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Why/How is this Wick's theorem?

+ 3 like - 0 dislike
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Let $\phi$ be a scalar field and then I see the following expression for the square of the normal ordered version of $\phi^2(x)$.

$$T(:\phi^2(x)::\phi^2(0):) ~=~ 2<0|T(\phi(x)\phi(0))|0>^2 $$ $$ ~+~ 4<0|T(\phi(x)\phi(0))|0>:\phi(x)\phi(0): ~+~ :\phi^2(x)\phi^2(0):$$

It would be great if someone can help derive the above expression - may be from scratch - and without outsourcing to Wick's theorem - and may be help connect as to why the above is related (equal?) to the Wick's theorem?

  • Isn't the above also known as OPE (Operator Product Expansion)? If yes, then is there at all any difference between OPE and Wick's theorem? Is there a systematic way to derive such OPEs?

  • Can one help extend this to Fermions?

This post imported from StackExchange Physics at 2015-02-02 13:25 (UTC), posted by SE-user user6818
asked Apr 20, 2012 in Theoretical Physics by user6818 (955 points) [ no revision ]
Apologies but this question seems to be asking "please teach me Wick's theorem" - and it moreover says "do it without teaching me Wick's theorem". Have you tried to study Wick's theorem? At least en.wikipedia.org/wiki/Wick%27s_theorem ? The standard pedagogical treatment answers all your questions. The Wick's theorem is the systematic way to construct such identities that you're looking for. Fermions differ by some signs only. I find it questionable whether copying/rephrasing sections from standard textbook material is a good way to use this server and people's time.

This post imported from StackExchange Physics at 2015-02-02 13:25 (UTC), posted by SE-user Luboš Motl

1 Answer

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As Lubos Motl mentions in a comment, for all practical purposes, the sought-for equation (v1) is proved via Wick's Theorem.

It is interesting to try to generalize Wick's Theorem and to try to minimize the number of assumptions that goes into it. Here we will outline one possible approach.

I) Assume that a family $(\hat{A}_i)_{i\in I}$ of operators $\hat{A}_i\in{\cal A}$ lives in a (super) operator algebra ${\cal A}$

  1. with (super) commutator $[\cdot,\cdot]$, and

  2. with center $Z({\cal A})$.

Here

  1. the index $i\in I$ runs over an index set $I$ (it could be continuous), and

  2. the index $i$ contains information, such as, e.g., position $x$, time instant $t$, annihilation/creation label, type of field, etc., of the operator $\hat{A}_i$.

II) Assume that $$ \forall i,j\in I~: \qquad [\hat{A}_i,\hat{A}_j] ~\in~Z({\cal A}). $$

III) Assume that there are given two ordering prescriptions, say $T$ and $::$. Here $T$ and $::$ could in principle denote any two ordering prescriptions, e.g. time order, normal order, radial order, etc. This means that the index set $I$ is endowed with two strict total orders, say, $<$ and $\prec$, respectively, such that

  1. The $T$ symbol is (graded) multilinear wrt. supernumbers.

  2. $ T(\hat{A}_{\pi(i_1)}\ldots\hat{A}_{\pi(i_n)})~=~\pm T(\hat{A}_{i_1}\ldots\hat{A}_{i_n} )$ is (graded) symmetric, where $\pi\in S_n$ is a permutation of $n$ elements.

  3. $ T(\hat{A}_{i_1}\ldots\hat{A}_{i_n} )~=~\hat{A}_{i_1}\ldots\hat{A}_{i_n}$ if $i_1 > \ldots > i_n$. [A similar condition should hold for the second ordering $(::,\prec)$.]

  4. In the special case where some of the $ i_1 , \ldots , i_n$ are equal${}^{\dagger}$ (wrt. the order <), then one should symmetrize in appropriate (graded) sense over the corresponding subsets. For instance, $$ T(\hat{A}_{i_1}\ldots\hat{A}_{i_n} )~=~\hat{A}_{i_1}\ldots\hat{A}_{i_{k-1}}\frac{\hat{A}_{i_k}\hat{A}_{i_{k+1}}\pm\hat{A}_{i_{k+1}}\hat{A}_{i_k}}{2}\hat{A}_{i_{k+2}}\ldots\hat{A}_{i_n}$$ if $i_1 > \ldots > i_k=i_{k+1}> \ldots > i_n$. [A similar condition should hold for the second ordering $(::,\prec)$.]

