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.
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How to recognize Feynman diagrams from the $S$-matrix expansion?
I'm studying scattering processes in QED and one usually have to compute first of all the Scattering matrix
$$\hat{S}=T\biggl (\exp\{-i\int d^{4}x:\bar{\psi}(x)\gamma_{\mu}\hat{A}^{\mu}(x)\hat{\psi}(x)...
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Symmetry Factor and Wicks Theorem
I have a problem with a particular kind of exercise. The question is:
Consider $\phi^4$-theory with $\mathcal{L}_\text{int}=-\frac{\lambda}{4!}\phi^4$. Give the symmetry factors of the diagram and ...
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Product of normally ordered exponentials as a normal ordering of product of exponentials
I want to simplify a product of normally ordered exponentials that are in the following form
$$:e^{x(\hat{a}^\dagger+\alpha_x^*)(\hat{a}+\alpha_x)}:\times :e^{y(\hat{a}^\dagger+\alpha_y^*)(\hat{a}+\...
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Applying Wick's theorem to mass-terms in a free field theory
I'm studying a free field theory in $d$ dimension which consists of two sets of $O(N)$-scalars
\begin{equation}
S = \int_{\mathbb{R}^d}d^dx \left( \frac{(\partial_\mu\phi^i)^2}{2} + \frac{(\partial_\...
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Wick contraction with exponential operator in 2D CFT
Consider a 2d CFT in radial quantization. Let $A(z)$ be some primary field and $Q$ a charge that can be written as $$Q = \oint dz \, z^{(h-1)}J(z)\tag{1}$$ for some holomorphic current $J(z)$. I will ...
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The vanishing of vacuum expectation value
I have some difficulty understanding why the vacuum expectation value vanishes. As illustrated in my notes, we can split the field into two parts:
$$
\phi(x) = \phi^+(x) + \phi^-(x),
$$
where $\phi^+(...
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Greiner´s Field Quantization question [closed]
I upload a screenshot of Greiner´s book on QFT. I don´t understand one step. I need help understanding equation (3), what are the mathematical steps in between?
Greiner, Field Quantization, page 245 (...
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Relationship with double summing of $a_{\ell m}$
I would like to convince myself of the following relationship in an astrophysical context:
\begin{aligned}
& \sum_{m}\sum_{m^{\prime}}\left\langle a_{\ell m} a_{\ell m}^* a_{\ell m^{\prime}} a_{\...
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Free fermion OPE
In Di Francesco's Conformal Field Theory, the propagator for the free Majorana fermion theory is given by
$$ \langle{\psi(z) \psi(w)}\rangle = \dfrac{1}{2\pi g} \dfrac{1}{z-w}$$
and the energy-...
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Matrix Elements Under BCS basis
I am learning BCS theory but I am stuck at a very beginning step in my deduction. The BCS wave function is
$$
|\rm BCS\rangle=\prod_{k>0}(u_k+v_ka_k^\dagger a_{\bar{k}}^\dagger)|\rm vac\rangle
$$
...
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Kardar "Statistical Physics of Fields" expectation value identity
In Eq. (5.7) of his book "Statistical Physics of Fields", M. Kardar proposes the identity
$$
\langle e^{\sum_i a_ix_i} \rangle
=\exp{\left[\sum_{ij}\frac{a_ia_j}{2}\langle x_ix_j \rangle\...
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Defining Wick/normal ordering beyond rearranging the order of annihilation and creation operators [duplicate]
Most introductory quantum field theory books define Wick ordering as rearranging a product of creation and annihilation operators such that all the creation operators appear to the left of any ...
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What justifies the interpretation of Feynman Diagrams of physical processes? [duplicate]
Feynman diagrams arise mathematically essentially as neat graphical ways of organising the terms in Wick's theorem for time-ordering. But at the same time we're supposed to interpret them as some sort ...
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Derivation of OPE in chapter four of Fradkin's book
In chapter four of Eduardo Fradkin's book "Field Theories of Condensed Matter Physics", he considers the Lagrangian $$\cal{L}=\frac{K}{2}(\partial \phi)^2 + u \cos(\phi).$$ He treates the ...
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Wick's contractions
According to Wick's theorem, we have
$${\left\langle {T\left[ {\hat A\hat B\hat C\hat D} \right]} \right\rangle _0} = {\left\langle {T\left[ {\hat A\hat B} \right]} \right\rangle _0}{\left\langle {T\...
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Normal ordering in Sine-Gordon model [duplicate]
I am studying Bosonization from Giamarchi's book (Quantum Physics in 1D), in Appendix E while doing RG analysis at second order he says (Eq. E.18) that we can NOT expand cosine directly because field $...
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Query in the proof of Wick's theorem
I am looking at the proof of Wick's theorem in the notes here:
https://www.imperial.ac.uk/media/imperial-college/research-centres-and-groups/theoretical-physics/msc/current/qft/handouts/...
