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Homework Help: Proof of Wick's Theorem for 3 fields

  1. Jul 4, 2017 #1
    1. The problem statement, all variables and given/known data

    Question attached:

    wick3question.png
    2. Relevant equations

    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 ## \phi(x)^- ## is the creation operator part.


    3. The attempt at a solution

    Assume non-trivially that ## z^0 > x^0 > y^0 ##

    Then ## T(\phi(z),\phi(x),\phi(y)) = \phi(z) T(\phi(x) \phi(y)) ##

    ##=(\phi(z)^+ + \phi(z)^-) T (\phi(x),\phi(y)) ##

    Since ##\phi(z)^-## is already normal ordered, look at the term multiplied by ##\phi(z)^+##:

    ##=\phi(z)G(x-y) + \phi(z)^+:\phi(x)\phi(y): ## (1)

    The term to be concerned with from

    ##\phi(z)^+:\phi(x)\phi(y):## is ##\phi(z)^+\phi(x)^-\phi(y)^-=\phi(x)^-\phi(z)^+\phi(y)^- +[\phi(z)^+,\phi(x)^-]\phi(y)^-= \phi(x)^-\phi(y)^-\phi(z)^+ +\phi(x)^-[\phi(z)^+,\phi(y)^-] + [ \phi(z)^+,\phi(x)^-]\phi(y)^-##

    So putting this with (1) I have

    ## T(\phi(z),\phi(x),\phi(y)) = : \phi(z) (\phi(x) \phi(y)): + [ \phi(z)^+,\phi(x)^-]\phi(y)^- +\phi(x)^-[\phi(z)^+,\phi(y)^-] +\phi(z)(G(x-y)) ##

    So looking at the solution the last term is right, but the other propagator terms , should have a factor of both the creation and annihilation parts of the field, ##\phi(y)^- + \phi(y)^+ ## and ## \phi(x)^+ + \phi(x)^- ## , multiplying the propagator? and should be multiplying the RHS of the propagator rather than the LHS ? I'm not sure what I have done wrong...

    Many thanks in advance.
     
  2. jcsd
  3. Jul 9, 2017 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
  4. Jul 10, 2017 #3
    got empty pockets again hun
     
  5. Apr 14, 2018 #4
    ok so i've figured out what i was doign wrong, and it's a pretty quick spot, and I assume perhaps a common sort of mistake , so I'm surprised noone replied but hey..

    basically my definition of 'normal-ordered' was as soon as you had any permutation of the operators that would cause either the bra or ket of the vacuum to vanish ,so either to a creation ladder on the lhs or a annihilating on the rhs , thee job was done. when instead you needed ALL creation operators on the left and all annihilating operators on the rhs. I'm actually not to sure why this is, the use of wicks theorem i've seen is when a bunch of ladder operators are sandwhiched between the vacuum bra and ket, and so this would suffice to cause it to vanish. However if I think about a proof by induction, adding more fields to it, it makes sense that you'd want it 'fully normal-ordered'...

    ta
     
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