Canonical Commutation Relations: Why?

[jjustinn]

Virtually every treatment of quantum mechanics brings up the canonical commutation relations (CCR); they go over what the Poisson bracket is and how it relates to a phase space / Hamiltonian mechanics, and then say "then, you replace that with ih times the commutator, and replace the dynamical variables with Hermitian operators, and TADA! You now have TEH QUANTUM MACHANICHS"...sometimes there's a vague attempt to explain it away by pointing to the Fourier-duality of the canonical variables (p/q), or if they really want to put the argument to rest, they say "by the Stone-von Neumann theory".

But...that doesn't really help me. Reading the Wikipedia article on Stone-von Neumann seems to have a lot of information, but I can't really make out a real explanation / derivation from it.

So -- can anyone explain this to a n00b? Here are a few points that I'm curious about in particular:

- what does a Poisson bracket have to do with a (anti-)commutator?

- I get how if you multiply a commutator by h, at a classical level it would look like the operators did indeed commute -- but why that particular value? And where does the imaginary unit come in?

- What does this have to do with the change from a finite-dimensional phase space to an infinite-dimensional Hilbert space, and the change from variables in this phase space to Hermitian operators on the Hilbert space?

...though these are by no means anywhere near the only questions -- they're more to just maybe get the conversation started. Also, any pointers to books / references would be great.

Thanks,
Justin

 

[jambaugh]

This has to do with the Noetherian association of observables (which may be conserved under a given dynamics) with generators of transformations (which may be symmetry transformations if the dynamics conserves the corresponding observable a la Noether's theorem).

In short there is a Lie group structure common to classical and quantum mechanics (the Lie group whose Lie algebra of generators provide possible components for the Hamiltonian which in turn is the generator of time evolution of the system).

In classical mechanics the representation of this transformation group is as action conserving flows of points in phase space, that is to say the dynamic flow from one state to another state. These will be (action area conserving) diffeomorphisms of phase space generated by the Poisson bracket action of the Hamiltonian function. The Poisson bracket is the representative of the Lie product for this representation.

In quantum mechanics one moves to a linear representation and the Lie product is necessarily the commutator product of operators in the associative algebra in which the group is represented.

It all comes down to the fact that e.g. momenta rotate in both quantum and classical mechanics as vectors. Angular momenta generate these rotations. Likewise momenta generate translations. Once you establish the dynamic group generated by the observables the main thing left to resolve is how these generators are represented and the information content in the measurements (spectrum and logic/probability inferences for subsequent measurements.)

 

[jjustinn]

...
In classical mechanics the representation of this transformation group is as action conserving flows of points in phase space, that is to say the dynamic flow from one state to another state. These will be (action area conserving) diffeomorphisms of phase space generated by the Poisson bracket action of the Hamiltonian function. The Poisson bracket is the representative of the Lie product for this representation.
...

Extraordinarily well-put. I think I at least understand the line of argument now.

You wouldn't be able to suggest a reference to could fill out the rest of the details, would you? Or even better, anything more broadly on this nexus of group/representation theory and mechanics (quantum or otherwise)? It seems everything I find on the topic is either totally abstract, or 100% specific to the development of the standard model. At the other extreme, books on advanced mechanics (in the model of Abraham / Marsden) that brush on this are a dime a dozen, but I haven't seen anything that breaks it down at that level (e.g. the Poisson bracket as a representation of the group of translations / the commutator as the unique linear representation of the same), let alone anything that goes into any depth on the topic.

 

[kith]

Sakurai introduces the momentum operator as the generator of translations and then derives the commutational relations from that. He doesn't go in depth, but at least they don't fall from the sky. (He does the same for the Hamiltonian which leads to the Schrödinger equation)

 

[jambaugh]

Extraordinarily well-put. I think I at least understand the line of argument now.

You wouldn't be able to suggest a reference to could fill out the rest of the details, would you? Or even better, anything more broadly on this nexus of group/representation theory and mechanics (quantum or otherwise)? ...

Hmmm... For me this kind of jelled out of understanding of QM and looking back at CM.
I built my picture up trying to parse "Quantization of Gauge Systems"
By Marc Henneaux, Claudio Teitelboim.

I doubt it is a good starting point. But the detailed discussion of constrained systems and general covariance in the classical and quantum mechanical setting provides an environment to build one's understanding. Possibly other more introductory references on constrained systems would be better. (this one is what I have).

Note that in the constrained systems one is working on a sub-manifold of phase-space and to keep to that manifold one modifies the Poisson bracket to a Dirac bracket.

You might start with the wikipedia article on the http://en.wikipedia.org/wiki/Dirac_bracket" , and its related links.

Maybe I ought to try to write up something like "An Introduction to Modern Physics, via Noether an Lie".

