# How do the vacuum bubbles cancel out in gell-mann-low formular? [duplicate]

In interacting quantum field theory we often want to calculate vacuum expectation values of the interacting theory using the gell-mann-low formular. Lets consider two interacting real scalar fields in $$\phi^3$$-theory up to order of $$\lambda^2$$. \begin{align} <\Omega|T\{\phi(x)\phi(y)\}|\Omega>= \frac{<0|T\{\phi(x)\phi(y)\exp(i\int d^4z\frac{\lambda}{3!}\phi^3(z))\}|0>}{<0|T\{\exp(i\int d^4z\frac{\lambda}{3!}\phi^3(z))\}|0>} \end{align} We can then calculate this vacuum expectation values of the free theory using Wick's theorem. Lets first have a look at the numerator: We get 6 different terms, each of which can be expressed as a feynman-diagram. Two of this diagrams contain "vacuum bubbles", that are connected subgraphs, that do not contain an external point. Now lets have a look at the denominator: Here we get only two non vanishing terms, which are exatly these vacuum bubbles. In the literature I can find, that these vacuum bubbles in the numerator can always be factored out and therefore cancel with the denominator. But I can not see why this is the case, because what we have is something like \begin{align} <\Omega|T\{\phi(x)\phi(y)\}|\Omega>= \frac{\sum_{\textit{connected}}+\sum_{\textit{containing bubbles}}}{1+\sum_{\textit{bubbles}}} \end{align} I see, that a single diagram, that contains one such bubble can be factored in a part containing just the bubble, and one part containing the other connected part, that contains the external points. But I can not see how the bubbles actually cancel out in the formular above, since only two of the 6 diagrams in the numerator contain these bubbles.

• Commented Jan 28, 2023 at 15:10