Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [wick-theorem]

A combinatoric procedure in QFT of reducing arbitrary products of creation and annihilation operators to sums of products of pairs of these operators. A string of such operators is rewritten as the normal-ordered product of the string, plus the normal-ordered product after all single contractions among operator pairs, plus all double contractions, etc., plus all full contractions.

0
votes
0answers
12 views

Euclidean Feynman Rules derivation using Wick Theorem

When studying perturbation theory and Feynman rules, the standard derivation seems to start from the S-Matrix element in the interaction picture, and expands it into some series, after which Wick's ...
2
votes
0answers
61 views

Error in proof of Wick's theorem

I am having a little trouble proving wick's theorem. I'll start from the last step that I know is correct. We define $$\left\langle \prod_{j=1}^{2m}x_{i_j}\right\rangle:=\left.\frac{\partial^{2m}}{\...
1
vote
1answer
66 views

Feynman diagrams in Gaussian integrals

I am looking for suggestions for material regarding Feynman diagrams for gaussian integrals. I am looking for something of the sort of: Pedro Vieira, Statistical Physics Applied to Quantum Field ...
0
votes
0answers
46 views

How does one calculate Fourier transform of Feynman propagator?

I am struggling with calculating the following integral on Sredinicki: How did he get the second line of (10.6)? That is, how did he calculate the Fourier transform of Feynman propagator?
0
votes
0answers
49 views

How to solve this particular problem of Wick's theorem?

So I know the basics of Wick's theorem, but unsure about how to solve this time ordered product of a term that involves normal ordering. Is it just simply the sum of all possible contractions, but no ...
0
votes
0answers
15 views

Gutzwiller renormalization factors

I am computing the expectation value of the kinetic term of a tight-binding model, respect to the Gutzwiller wavefunction, in the limit of infinite lattice-coordination, i.e using these constraints (...
0
votes
0answers
22 views

$\psi\psi \longrightarrow \psi\psi$ scattering in Scalar Yukawa model

In David Tong's lecture notes, Equation 3.48 In line 2, how is $|0\rangle \langle 0|$ introduced between $\psi^{\dagger}(x_1)\psi^{\dagger}(x_2)\psi(x_1)\psi(x_2)?$ Why is $\langle p_2',p_1'|\psi^{\...
0
votes
0answers
29 views

Expectation Value Of The Double Occupancy Operators' Product

I want to prove the relation \eqref{eq:Metz_relation} that i found in this article. \begin{equation} \left\langle\varPhi_0\right|\prod_{i} \hat{D}_i\left|\varPhi_0\right\rangle= \left\langle\varPhi_0\...
2
votes
0answers
92 views

Physical Meaning of the Gutzwiller Constraints

I have a doubt on the constraints for the expecation values obtained by Bünemann et all. First i want to introduce my notation To analytically solve a tight-binding model, \begin{equation} \hat{H}= ...
1
vote
0answers
75 views

What is radial ordering?

In my String theory lecture radial ordering was introduced and I don't understand what it is. My first guess was $$R(A(z)B(w)) = A(z)B(w)\Theta(|z|-|w|) + B(w)A(z)\Theta(|w|-|z|).$$ But then we have ...
1
vote
1answer
90 views

Expectation value of a path-ordered exponential

Let us define our path-ordered operator $\overrightarrow{U}\left(t_1,t_2\right)$: $$ \overrightarrow{U}\left(t_1,t_2\right)=\overrightarrow{\mathcal{P}}\exp\int_{t_1}^{t_2}dt\,\mathcal{O}\left(t\...
1
vote
1answer
297 views

Symmetry factor in $\phi^4$ theory

I'm having trouble while trying to understand what the symmetry factor of a Feynman diagram really is. From books I get that it is a geometrical factor that you get by the number of ways in which you ...
0
votes
0answers
59 views

Calculation of a 4-point function by path integrals

In Srednicki's book in chapter 8 a four-point function is computed as a sum of products of propagators: $$<0|T\phi(x_1)\phi(x_2) \phi(x_3)\phi(x_4)|0> = \frac{1}{i^2}[\Delta(x_1 -x_2)\Delta(...
0
votes
0answers
86 views

Time ordering, normal ordering and Wick's contraction

I'm reading chapter 4 of Peskin & Schröder, and I'm confused how they express the time ordering of two fields: let $T$ denote time ordering, $N$ the normal ordering and I use $C$ for the ...
2
votes
1answer
70 views

Kac-Moody algebra, proof of parameters calculation

I'm following the notes "Ginsparg - Applied Conformal Field Theory" (https://arxiv.org/abs/hep-th/9108028) and I'm stuck on a proof at page 140 about Kac-Moody algebras. I would like to prove that $\...
2
votes
0answers
65 views

