Physics 618 2020

March 24,2020

Wignerstheonemsymmetryin QM - preserves

probabilities of measurements

MY Map from Pune states

- > pure states

preserving overlaps

Map : PILL :-. {rank 1

TProjectors

HilbertPiH→Le

}space

Tr P, Pz

f : P- s FCP )

Tr f CP, )ffPz)=TrP , PzH P , ,PzG BLL

PILL is not linear

e.g. K= 62 Rfl=÷- s2

Auttfl ) - U#l ) IAUHL )

ueU(Ll ) ktull = 11411

ae AUCH ) is �1� - antilin

a 4,+4. ) = AH

, ) talk )

but ZEE a €4 ) = z*a(4 )

a is anti unitary if in addition

HAYK = 11411 ¥ fem.

Autcfl ) is a group .

\ 11 a°u(y ) 11=11%11 = 11411

/ :I⇐II÷I:* .tn,

but a,

. as is unitary because

( Z* )* = Z

.

¢ : Autfk ) ⇒' Zz ={±i }(si . se'

itssgaiitilw,

) ¢ is a grouphon.

Hua )→Autee)§ 2- 1

it : Aotffl ) - AUKQM )"

group

f :Rfl→pLlpreserving

overlaps

Icu ) : P - U Put = up a-'

¥=

it (a) : P -s a Pat = apoi'¥ I

If A is ¢ -

anlilinearptph( Atul

, ,4. ) = ( 4

, ,A4D*Compare A is �1�- linear

( Att, ,4 . ) = ( 4

, ,Ak )ut . u

' '

a+=a. '

By cydicity of truce :

Trfialpi) itcukpz ) ) - TrydRR )

:- utfk)±sAut(QM)→

1

Wigner .

.

This is {orjed-

ie. any symmetry of a quant.

System can be represented bya unitary or anti unity operator

on Ll.

Ker ⇐ ) = { z . I / Izki }I UC1)

HUH → Auth ) Is AUHQM)→1

In general we'll have dynamics

H+=H Hamiltonian

Somegroup

G

zetxTInsionsu@DGzg.g

.

11 1 te

1→£EHAo#H) - AUHQM) → 1

G- is a central extension of G-

*OU(g,)U(gz)=C§ygDU(g.

phase-

Linear operators are alwaysassociative

.

(UCGDUCGZDUCGD= Ucg , )(U§⇒U1g } ) )

-g ;gDC(gigugs )

= C(g , , gzg })c(gz÷Proj . Rep 's are ubiquitous in QM

QFT. - -

Quant.

of bosons } fermions:

Heisenberg group Cliffordgroup

Xp--

-

T -

Centrally extends Centrallyextendstranslations

on phase SP.

zenAnomalies in QFT

Kao M . .dy groups Virasonogroups

-- .

.

How To Classify Central Extensions

A - Abeliangroup

G -

any group

1→ A -2, G Is G→ 1

ZCA ) a ZCE )

• If sequence splits then

WGCA ) = a at A central

EE A×G

• Choosea section

S : G -7€ Ios = Id

Ointer )

a

* ( Sky, ) scgz ) stggzj'

) = 1

s$S(gz ) Scg,gj'=r§(g±

-

x@sCgi7sCgD-KfsCgi.gzDsCg.gz.t

for some function

fsz GxG→A@Scg , )¢gz) Kg} ) ) =§Cg, )s§z ) )S(gD

|| F g , ,gz , g }EGt.ae?ikYIk#

ocycle identity ,

sple_G.nsa.mg FCG,

I )=f(hg)=ffhDfcgg'

) - fcjig )

Def :

.

1.

A 2- cochair on G w/ values in A

f : GXG → A

TfaHi¥ { 2- adais }=dGA )

2.

A 2- #l= on G w/ values in A

is FeCYGA) sit. f satisfies

the cocyde identity .

{ 2- aqeks }

:=Z2(G

,A )

Rmki : Language from topology .

1212 C- CG,

A ) and ZYGA )are both Abelian

groups=⇐¥k;9z)

=ffggDfA,gp( f

, .f*)(g , ,gD=£G ,,gD.fdg , ,q )- #

prod . in A

Eggas

;±÷§HA -Et÷G - i

defines fse ZYGA )

choose a differ

.se#S?

how is fg related to fs ?

I §§ ) ) =#(Scg ) ) ⇒

Rg)=2Hg))s§€yforso.me_fvnctiont.G-A.mg

§, ,gz ) = fs ( g , ,g . ) tcg ,

)t$tIgi*=}A Abelian =) order

doesn't matter

Used here that it is a centretea

DEI wesay

two cocydes f and I

differ by a cob=dary if F

t : G - A sit .

Icg, ,g . ) =fCg,,gDt§)t§⇒Cggz)

Equiv . Rll : I ~ f if theydiffer by acacycle . It ] . equiv .

D¥ HYG,

A ) - group cohomologyset of equiv. classes of Gayles

.

Thereto Isomorphism classes ofcentral extensions of G by A are in

LT 1- I correspondence with elements

of HCGA ).

habE.sc#TEExtChoose a section -→ # ] indptofsection

.

Ext →

H4GA←4G.AT#t1m......

' '

a⇐FMIkn.

1

4 :I

→ E'

ison.

s'

= Yos is a section of t'

signs 'cgz)=iH(g, ,g, ) ) skgigz)

⇒ Same cocyde fs , = fs.

suppose If ]E HYG ,A )

£ = AXG as a set

withgroup structure

( a ,g , ) . ( augz ): = ( aiazfcgygz )

, gig . )you

check : This is a valid

group law. Associative -←sfi÷

ycle .

