Talk:Quantization (physics)

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The word "quantization" is not the exclusive property of field theory. Planck's law enforced the quantization of energy. Bohr postulated the quantization of angular momentum... --Ujm 08:30, 15 October 2006 (UTC)

[edit] Style collaboration proposal

I think this page is a very valuable resource for much of the QFT and HEP sections of the encyclopedia. Very commendable work. I have one stylistic complaint though. Please don't interpret this as belittling anyone's past efforts: I fully appreciate how difficult it is to write up material like this. So here goes —

The descriptions seem rather abstract. I'm a working physicist, and I had to stop and re-read parts every now and then. I wonder what fraction of graduate students in particle physics would recognize the descriptions in one reading without having to stop and think. As the first comment on this page indicates, we don't need to write this page for the amorphous public, but keeping science undergrads in mind would not be a bad idea. Would it be possible to start a collaboration to rework this material? Bambaiah 06:52, Jun 2, 2005 (UTC)

Sorry, there have been more mathematicians on WP than physicists to date. Are you saying you want to re-write, but want someone's help? I could certainly blast this into something more readable; unfortunately, it would also get a lot longer. Quantization is a hot topic these days. I was planning on expanding on both geometric and deformation quantization. linas 15:13, 2 Jun 2005 (UTC)

Well, I skimmed the article a bit more; what is it that you want to do to it? In many ways, this article is not bad; I guess what's missing is maybe an undergrad-level introduction; but note also that undergrad level quantization is totally ad hoc, and consists in mostly hacking on the Schreodinger equation and introducting the spherical harmonics. What can be said that would be "simple"? I guess one could say in simpler terms that the canonical momentum is and so ergo schreodinger and Klien eqn and Dirac eqn. Call this section "ad hoc quantization" or "first quantization" and then state that "canonical quantization" is the modern math formulation of ad hoc hamiltonian based quantization. Probably canonical quant should be made into its own article.

By contrast, second quantization already has its own article (albeit its nearly empty) and path integral has a large excellent article. BRST is rather throughly opaque and needs intro ... yikes! I did once read in BRST decades ago, but I didn't understand it then. I was going to expand on geometric quantization and also I've got a copy of an excellent 1977 paper on deformation quant that I was planning on studying, but I was planing on strecthing this all out for half a year or more :) WP is missing articles on some of the more basic background material. I am, ahem, an amateur having a mid-life crisis wishing I hadn't left academia; so this is spare time activity for me. linas 05:07, 3 Jun 2005 (UTC)

I tried to demonstrate. what I wanted to do through two major edits:

• The first was the edit of the section "canonical quantization". I created a more detailed article on that topic (which is still not complete) and reduced the section to an introduction to that article. I also moved the technical material in this section to a section of its own in the article on "canonical quantization". Perhaps the article on "second quantization" becomes redundant now and needs to be merged either with "Quantum field theory" or "canonical quantization".
• The second was the edit of the introduction and the introductory part of section 1. Many of the points touched upon there were either straight quantum mechanics, or philosophy of quantum mechanics. They can be reduced to a reference. I think it reads better now — clearly not npov :), so a second opinion on this would be very good to have.

As for me, I'm relaxing after a long computation, so I'll also not keep up this pace for long. Bambaiah 09:37, Jun 4, 2005 (UTC)

Reviewing edits now ... one big problem with the edits is that quantization isn't just about field theory; we don't even know how to reliably quantize simple systems. e.g. canonical quantization, geometric quantization and deformation quantization are about first quantization of simple finite-dimensional mechanical systems, not second quantization of fields. Case in point the Hilbert-Polya conjecture states that the Riemann hypothesis is the first quantization of a very very simple one-d Hamiltonian; numerous hints show that its somehow "simpler" than the harmonic oscillator. We neither know what that Hamiltonian is nor how to quantize it. So my interest is in number theory, not field theory per se; the quantization is to try to get at a basic conjecture of number theory.

This is why I mumbled about "ad hoc" quantization such as the Schroedinger equation. Schroedinger equation is a recepie for first quantization that sort-of works in some cases. The article you wrote on canonical quantization will need to be moved to canonical second quantization. These are distinct concepts having vastly different theories. linas 17:25, 4 Jun 2005 (UTC)

I appreciate your feedback. I'm not sure exactly what you mean by applications to number theory. I'll spend some time talking to number theorists about the example you have given. I appreciate your second point, and taken your critique into account by modifying the article on canonical quantization. Quantum mechanics and quantum field theory are distinct things, but not vastly different. Dirac's book [ISBN 0198520115] clarifies, I think. Bambaiah 07:23, Jun 6, 2005 (UTC)

I doubt your number theorist friends will have a clue; the relationship is pushed mostly by physicists, e.g. Sir Michael Berry. The relation is speculative, but it has a hold of my imagination. I don't know that there's more than one paper a year published on this; but I don't know. Some nuclear physics people studying Gaussian unitary ensembles may be aware of this. Ditto for any quantum-chaos types; the Riemann Hamiltonian looks to be "quantum chaotic", it has all the right statistics to be chaotic.

Mathematically speaking, QFT and QM are very very different. The techniques used to pose and solve the shroedinger eqn. e.g. hydrogen atom, are nothing at all like second quantization. Only to physicists do these things look alike. :) I'd like to have this article be about the topic of "how to quantize" and not about "what is perturbative field theory" which is the current standard treatment of second quantization. linas 14:51, 6 Jun 2005 (UTC)

I've been alluding to difficulty of quantizting simple systems. Here's an explicit example. Pick a riemann surface, any surface. They're all Kahler manifolds, they're all symplectic manifolds. They're all 2D phase spaces, with exactly one x and one p coordinate. Pick a Hamiltonian, any hamiltonian. For symplectic manifolds, a hamiltonian is a real-valued function on the phase-space/symplectic manifold. Just to keep things simple, pick the Hamiltonian H = p² / 2m, its just a pure kinetic term. Or pick the SHO p² / 2m + kx² / 2 if you wish. Now quantize it, by any means desired. Tell me what the eigenstates are, what the energy levels are. Hah. Not so easy, eh? And mind you, this is just a plain old 1-dimensional problem. It gets worse in higher dimensions, never mind infinite dimensions. linas 15:56, 6 Jun 2005 (UTC)

Ok, let's pick the SHO and quantize it by RCQ. Eigenstates? Really easy. Energy levels? Not a problem. (Hmm... But the energy levels turn out to be "incorrect": they start from zero. This is just because quantum Hamiltonian is defined differently.)

OK, Easy? Then explain it to me. Pick, for example, the Riemann surface . Tell me what p and x are on this surface, then tell me how to quantize either the pure kinetic or the SHO, and tell me what the eigenstates are. The only two reimann surfaces I know how to quantize are the flat plane (the eigenstates are plane waves) and the sphere (the eigenstates are sphere harmonics). I don't know how to do any surfaces of negatvie curvature, refernces to textbooks or papers that do the math explicitly would be great. linas 22:54, 7 Jun 2005 (UTC)

Here's another simple example that I find eminently confusing. Review the article geodesic. Note the prominent place of the Hamiltonian there. How do I quantize that Hamiltonian? I would like to know. linas 16:11, 11 Jun 2005 (UTC)