered the Fresnel bright spot, sometimes also named an Arago spot or Poisson spot (!). That's a characteristic phenomenon associated with the wave nature of light. Later, I will mention that he would also write down an important equation related to the wave equation in classical physics. Not bad for a wave hater.

But Poisson's purely positive contributions are even more impressive.

He would improve the Laplace equation by adding the right hand side:\[

\nabla^2 \varphi = \rho

\] Yes, it became the Poisson equation. However, he did numerous other things to deal with the Laplace equation. For example, he invented the Poisson kernel that may be used to solve – paradoxically – the Laplace equation with some boundary conditions.

In statistics, we cannot overlook the Poisson processes. Things may "beep" at any moment, with some frequency, independently from the previous moments. The "beeps" associated with the decaying radioactive nuclei are the most canonical example of a Poisson process. If you quantify the probabilities that you obtain \(n\) beeps after some time, you may derive the Poisson distribution.\[


\] A generalization of this distribution with two more parameters (for overdispersion and underdispersion) is known as the Conway–Maxwell–Poisson distribution. Another advanced invention in statistics that uses the Poisson distribution is the Poisson regression, a type of regression assuming that the \(y\) variable is Poisson-distributed.

A 19-year-old Poisson may be found on this picture from Paris. You may call it a higher society. At any rate, prostitutes are debating with their consumers.

Poisson has also invented a method to count cells etc. (if there are too many), the so-called method of Poisson zeroes (or the most probable number method). You dilute the sample enough (by a sufficient factor) so that you start to see holes, and the estimated number of copies is a calculable function of the dilution factor.

The Euler–Poisson–Darboux equation is some partial differential equation that is rather important in solving the classical wave equation. Given Poisson's opposition to the wave theory of light, that's quite an ironic contribution by Poisson, isn't it? But the history of physics is flooded with similar ironies. Quantum hater Einstein's contributions to the quantum theory of light, Bose-Einstein quantum statistics, and entanglement are among our beloved modern examples.

Poisson's ratio quantifies the compression and strain of materials.

When it comes to the Fourier transformation, Poisson would be the man who realized the Poisson summation formula\[

\sum_{n\in \ZZ} f(n) = \sum_{k\in \ZZ} \hat f(k)

\] which effectively says that the Fourier transform of the sum of delta-functions located at all integers is the same sum (up to possible stretching, shrinking, and/or factors of \(2\pi\) that may depend on conventions).

Finally, Poisson is the father of the Poisson bracket and the Poisson algebra generated by the brackets.\[

\{f,g\} = \sum_{i=1}^{N} \left(
\frac{\partial f}{\partial q_{i}} \frac{\partial g}{\partial p_{i}} - \frac{\partial f}{\partial p_{i}} \frac{\partial g}{\partial q_{i}}\right)

\] This rather contrived formula is one of the symbols of the culmination of advanced, abstract classical physics. Since the 20th century, we've known that this bracket is what the simple quantum mechanical commutator\[

\frac{1}{i\hbar} [F,G] = \frac{1}{i\hbar} (FG-GF)

\] reduces to in the classical, \(\hbar\to 0\) limit. You may see that the quantum formula for the commutator is vastly simpler than its classical counterpart, the classical Poisson bracket. The comparison of the Poisson bracket and the commutator is one of the important examples of the fact that quantum mechanics is sometimes much simpler and more straightforward than classical physics! The people who want to rewrite, modify, revolutionize quantum mechanics because it seems "more contrived" to them then classical physics haven't understood it. Quantum mechanics is a different theory – a more general one – and from a sufficiently technical, algebraic perspective, it's actually simpler than its classical limit. There is absolutely no need to "simplify" or "revolutionize" quantum mechanics. Making quantum mechanics more similar to classical physics – if it were possible, and it's not – would necessarily make it more awkward and less natural than it is.

To a certain extent, you might be shocked that Poisson (and Hamilton – who used the bracket in his Hamilton equations of motion; and I have forgotten which man did what even though I used to know this history in some detail, but only for a little while) was able to invent such a seemingly artificial construct that just happens to map to something extremely simple once we switch to quantum mechanics (which he couldn't have known due to his confinement in a wrong century).

