physicsgmoore/618spring2020/618 march 24 2020.pdfdei we say two cocydes f and i differ by a cob=dary...
TRANSCRIPT
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
.
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$S(gz ) Scg,gj'=r§(g±
-
for some function
fsz GxG→A@Scg , )¢gz) Kg} ) ) =§Cg, )s§z ) )S(gD
|| F g , ,gz , g }EGt.ae?ikYIk#
ocycle identity ,
sple_G.nsa.mg FCG,
I )=f(hg)=ffhDfcgg'
) - fcjig )
Def :
.
1.
A 2- cochair on G w/ values in A
f : GXG → A
TfaHi¥ { 2- adais }=dGA )
2.
A 2- #l= on G w/ values in A
is FeCYGA) sit. f satisfies
the cocyde identity .
{ 2- aqeks }
:=Z2(G
,A )
Rmki : Language from topology .
1212 C- CG,
A ) and ZYGA )are both Abelian
groups=⇐¥k;9z)
=ffggDfA,gp( f
, .f*)(g , ,gD=£G ,,gD.fdg , ,q )- #
prod . in A
Eggas
;±÷§HA -Et÷G - i
defines fse ZYGA )
choose a differ
.se#S?
how is fg related to fs ?
I §§ ) ) =#(Scg ) ) ⇒
Rg)=2Hg))s§€yforso.me_fvnctiont.G-A.mg
§, ,gz ) = fs ( g , ,g . ) tcg ,
)t$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 =
[email protected] 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.