# Symmetries in Sidney Colemans QFT script

• silverwhale
In summary, Coleman introduces the concept of symmetries and conservation laws in page 40 of his notes. He defines q^a(t) \rightarrow q^a(t,\lambda) for a transformation of the generalized coordinates and Dq^a \equiv \frac{\partial q^a}{\partial \lambda}. He writes that if no literature exists to explain what this derivative means, somebody can explain it to him. Finally, he thanks everyone for their help and ends the summary.
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!

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##.

It reminds of the equations for geodesic deviation in GR.
But is there a source where I can learn this stuff deeper?

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.

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!

## 1. What are symmetries in Sidney Coleman's QFT script?

Symmetries in Sidney Coleman's QFT script refer to the fundamental principles that dictate the behavior of physical systems under transformations. These transformations can be spatial, temporal, or involve changes in energy or other physical quantities. In QFT, symmetries play a crucial role in determining the properties and interactions of particles.

## 2. How are symmetries represented in Sidney Coleman's QFT script?

In QFT, symmetries are represented mathematically using a group theory framework. This involves using mathematical structures such as Lie groups and Lie algebras to describe the symmetries of a system. These representations provide a powerful tool for understanding the properties and behavior of particles in quantum field theory.

## 3. What is the significance of symmetries in Sidney Coleman's QFT script?

Symmetries play a crucial role in QFT as they provide a way to classify and understand the properties and interactions of particles. They also provide a guide for constructing and solving equations that govern the behavior of physical systems. In addition, symmetries can also provide insights into the underlying fundamental laws of nature.

## 4. How do symmetries affect the predictions of Sidney Coleman's QFT script?

Symmetries have a direct impact on the predictions of QFT. They can lead to conservation laws, which dictate that certain physical quantities remain constant during interactions. Symmetries can also determine the selection rules for particle interactions and can even predict the existence of new particles. Therefore, understanding symmetries is crucial for making accurate predictions in QFT.

## 5. Can symmetries in Sidney Coleman's QFT script be broken?

Yes, symmetries in QFT can be broken. This can occur under certain conditions, such as at high energies or temperatures, or due to interactions with other particles. The breaking of symmetries can lead to a change in the properties of particles and can even give rise to new phenomena. This is an active area of research in QFT, and understanding symmetry breaking is essential for accurately describing the behavior of physical systems.

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