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#