Much like the Poisson bracket (from the Hamiltonian formalism) simplifies to the commutator in quantum mechanics, the action (from the Lagrangian formalism) simplifies to the exponent in the Feynman path integral which allows one to directly write down the formula for the transition probability amplitudes (classical physics doesn't allow us to write compact formulae for the "answers" to questions about the evolution; all the answers are obtained "implicitly"; you're told to find a solution to some equations and to derive the answers from the solution). In both cases, the quantum formula is more straightforward than its classical limit (or "its classical counterpart", if you prefer to deny that classical physics is a limit of quantum mechanics).

The third example of the same simplification in quantum mechanics that I would remind you is the Noether theorem. Emmy Noether's original papers about that theorem were virtually unreadable and extremely contrived. In quantum mechanics, the Noether theorem – the one-to-one map between conservation laws and symmetries – is proven easily. The equation \([H,L]=0\) may be either interpreted by saying that the Hamiltonian is symmetric under symmetries generated by \(L\), i.e. as a symmetry of the dynamics; or it may be interpreted by saying that the Heisenberg-equation-computed time derivative of \(L\) vanishes, i.e. as a conservation law. These simple sentences replace dozens of unreadable pages from Noether's pencil.

Back to the Poisson brackets. Poisson realized that the brackets allow him to define the algebra which also obeys the Jacobi identity.\[

[F,[G,H]]+[G,[H,F]]+[H,[F,G]]= 0

\] In terms of the commutator, this identity is easy to be proven. Each of the the three double commutators may be decomposed to \(2\times 2 = 4\) terms with various signs, and these \(12\) terms may be seen to arrive in \(3!=\)six pairs of identical twins (\(FGH,FHG,GHF,GFH,HFG,HGF\): all permutations) with the opposite signs so everything cancels. But if you replace the commutator \[

[\dots, \dots]\to \{\dots,\dots\}

\] by the Poisson bracket, the proof becomes somewhat less elegant and natural. Second phase-space derivatives are needed to prove the identity, along with the Leibniz rule for the derivative of a product. Still, Poisson was able to see through such algebraic facts – explosive traces of quantum mechanics that may already be seen in the world of classical physics if you rewrite it in a sufficiently sophisticated form.

I have mentioned Poisson, Hamilton, Lagrange, and Noether as the "prophets of quantum mechanics" within the framework of classical physics. In this context, I can't resist to remind you that Ludwig Boltzmann was another "seer" of this kind. His understanding of classical statistical physics made it clear that it should ultimately be possible to count the states in the phase space (in the sense of integers and countable sets: see his tomb, it's clear that it's what he meant at the end) and indeed, quantum mechanics said that the phase space is effectively discrete, with one basis vector per cell of the phase space whose volume is a power of \((2\pi\hbar)\).

It's my belief that all these people would feel immensely satisfied if they could see quantum mechanics in its final form – as found in the mid 1920s – and the simplification of their seemingly convoluted insights that quantum mechanics brought to the plain sight. I am sure that they had to believe that all these discoveries of theirs had to have some deeper reason, something that shows why these apparently artificial structures in classical physics were actually natural and deep.

Too bad that no one has resuscitated Poisson, Hamilton, Lagrange, Noether, and Boltzmann yet. If you succeed, please let me know.


  1. Some may come out of their graves when INRJ returns. You may get to meet them. Just believe.

  2. Cloning sort of works but it would still be a new person with new likes and interests. If it could work at all for humans.

  3. Svik,


    The Romans didn't have a J. They were alphabetically challenged.

    But they did have cement and central heating.

    And a decent army most of the time. :)

    But back to poisson, which reminds me: there are two things that smell like fish. But only one of them is fish. :)

  4. OK INRI it is. I wonder what the Greek and Aramaic initials are?

  5. This history of math is extremely interesting. I would like to go back and review all stuff I studied and more in light of the history, engineering applications and quantum theory applications.

    Do you have any favorite book on this subject??

    In fact I would like to product a review myself but with the computing code added so we can demonstrate how the equations actually work and prove that we really understand it. With lots of graphs and fancy simulations to get people Interested.

    In school it seems formulas are just pullednout of the air with no relevance to anything.

  6. Every day I come here as a simple civil/sanitary engineer (ret). Most of the time I leave befuddled. But today you reveal to me that the method of most probable numbers, which I actually taught to sanitary microbiology lab students, came from Poisson.

    By the way, it's INRI not because the Romans didn't have a J but because the name is pronounced Yesu, or more likely, Yesuah.

  7. Greek: ΙΝΒΕ? That's a guess — I don't know. Anna V will though.

    Aramaic: I wouldn't have the foggiest.

