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7
votes
4answers
2k views

Why is normal ordering a valid operation?

Why is normal ordering even a valid operation in the first place? I mean it can give us some nice results, but why can we do the ordering for the operators like that? Is its definition motivated by ...
7
votes
1answer
327 views

Correlated three-particle Green Function

I know the relationship between normal and correlated two-particle Green Functions for fermions: $$G_c(1,2,3,4)=\Gamma(1,2,3,4)=G(1,2,3,4)+G(1,3)G(2,4)-G(1,4)G(2,3)$$ Also known as irreducible ...
6
votes
1answer
655 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 ...
5
votes
2answers
552 views

Gaussian integral of a function with nonzero mean (generalizing Wick theorem)

From the wikipedia article, for a Gaussian integral of an analytic function we have that This is equivalent to the Wick theorem when f(x) is a polynomial. Now I'm trying to obtain a similar ...
5
votes
1answer
208 views

Quick question regarding Wick's theorem

Let $T\{...\}$ denote time-ordering, $N\{...\}$ normal-ordering and $\left<ab\right>$ be the propagator. Wick's theorem states that $$ T\{ab\} = N\{ab\} + \left<ab\right>. $$ I now ...
5
votes
1answer
231 views

The basic equation of bosonization

[..quoting from Page 11 of Polchinski Vol2..] Given $1+1$ conformal bosonic fields $H(z)$ one has their OPE as, $H(z)H(0) \sim -ln(z)$ Then from here how do the following identities come? ...
5
votes
1answer
211 views

Klein factors and Conformal Field Theory

Consider the mode expansion of a (chiral) scalar field confined to a disc with circumference L: $$ \phi(x) = \phi_{0} + p_{\phi} \frac{2\pi}{L} x + \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} ...
5
votes
2answers
769 views

Teach me Wick's theorem the honest way

Generally speaking the average guy marginally acquainted with quantum field theory or advanced combinatorics describes Wick's theorem as some sort of correspondence between higher order differential ...
5
votes
1answer
288 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, ...
5
votes
1answer
377 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". ...
5
votes
0answers
169 views

Functional integral aproach for Feynman rules [duplicate]

I am familiar with the basic ideas of quantum field theory but I feel uncomfortable when I have to derive Feynman rules by myself for a given action (for example in non-linear sigma models or ...
4
votes
1answer
2k views

Wicks Theorem and Gaussian Integrals

I am trying to complete A. Zee's book QFT in a Nutshell and at page 15 he mentions Wick's theorem and Wick contractions. (apologies for the huge page-snip). Why does he mean by connecting the '$2n$ ...
4
votes
1answer
629 views

Wick Contraction

I am reading Quantum Field Theory in a Nutshell by A. Zee. Zee introduces the rationale/machinery behind Feynman diagrams in three steps: Baby -> Child -> "Real". The baby problem generates ...
4
votes
1answer
465 views

Wick Order and Radial Ordering in CFT

I am not so much familiar with the computations tools of conformal field theory, and I just run into an exercise asking to demonstrate the following formula (related to the bosonic field case): ...
4
votes
2answers
316 views

Chronological and normal ordering

I've realized I'm little bit confused when I want to treat elements like this $$\left<\phi_0|T\{a_p(t)a_p^+(t')V(t_1)V(t_2)\}|\phi_0\right>$$ with $$V(t)=\dfrac12 \dfrac{1}{(2\pi ...
4
votes
1answer
226 views

Free Vacuum vs Interacting Vacuum and Wick's theorem

I'm studying perturbation theory in QFT and I stumbled on a conceptual problem. My understanding of the interplay between LSZ reduction formula and the Gell-Mann & Low perturbation series is ...
4
votes
1answer
196 views

Virasoro TT OPE in Polchinski's book

I'm trying to understand eq. 2.2.11 in Polchinski's first book. He's computing $$:\partial X^\mu(z)\partial X_\mu(z): :\partial' X^\nu(z')\partial' X_\nu(z'):$$ Now, I understand why this ...
4
votes
0answers
118 views

Mode operators in the Virasoro algebra

This questions concerns Exercise 2.11 in Polchinski. We are asked to compute the commutator $$L_{m}(L_{-m}|0;0\rangle) - L_{-m}(L_{m} |0;0\rangle)$$ By plugging the mode expansions, we use the ...
3
votes
1answer
2k views

Why/How is this Wick's theorem?

