Questions about Peskin QFT

[ismaili]

Dear all,

While I was reading chap2 of Peskin, I got some questions.
(1) The vanishment of the commutator of fields $$[\phi(x),\phi(y)]=0$$ means that the measurements at $$x$$ and $$y$$ do not interfere at all. Is this a postulate? Is this the so-called micro-causality?

(2) How Peskin deform the contour of fig.2.3 ? Why the two contour integrals are the same?

(3) How to prove if $$x,y$$ are space-like separated, there is a continuous Lorentz transformation take $$x-y$$ to $$-(x-y)$$? i.e. I don't understand fig.2.4.

Thanks for anyone.

 

[hellfire]

While I was reading chap2 of Peskin, I got some questions.
(1) The vanishment of the commutator of fields $$[\phi(x),\phi(y)]=0$$ means that the measurements at $$x$$ and $$y$$ do not interfere at all. Is this a postulate? Is this the so-called micro-causality?

Yes, this is the microcausality condition.

(3) How to prove if $$x,y$$ are space-like separated, there is a continuous Lorentz transformation take $$x-y$$ to $$-(x-y)$$? i.e. I don't understand fig.2.4.

If they are spacelike separated you can define a spacelike vector V that connects them. Then, you can easily show that there exists a Lorentz transformation that transforms V into -V. This will be a rotation of 180 degrees. If you try the same procedure for two points within the light-cone, connected by a timelike vector, you will see that the transformation is not possible.

 

[ismaili]

Yes, this is the microcausality condition.

Thanks. I guessed this is a "postulate", however, the book didn't give a clear assertion that this is a postulate. So I doubt that this can be derived. Now I think it is a postulate of QFT.

If they are spacelike separated you can define a spacelike vector V that connects them. Then, you can easily show that there exists a Lorentz transformation that transforms V into -V. This will be a rotation of 180 degrees. If you try the same procedure for two points within the light-cone, connected by a timelike vector, you will see that the transformation is not possible.

Thanks, I got it. But it seems that the argument have to be slightly modified. If V is a spacelike vector, we need not only the rotation to transform V into -V. Because the temporal coordinate is flipped too, so I guess we need a boost also.

Thanks for the discussion!

 

[hellfire]

Thanks. I guessed this is a "postulate", however, the book didn't give a clear assertion that this is a postulate. So I doubt that this can be derived. Now I think it is a postulate of QFT.

I started a thread a time ago with a similar question. You may want to use the search function to find it. It seems it is actually a postulate: it can be derived for specific representations such as the Fock representation that, however, restricts itself to positive mass solutions. There is no general way to derive it.

Thanks, I got it. But it seems that the argument have to be slightly modified. If V is a spacelike vector, we need not only the rotation to transform V into -V. Because the temporal coordinate is flipped too, so I guess we need a boost also.

Yes, but I think that a boost will not do the work to completely transform V into -V if it is timelike.