How Does Causality Emerge in Quantum Field Theory Expressions?

[Neitrino]

Gentlemen,
Could you help me please, I am sure it is not even worthy of your attention, but anyway...

In Peskin, Schroeder - from expresion <0|\phi(x)\phi(y)|0> survives <0|a_p a_q^\dag|0> so it creates one-particle state |q> at position y and another one-particle state | p> at postion x. But how do I intuitively see that causuality/propagation of particle between these positions is imbeded and considered in that expresion?

And another question <0|\phi(x)|p>=...e^ipx formula 2.42

it's said that it is a position-space representation of the state |p> just as in NR QM <x|p>, so it should be projection of single-particle |p> state onto the <x| baisis and what vectors/basis that state is projected on? where is that <x| vectors in 2.42

P.S. I am sure i missed something very simple in understanding of above and that's why posting such "silly" questions

Thanks in advance

 

[snooper007]

(1) I think the expresion &lt;0|\phi(x)\phi(y)|0&gt;
survives &lt;0|a_p a_q^\dag|0&gt; means:
&lt;0|a_p^\dag a_q^\dag|0&gt;=0 and &lt;0|a_p a_q|0&gt;=0;
only &lt;0|a_p a_q^\dag|0&gt; survives, of course p and q are arbitary,
not single p and single q. the final result will be an integral over all possible p or q.

(2) &lt;0|\phi(x)=&lt;x|, this is a simple calculation.
there is no special physical significance here, the author, I guess, just mentioned NR
case to make the formula be easily understood.

 

[Neitrino]

(2) &lt;0|\phi(x)=&lt;x|, this is a simple calculation.

Dear Snooper007 thks for ur reply..
but &lt;0|\phi(x) it is a complex conjugation of \phi(x)|0&gt; (as u mentioned in QM forum). So &lt;0|\phi(x)=\int{\frac{d^3 p}{(2\pi)^3}\frac{1}{2E_p}e^{ipx}&lt;p|

but with &lt;0|\phi(x)=&lt;x| I am confused <-How/why it's that?