1

This is so confusing: books are saying that connected correlator is given by

$$\langle \phi(x_1) \phi(x_2) ... \phi(x_n) \rangle_C = \left.(-i)^{n-1}\frac{\delta }{\delta J(x_1)} \frac{\delta}{\delta J(x_2)}... \frac{\delta W}{\delta J(x_n)}\right|_{J=0} \tag{1}$$

evaluated at source $J=0$ but for $n=0$ point function I am getting

$$\langle 1 \rangle_C=iW[J=0]= \log(Z[J=0])\tag{2}$$

(by definition of cumulant generating functional $W$). For $n>0$ I get only connected diagrams (as expected).

  1. Is this formula valid only for $n>0$ for some reason?

  2. Also, usually by normalization $Z[J=0]$ is set to $1$ but if I actually evaluate it I get divergent sum of bubble diagrams. Why the discrepancy?

Qmechanic
  • 220,844
Nairit Sahoo
  • 375

1 Answers1

1

  1. OP is right: The graphical interpretation of OP's eq. (1) $$ \langle F[\phi]\rangle^c_J~=~F\left[ \frac{\hbar}{i} \frac{\delta}{\delta J} \right]\frac{i}{\hbar} W_c[J]~=~F\left[ \frac{\hbar}{i} \frac{\delta}{\delta J} \right]\ln Z[J]\tag{1}$$ is somewhat weird for a connected $0$-point function $$\langle 1\rangle^c_J~=~\frac{i}{\hbar} W_c[J]~=~\ln Z[J].\tag{2} $$ The connected $0$-point function (2) contains all connected diagrams, cf. OP's title question.

    The standard normalization of eq. (2) is $$\begin{align} Z_{g=0}[J\!=\!0]~=~&1,\cr \langle 1\rangle^c_{g=0,J=0}~=~\frac{i}{\hbar} W^c_{g=0}[J\!=\!0]~=~\ln Z_{g=0}[J\!=\!0]~=~&0.\end{align}\tag{3} $$

    The connected $0$-point function (2) should be compared to the definition of a (not necessarily connected) $0$-point function $$\langle 1\rangle_J~\equiv~\frac{Z[J]}{Z[J]}~\equiv~1.\tag{4}$$

  2. Concerning the normalization of $Z_{g=0}[J\!=\!0]$, see footnote 1 in my Phys.SE answer here.

Qmechanic
  • 220,844