Equal time contractions in Wick contractions

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In summary, the conversation discusses the use of equal time contraction terms in computing time ordered products of interaction hamiltonians in the Dyson expansion. It is stated that Wick's theorem will not generate any tadpole contributions in this case, as the loop starts and finishes at the same point. However, there is a question about the elimination of infinities through normal ordering, as Wick's theorem will still generate divergent bubble contributions. The normal ordering of the hamiltonian is explained as a means of renormalization to remove infinities arising from self-interactions. It is also mentioned that normal ordering can lead to unphysical results, such as non-conservation of 4 momentum, if not used. The questioner is seeking a
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CAF123
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One usually ignores equal time contraction terms when computing time ordered products of interaction hamiltonians arising in the Dyson expansion. E.g it is often said $$T(N(AB\dots)_{x_1} \dots N(AB \dots)_{x_n}) = T((AB \dots)_{x_1} \dots (AB \dots)_{x_n})_{\text{no E.T.C}}$$

Few questions

a)I think Wick's theorem applied to these expressions will not generate any tadpole contributions since there the loop starts at some x and finishes at the same x. Is that correct?

I've read that the S-matrix can be defined as the time ordered product of an exponentiated normal ordered interaction hamiltonian, the normal ordering a means of renormalisation eliminating infinities arising from such self-interactions.

b) Why is this statement completely true? Wick's theorem applied onto above will certainly give e.g a bubble contribution (comprising two propagators, one of which starts at some x and finishes at y and the other vice versa) which is divergent in four dimensions so it seems there is not a complete elimination of infinities.

c) Is there a more physical reasoning why we normal order the hamiltonian in the S-matrix? I've read it gives unphysical results such as non conservation of 4 momentum if we don't normal order but didn't understand this completely. Moreover, as far as I can see, the tadpole diagrams are not generated by normal ordering so it seems we lost contributions (or perhaps the whole process is for 1PI /amputated diagrams?)

Thanks!
 
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Basically I'd just like to understand the motivation behind using a normal ordered hamiltonian in the definition of the S-matrix. I don't see how this would give tadpole diagrams in any Wick expansion given the equation I cite in my OP (tadpole diagrams involve contractions between field at the same space time point). Anyone?
 

1. What are equal time contractions in Wick contractions?

Equal time contractions in Wick contractions are a mathematical tool used in quantum field theory to simplify the calculation of correlation functions. They involve contracting two field operators at the same time, and are represented by a special symbol called the Wick symbol.

2. Why are equal time contractions important in quantum field theory?

Equal time contractions are important because they allow us to calculate correlation functions in quantum field theory without having to consider all possible contractions between field operators. This greatly simplifies the calculations and allows us to make predictions about the behavior of particles and fields in a more efficient way.

3. How do equal time contractions affect the interpretation of correlation functions?

Equal time contractions affect the interpretation of correlation functions by removing the possibility of particles interacting with themselves at the same time. This is important because it ensures that the correlation functions are well-defined and physically meaningful.

4. Can equal time contractions be used in all quantum field theories?

Yes, equal time contractions can be used in all quantum field theories. They are a general technique that can be applied to any field theory, regardless of the specific interactions or symmetries involved.

5. Are there any limitations to the use of equal time contractions?

While equal time contractions are a powerful tool in quantum field theory, there are some limitations to their use. They are most useful in theories with a large number of fields, and may not be as efficient in theories with a small number of fields. Additionally, they may not be applicable in certain cases where the fields do not commute with each other.

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