Let $\phi$ be a scalar field and then I see the following expression (1) for the square of the normal ordered version of $\phi^2(x)$. $$T(:\phi^2(x)::\phi^2(0):) ~=~ 2<0|T(\phi(x)\phi(0))|0>^2 ...
3
votes
1answer
46 views

Feynman Propagator in Peskin & Schroeder

To prove Wick's Theorem, Peskin & Schroeder define the contraction of two fields: \begin{align} \text{Contract}[\phi(x)\phi(y)]\equiv \begin{cases} [\phi^+(x),\phi^-(y)] & \text{for ...
3
votes
3answers
237 views

Wick Theorem, ordering & CFT

I'm having a little trouble with correlation functions wick theorem and ordering in the context of OPE and CFT, for string theory. (1) My first question, the propagator is: $$<X(z) X(w)> = ...
3
votes
2answers
379 views

Vertex operator and normal ordering

The two point function, or propagator for a free massless boson, $\phi$ in 2 dimensions is given by, $$\begin{equation} \langle \phi (z,\bar{z})\phi(w, \bar{w})\rangle ~=~ ...
3
votes
0answers
137 views

A particlar normal ordering problem [duplicate]

Say we have an expression of the form: $$ \left<0\right|:\phi(x)^2: : \phi(y)^2:\left|0\right>, $$ where $\phi$ is some scalar field. I have heard the claim several times, that in evaluating ...
3
votes
0answers
1k views

How to prove Wick's Theorem (Zee's eq. I.2 (16)) via Gaussian integration?

I'm working through Zee's QFT in a Nutshell but there's an integral [I.2 (16)] I couldn't quite derive. The problem is to find $$\langle x_i x_j ... x_k x_l\rangle=\frac{\int ... \int dx_1 ... dx_n ...
2
votes
2answers
225 views

Identity of Operator Product Expansion (OPE)

I have one more s****d question in Polchinski's string theory book, Eqs. (2.3.14a) $$ j^{\mu}(z) :e^{ik \cdot X(0,0)}:~ \sim~ \frac{k^{\mu}}{2 z} :e^{ik \cdot X(0,0)}:,$$ where $j^{\mu}_a ...
2
votes
2answers
3k 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 ...
2
votes
1answer
145 views

Wick's Theorem: Why is the vacuum expectation value of uncontracted operators zero?

I'm am right now reading Chapter 4.3 (Wick's Theorem) in Peskin & Schroeder. It is said that In the vacuum expectation value, any term in which there remain uncontracted operators gives zero ...
2
votes
1answer
193 views

Operator product expansion in CFT

I'm on Polchinski's p39. Can someone please tell me the steps in the equivalence below? $$\exp\left[\frac{\alpha'}4\int d^2z_4 d^2z_5\ln|z_5-z_4|^2\frac{\delta}{\delta X^\mu(z_4,\bar ...
2
votes
1answer
1k views

Wick's Theorem examples

Does anyone know of websites or texts that have an abundance of examples of computing time-ordered products of fields using Wick's Theorem for both bosons and fermions? I'm not just talking about the ...
2
votes
1answer
523 views

Problem with Wick's theorem at first order

I've been struggling with a detail in Second Quantization which I really need to clear out of my head. If I expand the S-matrix of a theory with an interaction Hamiltonian $ H_I(x) $ then I have $$ S ...
2
votes
1answer
186 views

Wick's theorem again

Could someone please elaborate on the accepted answer to this mathoverflow post? I'm working on a problem that looks like this \begin{equation}I=\int d^{n} x\, f(\vec x)\, e^{-\frac{1}{2} \vec x ...
2
votes
1answer
142 views

Operator product expansion energy momentum tensor

We have the following equation from Polchinski (2.4.6) $$ T(z)X^{\mu}(0) \sim \frac{1}{z}\partial X^{\mu}(0) , \tag{2.4.6} $$ where $T(z)$ is defined as $T(z) = -\frac{1}{\alpha'} :\partial X^{\mu} ...
2
votes
0answers
37 views

Correlator $bc$ system [closed]

I have da doubt with bc system. Polchinski says (2.5.10) $$ b(z)b(0)~=~O(z). \tag{2.5.10} $$ I tried to compute the correlation function With eom, using eq (2.5.6b) by Polchinki, removing the ...
2
votes
0answers
82 views

Problem with OPE (from Polchinski) [closed]

I was reading Polchinski, Vol. 2 pag 12, while I found (10.3.12a): $$ e^{iH(z)}e^{-iH(z)}=\frac{1}{2z} + i\partial H(0) + 2zT^H_B(0) + O(z^2).\tag{10.3.12a} $$ Now I tried to do the OPE, what I ...
2
votes
0answers
173 views

Perturbation theory : quadratic external field

I'm trying to derive the explicit form of S-matrix of an interaction Hamiltonian $$H' = \frac{1}{2} \lambda \left[ \int d^3 x \rho({\vec x}) \phi({\vec x}, t)\right]^2\tag{1}$$ Even though the ...
2
votes
1answer
220 views

