Wick's Theorem for free fields only

In summary, Wick's theorem applies to Gaussian integrals in quantum field theory, where free fields correspond to these integrals. It is valid only if the initial state is given by a statistical operator of a certain form, and is often used in the interaction picture. The best source for learning about the Schwinger-Keldysh real-time formalism is Danielewicz's Quantum Theory of Nonequilibrium Processes II.
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I'm not quite following why Wick's Theorem only applies to free fields. What part of the argument depends on a free field assumption?
 
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  • #2
Wick's theorem is about Gaussian integrals. In quantum field theory, free fields correspond to Gaussian integrals.

http://www.colorado.edu/physics/phys7240/phys7240_fa14/notes/Week8.pdf
 
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  • #3
It can be used for that application, but the actual statement of the theorem just relates time-ordered operator products to normal-ordered operator products and contractions. I'm not seeing why the operators in question have to be free-field.
 
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You cannot introduce even a definition of normal ordering without a mode decomposition of a field in terms of annihilation and creation operators with respect to some single-particle basis. Such a decomposition only exists for free fields and that's why one uses the interaction picture.

In addition, it's important to note that Wick's theorem is valid if and only if the initial state is given by a statistical operator of the form ##\hat{\rho}=\exp(-\hat{A})##, where ##\hat{A}## is an appropriate single-particle operator. The special case of vacuum QFT, i.e., when ##\hat{\rho}=|\Omega \rangle \langle \Omega |## can be taken as the zero-temperature limit and zero-chemical potential(s) limit of the grand-canonical ensemble, ##\hat{\rho}=\exp(-\beta \hat{H})/Z## with ##Z=\mathrm{Tr} \exp(-\beta{\hat{H}})##, ##\beta=1/(k_B T)##.

For a very good introduction on these issues, see

Danielewicz, P.: Quantum Theory of Nonequilibrium Processes II. Application to Nuclear Collisions, Ann. Phys. 152, 305–326, 1984
http://dx.doi.org/10.1016/0003-4916(84)90093-9
 
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  • #5
Thank you, that puts me on the trail.
 
  • #6
Yes, this is the best source to start learning about the Schwinger-Keldysh real-time formalism (NB: If Keldysh is expected to be in the audience listening to your talk, better call it the "Keldysh real-time formalism" :-)).
 

1. What is Wick's Theorem for free fields only?

Wick's Theorem for free fields only is a mathematical tool used in quantum field theory to simplify the calculation of correlation functions of free fields. It states that the expectation value of a product of field operators can be expressed as a sum of products of the expectation values of individual field operators.

2. How is Wick's Theorem derived for free fields only?

Wick's Theorem for free fields only can be derived from the commutation relations of the field operators and the definition of the normal ordering of operators. It is essentially a consequence of the linearity and locality properties of quantum field theory.

3. What are the assumptions made in Wick's Theorem for free fields only?

Wick's Theorem for free fields only assumes that the fields are free, meaning they do not interact with each other. It also assumes that the fields are bosonic or fermionic fields, and that they are in their ground state.

4. Can Wick's Theorem be applied to interacting fields?

No, Wick's Theorem for free fields only cannot be applied to interacting fields because it assumes that the fields are free and do not interact with each other. However, there are extensions of Wick's Theorem that can be applied to certain types of interacting fields.

5. What are the benefits of using Wick's Theorem for free fields only?

Wick's Theorem for free fields only simplifies the calculation of correlation functions of free fields by breaking them down into products of simpler expectation values. This allows for easier and more efficient calculations in quantum field theory.

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