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Why are all the possible permutationss in the perturbative $S$-matrix calculations added together?
I have a question regarding the calculation of the $S$-matrix. During the calculation of second order term of the $S$-matrix for e.g. the Møller scattering $(e^
− + e^
− → e^
− + e^
−)$ $|i\rangle=|e^...
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Wick theorem for $N$-particle Matsubara green function: equal time contraction [duplicate]
I am wondering why, e.g., the book by Mahan, "Many-particle physics", mentions that contractions in Wick theorem for the $N$-particle Matsubara Green function between a pair of operators at ...
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Wick's theorem and Feynman propagator
(this is the image from book 'No nonsense QFT' by Jakob Schwichtenberg, page no, 426)
The quantity $[\phi_-(x),\phi_+(y)]$ is like an operator inside the bra-kets $\langle 0|$ and $|0\rangle$.
I'm not ...
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Two-point correlation function of two complex scalar fields
For a lagrangian:
$$
\mathcal{L}=\partial^\mu\phi_i^*\partial_\mu\phi_i-m_i^2|\phi_i|^2+\lambda(\phi_2^3\phi_1+\text{h.c.}).
$$
where summation over $i=1,2$ is understood.
I am trying to find the two ...
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Spin and scale dimension of canonical spin-1/2 fields in (1+1)d
I am reading the book "Non-perturbative methods in 2 dimensional quantum field theory" by Abdalla, Abdalla and Rothe and have some questions about the Chapter 2.4 "Bosonization of ...
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Multi-channel mean field theory
I have always been confused about the theoretical foundation of the mean field approximation. Below I follow the book Many-body Quantum Theory in Condensed Matter Physics by Bruus and Flensberg, ...
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Wick contraction in quantum field theory
I am reading Anthony Zee's "Quantum Field Theory in a Nutshell" (1st edition). On page 47, when evaluating the 4-point Green's function $G_{ijkl}^{(4)}$ to order $\lambda$ using Wick ...
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Time-ordering operators: Can we simplify expressions inside? [duplicate]
Clearly if we have two operators $\phi(t_1)$ and $\psi(t_2)$ and define a time ordering operator $T$ acting on operators such that $$T(\phi(t_1)\psi(t_2)):=\phi(t_1)\psi(t_2),~\text{if $t_1>t_2$ ...
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Using Wick's theorem on already normal-ordered functions
The book Quantum Field Theory of Many Body systems (X. G. Wen) defines Wick's theorem as follows.
I would like to employ this definition to simplify the 4-operator product $$\hat{O}\;=\;a_p^\dagger ...
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How one can use Wick's theorem for the product $A:\mathrel{B^{n}}:$?
I try to use Wick's theorem in the case that some products we deal with are already normal ordered.
My guess is that it could be something like
\begin{equation}
A:\mathrel{B^{n}}:~=~:\mathrel{AB^{n}}:+...
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Does normal ordering (not) conflict with canonical commutators?
Normal ordering is pretty useful to stop expressions from diverging in quantum field theory and works out perfectly fine regarding this, but there is this little problem: Consider for example an ...
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Wick's theorem on a four-point QED Green function in 2nd-order perturbation theory
Problem 13.1 in Mandl & Shaw's QFT. I need to calculate the second order contributions to the four point Green function
$$ \langle A^{\mu}(x_1)A^{\nu}(x_2)\psi(x_3)\bar{\psi}(x_4) \rangle, $$
...
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Calculating a four-point Green function using Wick's theorem (problem 12.1 in Mandl & Shaw)
In problem 12.1 in Quantum Field Theory, Mandl & Shaw the aim is to calculate the four point green function
$$ G^{\mu\nu}(x,y,z,w) = \frac{\langle 0 | T\big(A^{\mu}A^{\nu}\psi(z)\bar{\psi}(w)S\big)...
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Wick's Theorem Example from Stoof & Gubbels [duplicate]
I've been learning about Wick's theorem from a variety of sources when I came across this example from "Ultracold Quantum Fields" by Stoof et al. on page 148 (Here I have simplified the ...
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Computing $\langle 0|S |0\rangle$ in $\phi^4$ theory [closed]
$\newcommand{\bra}[1]{\langle #1|}$
$\newcommand{\ket}[1]{|#1\rangle}$
I have been reading David Tong's QFT notes. As part of an exercise, I am asked to examine $\bra{0} S \ket{0}$ to order $\lambda^2$...
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Normal ordered product of $4$ scalar fields $X^\mu$
I'm trying to get more familiarity with the conformal normal ordering used in Polchinski's String Theory vol. 1 and I'm currently trying to solve problem $2.2$ which asks to prove that the normal ...
