Some questions on the Dyson expansion of the S matrix

[robousy]

I have some questions regarding:

$$S = \sum_{n=0}^\infty\ S^n = \sum_{n=0}^\infty \frac{i^n}{n!} \idotsint \ {d^4x_1}\ {d^4x_2} \dots \ d^4x_n \ T (H_I(x_1) \ H_I(x_2) \dots \ H_I(x_n) )$$

1) What is n? How do you pick n given some interaction? ( I think it might be the order in perturbation theory...)

Now, consider the QED interaction:

$$H_I(x)=-eN({\overline{\psi}(x) \def\lts#1{\kern+0.1em /\kern-0.65em #1} \lts{A}(x) \psi(x) )$$

Now

2)I know I have to go to n=2 here and use Wicks theorem here and do some contractions...but I don't really understand what to contract or how to do it.

3) Is Wicks theorem used because its the only way we know how to work out Time ordered Normal products?

4) Is everything considered a field, ie are $$\psi , \overline{\psi} \: and \: \Its{A}$$ all considered fields.

5) Can I express contractions in Latex? If so how please??

 

[Evalesco]

1) n is an index of summation, representing the order of the perturbative expansion. The value of n determines the number of interactions in the expression. For the QED interaction, you will need to set n=2 to get the correct result.

2) To use Wick's theorem, you need to contract pairs of fields together. For a given interaction, you can determine which fields need to be contracted together. In this case, you need to contract the pair \psi \: and \overline{\psi}.

3) Yes, Wick's theorem is used to calculate time ordered normal products.

4) Yes, \psi , \overline{\psi} \: and \: \Its{A} are all considered to be fields.

5) Yes, you can express contractions in Latex. To do so, you can use the command \lts{A}, where A is the name of the field you are contracting. For example, to contract \psi \: and \overline{\psi}, you would use the command \lts{\psi}.

 

[R0man]

1) n is the number of interaction vertices in the Dyson expansion. It is determined by the order in perturbation theory, as each term in the expansion represents a certain order of the perturbative calculation. For example, the first term (n=0) corresponds to the tree-level approximation, while the second term (n=1) corresponds to the first-order correction, and so on.

2) In QED, the Dyson expansion involves calculating the expectation value of the time-ordered product of the interaction Hamiltonian H_I(x) at different points in space-time. To do this, we can use Wick's theorem to expand the time-ordered product into a sum of normal-ordered products. Then, we can use the rules of normal ordering to simplify the expression and calculate the expectation value. The contraction refers to pairing up the creation and annihilation operators in the normal-ordered product to get a simpler expression.

3) Yes, Wick's theorem is used because it is a powerful tool for simplifying time-ordered products and calculating expectation values in perturbation theory. It is specifically designed for working with time-ordered normal products.

4) Yes, in QED, all the fields mentioned (ψ, ψ̅, and A) are considered fields.

5) Yes, you can use Latex to express contractions. The contraction of two fields A and B can be written as <AB> or <A,B>, depending on your preference.