- #1

JD_PM

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- TL;DR Summary
- I was reading example 1.3.3 of Tong's lecture notes on QFT and I did not understand several points of it.

I have attached the PDF.

I am reading Tong's lecture notes and I found an example in which there are several aspects I do not understand.

This example is aimed at:

- Understanding what is the analogy in field theory to the fact that, in classical mechanics, rotational invariance gives rise to conservation of angular momentum.

Let's start then.

The infinitesimal form of Lorentz transformations is given as

$$\Lambda^{\mu}_{ \ \nu} = \delta^{\mu}_{ \ \nu} + \omega^{\mu}_{ \ \nu}$$

Where ##\omega^{\mu}_{ \ \nu}## is infinitesimal.

We know that the general Lorentz transformation condition is as follows (let's forget about priming indices in this post):

$$\eta^{\mu \nu} = \Lambda^{\mu}_{ \ \sigma} \Lambda^{\nu}_{ \ \tau} \eta^{\sigma \tau}$$

Thus, applying it to the infinitesimal case we get

$$(\delta^{\mu}_{ \ \nu} + \omega^{\mu}_{ \ \nu})(\delta^{\nu}_{ \ \nu} + \omega^{\nu}_{ \ \nu}) \eta^{\sigma \tau} = \eta^{\mu \nu}$$

Let me skip the Algebra; just note that I have dropped the term ##\omega^{\mu}_{ \ \sigma} \omega^{\nu \sigma}##. Thus we get

$$\omega^{\mu \nu} = - \omega^{\nu \mu}$$

Thus ##\omega^{\mu \nu}## is antisymmetric. I understand everything at this point.

Then Tong says we can make a check to see that the number of 4x4 antisymmetric matrices is equal to the total number of LTs: 4x3/2=6. This is a basic Linear algebra issue but I do not see it.

Now he shows that the transformation on a scalar field is given by

$$\phi(x) \rightarrow \phi'(x) = \phi(\Lambda^{-1}x) = \phi(x^{\mu} - \omega^{\mu}_{ \ \nu} x^{\nu}) = \phi(x^{\mu}) - \omega^{\mu}_{ \ \nu} x^{\nu} \partial_{\mu} \phi (x)$$

What I do not get is why

$$\phi(\Lambda^{-1}x) = \phi(x^{\mu} - \omega^{\mu}_{ \ \nu} x^{\nu})$$

We know that

$$\Lambda^{-1} = \Lambda^{\nu}_{ \ \mu} = \delta_{ \ \mu}^{\nu} + \omega_{ \ \mu}^{\nu}$$

Thus I get the same but with a positive sign

$$\phi(\Lambda^{-1}x) = \phi(x^{\mu} + \omega^{\mu}_{ \ \nu} x^{\nu})$$

He justifies the sign to be positive when we apply active transformations. Then I think he made a typo and it should be positive (I hqve to say I do not understand why we define two kinds of transformations though: active and positive).

Please let me attach an image of the rest of the example I want to comment:

From here I basically do not get the following:

- I do not get the last equality on 1.53 (i.e. why the last equality follows because ##\omega^{\mu}_{\mu}=0##; I do not see the antysimmetry argument).

- I do not see neither how to get 1.55 nor ##\partial_{\mu}(J^{\mu})^{\rho \sigma} = 0##

1) Why The number of 4x4 antisymmetric matrices is equal to the total number of LTs: 4x3/2=6.

2) Why ##\phi(\Lambda^{-1}x) = \phi(x^{\mu} - \omega^{\mu}_{ \ \nu} x^{\nu})##

3) How to get equation 1.53

4) How to get equation 1.55

Thank you very much.

This example is aimed at:

- Understanding what is the analogy in field theory to the fact that, in classical mechanics, rotational invariance gives rise to conservation of angular momentum.

Let's start then.

The infinitesimal form of Lorentz transformations is given as

$$\Lambda^{\mu}_{ \ \nu} = \delta^{\mu}_{ \ \nu} + \omega^{\mu}_{ \ \nu}$$

Where ##\omega^{\mu}_{ \ \nu}## is infinitesimal.

We know that the general Lorentz transformation condition is as follows (let's forget about priming indices in this post):

$$\eta^{\mu \nu} = \Lambda^{\mu}_{ \ \sigma} \Lambda^{\nu}_{ \ \tau} \eta^{\sigma \tau}$$

Thus, applying it to the infinitesimal case we get

$$(\delta^{\mu}_{ \ \nu} + \omega^{\mu}_{ \ \nu})(\delta^{\nu}_{ \ \nu} + \omega^{\nu}_{ \ \nu}) \eta^{\sigma \tau} = \eta^{\mu \nu}$$

Let me skip the Algebra; just note that I have dropped the term ##\omega^{\mu}_{ \ \sigma} \omega^{\nu \sigma}##. Thus we get

$$\omega^{\mu \nu} = - \omega^{\nu \mu}$$

Thus ##\omega^{\mu \nu}## is antisymmetric. I understand everything at this point.

Then Tong says we can make a check to see that the number of 4x4 antisymmetric matrices is equal to the total number of LTs: 4x3/2=6. This is a basic Linear algebra issue but I do not see it.

Now he shows that the transformation on a scalar field is given by

$$\phi(x) \rightarrow \phi'(x) = \phi(\Lambda^{-1}x) = \phi(x^{\mu} - \omega^{\mu}_{ \ \nu} x^{\nu}) = \phi(x^{\mu}) - \omega^{\mu}_{ \ \nu} x^{\nu} \partial_{\mu} \phi (x)$$

What I do not get is why

$$\phi(\Lambda^{-1}x) = \phi(x^{\mu} - \omega^{\mu}_{ \ \nu} x^{\nu})$$

We know that

$$\Lambda^{-1} = \Lambda^{\nu}_{ \ \mu} = \delta_{ \ \mu}^{\nu} + \omega_{ \ \mu}^{\nu}$$

Thus I get the same but with a positive sign

$$\phi(\Lambda^{-1}x) = \phi(x^{\mu} + \omega^{\mu}_{ \ \nu} x^{\nu})$$

He justifies the sign to be positive when we apply active transformations. Then I think he made a typo and it should be positive (I hqve to say I do not understand why we define two kinds of transformations though: active and positive).

Please let me attach an image of the rest of the example I want to comment:

From here I basically do not get the following:

- I do not get the last equality on 1.53 (i.e. why the last equality follows because ##\omega^{\mu}_{\mu}=0##; I do not see the antysimmetry argument).

- I do not see neither how to get 1.55 nor ##\partial_{\mu}(J^{\mu})^{\rho \sigma} = 0##

I know there are many questions here, so let's summarize:I know there are many questions here, so let's summarize:

1) Why The number of 4x4 antisymmetric matrices is equal to the total number of LTs: 4x3/2=6.

2) Why ##\phi(\Lambda^{-1}x) = \phi(x^{\mu} - \omega^{\mu}_{ \ \nu} x^{\nu})##

3) How to get equation 1.53

4) How to get equation 1.55

Thank you very much.