Symmetries in Sidney Colemans QFT script

[silverwhale]

Hello Everybody,

I am learning QFT using Sidney Colemans lecture notes (and the Peskin Schroeder book). They can be found here:

http://arxiv.org/abs/1110.5013

Now, in page 40, he introduces in the paragraph Symmetries and Conservation laws some definitions which I don't quite understand.

First he writes,

$$q^a(t) \rightarrow q^a(t,\lambda).$$
For a transformation of the generalized coordinates.

Second he defines,

$$Dq^a \equiv \frac{\partial q^a}{\partial \lambda}.$$

Now my question is just, does anybody recognize these definitions? is there some book where I can find, learn and play with them?
I never saw such a definition for a symmetry.
Well, if no literature exsits, can anybody explain to me what this derivative means?

Thanks alot!

 

[VantagePoint72]

It's a functional derivative. The idea is you have some curve parametrized by t. Now imagine you have a whole family of curves with the same endpoints where each curve is smoothly indexed by another parameter, $\lambda$. That is, if t is fixed then as you change $\lambda$ you smoothly trace a line through the whole family of curves. It doesn't matter exactly how you draw the family of curves since we only will care about the ones very close to the original trajectory, and we define everything so that $q^a(t,0)$ is just the original trajectory.

Now taking the partial derivative (evaluated at $\lambda=0$) of this new 2-parameter function with respect to lambda is the same as just varying the function using the usual tools of variational calculus. If you've done this before (say for the Lagrangian formulation of mechanics, deriving the Euler-Lagrange equations, etc.) you can just think of this as being the same as what you would normally write as as $\delta q^a$.

 

[silverwhale]

I understand your reply. Ifor sure did variational calculus.
It reminds of the equations for geodesic deviation in GR.
But is there a source where I can learn this stuff deeper?

Thanks for your reply!

 

[VantagePoint72]

Since it does, as you say, look a lot like geodesic deviation, I can only guess that it's standard formalism in differential geometry. So, I guess you could look there—but I'm not sure there's much "deeper" than what you see. In any case, the only time I've seen that notation besides Coleman's notes were in Michael Luke's QFT notes—which are just an abridgement of Coleman's.

 

[silverwhale]

Ok! got it! I'll check some GR books since to be honest I skipped the GD eq., so to get a feeling for it.

Well if needed I'll chekc some differential geometry books.

But your explanation was straight forward anyway!
So many thanks for your help!