IV) It then follows from assumptions I-III that the (generalized) contractions $$ \hat{C}_{ij}~=~T(\hat{A}_i\hat{A}_j)~-~:\hat{A}_i\hat{A}_j:~\in~Z({\cal A}) $$ belong to the center $Z({\cal A})$. [By the way, physicists will often casually refer to the operators in the center $Z({\cal A})$ as $c$-numbers.]

V) It is now a straightforward exercise to establish a corresponding Wick's Theorem, meaning a rule for how to re-express one ordering prescription $T(\hat{A}_{i_1}\ldots\hat{A}_{i_n})$ in terms of the other ordering prescription $::$ and (multiple) contractions $\hat{C}_{ij}$. And vice-versa with the roles of the two orderings $T$ and $::$ interchanged. Such Wick's Theorems can now be applied successively to establish nested${}^{\ddagger}$ Wick's Theorems. These Wick's Theorems may be extended to a larger class of operators than just the $(\hat{A}_i)_{i\in I}$ family through (graded) multilinearity.

VI) Let us now assume that the operators $\hat{A}_i$ are Bosonic for simplicity. A particular consequence of a nested Wick's Theorem is the following version of the sought-for equation

$$T(:\hat{A}^2_i::\hat{A}^2_j:) ~=~ 2\hat{C}_{ij}^2 + 4 \hat{C}_{ij}:\hat{A}_i\hat{A}_j: + :\hat{A}^2_i\hat{A}^2_j:~.$$

Finally, let us mention that Wick's Theorem, radial order, OPE, etc., are also discussed in this and this Phys.SE posts.

--

${}^{\dagger}$ Being equal wrt. an order is in general an equivalence relation, and it is often a weaker condition than being equal as elements of $I$.

${}^{\ddagger}$ A nested Wick's Theorem (between radial order and normal order) is briefly stated on p. 39 in J. Polchinski, String Theory, Vol. 1. Beware that radial order is often only implicitly written in CFT texts. I'll update the answer if I find a better reference.

This post imported from StackExchange Physics at 2015-02-02 13:25 (UTC), posted by SE-user Qmechanic
answered Apr 21, 2012 by Qmechanic (2,790 points) [ no revision ]
Most voted comments show all comments
Having gone through may be 2 or 3 quite advanced courses in QFT, this still looks to me to be something quite unfamiliar and hence quite surprised at Lubos's comment. What you call as a "straightforward excercise" doesn't seem to be so at all - can you kindly fill in a few more steps!

This post imported from StackExchange Physics at 2015-02-02 13:25 (UTC), posted by SE-user user6818
Your definition of a "contraction" also looks new - I would think that a "contraction' of two fields means the vacuum expectation value of the time-ordered product. And by Wick's theorem, I would understand the statement that the difference between the time-ordered product and the normal odered product is a sum over all possible contractions. So how is what you are doing the same as Wick's theorem? (..as Lubos seems to be claiming..)

This post imported from StackExchange Physics at 2015-02-02 13:25 (UTC), posted by SE-user user6818
I plan to update the answer soon.

This post imported from StackExchange Physics at 2015-02-02 13:25 (UTC), posted by SE-user Qmechanic
Thanks a lot! I am eagerly looking forward to a detailed update from you.

This post imported from StackExchange Physics at 2015-02-02 13:25 (UTC), posted by SE-user user6818
I like this answer just the way it is, as it makes it very clear that Wick's theorem is not a theorem about QFT, but is rather a completely general theorem about algebras obeying a few simple axioms. The standard textbook treatment with fields and VEV's obfuscates what is really just basic algebra.

This post imported from StackExchange Physics at 2015-02-02 13:25 (UTC), posted by SE-user Jonathan
@QMechanic Thanks for the answer. I think I got the identity by doing the usual Wick's theorem on the $:\phi^2(x)\phi^2(0):$ and then rearranging the terms. It kind of had to be approached in the "reverse". I tried doing similar stuff with the fermionic OPEs like the one needed for current-current correlators in Bjorken scaling analysis and there too I could get the right answers. I am just wonder if there is a general framework in this kind of "converse" thinking with the Wick's theorem fits in. It looks a bit ad-hoc this way.

This post imported from StackExchange Physics at 2015-02-02 13:25 (UTC), posted by SE-user user6818
@Qmechanic I have earlier seen this reference to Polchinski's chapter. I guess I will take a fresh look at that. Thanks for the discussion. I have accepted your answer.

This post imported from StackExchange Physics at 2015-02-02 13:25 (UTC), posted by SE-user user6818

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