Most recent edit on 2008-05-05 08:51:30 by CharlesFrancis
Additions:
This is evaluated as a contour integral» using the residue theorem», and noting that the integral on C2 vanishes in the lower half plane if x0 > 0, and in the upper half plane if x0 < 0.
Deletions:
This is evaluated as a contour integral» using the residue theorem», and noting that the integral on C2 vanishes in the lower half plane if x0 > 0, and in the upper half plane if x0 < 0.
Edited on 2008-02-06 02:58:05 by CharlesFrancis
Additions:
This is evaluated as a contour integral» using the residue theorem», and noting that the integral on C2 vanishes in the lower half plane if x0 > 0, and in the upper half plane if x0 < 0.
Using the above lemma,
where 
if 
and 
if 
is the momentum from j to i. Provided that the integrals converge at 
, the step functions can be summed to unity in the integrand.
For Dirac particles, the creation and annihilation operators obey
Hence, using the above lemma,
where 
if 
and 
if 
is the momentum from j to i (in the direction of the arrow). Provided that the integrals converge at 
, the step functions can be summed to unity in the integrand.
For a Dirac particle,p0 > 0 , and we can simplify the denominator by shifting the pole under the limit, replacing 2ip0 + ε2 with iε. Thus, the Dirac propagator arrowed from j to i is
Deletions:
This is evaluated as a contour integral» using the residue theorem», and noting that the integral on C2 vanishes in the lower half plane if x0 > 0, and in the upper half plane if x0 < 0.
Using the (lemma above),
where if and if is the momentum from j to i. Provided that the integrals converge at , the step functions can be summed to unity in the integrand.
It has been shown that, for Dirac particles, the creation and annihilation operators obey
Hence, using the lemma (above),
where if and if is the momentum from j to i (in the direction of the arrow). Provided that the integrals converge at , the step functions can be summed to unity in the integrand.
For a Dirac particle,p0 > 0 , and we can simplify the denominator by shifting the pole under the limit, replacing 2ip0 + ε2 with iε. Thus the Dirac propagator arrowed from j to i is
Edited on 2008-02-06 02:38:28 by CharlesFrancis
Additions:
This is evaluated as a
contour integral» using the
residue theorem», and noting that the integral on C2 vanishes in the lower half plane if
x0 > 0, and in the upper half plane if
x0 < 0.
Deletions:
<img title="contour for x^0<0" class=right alt="Propagators-9" src="images/feynman/Propagators-9.gif">This is evaluated as a [[http://en.wikipedia.org/wiki/Methods_of_contour_integration contour integral]] using the [[http://en.wikipedia.org/wiki/Residue_theorem residue theorem]], and noting that the integral on C2 vanishes in the lower half plane if <span class=math><i>x</i><sup>0 > 0, and in the upper half plane if <span class=math><i>x</i><sup>0 < 0"".
Oldest known version of this page was edited on 2008-02-06 02:37:31 by CharlesFrancis []
Page view:
Derivation of Propagators
Lemma:
Proof:
Since
,
So,
<img title="contour for x^0<0" class=right alt="Propagators-9" src="images/feynman/Propagators-9.gif">This is evaluated as a [[http://en.wikipedia.org/wiki/Methods_of_contour_integration contour integral]] using the [[http://en.wikipedia.org/wiki/Residue_theorem residue theorem]], and noting that the integral on C2 vanishes in the lower half plane if <span class=math><i>x</i><sup>0 > 0, and in the upper half plane if <span class=math><i>x</i><sup>0 < 0.
====<a name="
ThePhotonPropagator"></a>The Photon Propagator====
Using the ([[Propagators lemma above]]),
<img title="The photon propagator" alt="Propagators-10" src="images/feynman/Propagators-10.gif" align="texttop" vspace="2">
<img title="substitute" alt="Propagators-11" src="images/feynman/Propagators-11.gif" align="texttop" vspace="1">
<img title="Use the lemma" alt="Propagators-12" src="images/feynman/Propagators-12.gif" align="texttop" vspace="2">
where <img title="tilda’d quantity" alt="Propagators-13" src="images/feynman/Propagators-13.gif" align="texttop" vspace="0"> if <img title="positive time" alt="Propagators-14" src="images/feynman/Propagators-14.gif" align="texttop" vspace="0"> and <img title="tilda’d quantity" alt="Propagators-15" src="images/feynman/Propagators-15.gif" align="texttop" vspace="0"> if <img title="negative time" alt="Propagators-16" src="images/feynman/Propagators-16.gif" align="texttop" vspace="2"> is the momentum from <span class=math><i>j to <span class=math><i>i. Provided that the integrals converge at <img title="equal time" alt="Propagators-17" src="images/feynman/Propagators-17.gif" align="texttop" vspace="2">, the step functions can be summed to unity in the integrand.
<img title="sum the step functions" alt="Propagators-18" src="images/feynman/Propagators-18.gif" align="texttop" vspace="2">
<a href=
http://www.teleconnection.info/rqg/FeynmanDiagrams#TheFeynmanPropagator>Return</a>"»"
The Dirac Propagator
It has been
shown that, for Dirac particles, the creation and annihilation operators obey
and
.
Hence, using the lemma (
above),
where
if
and
if
is the momentum from
j to
i (in the direction of the arrow). Provided that the integrals converge at
, the step functions can be summed to unity in the integrand.
For a Dirac particle,
p0 > 0 , and we can simplify the denominator by shifting the pole under the limit, replacing
2ip0 + ε2 with
iε. Thus the Dirac propagator arrowed from
j to
i is
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