  8. "By the way, it's INRI not because the Romans didn't have a J ..."

    Jawohl, that's correct. But then no one here said it was.

    "By the way, it's INRI not because the Romans didn't have a J but because the name is pronounced Yesu, or more likely, Yesuah.

    Never mind the Jesus — Svik got that part right. It's the Jews he got wrong.

    Check your plumbing.



    You could be right after all! I.e. INRJ is just as good as INRI. Maybe.

    Indeed, I now have a vague recollection of seeing it in the form INRJ but with the tail on the J drooped somewhat, as if it had been on the beer. :)

    So I looked up the letter J on wiki:
    "The letter 'J' originated as a swash character, used for the letter 'i' at the end of Roman numerals when following another 'i', as in 'xxiij' instead of 'xxiii' for the Roman numeral representing 23."

    OK, that's just talk about numerals but, given that the Romans were using it in this manner, I don't see why they wouldn't use it at the end of a set of initials too.

    So I take it all back. :)

  10. Yes but if you search for INRI images on goggle u get lots of crosses bit hardly even one on a inrj search so Inri is more correct.

  11. "A 19-year-old Poisson may be found on this picture from Paris. You may call it a higher society. At any rate, prostitutes are debating with their consumers."

    You had me going there for a while, Luboš, but I've finally just twigged, maybe. Does your mind run around about two foot six above the ground like mine? :)

    "...may be found..."

    Hmmm. :) That looks like a probability statement to me, and a good one — indeed, the chances definitely are! It's a safe be all right!

    OK. I think I can now answer my own question. Welcome to the club, old boy! The whitebait here is very nice I'm told, but maybe like me you prefer the salmon? :)

  12. Haha! It seems we've swapped places! :)

    I just can't now get rid of the feeling that I've seen it with a long, droopy' J somewhere. But that must have decades ago, maybe fifty years or so.

  13. Sorry, this blog post wasn't meant to be dedicated to religious nuts and the propagation of their feelings. It's about an eminent French mathematician and physicist Poisson.

  14. Yes I did edit it. But I only do in the first 10 minutes or so. But you probably responded too quick. Sorry.

    I did some searches and figure I might as well fill in the answers.

    You are not loosing it. Don't worry be happy.

    By the way where did you learn your German. Were you a teacher?

    There is a good joke about an English pilot flying into Germany in the sixties. He was having trouble reading to map for ground navigation to the gate. So the German controller got on the radio and barked. Have you never been to Germany before?
    So the English pilot answered yes I have but the last time I never had to stop.

  15. I am going to resuscitate them in a minute. Let me just finish my proof of the Riemann hypothesis real quick.

  16. You are stupid like the things you write in your blog that won't ever deserve the respect of any scientist. The fact that you're joking about resurrection in the end make you also ridiculous as a person

  17. You are really stupid.

  18. And all your records discredited you

  19. Your past records discredited you: you were fired from Harvard Physics
    Department for your unprofessional activity. Blame yourself.
    My prediction for you: your reputation will be even more damaged after these sort of things

  20. In greek Jesus is written with an iota, Iesous.

  21. You're not even able to reply to a person, because you don't have logical reasoning. In fact also your articles are a waste of time and will never be appreciated by the mainstream community. Ah ah ah LOL

  22. just a suggestion about the best thing for the two of us: ban all the phoney baloney nonsense that you are writing here

  23. Try to resuscitate the neurons of your brain, Lubos, Lubof, Lubosh.... sh.... Let me know if when you can do that,

  24. Your past records discredited you: you were fired from Harvard for your unprofessional activity.
    Blame yourself for your sloppy string theory.
    My prediction for you: your reputation will be even more damaged soon.

  25. And the Septuagint is still the true Christian Bible.

  26. Boy what trouble a little J can cause. Sorry for starting it all. At least J. Archer let me off the hook.

    I love your math insights. Specially stuff like the poissom bracket. It shows that Q.M. is the simpler and better theorem as per Occam's razor.

    I just read a book by a chemist who was determined to understand q.m. properly.
    There was this guy who asked schroodinger "so where is the wave equation" so he went of with the unnamed girl friend and derived it from the standard wave equation. The derivation is quite simple. And the zuprise is that the equation still holds after all these years.

    In fact he worked both the standard and dirac versions.

    I have dug out my old quantum text book and other physics books as I will be needing for work soon. Fortunately the mice did not get into those books.