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Chapter V
Interacting Fields
Lecture 4
Books Recommended:
Lectures on Quantum Field Theory by Ashok
Das
 Advanced Quantum Mechanics by Schwabl
Wick’s Theorem
Perturbative expansion of S-matrix contains
Time ordered product of interaction
Hamiltonian.
To find transition amplitude one need to find
.
In the perturbative expansion of S matrix,
Those terms will contribute in which
Annihilation operator of inital field particle
act on the initial state |i>.
Time ordering places the operators occurring
earlier in time on right of operator occurring later
time. Creation operator occurring earlier in time
will be placed to right of annihilation operators.
This is opposite to normal ordering in which
annihilation operators are placed to write
of creation operators.
Wick’s theorem gives the relation between time
ordered product and normal ordered product
of the field operators.
Consider the case of real scalar field
Consider case
If
and calculate
We define contraction for two fields as
Which is nothign what definiton of propagtor
Thus, we have
For complex fields
Note
In general for a string of fields, we write
Example
Note
Wick’s 1st theorem
Wick’s 2nd theorem
In above contractions cannot be between the
fields within same normal ordered product.