Klein-Gordon Field: Understanding Eq. (1)

[grimx]

Hi everyone! Im' a new member and I'm studying Quantum Field Theory.

I read this:

"The interpretation of the real scalar field is that it creates a particle (boson) with momentum p at the point x."

and :

\phi\left(x\right) \left|0\right\rangle = \int \frac{d^3p}{(2\pi)^3(2\varpi_p)} e^{-ipx} |p\rangle (1)

but I didn't understand this equality... i know that:

\phi (x) = \int \frac{d^3p}{(2\pi)^3(2\varpi_p)} (a_p e^{ipx} + a^+_p e^{-ipx}) (2)

So... where it goes the term a_p e^{ipx} in the expression (1) ?

Can someone kindly show me all the steps?
I know it's a stupid question, but I can not understand.

thank you very much!

 

[WannabeNewton]

The vacuum is annihilated by $a_p$ by definition.

 

[grimx]

Thank you for your reply.
But in theory... a_p it should not destroy the particle created by a^+_p??

What am I doing wrong?

Thank you.

 

[WannabeNewton]

We don't have $a_p a^{\dagger}_p$ in the free KG field. We have $a_p$ attached to the negative frequency modes and $a^{\dagger}_p$ attached to the positive frequency modes so they act independently of one another.

As such $\phi(x)|0\rangle$ simply creates a particle at $x$.

 

[grimx]

Thanks! :)