Dirac Contraction: Evaluating Propagators at Same Point

[ghotra]

I don't have pstricks...so I am going to use words.

$$\text{contraction}\{ \overline{\psi}(x_1) \psi(x_1) \}$$

My question: Propogators are usually dealing with different points...but what is the contraction of two quantities evaluated at the same point.

Thanks.

 

[ghotra]

Wick's theorem doesn't apply when you have equal-time contractions. I haven't learned this yet, but these equal-time contractions are somehow lumped into the interacting vacuum state.

Is this correct?

 

[Physics Monkey]

Wildly infinite. Seriously though, the the fact that the anticommutation relation between the field and its conjugate momentum is proportional to a delta function should make it clear that the product of two field operators evaluated at the same spacetime point is divergent.

 

[Physics Monkey]

And yes, propagators evaluated at the same space time point can be associated with vacuum fluctuations. They generally have the meaning of some kind of self energy. For example, in $$\phi^4$$ theory you can have disconnected figure eights and so forth, these are vacuum fluctuations. You can also have a line with a loop attached in the middle at a single point, such diagrams represent self interaction.

 

[abszero]

So on a related question, if I were trying to evaluate some sort of thermodynamic potential, what would the disconnected diagrams contribute, if anything? What would the physical interpretation be?

 

[Physics Monkey]

In short: Disconnected diagrams always contribute nothing to thermodynamic functions.

In detail: The linked cluster theorem guarantees that such diagrams exponentiate and factorize. The partition function is determined by all the diagrams, connected or not. However, thermodynamic information is contained in the log of the partition function. This log has a special name (besides being the free energy): it is the cumulant generating function (the partition function is the moment generating function). The linked cluster theorem tells you that disconnected diagrams always cancel when calculating the cumulants (which contain the thermodynamic information).

 

[D3mo]

This is off topic, but Physics Monkey, have you read the private message I sent you. Sorry to disrupt anything.