Conformal transformation of a vertex operator before normal ordering

Let us consider a free scalar boson $\varphi(z,\bar{z})$ on the complex plane and assume the following two-point correlation function \begin{eqnarray} \langle\varphi(z,\bar{z})\varphi(w,\bar{w})\...
1
vote
0answers
83 views

Apparent problem in using Wick's theorem to calculate matrix elements of two body operators

In the second quantized notation, a two body operator $\hat{O}$ can be written as $$\hat{O} = \sum\limits_{x_1,x_2,x_3,x_4} O_{x_1,x_2,x_3,x_4} a^\dagger_{x_1}a^\dagger_{x_2}a_{x_4}a_{x_3} ,$$ where ...
1
vote
0answers
90 views

Operator product expansion of more than 2 operators in CFT

I’m confused about OPE in 2d CFT. I’ve found difficulties in taking OPE of the product of 3 operators. Consider the following operator product. \begin{align*} O_1(z) :O_2(w) O_3(w): = O_1(z)\frac{1}{...
1
vote
0answers
106 views

What exactly are we doing when we “invent” Feynman Diagrams?

So, I am trying to derive the Feynman rules for Yukawa theory (following the section in Peskin). Specifically, for the process 2 fermions $\rightarrow$ 2 fermions. To second order, I then have that ...
0
votes
0answers
115 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 $c$ ...
4
votes
2answers
138 views

Radial ordered commutation relation

In the book Conformal Field Theory of Francesco, Mathieu and Sénéchal, in Sec. 6.1.2, the authors state that the integral $$ \oint_w \mathrm{d}z~ a(z)b(w) ~=~ \oint_{C_1} \mathrm{d}z~ a(z)b(w) - \...
18
votes
2answers
568 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 $\...
3
votes
2answers
130 views

OPE double-contractions between $T$ and $e^{ikX}$

I am reading David Tong's lecture notes chapter 4 http://www.damtp.cam.ac.uk/user/tong/string.html On the top of page 82 in the eq. before eq. (4.27), we are computing the OPE between $T$ and $e^{ikX}...
0
votes
1answer
65 views

What is the calculation rule of the normal ordering operator?

Here $\phi_I$ is just the free Klein-Gordon field. So, this field is decomposed of two components shown above. Now let $N$ be the normal ordering operator. Then, I think that $N(\phi_I^+(x)\phi_I^-(y))...
2
votes
1answer
362 views

What is a contraction in QFT?

I have been reading QFT and I am stumbling upon the idea of Wick's theorem. The correlation functions have something to do with "contractions". I want to understand what the physical meaning of a ...
1
vote
1answer
168 views

Wick contraction and propagator confusion

I am having trouble understanding the how the Wick contraction leads to the Feynman propagator for scalar fields. The Feynman propagator can be written as $$ D_F(x-y)=\langle 0 | T(\phi(x) \phi(y)) | ...
-1
votes
1answer
100 views

Computing the OPE of $T : \mathrm{e}^{ikX} : $ [closed]

I've hit a stumbling block where I'm just not seeing how to get from line to line in the following calculation from David Tong's strings notes. Can someone spell out how line 1 becomes line 2 in the $\...
0
votes
1answer
151 views

Why do we have $[\phi_1^+,:\phi_2\phi_3:]=:[\phi_1^+,\phi_2^-]\phi_3:+:\phi_2[\phi_1^+,\phi_3^-]:$?

How $$[\phi_1^+,:\phi_2\phi_3:]=:[\phi_1^+,\phi_2^-]\phi_3:+:\phi_2[\phi_1^+,\phi_3^-]:$$ with $\phi_i=\phi(x_i)$ field operators ($\phi_i^+$ is the annihilation part while $\phi_i^-$ is the creation ...
1
vote
1answer
141 views

Time ordered product of bilinear functions for Dirac-field

If we have two Observables (bilinears of Diracfield $\psi(x)$) $O_1(x)=\bar{\psi}(x)\Gamma_1\psi(x)$ and $O_2(y)=\bar{\psi}(y)\Gamma_2\psi(y)$ and if we calculate their time ordered product $T(O_1(x) ...
3
votes
1answer
392 views

Wick Theorem: Performing contractions in the right order

The first line is one of four terms that one gets after applying Wick theorem to the time-ordered product of these field operators and as far as i understand it is just a short-hand notation for which ...
1
vote
0answers
223 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
vote
1answer
139 views

Fermion density with Wick's theorem

I want to calculate the expectation value \begin{equation} \langle\textrm{F}\rvert\Psi^\dagger_{m_1}(x_1)\Psi_{m_1}(x_1)\Psi^\dagger_{m_2}(x_2)\Psi_{m_2}(x_2)\lvert\textrm{F}\rangle \end{equation} ...
2
votes
0answers
62 views

