Proof of Wick Decomposition: Expectation Values of Density Matrices

[thatboi]

Hey all,
I am currently looking at a proof on the Wick Decomposition from this paper: https://www.sciencedirect.com/science/article/pii/0003491684900927

Specifically, the part that proves if a state satisfies the Wick Decomposition, then it has a density matrix of a specific exponential form (starts from equation A.11). Can someone explain how (A.13) implies that the expectation values for \rho and \rho' end up being equivalent? I am confused here.
Thanks!

 

[eljose79]

Hi there,

Thank you for sharing your question on the forum. The Wick Decomposition is a well-known method in quantum field theory that allows us to express the expectation value of a product of field operators in terms of the expectation values of individual operators. This is useful because it simplifies calculations and makes it easier to analyze quantum systems.

In the proof you are referring to, the authors use the Wick Decomposition to show that if a state satisfies the decomposition, then it has a density matrix of a specific exponential form. This is shown in equation A.11, where the authors use the Wick Decomposition to rewrite the expectation value of a product of field operators in terms of the expectation values of individual operators.

In equation A.13, the authors then show that the expectation values for the density matrix \rho and the transformed density matrix \rho' are equivalent. This is important because it shows that the Wick Decomposition preserves the equivalence of expectation values for these two density matrices.

To understand this, we need to look at the definition of the transformed density matrix \rho'. As shown in equation A.10, \rho' is obtained by applying the Wick Decomposition to the original density matrix \rho. This means that \rho' is expressed in terms of the expectation values of individual field operators, just like \rho. Therefore, when we compare the expectation values of \rho and \rho', they turn out to be equivalent.

I hope this explanation helps to clarify your confusion. If you have any further questions, please feel free to ask. Good luck with your research!