Download - But the integral is not Lorentz invariant!
![Page 1: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/1.jpg)
n
jjjzjy
n
jjx pdMdpdpdpMd
2
32
2
2
But the integral is not Lorentz invariant!
A particle one decay: n 321
![Page 2: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/2.jpg)
33
3
3
222 22
1
22
1
pd
E
pd
cmp
We make it more complicated by allowing an indefinite p0 and then fixing it:
0222
4
4
22
pcmppd
But in this form, we can be sure it is Lorentz invariant!We can perform the p0 integration to recover the 3 space form.
22220222 cmppcmp
)(2
122 axaxa
ax
2220
222
022220
2
1cmpp
cmppcmpp
This is Lorentz invariant.
![Page 3: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/3.jpg)
n
jjpdMd
2
32
n
jjj
n
j j mcpE 2 2222 2
1
2
1
n
j
j
j
pd
EMd
23
32
22
1
npppp 321442
No matter what, the overall 4-momenta are conserved!
n
j
j
jn
pd
EppppMd
23
3
321442
22
12
![Page 4: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/4.jpg)
n
j
j
jn
pd
EppppMd
23
3
321442
22
12
But M2 is still not Lorentz invariant.Γ transforms like 1/t1. t1 transforms like E1.
2
2
1cv
t
2
2
2
1cv
mcE
1
1
E
n
j
j
jn
pd
EppppM
Ed
23
3
321442
1 22
12
1
Now we can be sure M2 is Lorentz invariant. It’s called Feynman Amplitude.
![Page 5: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/5.jpg)
Now we apply this to pion two photon decay.
20
Choose the rest frame of pion:
21 cmE 01 p
3
33
332
3
2321
442
1 22
1
22
12
1
pd
E
pd
EpppM
Ed
![Page 6: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/6.jpg)
![Page 7: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/7.jpg)
Two body scattering: n 321
n
j
j
jn
pd
EppppMd
33
3
321442
22
12
as you would expected If we consider only Lorentz transformation along the 1-2 colliding axis, the cross section is invariant! But we do want to pull out a Lorentz invariant factor that reflects the inverse flux that must appear in cross section:
![Page 8: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/8.jpg)
2221
2214 cmmpp
In the rest frame of particle 2
11212
222
1212
2221
221 444 vEEpcmcmEcmcmmcmE
112 v
n
j
j
jn
pd
EppppMd
33
3
321442
22
12
2221
2214
1
cmmpp
n
j
j
jn
pd
EppppM
cmmppd
33
3
321442
2221
221
22
12
4
1
![Page 9: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/9.jpg)
Two body scattering
![Page 10: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/10.jpg)
![Page 11: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/11.jpg)
Feynman Rules
To evaluate the Lorentz Invariant Feynman Amplitude MComponents of Feynman Diagrams
External Lines
Internal Lines
Vertex
ip
iqpropogator
Particle Content (their masses and spins)
Interactions
![Page 12: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/12.jpg)
For a toy ABC model
Three scalar particle with masses mA, mB ,mC
Factors
External Lines
Internal Lines
Vertex
ip
iq
-ig
1
222 cmq
i
jj
321442 kkk
A B
C
1k
2k
3k
![Page 13: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/13.jpg)
Draw all diagrams with the appropriate external lines
Integrate over all internal momenta iqd 442
1
Take out an overall momentum conservation. npppp 321
442
That’s it!
It’s so simple.
![Page 14: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/14.jpg)
Consider first the diagrams with the fewest number of vertices.
CBA
Each extra vertex carries an extra factor of –ig, which is small.
![Page 15: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/15.jpg)
Scattering
To the leading order, there could be more than one diagrams!In the diagrams, the B lines in some vertices are incoming particle.
The striking lesson: Lines in a vertices can be either outgoing or incoming, depending on their p0.p0 for an observed particle is always positive.So this vertex diagram is actually 8 diagrams put together! That’s the simplicity of Feynman diagram.
![Page 16: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/16.jpg)
It’s similar to electron-electron scattering!
![Page 17: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/17.jpg)
![Page 18: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/18.jpg)
24 pp
The momentum conservations can be carried out immediately in the beginning!
The result can be written down right away!
This page also teaches us an important lesson:
After momentum conservation is enforced, the momentum of internal particle c is
24 pp
It does not satisfy momentum mass relation of a particle! 22224 cmpp c
The internal particle is not a real particle, it’s virtual.
![Page 19: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/19.jpg)
媒介粒子
交互作用是透過交換媒介粒子來進行
媒介粒子媒介一個交互作用
γ
![Page 20: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/20.jpg)
Feynman’s Diagram
![Page 21: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/21.jpg)
n
p
p
n
交互作用不只改變粒子的動量,也可能改變粒子的身分!
