in the rest frame of the spin-½ particle: spin up electron spin down electron ?? is the e= mc 2...
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In the rest frame of the spin-½ particle:
spin upelectron
spindown
electron
? ?
Is the E=mc2 unphysical? Meaningless?
Can we enforce B always be zero?
ue rpEti )(
2242 cpcmE )()( tEtEtE
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1932 Carl Andersonpublisher’s thiscloud chamberphotograph.
Droplet density (thickness) of track identifies it as that of an electron?????????
Curvature of track confirms the charge to mass ratio (q/m) is that of an electron?????????
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B-field into page
Direction of curvature
clearly indicates it is
POSITVELY charged!
The particle’s slowing in its passage through lead foil establishes its direction
( UP! )
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Additional comments on Matter/Antimatter Production
e+eParticles are created in pairs e+
e
and annihilate in pairs
Conserves CHARGE, SPIN(and other quantum numbers
yet to be discussed)
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p+pp+p+p+p
Lab frame (fixed target) Center of Momentum frame
a b a c db
a b
at thresholdof production final state
total energyEalab Eb
lab=mc2
palab pb
lab=0
So conservation of energy argues: EaCOM+Eb
COM=4mc2
= 4mprotonc2
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By conservation of energy: EaCOM+Eb
COM=4mc2
and
by the invariance of the inner produce of the 4-vector pp
(EaCOM+Eb
COM)2 (paCOM + pb
COM)2c2
=(Ealab+Eb
lab)2 (palab + pb
lab)2c2
paCOM + pb
COM = 0 mc2 0
( 4 mc2 )2 = m2c4 + 2Ealabmc2+ m2c4
16mc2 = 2m2c4 + 2Ealabmc2
Ealab = 7mc2 = 6.5679 GeV
(using mp=938.27231 MeV/c2)
(EaCOM+Eb
COM)2 = (Ealab+ mc2)2 (pa
lab)2c2
= (Ealab )22Ea
labmc2+ m2c4(palabc)2
= {m2c4+(palabc)22Ea
labmc2+ m2c4(palabc)2
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BevatronBeam
Carbon Target
M1
Q1
Shielding
S1
Q2 M2
C1
C2
C3
S2
S3
1955 - Chamberlain, Segre, Wiegrand, Ypsilantis
Berkeley BEVATRON accelerating protons
up to 6.3 GeV/c
10 ft
magnetic steeringselects
1.19 GeV/c momentum negatively
charged particles
Čerenkov counters
thresholdsdistinguish > 0.75 > 0.79
scintillation countersmeasure particle“time of flight”
1.19 GeV/c s: 0.99c 40nsec Ks: 0.93c 43nsec
ps: 0.99c 51nsec
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0.5 1.0
Ratio: m/mproton
Selecting events with TOF: 401 nsec
and 0.79<
0.148=m/mp
Selecting events with TOF: 511 nsecand 0.75<<0.79
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2 3 4 5 6 7 8 GeV
Anti-proton production rate(per 105 ) vs beam energy
2.0
1.0
The Fermi energyof the confinedtarget protons
smears the turn-on curve.
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4.0 5.0 6.0 7.0
Ant
i-pro
tons
per
105
- s
proton kinetic energy GeV
The Fermi energy of the confinedtarget protons smears the
turn-on curve.
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0))((
mcimci
We factored the Klein-Gordon equation into
then found solutions for:
0)( mci
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Free particle solution to Dirac’s equation
(x) = ue-ixp/h
u(p)
1
0
cpz
E+mc2
c(px+ipy)E+mc2
0
1
c(pxipy)E+mc2
cpz
Emc2
1
0
cpz
Emc2
c(px+ipy)
Emc2
1
0
c(pxipy)Emc2
cpz
Emc2
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0)( mci
What if we tried to solve:
We would find 4 nearly identical Dirac spinors with the uA, uB (matter/antimatter entries) interchanged:
E+mc2 Emc2
0))((
mcimci
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xax In general, any ROTATION or LORENTZ Transformation mixes vector components:
33221100 xaxaxaxaxa
space-time coordinates
not the spinor components!
a = sin, cos, 1, 0 for R
= , , 1, 0 for
If we want to preserve “lengths” and “distances”
33221100 xxxxxxxxxxxx
xxaaxaxa ))(( aa
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aa
Now watch this:
)(
1
1
aa
aa
aa
taa 1 The transformation matrices must be ORTHOGONAL!
axx ' 'xax tSo must mean
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axx ' 'xax tSo must mean
xxaa
xaaxa
xaaxa
xax
'
)(')(
'
1
11
xxa '
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xxa '
xax'
a
xx
x
xx
''chain rule (4 terms!)
xa
x
' x
ax
'
or
Finally