OPE of fermionic field bosonization in string theory, in Polchinski 10.3.12

In Polchinski's String Theory Vol. 2, equations 10.3.12 are $$e^{iH(z)}e^{-iH(-z)}~=~\frac{1}{2z}+i\partial H(0)+2zT_B^H(0)+O(z^2)\tag{10.3.12a}$$ ...
1
vote
1answer
313 views

Compute the central charge of $bc$ conformal field theory

I have a s****d question, how to calculate the central charge of $bc$ conformal-field theory in Polchinski's string theory, Eq. (2.5.12)? For a $bc$ CFT given by $$S=\frac{1}{2\pi } \int d^2 z \,\,b ...
1
vote
1answer
87 views

0+0 self interacting QFT - $ e^{-\sin^2 x}$ type integral — Bessel function expansion around infinity

In a physics paper (here) I found this variant of the Bessel function of the first kind. $$ \tag{1} Z(g) ~=~ \frac{1}{\sqrt{g}} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} e^{-\frac{1}{2g} \sin^2 x} \, dx ...
1
vote
1answer
57 views

Time-ordered product of two normal-ordered products of fields

Suppose you have a scalar field theory with field operators $\phi(x)=\phi(x)_+ + \phi(x)_- $ that can be decomposed into terms of annihilation and destruction operators. Let $$ D(x-y) = ...
1
vote
1answer
1k views

Wick's theorem proof details

Let's have Wick's theorem in following form: for fields $$ A_{i}(x) ~=~ A_{i}^{+}(x) + A_{i}^{-}(x), $$ where first summand contains creation operator and the second contains destruction one, is a ...
1
vote
1answer
76 views

Normal Ordering in String Theory: Polchinsky vs. all others

Polchinsky defines normal ordering in string theory as: $$:X^\mu(z,\bar z)X^\nu(w,\bar w): = X^\mu(z,\bar z) X^\nu(w, \bar w) + \frac{\alpha'}{2} \eta^{\mu\nu} \log |z-w|^2$$ and for more ...
1
vote
0answers
62 views

Time ordering of normal ordered product

I would like to calculate $$<0|T(:x^4::y^4:)|0>$$ for scalar fields $x$, $y$ "by hand", but I don't understand yet how. With Wicks theorem I'd say this is strictly 0. Is this correct? By hand I ...
1
vote
2answers
100 views

Central charge in energy-momentum tensor OPE

I think that general point of view about central charge in books is considering OPE $T(z) T(w)$ for different field theories and finding that general expression for the most singular term is about to ...
1
vote
1answer
75 views

Operator Product Expansion

I wonder why in OPE in CFT terms like $$ \frac{:O(z) O(w):}{(z-w)^2} $$ occur, for example in the OPE of Energy-momentum tensor with itself: $$T(z) T(w) = \frac{c/2}{(z-w)^4} + \frac{T(z)}{(z-w)^2} ...
1
vote
0answers
68 views

Linear Dilaton CFT

I´m doing the exercises on the Tong lectures of String Theory, in particular Problem Sheet 2: Consider the tensor: $T(z) = \frac{-1}{\alpha '} :\partial X(z) \partial X(z): - Q \partial^2 X$. By ...
1
vote
0answers
49 views

Wick's theorem: Classical Version. Derivation question and what happens with odd moments? [closed]

I am trying to prove the classical version of Wick's theorem: For a set of random variables ${a_i}$, with covariance matrix $M$ and $\rho(\vec a)$ a Gaussian probabilitiy density: $$\langle a_j a_k ...
1
vote
0answers
47 views

Calculating OPE of Graviton Vertex Operator [duplicate]

Consider Exercise 2.8 in Polchinski's String Theory book. We are asked to compute the weight of $$f_{\mu \nu}:\partial X^{\mu} \bar{\partial}X^{\nu}e^{ik\cdot X}:$$ I have carried out the usual ...
1
vote
0answers
99 views

Wick's Theorem For Product of Fields [closed]

I am trying to write an expression for $$\langle (\phi(x,t))^m (\phi(x',t'))^n \rangle$$ where $n$ and $m$ are even with respect to a real Gaussian action, in terms of $$\langle \phi(x,t) ...
1
vote
0answers
82 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 ...
1
vote
0answers
92 views

Wick theorem applying to partly ordered operator

I symbolize $T$ as the time-ordered operator and $::$ as normal order symbol. I know that in quantum field theory generally we have: $$T\phi_1(x_1)\dots\phi_n(x_n)=:\phi_1(x_1)\dots\phi_n(x_n):+A$$ ...