Field Strenght tensor in QED

[malawi_glenn]

Hi, I was reading earlier today in Peskin about QED field strenght tensor:

equation 15.15 and 15.16
$$[D_\mu, D_\nu ] \psi= [\partial _\mu , \partial _\nu ] \psi + ([\partial _\mu, A_\nu] - [\partial _\nu, A_\mu]) \psi + [A_\mu,A_\nu] \psi$$


Where A is the gauge field...

That part, I have control over.

Now I know that the first and last commutator is zero (abelian gauge theory and partial derivatives commute), but the middle one is really bothering me!

I obtain:

$$(\partial _\mu A_\nu - \partial _{\nu} A_\mu) \psi + (-A_\nu\partial _\mu + A_\mu \partial _\nu) \psi$$

And that last $(-A_\nu\partial _\mu + A_\mu \partial _\nu) \psi$ should be ZERO, so that:

$$F_{\mu \nu} = [D_\mu, D_\nu ] = \partial _\mu A_\nu - \partial _{\nu} A_\mu$$

BUT I don't know why ...

Any help or insight would, I would be very thankful of :shy:

 

[malawi_glenn]

I found an older thread "Yang Mills field stress tensor" where George Jones gave an excellent answer!

:-)

 

[sir-pinski]

The middle term in the commutator, ([\partial _\mu, A_\nu] - [\partial _\nu, A_\mu]) \psi, is not zero because the gauge field A_\mu is not a constant, it is a function of space and time. This means that the derivative of A_\nu with respect to \partial _\mu is not necessarily equal to the derivative of A_\mu with respect to \partial _\nu. In other words, the order in which the derivatives act on the gauge field matters. This is why the middle term is not zero and leads to the field strength tensor F_{\mu \nu}.

To understand this better, let's look at the components of the field strength tensor:

F_{\mu \nu} = \partial _\mu A_\nu - \partial _{\nu} A_\mu

In the first term, we have the derivative of the gauge field with respect to the first index \mu, and in the second term, we have the derivative of the gauge field with respect to the second index \nu. This shows that the field strength tensor takes into account the non-commutativity of the derivatives acting on the gauge field.

So why is the last term, [A_\mu,A_\nu] \psi, also zero? This is because in QED, the gauge field A_\mu is a vector field, and vector fields commute with each other. This means that the commutator of two gauge fields is zero, and therefore the last term in the commutator is also zero.

I hope this helps to clarify the role of the field strength tensor in QED and why the middle term in the commutator is not zero. Keep in mind that the field strength tensor is crucial in understanding the interactions between particles in QED and plays a central role in the theory.