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72 views

Perturbation expansion with path integrals

This is from Hugh Osborn's 'Advanced Quantum Field Theory' notes, Lent 2013, page 15. I want to evaluate the expression $$ Z = \exp\Big(\frac{1}{2} \frac{\partial}{\partial \underline{x}} . A^{-1} \...
1
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1answer
53 views

Problem with expansion of normal ordering

I am reading normal ordering..and far now I'm able to understand. I am stuck in third line from second expression in the book Lectures On Quantum Field Theory By Ashok Das in page no. 237. It is ...
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0answers
145 views

Problem with Wick's theorem (normal ordering of a contraction)

Taking the example of two bosonic fields, Wick's theorem is \begin{equation} T\{\phi(x_1)\phi^\dagger(x_2)\} = N\{\phi\phi^\dagger\} + N\{(\phi\phi^\dagger)_c\} \end{equation} where the subscript $c$ ...
19
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2answers
625 views

There are too many Wick's Theorems!

This is a follow-up question to QMechanic's great answer in this question. They give a formulation of Wick's theorem as a purely combinatoric statement relating two total orders $\mathcal T$ and $\...
1
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0answers
249 views

Mean field approximation in BCS theory

Bardeen, Cooper and Schrieffer's (BCS) theory describes spinful Fermions that mutually interact via an attractive contact interaction. The general Hamiltonian reads in second quantization $$H = \sum_{...
1
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3answers
656 views

Normal ordering of the commutator between annihilation and creation operator

According to the commutation relation of annihilation and creation operators, $$[a,a^{\dagger}]=1. \tag{1}$$ I would like to calculate the vacuum expectation value of the normal order of this ...
2
votes
1answer
2k views

Proof of Wick's theorem

I'm tackling proof of Wick's theorem. By induction. Let us suppose we have already proved $$ C_2 \cdots C_n = N(C_2 \cdots C_n + (\text{all possible contractions}) ) \quad (C_i=a\,\, \text{(...
1
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1answer
315 views

Wick theorem on correlator in quantum mechanics

I have an exercise to calculate the following one dimensional integral explicitly using the Wick theorem: $$<q(t_1)q(t_2)q(t_3)q(t_4)q(t_5)q(t_6)>=\frac{\int Dq q(t_1)q(t_2)q(t_3)q(t_4)q(t_5)q(...
2
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0answers
210 views

Wick contractions with respect to an arbitrary state

I've been working through the following set of slides related to Wick's theorem: http://www.euroschoolonexoticbeams.be/site/files/2008_JDobaczewski_lecture.pdf From slide 19 onwards the following is ...
0
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0answers
69 views

Truncation of the product of operators

Given that the operators $A,B $ and $C$ commute with each other, how can we justify the following approximations: $<ABC> ≈ <A><BC> + <AB><C> + <AC><B>$ - 2$&...
2
votes
2answers
214 views

Quantum Operators: An Identity

I came across the following neat property: For an operator $\hat{A}$ which is a linear combination of creation and annihilation operators, we have: $$ \langle e^{\hat{A}} \rangle = e^{\langle \...
1
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0answers
212 views

How to apply Wick's theorem in 2nd quantization for Spin Density Operators?

I am trying to work out a correlation function consisting of two spin density operators. Once I rewrite everything in 2nd quantized form, I am unsure of how to apply wicks theorem because the paul ...
5
votes
1answer
556 views

Wick's theorem for calculating OPE

I am trying to understand a calculation using Wick's theorem. Let $T(z)$ be the analytic part of a stress-energy tensor, and $\phi(z)$ a free boson field. Now, $$T(z)\partial_{w}\phi(w)=-2\pi:\...
5
votes
2answers
715 views

Why is the Wick contraction in HFB or BCS equal to a single-particle density?

I'm trying to understand how in Hartree-Fock-Bogoliubov (HFB) or BCS theory we can write a product of creation/annihilation operators as single-particle densities under the guise of "Wick's theorem". ...
8
votes
1answer
2k views

Time-ordering vs normal-ordering and the two-point function/propagator

I don't understand how to calculate this generalized two-point function or propagator, used in some advanced topics in quantum field theory, a normal ordered product (denoted between $::$) is ...
4
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2answers
6k views

When can I use Wick's theorem?

Wick's theorem means that for fermions, a four point correlation function (for example) can be written in terms of two point correlation functions: \begin{equation} \langle b_l^\dagger b_l b_m^\...