Ambiguity in free field operators

I am interested in the ambiguities which exist in defining the composite free field operators--i.e., operators corresponding to monomials of the fundamental field operator (and their derivatives). In ...
1
vote
1answer
366 views

Question on Wick's theorem for fermions

I have a guilty suspicion this should be obvious. What is the difference between these two expectations taken over the same measure ($\int \mathrm{d}\mu(\bar\psi,\psi)\exp{\sum \bar\psi A\psi}$ for ...
3
votes
1answer
279 views

Wick contraction corresponding to a connected diagram in $\phi^4$-theory to second order

I am trying to understand the diagrams that comes from a two-point correlation function, $$\langle \Omega|T\{\phi(x)\phi(y)\}|\Omega\rangle$$, in $\phi^4$-theory. The zeroth order contribution, i.e. $\...
5
votes
0answers
217 views
+50

Questions about the equivalent forms of Wick's theorem?

NOTE: The problems have been editing with more details. I have met Wick's theorem first in this book fundamentals of many-body physics when talking about the perturbation expansion of zero ...
0
votes
0answers
182 views

Feynman diagrams - swapped only by vertex label

Consider as an example $\phi^3$ theory, which contains at second order both the contractions: $$\newcommand{\mean}[1]{\langle #1 \rangle} \mean{\hat a_q \phi(\color{red}{x})}\mean{T\phi(x)\phi(x)}\...
2
votes
1answer
96 views

Operator product expansion involving derivatives

I have questions regarding the equation (2.2.4) in Polchinski Vol 1: $$ X^\mu (z_1,\bar{z}_1) X^\nu(z_2,\bar{z}_2) = -\frac{\alpha'}{2}\eta^{\mu\nu} \ln|z_{12}|^2 + \sum_{k=1}^\infty \frac{1}{k!}\...
2
votes
1answer
271 views

Different consequences of Wick's theorem in fermionic and bosonic condensed matter systems

Based on Wick's theorem, the time-ordered product of operators can be written as a sum of normal-ordered product and products involving all types of contractions. Upon taking the ground state ...
1
vote
1answer
557 views

Proof of Wicks Theorem, 3 Fields

The problem statement, all variables and given/known data Question attached: Relevant equations Using the result from two fields that $ T(\phi(x) \phi(y))= : \phi(x) \phi(y) : + G(x-y)$ Where $G(...
3
votes
1answer
131 views

Normal order vs Time order for fermions

For a conformal field $X$, Polchinski gives a relation between the time ordering $T$ (or equivalently the radial ordering ${\cal R}$) of a functional of identical fields and the normal ordering, which ...
3
votes
1answer
224 views

Calculating vertex factor for scalar field theory

I am practising basic QFT and am having some trouble with calculating the vertex factor of an interacting theory involving two real scalar fields, $\phi_{1}$ and $\phi_{2}$. If I create a generic ...
1
vote
1answer
282 views

How to use Wick's theorem to compute this matrix element?

I wanted to see how to use Wick's theorem in practice (I know with Feynman diagrams it is better, but here I want to do this with Wick's theorem only), so I considered computing the matrix element for ...
4
votes
0answers
223 views

Appending a Noether current to a Feynman rule

Background Typically in QFT one derives the Feynman rules by differentiating certain terms in the Lagrangian w.r.t the relevant fields. So for instance if our term is $\mathscr{L} =\phi_1\phi_2\phi_{\...
6
votes
2answers
1k views

Actually calculating something using Wick's Theorem

I am still struggling to get my head around QFT and whilst I think I understand the method of generating functionals to compute correlation functions (as in my question here), my course notes often ...
1
vote
3answers
595 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 ...
1
vote
0answers
149 views

Wick's Theorem for Calculating the Vacuum Functional

The vacuum functional of a theory of free fermions is the overlap between the bare vacuum and the interacting vacuum (i.e. the true groundstate of the Hamiltonian). If the theory preserves particle ...
2
votes
0answers
802 views

Symmetry factor for 1PI Feynman diagrams in $\phi^4$ theory

I am trying to understand the various factors that the Feynman amplitude will carry corresponding to the Feynman diagrams of Fig. 1 of this reference. I understand that the $n^{th}$ diagram containing ...
7
votes
0answers
350 views

Wick theorem and OPE

I'm trying to work out in detail how the Wick theorem is used for constructing OPEs in CFT. One of the first things which bothers me is the difference in definitions of normal ordered product and ...
1
vote
0answers
130 views

OPE of normal ordered operators

In what follows I use $\mathcal{N}\{\ldots\}$ for normal ordering, $\langle\ldots\rangle$ for contraction and $\operatorname{Reg}\{\ldots\}$ for the complete sequence of regular terms which is ...