![Page 22: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/22.jpg)
So heavy particles, though decay fast and don’t existent in nature, do matter. (they could be virtual)
媒介粒子
媒介粒子只有在交互作用進行中存在,如果只觀察物體前後變化,就不會看到它。它幾乎不是一個實在的粒子,而是一個虛粒子。
虛粒子即使不穩定地存在,也會透過它所媒介的交互作用,影響實在的世界
![Page 23: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/23.jpg)
24 pp
It does not satisfy momentum mass relation of a particle! 22224 cmpp c
The internal particle is not a real particle, it’s virtual.
24 pp
In a sense, the relation is only for particles when we “see” them!
Internal lines are by definition “unseen” or “unobserved”.
It’s more like:
It’s more a propagation of fields than particles!
![Page 24: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/24.jpg)
24 pp
Field fluctuation propagation can only proceed forward that is along “time“ from past to future.
This diagram is actually a Fourier Transformation into momentum space of the spacetime diagram.
The vertices can happen at any spacetime location xμ and the “location” it happens need to be integrated over. After all you do not measure where interactions happen and just like double slit interference you need to sum over all possibilities.
![Page 25: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/25.jpg)
B
A
B
A
For those amplitude where time 1 is ahead of time 2, propagation is from 1 to 2.
C1
2
B
A
B
A
For those amplitude where time 2 is ahead of time 1, propagation is from 2 to 1.
C
1
2
24 pp is actually the sum of the above two diagrams!Feynman diagram in momentum space is simpler than in space-time!
![Page 26: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/26.jpg)
24 pp
In this diagram, particles seem to have certain momenta. According to uncertainty principle, their position uncertainty is infinity!This diagram is only an approximation of more precise treatment using wave packet!
You need to “superpose” similar diagrams with slightly uncertain momentum t produce incoming and outgoing wave packets.
Lesson 3:
![Page 27: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/27.jpg)
![Page 28: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/28.jpg)
Δk Δx
In most situations, the Feynman amplitudes are not very sensitive to small uncertainties in momenta.We can approxiamte the amplitude for wave packets with a momentum distribution by that for plane waves with the central momentum value:k0
![Page 29: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/29.jpg)
Lifetime of A
CBA
321442 ppp igiM
gM
20 for m
![Page 30: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/30.jpg)
![Page 31: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/31.jpg)
![Page 32: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/32.jpg)
![Page 33: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/33.jpg)
If all the momenta in the diagrams can be determined through momentum conservation, the diagram has no loop and is hence called tree diagram!
24 pp
If there is a loop in the diagram, some internal momentum is not fixed and has to be integrated over!
![Page 34: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/34.jpg)
![Page 35: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/35.jpg)
The WorldParticle content
![Page 36: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/36.jpg)
qe,
qe,
Interactions
![Page 37: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/37.jpg)
![Page 38: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/38.jpg)
Schrodinger Wave Equation
He started with the energy-momentum relation for a particle
made Quantum mechanical replacment:
How about a relativistic particle?
![Page 39: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/39.jpg)
The Quantum mechanical replacement can be made in a covariant form:
As a wave equation, it does not work.It doesn’t have a conserved probability density.It has negative energy solutions.
![Page 40: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/40.jpg)
The proper way to interpret KG equation is it is actually a field equation just like Maxwell’s Equations.
Consider we try to solve this eq as a field equation with a source.
)(2 xjm
We can solve it by Green Function.
')',()',( 2 xxxxmxxG
G is the solution for a point-like source at x’.
By superposition, we can get a solution for source j.
)'()',(')()( 40 xjxxGxdxx
![Page 41: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/41.jpg)
Green Function for KG Equation:
')',()',( 42 xxxxmxxG
By translation invariance, G is only a function of coordinate difference:
)'()',( xxGxxG
The Equation becomes algebraic after a Fourier transformation.
)(
~
2)'( )'(
4
4
pGepd
xxG xxip
1)(~22 pGmp
)'(4
44
2)'( xxipe
pdxx
22
1)(
~
mppG
This is the propagator!
![Page 42: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/42.jpg)
'x x
Green function is the effect at x of a source at x’.
That is exactly what is represented in this diagram.
KG Propagation
The tricky part is actually the boundary condition.
![Page 43: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/43.jpg)
B
A
B
A
For those amplitude where time 1 is ahead of time 2, propagation is from 1 to 2.
C1
2
B
A
B
A
For those amplitude where time 2 is ahead of time 1, propagation is from 2 to 1.
C
1
2
is actually the sum of the above two diagrams!
To accomplish this, 22
1)(
~
mppG
imp
pG
22
1)(
~
![Page 44: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/44.jpg)
We need:
![Page 45: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/45.jpg)
Dirac propose it could be true for matrices.
![Page 46: But the integral is not Lorentz invariant!](https://reader031.vdocuments.net/reader031/viewer/2022012916/568138ad550346895da06b09/html5/thumbnails/46.jpg)