If ] =[ I ] ←

then F isomorphism of extension

I

→A2→EIG - ) 1

it:@,gj→g

2 : a - Can )

Ex 'Ykm F- HYGA )' P

these are

inverses�1� Central extensions k

, projective reps .

pcg , ) pcgz ) - c(g,,gz)@§,gD×atisfies cocyderelation

i. define xuiixa 49$are valued1 → UH ) - E → G- ,

in Ua )e.

-

using thiscocycle .a

z, g) =

zpcg )

is a TRUE REPRESENTAION

Of E.

You thinkyou have a symmetry

group G. start to define

operators U (g) on Ll discover

Ucg, )U(gD=c§ygz)U(g→can't be removed by phaseredefinitionof Uq )

Truesymmetry group is E

,the

Central extension.

Expl= Rotationgroup SOB )

Yz -

spin rep's projective .

Truesymmetry group is SUH )

.

Ui ).

→ E - ssocs ) - si

¥Zj-s SUK ) → SOG ) - ) I

C c- ± ,

�2� HZCG ,A ) this in fact a

group .

[ f,] . [ f. ] = [ f

,.f

,]

"

check this is well - defined 4

LAI . #=#Eg- AYE

,"sG→i

1→AI , FTC → ,

1→A×A→EXGT -G÷ PaAxa ⇒ pullback → G

�3� Trividovvdzable

Trivial coeyde f@,sgz) = 1-

Trivializabltcocyde [ f ] - [ 1 ]

f §, ,g , ) =

491>+152 ) for-

sometlggz ) tt is called the "

trivialization .tt"

£,

,tz are two trivializations

t ,¥=tzcg)¢§ ) ¢ :G→A

is a grouphomomorphism

.

�4� Analogy to Gauge Theory

Ain = And - ijtxifngcx )

g : spacetime → UCI )

£ vs. f change of gauge.

Fm =

fnAr-dvAmFtta.F.dxrndxuCfTdtAAsIFw-o-Amis@DgqseEI.If Jc spacetime is a

closed loop1

§A := S Adxetsihdtfttdt0

t.DE#)ifalsgxee

It can happen that F= O

but Holt,

A ) is not =1

(~ solenoid ' '

spacetime"

y#|-=NRt{o}

Ditties.

§A= 2+13

e×p(2#iB) Betz

theorem Gauge equiv . class FAis determined by the function

Hal : closed loops → W )

�5� Often useful to choose gaugesto simplify a problem .

Let's use co boundaries to simplifya couple . te ZYG ,

A ) ⇒

f ( hg )=f( g ,1 ) = fan )

change by a co boundary t

f' ' '

( i,

, ) = f( i. , )t±¥il )

= f 11,1 ) Hl )So if we choose tcy =f( 1,15

'

then we set

I ' '

( 1.1 ) - f' ' '

( ng ) - f "G , D= 1

"normalized cocycle

"

further simplify .

so long as

El ( D= I. partial choice ofGage

ftkg ,gy=fEg,gyt%tE÷8.5's

= ff'g,jyEgtYj '

,

G = Innu.I Nontnv

.

{ glory {

"

glgzti }~ ,

Noninv = S,

IS,

Yno two elements ofS

,

⇐are related by gtg

'

choose EYG ) ge s, so that

IYG,g'1=1 tgeNoni

Ifge Inv

He's.gg?.=fYg.gstfYB:

(gg , EcgjWe haven't fixed I on Iuvyet .

If I 'Lg, g) is Not a perfectsquare in the

group we cannot

gauge it to 1.

Example of a

gauge invariant

obstruction to putting f → 7.

If Ag , g) is not a perfectSquare for some involution

GEGthen f is a nontrivial couple

.

Example 1Central

=: Extensions of Zz by Zz

run

± 2 T

1 →zz-E.->zoall

{ ho, } { to }

0,? i Fil

WLOG f( 1,1 )=f( 1,47-5-(0*1)=1f ( a ,r . ) ? Eti

f ⇐, . ) = 1 Trivial cocyck.

17 2h → zz*z.

→ 2<+1

f ( a, E) = o

,0

,is not a

perfect square !

E -3214

( 1,02 ) . ( l, I ) = ( far )

,rial )

= ( f ( a ,oi ),

1 )

= ( a, ,

I ) order 2

so ( I , E) has order 4 and

generates the whalegroup .

÷×ample= Ext 's of Zp by Zp

p -

prime) Zp → G 7 Zp e- 1

Methods of topology show that

HY zp ,Z

, ) ± ZpBut - > puzzle :

Sylow + classeq . → there

are only two isomorphism types of

groups of order p2 ! !

1 → Zp→ G- → zp → 1

then € I Zp×Zp

or E =xp .

} %possibilities

How can there bep different

isomorphism classes of central

extensions forp an odd prime ?

Answer : Friday.

÷aranor - Bohm effect

.

Inq.vn . the phase picked up

by a changed particle travelingin an electromagnetic field is

exp ifye Ae = eclectic

change .

HE+3=0K

Reps £411

ostdiedneoid peez

solenoid.

/4

,

=

explifea) 4

.

=) nontrivial interference

4, th ABeffect .

Flatgauge fields ( flat :Fµ=o )

Can nevertheless have nontrivial

holonomy.