tt properties ÷i÷m=ir÷:: · for example in' = i'rn y = tiz rude, y = mouthy = mojo t...
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33.DiracFieldBiliue1. Definition :
¥ingrrr3"
Ita )with @5)t -- Tt ,
(2532=1,{85,893=0 for µ -
- o.az, ]
Tluplies [yes ,Sw ] .
- o → (Eco ) ⑤ (QE),2
. The following bilinear leave definite transformation properties under the Lorentza- group :
M = 1 scalar x1 Sot@e3 )
÷i÷m=ir÷::: I :÷÷:! ":÷:→ →
TMNT pseudo - vector x 4c -7 - c
25 pseudo -scalar x i [' = F) x F
For example ,
in'
= I'rn y ' = Tiz'
rude,Y = Mouthy = Mojo
T°→ JM is conserved Noether current corresponding to the continuous symmetry
4 → eid 4 of the Dirac Lagrangian
3.4.Quautizahuuaffkedire.CI#
1. Lagrangian :
F- 4-( irade - m) 4 = 4-(if- m) y
2.
Canonical momentum : ITA = §¥p.
= i 4ft
3. Hamiltonian H=fd3× 4+1- i xD turf] 4 with 1=20 ,I = NOT
?
-
= Hp
→ Expand 4 in eigeucuodes of HD to diagonalize H
4. Eigeuueodes : Hp Us#lei
?= Epa - n - and H, use
') e-i ⇒
=- Ep, - n -
[ Cirodo tir't - m ) 4=0 ]
T.tlodeexpausiceu.nu/cxT--gfdIE,fzep#Ca.suscpTeiF'⇒t h use, e- IFI' ] a.m,
6.
Use
Hokey - g f EET Capsule're "⇒ - h¥us⇐, e- IFE ]then ( orthonormalit-1 relations (3,9 ) )
H -- fax 4TH ,> 4 E g f EF laptops - bsptbp )
T-irsttoy.co#tafor
7-.
Canonical quantization with equal- bine commutators:
[Ya#t.IT#1T--i8asoh1Cx'- I') ⇐ [14*1,4543]=5*8*-57 (3. ez,
[Yuli't 14¥13 - O
8 →° Made algebra
[Api, aaft ] = [ SET , bit ] = Carp Srs or (F - E)
[Open, Sgt't' ] a- o
M show : Gml and spin sans knot → Girl, } [4. IT, lusty, ]→ ioreducille Representation = Ei Foch space =4Ut4
u-soo
9.Poesie (bist)"
107 has energy - u Ep → - •
-7 Uo stable vacuum state
10. Fix C? ) : best
e) CH, 6,5+3 = Ep Spt → ftp.t
ii ::÷i÷÷÷:÷÷:÷.is/:::::ia:iii:c) [ftp.bsqit ] = - HTP 8" s E'-I')d) H= . . .la#aEi-bsptSf. ) + court
f)But 11 BEE lo> 112 = Lol CBE, bit ] to> = - Gaps Co) s O
→ negative worm States
11.com#iou:C3.pis
• either an instability of flee vacuum
• or a loss of unitarily
→ no consistent quantization possible !
-
Second toy : Auficocuusofatoos-
7. Canonical quantization with equal- time commutators :
{4EhUst#i}=8as8③⇐-57aud{Ua#h4E7}8
.
°→ Mode algebra :
ftp.asqstf-fbp.SI#lg--l2ct5sstss31E-E'1audfap,sgtH2g--0(
fermionic Foch Space→ irreducible Representation =-
9.Tootie b'pit (G) has energy - Eps & infinite sum over momenta
→ still no stave vacuum state
10. Fix C? ? ) : best
" "amino
invariant under b.← St !b) The mode algebra (3,73) is-
→ Unitarily is preserved and Hamiltonian is bounded from below
tHeiseubeogpictuif :* eilttap, e- i Ht z ape, e-
iEftand eitttgp, e - iHt ± Sps, e-
iEat
and 4 =eittt yes, e- IH t we feed
:÷:÷÷÷÷÷⇐i÷÷::÷.int#:::::::i:SContinuous symmetries E- Conserved charges~
• Time translation → Hamiltonian
• Spatial translation → momentum operator
F' ± Idk 44in, y ± I Ftaistaptbpitssp )
→
• Spatial rotations → angular lceacueufucy operator J
=p fax lutfxxc- it) + I ) 4 wine I -_ (% fs.eu,@i'4 °→ conserved current jM= 4-VM 4 -3 conserved charge :"""
:÷÷*÷÷:÷÷÷÷÷÷÷÷÷÷Excitatiaue.Parb.cl#dpt1o) : Fermion with Energy EF
bast 10) : Antifermion with Energy EFmomentum I
'
momentum p→
Spin 3=12 (polarizations )spin 3=12 ( polarization opposite to 5)and charge Q = tl
and change Q =- l
Note3
• The two state , for 5=1,2 suggests a spin - E representation
• To show this ,the action of J ( see 9,14) ) on one-particle States must be studied
• One fiudl for particle , at test :
Tz doit = ± E aft and Iz b to> = 't E soit lo>T
+ gsm.,- 5--2=191
horeutttraueformatr.eu#
1. ¢ Lorentz transformation 1 E 50443) on single particle state IF, s > = 2EF GET to>
His > → UN E'is >
UX ) : representation of 50473) on Foca space
<→2
. Special case : quantization axis parallel to boost and/or rotation axis ④→ spin polarizations do aof mix :
U@laps, U - n =EEF a in general : mixing\of age 9,5 )
3.Then :
stair > = 2 EIGHT 84145'- I') S" = E. slut# UX ) r )- -
Looeutt invariant =L
→ UN is unitary
4.Now we have 3 representations :
I acts on 4- vectors in 112%3 D= 4 not unitaryi÷::::÷::÷:÷:÷¥:÷::::::÷
F.Action by conjugation ace field operators :
→ UXUXU-nxi-li.io#
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