Complex scalar field propagator evaluation.


by Ace10
Tags: compex, evaluation, field, greens function, klein gordon field, propagator, quantum field theory, scalar
Ace10
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#1
Oct29-13, 12:12 PM
P: 6
Good afternoon fellow scientists,i have a small problem in evaluating the propagator for the complex Klein-Gordon field. Although the procedure is the one followed for the computation of the propagator of the real K-G field, a problem comes up:

As known: <0|T[itex]\varphi^{+}(x)\varphi(y)[/itex]|0> = [itex]\Theta(x^{0}-y^{0})[/itex]<0|[itex]\varphi^{+}(x)\varphi(y)[/itex]|0> + [itex]\Theta(y^{0}-x^{0})[/itex]<0|[itex]\varphi(y)\varphi^{+}(x)[/itex]|0>

and <0|[itex]\varphi^{+}(x)\varphi(y)[/itex]|0>=<0|[itex]\varphi(y)\varphi^{+}(x)[/itex]|0>

But if we try to verify that one of the above correlation functions is a green's function of the K-G equation we hit the obstacle: [itex]\partial_{x}[/itex]<0|[itex]\varphi^{+}(x)\varphi(y)[/itex]|0> . And I refer to it as an obstacle because of the commutation relation [[itex]\varphi(x),\pi^{+}(y)[/itex]]=0..How could i deal with this calculation..? Thanks in advance.
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Ace10
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#2
Oct29-13, 12:16 PM
P: 6
sorry about the equation faults, if something is not clear or needs correction, please let me know.
Avodyne
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#3
Oct29-13, 01:22 PM
Sci Advisor
P: 1,185
Try putting the entire equation in tex. The tex symbols for left and right angle brackets are \langle and \rangle.

Crake
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#4
Oct29-13, 01:55 PM
P: 65

Complex scalar field propagator evaluation.


Quote Quote by Ace10 View Post
Good afternoon fellow scientists,i have a small problem in evaluating the propagator for the complex Klein-Gordon field. Although the procedure is the one followed for the computation of the propagator of the real K-G field, a problem comes up:

As known: <0|T[itex]\varphi^{+}(x)\varphi(y)[/itex]|0>=[itex]\Theta(x^{0}-y^{0})<0|[/itex]\varphi^{+}(x)\varphi(y)|0>+[itex]\Theta(y^{0}-x^{0})<0|\varphi(y)\varphi^{+}(x)[/itex]|0>

and <0|[/itex]\varphi^{+}(x)\varphi(y)|0>=<0|\varphi(y)\varphi^{+}(x)[/itex]|0>

But if we try to verify that one of the above correlation functions is a green's function of the K-G equation we hit the obstacle: [itex]\partial_{x}[/itex]<0|[/itex]\varphi^{+}(x)\varphi(y)|0> . And I refer to it as an obstacle because of the commutation relation [\varphi(x),\pi^{+}(y)]=0..How could i deal with this calculation..? Thanks in advance.
Hello,

You should take a look at this link. I know it's quite hard to find, but it gives a pretty good introduction on mathjax/latex.
Ace10
Ace10 is offline
#5
Oct30-13, 06:58 AM
P: 6
I think its ok now..Good afternoon fellow scientists,i have a small problem in evaluating the propagator for the complex Klein-Gordon field. Although the procedure is the one followed for the computation of the propagator of the real K-G field, a problem comes up:

As known: <0|Tφ + (x)φ(y) |0> = Θ(x 0 −y 0 ) <0|φ + (x)φ(y) |0> + Θ(y 0 −x 0 ) <0|φ(y)φ + (x) |0>

and <0|φ + (x)φ(y) |0>=<0|φ(y)φ + (x) |0>

But if we try to verify that one of the above correlation functions is a green's function of the K-G equation we hit the obstacle: ∂ x <0|φ + (x)φ(y) |0> . And I refer to it as an obstacle because of the commutation relation [φ(x),π + (y) ]=0..How could i deal with this calculation..? Thanks in advance.
As for the problem itself, any help?
andrien
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#6
Oct30-13, 07:14 AM
P: 976
See if this helps,you can see it further in Peskin and Schroeder's book.
Ace10
Ace10 is offline
#7
Oct30-13, 08:40 AM
P: 6
Thanks adrien,I have in mind the corresponding paragraph in Peskin and Schroeder's book but I'll check this out too, it's quite helpful.


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