Casual talk. Constrained Hamilton systems. Dirac brackets.

[jostpuur]

Casual talk. Constrained Hamilton systems. Dirac-Poisson brackets.

Casual talk. Constrained Hamilton systems. Dirac-Poisson brackets.

Hi guys, I think I have finally succeeded in understanding the ideas which Dirac explained in the two first chapters of his book "Lectures on Quantum Mechanics". I'm not saying that I would have fully studied everything covered there, but I believe that I have now understood what it is all about mostly, and it is amazing stuff.

Do I belong to an exclusive club now?

I don't have any technical questions about the topic now, so I don't think that this would belong to the Classical Physics or Quantum Physics sections of this forum. I merely wanted to chat about few related things.

Something negative: I'm slightly frustrated that it took this long eventually. I reached the cursed age of 30 years before managing to understand this topic. Certainly my "physics career" would have proceeded more smoothly if I had learned the Dirac brackets when 20 years old for example.

There is a joke (or lore) related to physics fantasies of young students. First they fantasize about specializing in quantum gravity and similar stuff, because they have learned their "science" from popular documentaries, but during few years in university they learn to stay away from quantum gravity, and instead specialize in something less ambitious. Although it is true that in most cases leaving quantum gravity alone is probably a sign of certain maturity, in my opinion the joke doesn't end there. Do these people really know why they were unable to continue towards the quantum gravity? If they are asked that what would they study next, if they still wanted to proceed towards quantum gravity, would they know the answer?

Isn't it precisely the quantization of constrained Hamiltonian systems, that ends up being a critical barrier? If a student has never heard about the entire thing, then his/her thoughts concerning the quantum gravity will remain in extreme fog?

Since the quantum gravity is still an open problem, the full correct approach isn't known of course, but isn't it obvious at this point that the quantization of constrained Hamiltonian systems must be at the beginning of the path? I mean that I cannot tell what is the correct approach to quantum gravity, but I can tell you that if the approach doesn't include the quantization of constrained Hamiltonian systems, or something similar or equivalent, then the approach is hopeless for sure.

I know there are people who have attempted to specialize in string theory without ever hearing about the constrained Hamiltonian systems or the Dirac's book. What are your thoughts on that? To me it looks like a failure on the part of lecturers and advisors. Or is the string theory so broad topic that you can do that?

Is it not so, that overall this topic is not very well known? I have some personal experience of situations where I have attempted to ask something about the Dirac brackets from some very big authoritative guys in mathematical physics, and they have responded something like "this is the thing explained in the Dirac's book!" (waving hands to depict the small physical size of the concise book), and having then wanted to change the topic quickly. You can sense that people are not very confident with this.

I originally attempted to study QFT from the Peskin and Schroeder, and I still have the book. To me one of the most suspicious things in theoretical physics has been that the way Peskin and Schroeder explain the quantization of Dirac's field seems to reveal that they don't know the constrained Hamilton systems either. Am I still alone with this opinion? They claim that something, which clearly is not a harmonic oscillator, would "almost" look like a harmonic oscillator, and it makes no sense. The topic is tricky, because the system, which clearly is not a harmonic oscillator, happens to a be non-trivially constrained system, and hence beyond the understanding of most readers.

If there are people here who have been taught the Dirac brackets by some lecturer, I would be interested to know where and how that has happened. Perhaps the best and biggest universities have some courses on this? I know for sure that most universities don't have.

Since it has been difficult to find pedagogical material on this, and since I feel like having survived the initial barrier, I cannot help thinking about at some point writing some pedagogical tutorial on this topic. I'm not ready to start yet, but I'm planning anyway, and that's why I'm interested in inquiring about what other people think about studying this.

 

[atyy]

The Dirac ket |a> is a vector, and the Dirac bra <a| is a dual vector. The bra-ket <a|b> is the inner product, more usually written (a,b). The notation is especially useful when dealing with more than one particle, because the Hilbeert space (which is essentially a vector space with an inner product) of two particles is the tensor product of the Hilbert spaces of each particle.

The constrained Hamiltonain formalism is extensively used in one approach to quantum gravity - Loop Quantum Gravity. That approach takes seriously the idea that no new degrees of freedom are needed to quantize gravity. String theory's presumption is that new degrees of freedom must be used to quantize gravity, such that the gravity we see is not a fundamental force, but only a low energy emergent field. String theory more often uses the path integral, rather than the Hamiltonian formalism. The gauge redundancy which requires the constrained Hamiltonian formalism is treated using BRST in the path integral formalism.

An introduction to the bra-ket notation is found in section 3.1 of these notes http://arxiv.org/abs/1007.4184.

 

[jostpuur]

I meant the generalized Poisson bracket by "Dirac bracket"! This: http://en.wikipedia.org/wiki/Dirac_bracket I never realized the possibility of ambiguity. The name apparently gets ambiguous if the context isn't mentioned clearly. :tongue:

 

[atyy]

You might like to look at http://books.google.com/books?id=2FAuAKEKFyYC&source=gbs_navlinks_s.

 

[jostpuur]

You might like to look at http://books.google.com/books?id=2FAuAKEKFyYC&source=gbs_navlinks_s.

Yes this it true. I have had this book in my shelf for some time. It begins with the same things as Dirac's book, but then continues further into some amazing stuff. Since I never (before very recent success) had got through the very beginning, the contents had remained beyond my reach.

 

[atyy]

I'm a biologist, and have no detailed mastery of the formalism. But I have been curious enough to try to get the general ideas behind them. One exposition that seemed quite approachable to me was that in Dona and Speziale's http://arxiv.org/abs/1007.0402.

 

[jostpuur]

I interpret this silence as a sign of that people here are not very comfortable with Dirac-Poisson brackets.