- #1
latentcorpse
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So we have the definition of the correlation (or Green's) function as
[itex]G^{(n)}(x_1, \dots , x_n) = \langle \Omega | \phi(x_1) \dots \phi(x_n) | \Omega \rangle[/itex]
Now I need to show that they are Lorentz invariant i.e. that
[itex]G^{(n)}(x_1',\dots,x_n')=G^{(n)}(x_1, \dots , x_n)[/itex]
However, we know that under a Lorentz transformation [itex]x_i \rightarrow x_i'=\Lambda x^i[/itex], the scalar field [itex]\phi[/itex] transforms as [itex]\phi(x) \rightarrow \phi'(x)=\phi(\Lambda^{-1}x)[/itex]
So if I make a Lornetz transformation, surely I would change all the [itex]x[/itex]'s to [itex]\Lambda^{-1}x[/itex]'s in the formula [itex]G^{(n)}(x_1, \dots , x_n)[/itex] but this would lead to
[itex]G^{(n)}(\Lambda^{-1} x_1 , \dots \Lambda^{-1} x_n)[/itex] which doesn't seem to be right?
Thanks for any help.
[itex]G^{(n)}(x_1, \dots , x_n) = \langle \Omega | \phi(x_1) \dots \phi(x_n) | \Omega \rangle[/itex]
Now I need to show that they are Lorentz invariant i.e. that
[itex]G^{(n)}(x_1',\dots,x_n')=G^{(n)}(x_1, \dots , x_n)[/itex]
However, we know that under a Lorentz transformation [itex]x_i \rightarrow x_i'=\Lambda x^i[/itex], the scalar field [itex]\phi[/itex] transforms as [itex]\phi(x) \rightarrow \phi'(x)=\phi(\Lambda^{-1}x)[/itex]
So if I make a Lornetz transformation, surely I would change all the [itex]x[/itex]'s to [itex]\Lambda^{-1}x[/itex]'s in the formula [itex]G^{(n)}(x_1, \dots , x_n)[/itex] but this would lead to
[itex]G^{(n)}(\Lambda^{-1} x_1 , \dots \Lambda^{-1} x_n)[/itex] which doesn't seem to be right?
Thanks for any help.