What is generating functional and vacuum-to-vacuum boundary conditions in QFT?

[VietBrian]

Hello everyone :) I'm reading the book QFT - L. H. Ryder, and I don't understand clearly what are the generating functional Z[J] and vacuum-to-vacuum boundary conditions? Help me, please >"<

 

[Polyrhythmic]

I don't know that book, but if it is an introduction to QFT, it should definitely explain those terms. The generating functional is given by

$Z \left[ J \right]=<0|0>$,

which can be expressed in terms of a path integral. It is called "generating" functional because you can apply functional derivatives with respect to J on it in order to gain vacuum expectation values for operators. What they mean by vacuum-to-vacuum boundary conditions could depend on the context, but it is probably the normalization

$Z \left[ 0 \right]=0$.

 

[VietBrian]

Thank you very much! ^^

I had carefully read the book again. The vacuum-to-vacuum boundary conditions turned out to be $\psi(t_i) = \psi_i$ and $\psi(t_f) = \psi_f$.

:D And, are the operators you talk about above the field operators?
$\dfrac{\delta Z[J]}{\delta J(t_1)\ldots \delta J(t_n)} = i^n \bra 0 \lvert T(q(t_1)\ldots q(t_n)) \rvert 0 \ket$

 

[Polyrhythmic]

Ah, I see.

Yes, that's exactly what I meant!

 

[VietBrian]

Oh, yeah, it's now clearer for me ^^ Thank you!

 

[Polyrhythmic]

$Z \left[ 0 \right]=0$

It's meant to be
$Z \left[ 0 \right]=1$,
sorry!

 

[VietBrian]

Does it mean vacuum is still vacuum if there is no source ?

 

[Polyrhythmic]

I guess you could put it like that.