How to draw Feynman diagrams

[chern]

I am reading the book tilted "quantum field theory in a nutshell(second version)" by A.Zee. On the page 45, for example there is a term [tex]\frac{1}{m^2}λJ^4[/tex]. The question is that how to related it to the Figure 1.7.1. or that, How can I draw the three diagrams in Figure 1.7.1 from the term [tex]\frac{1}{m^2}λJ^4[/tex]? Many thanks!

 

[chern]

Maybe I can relate the term to the figure 1.7.1 (a). Because to get [tex] λJ^4[/tex]. It needs one λ, therefore with a differentating operator [tex](\frac{d}{dJ})^4[/tex], this means one λ eliminates four "J". In the Figure 1.7.1 (a), which is just the vertice. But how to get the other diagrams (b) and (c) in the Figure 1.7.1. what is the physical meaning for the two diagrams?

 

[PeterDonis]

there is a term

1m2λJ4​

Actually the term is

$$
\left( \frac{1}{m^2} \right)^4 \lambda J^4
$$

Notice that the ##1 / m^2## is also raised to the fourth power. Each power of ##1 / m^2## is a line; each ##\lambda## is a vertex; and each ##J## is a source (i.e., a line that ends at an "edge" of the diagram, not connecting to anything else--the Figures mark these with a ##J## to make it clearer).

Figure I.7.1 (a) is easy to relate to what I've just said: there is one vertex with four lines going from it to four sources. But if you look at (b) and (c), you will see that they also meet that definition: there are four ##J##'s, one vertex (marked with a ##\lambda##), and four lines. In (b), one line, on the left, connects two ##J##'s; the second and third lines connect the two sources on the right with the vertex; and the fourth line is the loop that connects the vertex with itself. In (c), two of the lines connect pairs of sources; the vertex and other two lines are the figure 8 on the right, disconnected from the sources.

As for the physical meaning of the diagrams, you have to be careful, because the diagrams don't really represent separate physical processes. They are just a way of organizing various terms in an equation that describes the total interaction involved and allows us to calculate probabilities. But if you bear in mind that the following is just heuristic and should not be taken too literally, here is how one might describe the three diagrams:

(a) is two particles being emitted by two sources, interacting (at the vertex), and separating again to be absorbed by two sinks. (A "source", i.e., a ##J##, can either emit or absorb particles.)

(b) is a particle being emitted by a source and going directly to a sink to be absorbed, with nothing else happening to it; plus a second particle being emitted by a source, interacting with itself by emitting a second particle and then absorbing it immediately (the vertex and loop), and finally being absorbed by a sink.

(c) is two particles each emitted by a source and going directly to a sink to be absorbed, with nothing else happening to them; plus, off somewhere else, a pair of particles popping out of the vacuum, interacting with each other, and then disappearing into the vacuum again (the vertex and figure eight). You can see why you can't take these descriptions too literally.

More concretely, the physical meaning of the three diagrams is that, if all we know (i.e., all we can actually measure) is that we have two sources and two sinks--i.e., two particles are emitted and two particles are absorbed--then these diagrams are all the ways those two particles could potentially interact in between, and so must be taken into account in computing the probability of observing two particles being emitted and two particles being absorbed. But remember, these are just three of an infinite number of possible diagrams; other diagrams can have different numbers of lines, vertexes, or sources.

 

[chern]

To Peter Donis ,Thank you so much!

 

[naima]

the same diagrams in ##\phi^4## are also valid in statistical mechanics. In QFT we can say that a particle is created and is later annihilated. In statistical Mechanics there is not a time dimension in the plane of the diagram. How can we describe what "happens" ?

 

[PeterDonis]

In statistical Mechanics there is not a time dimension in the plane of the diagram. How can we describe what "happens" ?

What does the ##\phi## field represent in statistical mechanics?

 

[naima]

It may be a spin field the probability to get ## \phi (x1 x2 x3 ...)##
is something like ## e^{-^T \phi A \phi + \phi^4}##
See Ginzburg-Landau it seems that it comes from a wick rotated theory but at the end there is no Wick rotation. All remains euclidean.

 

[naima]

There is a paragrah (9.3) in Peskin and Schroeder on this subject:
The analogy between QFT and Statistical Mechanics.