Understanding Central Charges in Weinberg's Book

[Petr Mugver]

Hi all,

I'm having trouble solving problem 3 at page 105 in Weinberg's book, The Quantum Theory of Fields, Vol. 1:

Derive the commutation relations for the generators of the Galilean group directly from the group multiplication law (without using our results for the Lorentz group). Include the most general set of central charges that cannot be eliminated by redefinition of the group generators.

I can do the first two steps of the problem, but I can't figure out how to see whether a central charge can be eliminated or not, and how. I also know what the final result is, i.e. the only central charge left after redefinition of the generators must be

$$[K_i,P_j]=im\delta_{ij}$$

where m is a parameter that identifies the irreducible representation, the P's are the spatial translations generators, and the K's are the velocity transformations generators.

Any hint?

 

[strangerep]

Any hint?

Read Ballentine... ;-)

 

[Petr Mugver]

Read Ballentine... ;-)

Thanks a lot, that is EXACTLY what I was looking for!

The downside is that it's going to be my 32nd 600-page book on QM!

 

[Kevin_Axion]

No one understands Quantum Mechanics, don't feel bad, I'm sure many Theoretical Physicists have read more books than you on Quantum Mechanics and haven't found any deep intuition than they previously had, Quantum Mechanics is just; an aberration from what we see.

 

[vanhees71]

No one understands Quantum Mechanics, don't feel bad, I'm sure many Theoretical Physicists have read more books than you on Quantum Mechanics and haven't find any deep intuition than they previously had, Quantum Mechanics is just; an aberration from what we see.

The contrary is true! Quantum mechanics is the only self-consistent description of nature that is compatible with what we see.

 

[Kevin_Axion]

Yes, but my eyes only perceive the macroscopic world described by Classical Mechanics, of course the description of light is Quantum Mechanical in nature, but it isn't necessary at these length scales, hence it is an aberration from what we "see" - without instruments.

 

[MathematicalPhysicist]

Thanks a lot, that is EXACTLY what I was looking for!

The downside is that it's going to be my 32nd 600-page book on QM!

And you still not going to understand it.
:-)

 

[strangerep]

[...]
The downside is that it's going to be my 32nd 600-page book on QM!

And you still not going to understand it. :-)

Well, at least Ballentine's modern development does a reasonable job of
demolishing old interpretational bunkum like wavefunction collapse. :-)