Proof of Wick's Theorem for 3 fields

[binbagsss]

Homework Statement

Question attached:

Homework Equations

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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.

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.

 

[binbagsss]

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 no one 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