What is a contact transformation according to Dirac's Lectures on QFT?

[Sonty]

During the last week I've been reading Dirac. I browsed through his Quantum Mechanics and now I'm reading his Lectures on QFT. In both I found the term "contact transformation" and it seems he's the only one to use it because a few people at the faculty have heard about it but they can't give details as they didn't consider it very important. I don't eather because in Lectures on QFT he refers to the normal time evolution in classical mechanics (the canonical transformation generated by the hamiltonian) as being a contact transformation. Does anyone remember or should I just find his book on classical mechanics and see what he means by that and let you know?

 

[pmb]

Originally posted by Sonty
During the last week I've been reading Dirac. I browsed through his Quantum Mechanics and now I'm reading his Lectures on QFT. In both I found the term "contact transformation" and it seems he's the only one to use it because a few people at the faculty have heard about it but they can't give details as they didn't consider it very important. I don't eather because in Lectures on QFT he refers to the normal time evolution in classical mechanics (the canonical transformation generated by the hamiltonian) as being a contact transformation. Does anyone remember or should I just find his book on classical mechanics and see what he means by that and let you know?

It's just another name for the more modern term "Canonical Transformation"

This is explained in Goldstein where the term "Contact transformation" is explained. See"Classical Mechanics 2nd Ed.," Goldstein, Addison Wesley, (1980) page 382

I.e. the terms "contact transformation" and "canonical transformation" are fully synonymous. The former appears more in the older literature and the later appears more in the modern literature.

For a nice and thorough explanation of canonical transformations see "Classical Mechanics 3rd Ed.," Goldstein, Safko and Poole, Addison Wesley, (2002) Chapter 9

Pete

 

[Tipi]

It's just another name for the more modern term "Canonical Transformation"

This is explained in Goldstein where the term "Contact transformation" is explained. See"Classical Mechanics 2nd Ed.," Goldstein, Addison Wesley, (1980) page 382

I.e. the terms "contact transformation" and "canonical transformation" are fully synonymous. The former appears more in the older literature and the later appears more in the modern literature.

For a nice and thorough explanation of canonical transformations see "Classical Mechanics 3rd Ed.," Goldstein, Safko and Poole, Addison Wesley, (2002) Chapter 9

Pete

Hi,

This is a very old thread... I'm sorry, but there is not a lot about contact transformation on the web. So, if Pete is still here, I have a question for him:

In my french edition of Dirac's book, I can read:

"Les passages d'une représentation à une autre, que nous avons examinées, peuvent être appelés transformations canoniques. Il faut prendre garde de ne pas les confondre avec les transformations de contact..."

which in english can be read as :

"The passages from of a representation to another, which have been examined, can be called canonical transformations. Guard should be taken not to confuse them with the contact transformations".

I understood that canonical transformation is something like a unitary transformation, which transform some observables in some other observables which represent the same physical quantity (The olds and new operators have sames eigenvalues, eigenvectors, etc...). But the contact transformation switch from one system of observables, to another system of observables (The olds and new operators have not the same eigen values nor they have the same eigenvectors).

Am I wrong? I mean, are you sure that "the terms "contact transformation" and "canonical transformation" are fully synonymous"?

Regards,

Tipi

 

[Tipi]

Hi,

the contact transformation is what we call a change of basis, or of coordinates system.

The canonical transformation is the same as the representation transformation.

Because a change of representation is different from a change of basis, even if the mathematical formulation of these transformations are similar, this is a mistake:

the terms "contact transformation" and "canonical transformation" are fully synonymous

An example of a contact transformation is a rotation about an axis. This, in the position representation, express the fact that the description of the system is independent of the coordinates system choosen.

An example of canonical transformation is a switch from the position representation to the momentum representation.

Remark that classical physics do not study the representations, so Dirac mentionned that canonical transformation (as described in his book) have no counterpart in classical physics.Tipi

 

[PhilDSP]

E. T. Whittaker explained what a contact transformation is in the book "A Treatise on the Analytical Dynamics of Particles and Rigid Bodies". This was an apparently important book for Dirac and it develops the basis for Canonical Quantization - Poisson Brackets wherein a contact transformation is involved.

From memory, what he says is that two topological objects (such as circles in 2D) touch at a point. If those objects are translated into a space with a different basis then the point is translated also. The translation algorithm is the "contact transformation". I believe it must be more than a change of coordinates in Whittaker's case because it involves the representation of the two non-point objects that touch also. I'm not sure either if the basis is presumed to be linear (an abelian group).

 

[Tipi]

Hi,

thanks for the reference and explanations. I would just first say that what I wrote before about contact tranformation is surely false. Now, I think that change of representation and change of basis are the same, and I still don't really understand what is contact transformation. That's why your post help me.

Your explanation guided me to a book that I haven't read yet: Applications of Lie Groups to Differential Equations by Peter Olver. He refer to Anderson and Ibragimov and to Bluman and Kumei. He mentionned too that this transformation was introduced by Lie.

I'm really busy now, so I will put this misunderstanding on the pile with the others. But thanks again for your help.

Tipi

 

[tryhuk.v]

May be interesting for you ??

During the last week I've been reading Dirac. I browsed through his Quantum Mechanics and now I'm reading his Lectures on QFT. In both I found the term "contact transformation" and it seems he's the only one to use it because a few people at the faculty have heard about it but they can't give details as they didn't consider it very important. I don't eather because in Lectures on QFT he refers to the normal time evolution in classical mechanics (the canonical transformation generated by the hamiltonian) as being a contact transformation. Does anyone remember or should I just find his book on classical mechanics and see what he means by that and let you know?

See, e.g., http://www.volny.cz/tryhuk/
Publications:
Chrastinová, V. and Tryhuk, V., Generalized contact transformations, Journal of Applied Mathematics, Statistics and Informatics (JAMSI), 3 (2007), No. 1, 47-62. PDF