I was reading the Chapter 14 of the textbook by Philip Phillips on Advanced Solid State Physics, when he introduced the mean-field treatment of the quantum rotor model, he used the method first used in this paper: Quantum fluctuations in two-dimensional superconductors.
Here my question is related to the Hubbard-Stratonovich transformation (H-S transformation) used here to decouple the "spin" interaction:
Tτexp(∫β0dτ ∑kJkSk(τ)⋅S∗k(τ))
here the Sk(τ) is the operator written in the interaction picture, then he introduced the auxiliary field ψk:
Tτexp(∫β0dτ ∑kJkSk(τ)⋅S∗k(τ))=Tτ∫Dψk(τ) e−∫dτ∑kψ∗k(τ)ψk(τ)−2∫β0dτ∑kψk(τ)⋅S−k(τ)
because I learned H-S transformation in the context of path integral, where we write the partition function in terms of functional integral over the fields ("numbers"). So here I want to ask:
- Mathematically if it is well defined to do the H-S transformation on top of operators?
- It seems that the imaginary time ordering operator does not affect the auxiliary field ψk(τ) (can be seen from later on derivations in the book), what's the reason for that?
Answer
Well-defined mathematically depends on how much one wants to be rigorous. Here, I think a physicist would be satisfied with an argument like : working with eigenvectors of Sk, one shows that the formal manipulation with operators is true when working with the eigenvalues, so it is fine to do that in terms of the operators. I don't know how much effort is needed to make that satisfying for a mathematician.
Concerning the second question : the auxiliary field are (complex) numbers, so they all commute. Therefore, there isn't any ambiguity in their ordering. By construction, in a path integral formalism, the fields are always "time-ordered".
Update : to give a bit more details on the HS transformation for operators, I will use a toy model. Let ˆA be an operator, with eigenstates |a⟩ associated to the eigenvalue a. Then e12ˆA2|a⟩=e12a2|a⟩,=∫dxe−12x2+ax|a⟩,=∫dxe−12x2+ˆAx|a⟩.
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