What is Wick's theorem: Definition and 18 Discussions

Wick's theorem is a method of reducing high-order derivatives to a combinatorics problem. It is named after Italian physicist Gian-Carlo Wick. It is used extensively in quantum field theory to reduce arbitrary products of creation and annihilation operators to sums of products of pairs of these operators. This allows for the use of Green's function methods, and consequently the use of Feynman diagrams in the field under study. A more general idea in probability theory is Isserlis' theorem.
In perturbative quantum field theory, Wick's theorem is used to quickly rewrite each time ordered summand in the Dyson series as a sum of normal ordered terms. In the limit of asymptotically free ingoing and outgoing states, these terms correspond to Feynman diagrams.

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  1. PeroK

    Wick's Theorem and Bose Operators

    This is problem 18.3 from QFT for the gifted amateur. I must admit I'm struggling to interpret what this question is asking. Chapter 18 has applied Wick's theorem to calculate vacuum expectation values etc. But, there is nothing to suggest how it applies to a product of operators. Does the...
  2. Glenn Rowe

    A Scalar-scalar scattering Feynman diagram

    I'm trying to work out the Feynman diagrams for scalar-scalar scattering using the Yukawa interaction, as given in Chapter 6 of Lahiri & Pal's A First Book of Quantum Field Theory. The interaction hamiltonian is $$\mathscr{H}_{I}=h:\overline{\psi}\psi\phi:$$ where ##\psi## is a fermion field and...
  3. binbagsss

    Wick's theorem: 4 field correlator, 2 different fields

    Homework Statement ##T(\phi_1\Phi_2\phi_3\Phi_4)## where ## \phi_1## is ##\phi(x_1)## and ##\phi## and ##\Phi## are two different fields. By Wicks theorem ##T(\phi_1\Phi_2\phi_3\Phi_4)= : : + contracted terms.## QUESTION Are the fully contracted terms (apologies for the bad notation I'm...
  4. Ken Gallock

    I Wick's theorem and Nucleon scattering

    Hi. My question is about nucleon-nucleon scattering. In David Tong's lecture note, he discusses Wick's theorem and nucleon scattering (page 58-60). My problem is that I don't know how to calculate the second line of eq(3.48): \begin{equation} <p'_1, p'_2|:\psi^\dagger (x_1) \psi (x_1)...
  5. binbagsss

    Proof of Wick's Theorem for 3 fields

    Homework Statement Question attached: Homework Equations [/B] Using the result from two fields that ## T(\phi(x) \phi(y))= : \phi(x) \phi(y) : + G(x-y)## Where ##G(x-y) = [\phi(x)^+,\phi(y)^-] ## ## : ## denotes normal ordered and ##\phi(x)^+ ## is the annihilation operator part , and...
  6. M

    A Expert on Wick's Theorem needed

    Hi everyone, I use Wick's theorem to decompose expectation values of a string of bosonic creation and annihilation operators evaluated at the vacuum state. This can only be done when the time evolution is driven by a Hamiltonian of the form: H=\sum_{i,j}{\epsilon_{i,j} c^{\dagger}_{i}c_{j}}...
  7. C

    I Symmetry factor via Wick's theorem

    Consider the lagrangian of the real scalar field given by $$\mathcal L = \frac{1}{2} (\partial \phi)^2 - \frac{1}{2} m^2 \phi^2 - \frac{\lambda}{4!} \phi^4$$ Disregarding snail contributions, the only diagram contributing to ## \langle p_4 p_3 | T (\phi(y)^4 \phi(x)^4) | p_1 p_2 \rangle## at...
  8. H

    Vacuum diagrams vs. tree diagrams vs. loop diagrams

    Could someone please tell me the difference between tree diagrams and loop diagrams? If I'm thinking correctly tree diagrams are before contracting? Also how do vacuum diagrams fit into the picture? Thanks!
  9. N

    Evaluating Time-Ordered Product with Wick's Theorem

    Homework Statement [/B] Consider a real free scalar field Φ with mass m. Evaluate the following time-ordered product of field operators using Wick's theorem: ∫d^4x <0| T(Φ(x1)Φ(x2)Φ(x3)Φ(x4)(Φ(x))^4) |0> (T denotes time ordering) Homework Equations Wick's theorem: T((Φ(x1)...Φ(xn)) = ...
  10. E

    Wick's Theorem for free fields only

    I'm not quite following why Wick's Theorem only applies to free fields. What part of the argument depends on a free field assumption?
  11. Breo

    About Wick's Theorem, Time Order Operator, Normal Ordering and Green's Function

    So if I understood well, Normal ordering just comes due to the conmutation relation of a and a⁺? right? Is just a simple and clever simplification. Wick Theorem is analogue to normal ordering because it is related to the a and a⁺ again (so related to normal ordering, indeed). However I do not...
  12. G

    Problem with Wick's theorem at first order

    Hey there, first post here! 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 - 1= \int^{+\infty}_{-\infty} d^4 x H_I(x) +...
  13. M

    Wick's theorem for other statistics

    Hi all! I've got a question concerning Wick's theorem. I followed the proof in the book by Fetter and Walecka and it works well for particles with "normal" statistic, that means for bosons and fermons (commuting or anticommuting). But what about anyons, particles which don't commute just with a...
  14. J

    Wick's Theorem Proof (Peskins and Schroder)

    I'm having a bit of trouble working through the induction proof they give in the book. The step I don't understand is: (page 90 in the book, halfway down) N(\phi_2...\phi_m)\phi_1^+ + [\phi_1^+,N(\phi_2...\phi_m)] = N(\phi_1^+\phi_2...\phi_m) + N([\phi_1^+,\phi_2^-]\phi_3...\phi_m +...
  15. V

    Simplifying Feynman Diagram for Nasty 10 Point Green's Function?

    Why is the Feynman diagram for the following nasty 10 point Green's function so simple: I mean it only has two external points, one vertex, and one loop: Here is the offending function: \int d^4y_1 d^4y_2 <0|T[\phi (x_1) \phi (x_2) \phi^4 (y_1) \phi^4 (y_2)]|0> which I am assuming is simply...
  16. V

    The expectation of an expection (relating to Wick's Theorem)

    Hi: If we want to work out the expectation of: <0|T(φ1φ2)|0> ie. <0|<0|T(φ1φ2)|0>|0> apparently it is acceptable to pull out the <0|T(φ1φ2)|0>: So <0|<0|T(φ1φ2)|0>|0>=<0|T(φ1φ2)|0><0|I|0> I do realize this is a really stupid question, but I want to be 100% sure. Is this simply...
  17. D

    Generalized Wick's theorem

    I have the following contour integral form of Wick's theorem (C indicating contraction): C[A(z):BC:(w)]=\frac{1}{2 \pi i} \int _w \frac{dx}{x-w} C[A(z)B(x)]C(w) + B(x)C[A(z)C(w)] Does anybody know how to evaluate contractions like C[:AB:(z)C(w)]?
  18. E

    Peskin & Schroeder's proof of Wick's Theorem

    Hi, I am stuck on a step Peskin & Schroeder give in their proof of Wick's Theorem (Intro to Quantum Field Theory, p 90). In the middle of the page when they consider the term with no contraction, it seems like in between the 1st and 2nd lines they somehow factor out the normal ordering...
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