Lecture 2. Why BEC is linked with single particle quantum behaviour over macroscopic length scales
Lecture #1: Quantum electrodynamics
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Lecture #1: Quantum electrodynamics QED Lagrangian and Hamiltonian, equation of motion, scattering matrix, perturbation theory, Feynman rules, probability and effective cross section. Let us recall certain facts. Free electron, Dirac equation: 0 ) ( = − ∂ ∂ x m x i ψ γ μ μ , or ( ) 0 ) ( = − / x m p ψ , where μ μ μ μ γ x i p p p ∂ ∂ = = / , Solution for a free electron - flat wave*spin factor, corresponding to different helicity states (λ/2): ipx e p u x − = ) ( ) ( λ ψ , where + + = λ λ λ χ ε λ χ ε m p m p u | | ) ( , = = − + 1 0 , 0 1 χ χ More importantly (and very usefully!) ∑ = ± / = ± ± 2 , 1 ) ( ) ( λ λ λ m p p u p u Dirac equation for interacting electron: A e p p / + / → / , ) ( μ μ μ μ μ μ μ γ γ γ eA x i A e x i + ∂ ∂ = + ∂ ∂ = 0 ) ( ) ( = − + ∂ ∂ x m x eA x i ψ γ μ μ μ Photon- quantum of EM field: νμ μν F F − = , l ikl ik H F ε = , i i iE F − = 4 Or ν μ μ ν μν x A x A F ∂ ∂ − ∂ ∂ = In vacuum: 0 ) ( = ∂ ∂ x F x μν ν Solution for free photon – flat wave*spin factor (s=1): ikx e k e A − = ) ( μ μ , where μν ν μ μ δ = = e e k e * , 0 Interacting photons == Maxwell’s equations: (HW: show that this indeed is the case): ) ( ) ( x j x F x μ μν ν = ∂ ∂ ) ( ) ( ) ( x x e x j ψ γ ψ μ μ − =
Transcript of Lecture #1: Quantum electrodynamics
Lecture #1: Quantum electrodynamics QED Lagrangian and Hamiltonian, equation of motion, scattering matrix, perturbation theory, Feynman rules, probability and effective cross section. Let us recall certain facts. Free electron, Dirac equation:
0)( =⎟⎟⎠
⎞⎜⎜⎝
⎛−
∂∂ xmx
i ψγµ
µ , or ( ) 0)( =−/ xmp ψ , where µ
µµµγ xippp∂∂
==/ ,
Solution for a free electron - flat wave*spin factor, corresponding to different helicity states (λ/2):
ipxepux −= )()( λψ , where ⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
+
+=λ
λ
λ χελ
χε
mpmpu ||)( , ⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛= −+ 1
0,
01
χχ
More importantly (and very usefully!) ∑=
±/=±±2,1
)()(λ
λλ mppupu
Dirac equation for interacting electron:
Aepp /+/→/ , )( µµ
µµµµ
µ γγγ eAx
iAex
i +∂∂
=+∂∂
=
0)()( =⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−⎟⎟⎠
⎞⎜⎜⎝
⎛+
∂∂ xmxeAx
i ψγ µµ
µ
Photon- quantum of EM field: νµµν FF −= , liklik HF ε= , ii iEF −=4
Or ν
µ
µ
νµν x
AxAF
∂
∂−
∂∂
=
In vacuum:
0)( =∂∂ xFx µνν
Solution for free photon – flat wave*spin factor (s=1): ikxekeA −= )(µµ , where µννµµ δ== eeke *,0
Interacting photons == Maxwell’s equations: (HW: show that this indeed is the case):
)()( xjxFx µµνν
=∂∂
)()()( xxexj ψγψ µµ −=
Lagrange equation for field φ :
,0,
=∂∂
−∂∂
∂∂
φφ µµ
LLx
where µµ
φφ ,=
∂∂x
We need a Lagrangian that will give us both Dirac and Maxwell equations (HW1):
ψγψ µµ
µµυ
µυ⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−+∂∂
+−= meAx
iFFL )(41
To decouple free particle pieces and interaction pieces let us rewrite it as
ψγψψγψ µµµ
µµυ
µυ eAmx
iFFL +⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−∂∂
+−=41
Interaction Lagrangian: µµµµ ψγψ AjeALI −==
Interaction Hamiltonian:
∫−= xdLH II3
Scattering matrix )(),()( −∞Φ−∞∞=∞Φ S
Perturbation theory tells us:
),()(),( 00 ttStHttSt
i I=∂∂
The solution in its exact form:
∫∫ =⎟⎟⎠
⎞⎜⎜⎝
⎛−=
+∞
∞−
))(exp()(exp 4xdxLiTdttHiTS II
Please note that in its exact form the S-matrix is always unitary: ),exp(iR where R is real. If the interaction is weak we can use the perturbative expansion of the exponent:
))()...()((...! 21
42
41
4)(nIIIn
nn xLxLxLTxdxdxd
niS ∫∫∫=
Thus, knowing (effective) Lagrangian one is able to write the scattering matrix. Or, closer to home
∫ /××/×/−
= nnnn
nn xdxdxdxxAxxxAxxxAx
nieS 4
24
14
222111)( ...)()()(...)()()()()()(
!)(
ψψψψψψ
Matrix element for process iàf can be written as fSiM n
if || )(= Each fermion’s wave function can be written as a sum of creation of fermions with certain quantum numbers ( sa )and destruction of antifermions ( sb+ ). Similarly, each antifermion’s wave function can be written as a sum of creation of antifermions with certain quantum numbers ( sb ) and destruction of fermions ( sa+ ):
)(0|)(|1),(1|)(|0
),(0|)(|1),(1|)(|0
)()()(
;)()()(
)()(
)()(
)()(
)()(
xxxx
xxxx
xbxax
xbxax
rrrrrr
ssssss
rrr
sss
rrr
sss
−−−−−−
++++++
−++
−++
==
==
+=
+=
∑∑
∑∑
ψψψψ
ψψψψ
ψψψ
ψψψ
Now, in the expression for S we need to move all creation operators to the right and destruction operators to the left. Mind that bosons (photons) commute, while fermions (electrons) anticommute, thus you get a negative sign every time two fermions change place. Normally it does not matter, because we are interested in the square of the matrix element. It does matter when we’ll need to consider interference between two or more diagrams. Need to be careful with the relative sign of permutations. Field operators in momentum representation:
Inner photon line (Fourier transformation of the Green’s function))0(
)( 2 ikig
kD+
−= µν
µν
Massive boson )0()/(
)( 22
2
iMpMppgi
pD+−
−−= νµµν
µν
Inner electron line 0)(
0)( 22 imp
mpiimp
ipS+−
+/=+−/
=αβ
Photon operators )(1|)(|0 λµλµλ exA =
*)(0|)(|1 λµλµλ exA =
Electron operators )(1|)(|0 )()( pux µµµ ψ =++
)(0|)(|1 )()( pux µµµ ψ =++
)(1|)(|0 )()( pvx −=+− µµµ ψ
)(0|)(|1 )()( pvx −=−− µµµ ψ
Photon-fermion vertex µγie - energy momentum conservation at each vertex. But… Internal lines do not have to be “on mass shell”. 22 mp ≠ !!! Useful gamma-matrix trivia:
)(4)( ,0)#(4)( ,41
,
)( ,
0000
0
000
νσµδνδµσσδµνδσνµ
µννµ
µµµµ
γγγγγ
γγ
γγγγγγγγ
γψψ
γγγ
ggggggTroddTrgTrTr
I
+−==
==
==
=
==
++
+
+
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