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Trying to derive Polchinski's equation $(2.2.11)$
I'm having trouble to derive Polchinski's equation $(2.2.11)$ which is an example of product of normal ordered product of $\partial X^\mu$. Precisely, Polchinski defines the product of normal ordered ...
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Scalar derivative couplings: Effects on S-matrix and Feynman Rules
In Schwartz's field theory book ch. 7.4.2 he claims that interaction Lagrangians like
$${\cal L}_{\rm int} = \lambda \phi_1(\partial_{\mu}\phi_2)(\partial_{\mu}\phi_3)\tag{7.101}$$
lead to the Feynman ...
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Why is the time evolution operator not Weyl ordered?
I am reading the book "Structural Aspects of Quantum Field Theory" by Gerhard Grensing. After introducing the Weyl formalism of symmetrizing the coordinates and momenta, he starts with a ...
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Question on Operator Product Expansion of Stress-Energy Tensor with Itself
The stress-energy tensor is given by
$T(z) = \frac{1}{\alpha'} :\partial X(z) \partial X(z):$ , the normal-ordered product of the two given operators.
Thus taking the OPE of the stress-energy tensor ...
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Is bosonic normal-ordering equal to fermionic normal ordering?
I'm trying to understand Jan von Delft's "Bosonization for Beginners — Refermionization for Experts", in which he uses boson normal ordering and fermion normal ordering interchangeably, and ...
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What's the contraction for non-adjacent fields?
In section 8.2 of Coleman's QFT lectures, he introduces the definition of contraction of two fields,
where $T$ denotes time ordering and the colons normal ordering.
Then he proceeds to contraction in ...
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How to avoid paradoxes about time-ordering operation?
(Original title: is time-odering operator a linear operator?)
I'm confused with two formulas, one of which is
$$
\mathcal{T} \exp \left [-\frac{\mathrm{i}}{\hbar} \int_{t_0}^t \mathrm{d} t' \hat{H}_I(...
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Are self-loop diagrams zero if $L_\text{int}$ has a derivative term?
I was looking at the theory with interaction Lagrangian $L_\text{int}=\phi^3 \cdot \partial_{\mu}\phi$.
I was computing the following self-loop diagram
\begin{equation}
\langle
\phi_x \phi_z^3 \cdot \...
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Why do we "normal-order" instead of just subtracting off vacuum energy?
The Hamiltonian is arbitrary upto a constant anyway. Why don't we just subtract off the vacuum energy? The Hamiltonian was always observable only upto a constant.
Instead, we do normal-ordering, in ...
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Normal Ordered Product in Operator Product Expansions
In an example of operator product expansion applied to $\phi^4$ theory of the book QFT an integrated approach, where the Eulerian Lagrangian is
$$\mathcal{L}=\frac{1}{2}\left(\partial_{\mu} \phi\right)...
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Is the Chronological product defined in Zavialov's book coherent?
In Zavialov's book "Renormalized Quantum Field Theory" he defines the Chronological Product for arbitrary operators as follows (I-47):
Given two operators in the form
$$A(\phi) = \sum_{n=0}^\...
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Wick Theorem: number of contractions [closed]
I have to prove that the number of contractions in Wick's Theorem
is equal to:
$$\frac{n!}{(n/2)! \ 2^{n/2}} \ \ \ where \ \ n \ \ is \ even$$
I don't know how to start, if someone can help.
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Proof of Wick's theorem for general Gaussian states
For boson modes $a^\dagger_i, a_i$, consider the density matrix which is an exponential of quadratic operators in $a^\dagger_i$ and $a_i$:
$$\rho = e^{-H_{ij} a^\dagger_i a_j + (K_{ij} a^\dagger_i a^\...
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Defining normal-ordering in an OPE calculation
Suppose $\phi_1, \phi_2, \psi$ and $\bar{\psi}$ are free fields of a two-dimensional CFT with propagators on the plane given by
$$\phi_1(y)\phi_1(z) \sim -\log(y-z),\quad\phi_2(y)\phi_2(z) \sim -\log(...
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Normal ordering of number operators $n$th power
In resources I keep seeing the normally ordered form of the number operator to the $n$th power,
$${(a^\dagger a)}^n=\sum_{k=1}^n S(n,k){(a^\dagger)}^ka^k.$$
Why are we interested in the number ...
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Wick's theorem: From operators to fields
I understand Wick's Theorem when operators are involved to be,
$$\mathcal{N}(f(a,a^\dagger) = :\!\sum\textbf{All contractions}\!:$$
But I'm getting slightly confused when this is expanded to fields, I'...
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Wick's Theorem and Functional Derivative
In the Quantum Field Theory An Integrated Approach, Fradkin, the author derived the partition functional for a free scalar field (after analytic continuation to imaginary time ) as
$$Z_{E}[J]=Z_{E}[0] ...
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