What is the meaning of 'canonical' in physics?

[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.

 

[pmb_phy]

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'

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.

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?

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

 

[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.

 

[pmb_phy]

In mathematics and physics, the term "canonical" refers to something that is standard, fundamental, or essential within a specific context. It often implies a natural or preferred choice of representation, coordinates, or basis that simplifies the mathematical or physical description of a system. Here are a couple of common contexts in which the term "canonical" is used:

1. Canonical Coordinates: In classical mechanics, particularly in Hamiltonian mechanics, "canonical coordinates" refer to a set of generalized coordinates and their associated momenta that can be used to describe the state of a mechanical system. These canonical coordinates are chosen in such a way that they satisfy the Poisson bracket relations, which are fundamental in Hamiltonian dynamics. The choice of canonical coordinates simplifies the formulation of the equations of motion.
2. Canonical Transformations: In classical mechanics and Hamiltonian dynamics, "canonical transformations" are changes of variables (coordinates and momenta) that preserve the form of Hamilton's equations of motion and the Poisson bracket relations. Canonical transformations are essential for simplifying the analysis of mechanical systems and are often used to describe symmetries and conservation laws.
3. Canonical Ensemble: In statistical mechanics, the "canonical ensemble" is one of the standard ensembles used to describe the statistical properties of a system in thermal equilibrium. In this ensemble, the temperature is fixed, and the system exchanges energy with a heat reservoir. It is the ensemble of choice when studying systems with fixed temperature.
4. Canonical Form: In linear algebra and matrix theory, the "canonical form" of a matrix or linear transformation is a standard or simplified representation of that matrix. Examples include the Jordan canonical form and the Smith normal form.
5. Canonical Quantization: In quantum mechanics and quantum field theory, "canonical quantization" is a procedure used to promote classical fields and their conjugate momenta to quantum operators. This quantization method ensures that quantum commutation relations match the classical Poisson bracket relations, preserving the symplectic structure of the theory.

In these contexts and more, the term "canonical" indicates that a particular choice or representation is especially well-suited to the problem at hand, often simplifying mathematical expressions or highlighting important physical principles. The use of canonical methods and coordinates is a common practice in mathematical and physical sciences to make complex problems more manageable.

 

[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

footnote - Canonical is used here presumably in the sense of designating something a simple, general set of standard equations. It appears that the term was first introduced by J.G. Jacobi in 1837 (...) but in a slightly different context referring to an application of Hamilton's equations of motion to perturbation theory. Although the term rapildy gained common usage, the reason for its introduction remaind obscure even to contemporaries. By 1879, only 45 years after Hamilton explicitly introduced his equations, Thomas (Lord Kelvin) and Tait were moved by the adjective "canonical" to exclaim: "Why it has been so called would be hard to say."

I hope that clears this up a bit.

Pete

 

[NeutronStar]

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

Pete

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.

 

[pmb_phy]

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

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.

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.

Quantum mechanics spins off from analytical mechanics. For that reason the terminology is imported over too.

Pete

 

[NeutronStar]

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.
Pete

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.

 

[Galileo]

I hope that clears this up a bit.

Pete

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

 

[marcus]

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

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

 

[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.

 

[Kea]

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