Generalized Wick's theorem

In summary, the conversation is about evaluating contractions in the form of Wick's theorem. The integral form of the theorem is given and a question is asked about evaluating contractions in a specific form. The answer is provided as a reference to a book and it is mentioned that the topic was part of a takehome midterm exam at a university in the Netherlands.
  • #1
da_willem
599
1
I have the following contour integral form of Wick's theorem (C indicating contraction):

[tex]C[A(z):BC:(w)]=\frac{1}{2 \pi i} \int _w \frac{dx}{x-w} C[A(z)B(x)]C(w) + B(x)C[A(z)C(w)][/tex]

Does anybody know how to evaluate contractions like C[:AB:(z)C(w)]?
 
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  • #2
Yes, it is outlined in Di Francesco's book "Conformal Field Theory" page 189: I'll give you a link to google books since there is a free preview of that chapter :

http://books.google.nl/books?id=keU...X&oi=book_result&ct=result&resnum=7#PPA189,M1

I imagine you found this in a takehome exercise sheet for a String theory course in the Netherlands (it was a takehome midterm exam at UvA)

i also know this is a very late reply but ... oh well :D
 
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  • #3


Generalized Wick's theorem is a powerful tool in the field of quantum field theory, which allows us to simplify complex integrals involving contractions. In this case, the contour integral form of Wick's theorem provides a way to evaluate contractions involving three operators, namely A, B, and C.

The first term on the right-hand side of the equation involves the contraction of A and B, while the second term involves the contraction of A and C. This integral form allows us to evaluate the contractions by integrating over the variable w, which represents the position of the operator C.

However, the question arises on how to evaluate contractions involving more than three operators, such as C[🆎(z)C(w)]. This is a more complex case and requires the use of higher-order Wick's theorem, which involves multiple contractions and integrals.

To evaluate such contractions, we need to use the properties of the operators involved and apply the Wick's theorem recursively. This involves breaking down the contraction into smaller contractions and using the integral form of Wick's theorem to evaluate each of them separately.

In conclusion, while the contour integral form of Wick's theorem is useful in evaluating contractions involving three operators, higher-order Wick's theorem is required to evaluate more complex contractions involving multiple operators. It is a powerful tool in the study of quantum field theory and has many applications in theoretical physics.
 

What is Generalized Wick's theorem?

Generalized Wick's theorem is a mathematical formula used in quantum field theory to simplify the calculation of expectation values of field operators. It allows for the expansion of a product of field operators into a sum of terms that can be more easily evaluated.

Who developed Generalized Wick's theorem?

Generalized Wick's theorem was first developed by the physicist Gian-Carlo Wick in the 1950s. However, the concept of using normal ordering to simplify calculations in quantum field theory was first introduced by the mathematician John von Neumann in the 1930s.

What is the difference between Generalized Wick's theorem and Wick's theorem?

Wick's theorem is a special case of Generalized Wick's theorem, where the field operators in the product are all of the same type. Generalized Wick's theorem allows for the expansion of products with different types of field operators.

How is Generalized Wick's theorem used in practice?

Generalized Wick's theorem is used in theoretical physics, specifically in quantum field theory, to simplify calculations of expectation values and to derive Feynman diagrams. It is also used in other areas of theoretical physics, such as condensed matter physics and statistical mechanics.

What are the applications of Generalized Wick's theorem?

Generalized Wick's theorem has many applications in theoretical physics, including the calculation of scattering amplitudes, vacuum energy, and correlation functions. It is also used in the study of phase transitions and critical phenomena, as well as in the development of new mathematical methods in quantum field theory.

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