In Quantum field theory and the standard model from Schwartz, it is written on page 91 that bubbles are "diagram that have connected subgraphs not involving any external point"

And they say that such diagram will not contribute to the Gell-Mann-Low formula because they will simplify with the denominator.

But I don't understand this.

Indeed, if I only have the condition: "diagram with connected subgraph not involving external points", I could end up with a graph that involves 2 subgraphs.

  • One where the external points are connected to some of the internal ones
  • One where there are only internal points

It fullfill the condition as the second graph doesn't involve external points. And if they in fact meant that all the connected subgraphs musn't involve external points, for me it is not possible as I will always have external points involved in the numerator of Gell Mann-Low.

And in such a case because the first subgraph mixes internal & external points it will never be able to cancel with the denominator of Gell Mann Low?

Shouldn't be the definition of bubbles the following:

A graph is a bubble if all the external points are not connected to internal points. And in this case it will be simplified by the denominator of Gell Man Low when we will do the perturbative expansion.

What do I misunderstand?


1 Answer 1


You just need to read the statement on page 91 more carefully. Schwartz says:

To see the cancellation, note that the extra diagrams both include bubbles. That is, they have connected subgraphs not involving any external point. [emphasis present in the original]

He's not saying that diagrams with connected subgraphs that involve no external points are bubbles, he's saying they include bubbles. A bubble is indeed a diagram with no external points. The cancellation comes about by the denominator being the sum over bubble and the numerator factoring into the sum over diagrams with no bubble multiplied by the sum over bubbles, because disconnected diagrams factor into the product of their connected components.

  • $\begingroup$ Ok thank you. Yes I probably misunderstood what he said. Just to be sure, when you say "The cancellation comes about [...] the numerator factoring into the sum over diagrams with no bubble multiplied by the sum over bubbles". Do you agree with me if I say that the "no bubble" subgraph of the total diagram must only contain external points to be able to cancel with the denominator ? If it contains a single internal points it wouldn't be able to cancel. (Because as you said a bubble doesn't contain external points. So a "non bubble" can mix external & internal) $\endgroup$
    – StarBucK
    Oct 28, 2017 at 13:25
  • 1
    $\begingroup$ @StarBucK Yes, "no bubbles" means no subgraphs that do not contain at least one external point. $\endgroup$
    – ACuriousMind
    Oct 28, 2017 at 13:54

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