My MS in Math was in non-linear PDE's looking at symmetry group methods and my PhD work was in physics looking at Lie group deformations, trying to find new physics in altering the implicit group structure. So I kinda have a mathematical group centered perspective.

 

[jjustinn]

Sakurai introduces the momentum operator as the generator of translations and then derives the commutational relations from that.

...I built my picture up trying to parse "Quantization of Gauge Systems" By Marc Henneaux, Claudio Teitelboim.

...

You might start with the wikipedia article on the http://en.wikipedia.org/wiki/Dirac_bracket" , and its related links.

Thanks to both of you for the suggestions...I'll check them out.

Maybe I ought to try to write up something like "An Introduction to Modern Physics, via Noether an Lie".

That sounds like something that would be both interesting and invaluable. It seems like there's a void in this area -- that is, a rigorous and complete introduction to mechanics (quantum or otherwise) via infinitesimal transformations / Lie theory. It seems like everyone either glosses over it, using bits and pieces of results as intermediate steps in proofs, or assumes that you already understand it completely...

 

[sampo]

Hi, I have a basic question regarding the commuter brackets. Basically, what does the syntax mean?
I understand that in general, [a,b]=ab-ba, but I am confused as to how to make sense of that with fields. I am currently reading up on the quantization of the Klein-Gordon Equation, and I ran into such a problem. I attached a jpg file that has the rest (the book I got the equation from leaves out h-bars). Any help you can give will be most appreciated! Thanks.

 

[jambaugh]

Hi, I have a basic question regarding the commuter brackets. Basically, what does the syntax mean?
I understand that in general, [a,b]=ab-ba, but I am confused as to how to make sense of that with fields. I am currently reading up on the quantization of the Klein-Gordon Equation, and I ran into such a problem. I attached a jpg file that has the rest (the book I got the equation from leaves out h-bars). Any help you can give will be most appreciated! Thanks.

I think there was a typo, it should have read: $[\phi(x,t),\pi(y,t)]=i\delta(x-y)$

This is the field commutation relations where one is "second quantizing" i.e. quantizing a field.

You can think of the classical $\phi(x,t)$ as a abstract displacement over time of a model system located at position x. Dual to that displacement is a canonical momentum $\pi(x,t)$. You can think of the coordinates x (or y) here as a continuous index just as for an n dimensional system with $[q^k, p_j] = i\delta^k_j$ has index j or k.

This comes from modeling say a bosonic field, in terms of an array of harmonic oscillators located at each point of space. Each harmonic oscillator has its own coordinate and momentum. The coordinate value for the harmonic oscillator at x is $\phi(x)$ and its momentum is $\pi(x)$. You then have a field of classical systems which you then quantize to get a quantum field. Local coupling between these systems at each point allow them to propagate the field waves which when quantized become the bosons.

It is important to realize that the [tex]\phi(x,t)[/itex] here never was nor is a wave-function. It starts as a classical field (note it first appears in a classical Lagrangian) in a model which is quantized to yield the quantum field.

 

[sampo]

Thank you! that helps.

 

[The_Duck]

Ballentine's QM textbook works out some justification for this along the lines suggested above.

 

[Polyrhythmic]

I recently found a good explanation in "Quantum Field Theory - A Modern Perspective" by Nair.

 

[sampo]

I am trying to understand the canonical quantization of the Klein-Gordon equation and I am quite mixed up. I have been going over the same equations for over a week now and getting nowhere. My main textbook is Quantum Field Theory: a Modern Introduction by Michio Kaku, but I have been scouring the web as well. Attached is my work. If anyone can help point out where I am going wrong, it will be MOST appreciated. Thanks.

 

[sampo]

I could only attach three images, so here are the other two...

 

[sampo]

Apparently my first three images didn't attach. woops.

 

[PhilDSP]

That sounds like something that would be both interesting and invaluable. It seems like there's a void in this area -- that is, a rigorous and complete introduction to mechanics (quantum or otherwise) via infinitesimal transformations / Lie theory. It seems like everyone either glosses over it, using bits and pieces of results as intermediate steps in proofs, or assumes that you already understand it completely...

"Rotations, Quaternions, and Double Groups" by Simon Altmann provides an exceptionally thorough and all encompassing description from a geometric perspective (the author is a mathematician) Spinors are well described along with their relationships to quaternions, the Pauli matrices, infinitesimal transformations, Lie theory, representations and projections.

 

[sampo]

Thanks, I just ordered the book off of Amazon. Okay, I think I am comfortable with the mathematical derivation of the creation and annihilation operators. But I am still having problems understanding how both the position and momentum fields, and the creation and annihilation operators commute. I guess my question boils down to functionality. If all of these are functions onto the complex plane, how come their values don't commute? Attached are my questions in more explicit detail. Thanks.