# What does canonical mean?

1. Dec 20, 2004

### Galileo

I got this question today which I couldn't answer.
I said it something like 'something that is canonical is standardized in a way'.
I didn't even know what I meant with that, but looking up the definition, it seems I was partly right.

Canonical:
'Conforming to orthodox or well-established rules or patterns, as of procedure'

The context was in a question: 'derive the canonical commutation relations ([x,p],[x,y] etc.)'.

Why is it called canonical and what does it mean if used in physical contexts?
Why is $\frac{\hbar}{i}\frac{\partial}{\partial x}$ called canonical momentum? (In Langrangian mechanics $\frac{\partial L}{\partial \dot q}$ is also called the canonical momentum. I've learned it as generalized momentum though).
It's also used in statistical mechanics (canonical ensemble) where I am equally uncertain what it means.

2. Dec 20, 2004

### pmb_phy

What the term "canonical" means depends on context its used in. It also depends on the dictionary you use to look that darn term up in.

Nobody knows why. This is a mystery that has yet to been solved.

I recall that Goldstein mentioned that the term was first used in such and such a place but it was not clear why. The term "canon" refers to something religions like as in "cannon law". So if you want to give a name to something which you consider super imporant then you might for example call a type of imporant momentum in Lagrangian mechanics "canonical momentum" etc.

However the question of this subject is "What does canonical mean?" That has an answer. See
http://www.geocities.com/Athens/Styx/5478/canonical.html

It pertains to religion.

Pete
Pete

Last edited: Dec 20, 2004
3. Dec 20, 2004

### NeutronStar

With respect to physics and mathematics I've always taken the word canonical to basically mean generalized.

Like canonical coordinates would simply be generallized coordinates, canonical momentum would be generalized momentum.

A canonical system would simply be a generalized system.

In other words, whenever I see the word canonical I usually read it as generalized. These two words are completely interchangeable in my mind.

See http://en.wikipedia.org/wiki/Canonical for more specialized meanings of the word with respect to mathematics.

4. Dec 20, 2004

### pmb_phy

That isn't quite true. For example, there is quite a bit of difference between "generalized transformation" and "canonical transformation".

Pete

5. Dec 20, 2004

### pmb_phy

Let me quote what Goldstein et al say in Classical Mechanics - 3rd Ed, Goldstein, Safko and Poole (2002) at the bottom of page 358 pertaining to canonical equations of Hamilton
I hope that clears this up a bit.

Pete

6. Dec 20, 2004

### NeutronStar

I still take it to mean generalized but within the context of the generalization.

In other words, generalized within the specific rules and restrictions of the formalism under consideration,...

That would be opposed to being generalized in general.

I've seen the term used with respect to variational mechanics (i.e. Lagrangian and Hamiltonian dynamics). And I've also seen it used with respect to concepts in quantum mechanics. But in all of those cases I still just saw it as a generalization of the concept under consideration within those formalisms.

In other words, the term canonical appears to have a lot of semantic flexibility for various authors.

7. Dec 20, 2004

### pmb_phy

If by this you mean that you use it as a synonym then in some cases that's quite a common thing to do in certain cases. I don't see how you'd apply it to the term canonical transformation though. But if you don't mean it as a synonym then it seems that it could get confusing. Suppose you were asked "what is the generalized linear momentum of a charged particle in an EM field". That can refer to the quantity mv or to the quantity mv + qA if you're not using it as a synonym for canonical momentum.
Quantum mechanics spins off from analytical mechanics. For that reason the terminology is imported over too.

Pete

8. Dec 20, 2004

### NeutronStar

Alight, I see what you are saying. The word has value. I suppose that's why they use it. But I still see it as meaning a generalization within a specific context or formalism. I've actually seen it used outside the framework of variational mechanics as well. Like referring to other mathematical systems that have specific generalized constraints.

So I suppose I really read the word canonical to mean, generalized within the framework under consideration. So you’re right the words generalized and canonical aren't technically perfectly interchangeable. There is an important difference in their precise meanings.

9. Dec 22, 2004

### Galileo

Yes, thank you veyr much pmb.
It's exactly the answer I wasn't hoping for, though. :rofl:

10. Dec 22, 2004

### marcus

actually the meaning of "canonical" is roughly halfway between
kosher and organic
with just a touch of the connotation "pesticide-free"

11. Dec 22, 2004

### krab

I think "generalized" is not appropriate here. For given choice of space coordinates, the canonical momenta conjugate to these coordinates are very particular momenta. These canonical momenta, together with the position coordinates have very beautiful mathematical properties not available if momenta are defined differently. For example, it allows for the existence of a function of the variables called the Hamiltonian, which is the generator of their time evolution. Also, these variables, together with the bilinear operator we usually call Poisson brackets form a Lie algebra. You may not know what a Lie algebra is, but what it comes down to is that you have many more mathematical tools at your disposal.

12. Dec 22, 2004

### Kea

I think Krab is right. To me 'canonical' usually means 'the natural
best choice' of possible quantity.