Why Do Only Certain Equal Time Contractions in Quantum Field Theory Not Vanish?

[Vic Sandler]

In the second edition of Mandl & Shaw, page 271 it says that the only equal time contractions in eqn (12.113) that do not vanish are of the form \overline{\psi}(x_i)\not{A}(x_i)\psi(x_i). Is he referring to the fact that if you contract everything in \overline{\psi}(x_i)\not{A}(x_i)\psi(x_i)\overline{\psi}(x_i)\not{A}(x_i)\psi(x_i), you get eqn (7.5f) on page 102, and this vanishes as stated near the bottom of page 109? If not, how do we know that these equal time contractions vanish?

And how should I create a Feynman slash in latex?

 

[andrien]

Actually the diagram which you get by that term is shown on the next page and it is known as the tadpole diagram.
http://en.wikipedia.org/wiki/Tadpole_%28physics%29
It does not vanish automatically.However this diagram arises in higher order perturbation theory where the loop is coupled via a virtual photon to an electron line.This gives a self energy contribution but it turns out that this diagram does not have any physically observable consequence since it's size is independent of momentum of electron.Unlike the self energy correction, it can be fully absorbed into a renormalization constant which at the end drops out of calculation.It can be simply acheived if we just leave out the contribution from this tadpole diagram.
Also\not A

 

[Vic Sandler]

The sentence in the book embodies two facts. One is that the contribution of the tadpole which represents the contraction of the fermions in \overline{\psi}(x_i)\not{A}(x_i)\psi(x_i) is non-zero. The other is that the contribution from any other equal time contraction vanishes. It is this second fact that I am asking about. An example of another equal time contraction might be \overline{\psi}(x_i)\not{A}(x_i)\psi(x_i)\overline{\psi}(x_{i+1})\not{A}(x_{i+1})\psi(x_{i+1}). In this case, x_i^0 = x_{i+1}^0.The diagram for this appears on page 110. Is this what the author refers to when he says that the tadpoles are the only equal time contractions that do not vanish?

Actually, it just hit me that he might simply mean that the contractions of the photon A with either of the fermions \psi vanish and that the case x_i^0 = x_{i+1}^0 is excluded.

 

[andrien]

The sentence in the book embodies two facts. One is that the contribution of the tadpole which represents the contraction of the fermions in \overline{\psi}(x_i)\not{A}(x_i)\psi(x_i) is non-zero. The other is that the contribution from any other equal time contraction vanishes. It is this second fact that I am asking about. An example of another equal time contraction might be \overline{\psi}(x_i)\not{A}(x_i)\psi(x_i)\overline{\psi}(x_{i+1})\not{A}(x_{i+1})\psi(x_{i+1}). In this case, x_i^0 = x_{i+1}^0.The diagram for this appears on page 110. Is this what the author refers to when he says that the tadpoles are the only equal time contractions that do not vanish?

Actually, it just hit me that he might simply mean that the contractions of the photon A with either of the fermions \psi vanish and that the case x_i^0 = x_{i+1}^0 is excluded.

No,you got it wrong.Those vacuum diagrams does not vanish simply.In the massless limit you can show as the author has shown just below the tadpole graph that using the dimensional regularization that diagram vanish because integrand is an odd function but if there will be mass then you will have (/p+m) type thing in numerator and in this if you see this factor m times the integrand you will get a divergent result at short distances.If you will include a regulator with a cutoff then the integrand will diverge as λ²,which will become infinite in the limit of λ→∞.All those vacuum diagram gives these divergent results.There are an infinite numbers of them,these all infinities can be absorbed into a renormalization constant which at the end drops out of calculation.So the sum is over all connected diagrams leaving those vacuum diagram because they just come with infinity and cancels and give no observable effects.So you just omit them.If you read path integral formalism,then you will see that vacuum to vacuum amplitude is given by the sum of all connected diagrams i.e. e^(iW) where W is the sum over connected diagrams,every vacuum diagram is LEFT OUT.