COVARIANT HOMOGENEOUS NETS

OF STANDARD SUBSPAES

(n

Vincenzo Marinelli :

UNIVERSITY OF ROME TOR VERGATA

1st Virtual LQP WORKSHOP

fointongàng work with te .- H .

Neeb

°

18"

fune 202

Li localiTy Vs Synnetry in QFT

U THE Bisognano-Victim anni PPOPERTY IDENTIFY GEOMETRIE

AND ALGEBRAIC OBJECTS in a AQFT .

↳ THE ALGEBRAIC CONSTRUCTION OF THE FREEFIELDS STARTS FROM

THE GEOMETRIE DATE TODEFINE LOCAL ALGEBRAS ( R"

,

si,ds")

✓ THEE MOMENTS IN THIS CONSTRUCTION

¥ IDENTIFICATION : BOOST - WEDGE REGIONS

* REPRESENTATION THEORY → FIRST QUANT'ZATION

* SECOND QUANTIZATION ⇒ AOFT

Lo GIVEN ALLE GROUP C ( NO SPACETIMENEEDED) WE INVESTIGATE

A POSSIBLE DEFINITION OF A QFT STARTING FROM AN

A-BSTRACTNOTION OF WEDGE REGIONS

OUTLINEI.

PRELIMINARIES

II. BRUNETTI - GUIDO - LONGO CONSTRUCTION

FREE FIELD

111 . COVARIANT NETS OF STANDARDSUBSPACES

•PELIMINARIES

Pinesb FEATURES of WEDGE REGIONS CAN BE DETERMINEDAT LE ALGEBRA level .

↳ Minkowski ( R? g drag ( 1 ,-1

.. .

,-1 ) )

SYMMETRY GROUP : POINCARÉGroup p! = Ray L!

° WEDGE REGIONS ARE CHARACTERRED By THEIR SYMMETRIESx.etA

wafer" ix.lui }

"

÷!! ! a:\ nwaltiwnewr

ONE-PARAMETERWEDGE 1.1 i

RegionsNt a W - gw, a- Mj gnw . g-

'e Pa Group s of

g. e PIBoots

PÈRE FURTHEREXAMPLES

↳ di = psll.IR) no ds ! (⇒ I E. Eni )

÷WEDGES : ¥4 ,

a- tw Boost

o Mida PSLCL? ) a 0 s'e TI - CHIRALITY

HEDGES : INTERVALS I È Il ft'

DILATIONS

Iix. èx

①-- RO

↳ WE CAN DEFINE A COVARIANT SET OFWEDGES JUST START ING FROM

THE LIEALGEBRA ofof The SYMMETRY Group G. (=P ! ,

LÌ,

Moto.)

1-1 × 1-1

+weg → hwh.eu

'_ W

Xgw = Gxw g-'

← gli

Pines

to THE compenetri WEDGE : xw . = - xwo-ahi-N-j.no W"

R' ts :

ws'

: v'⑦ V

{Xw } is a Locale C- covarianti SETOF LE GENERATORI

FURTHERSimn-TRESUPOSITIOU.ITme REFLECTION ( PT Synnezry )

IIII:[ III. : sina.tt .

- x. ix. xD io v.

-

In

se : senz'-E ci

siThen fin 15 DEFINEDBY COVARIANCE

i

PROPERTIES 1 i 2 .

-* yww = W

'

* ⇐ Ancia)

Pines

1¥ IFTHERE Exists a POSITIVE come C cof

sentirai g Cg' EC

, 7⇐ G ( NEGATIVE GE - C )

e.g. vi.CITÉ"

, G.csl# I

THEN INCLUSIONS ARE OBTAINED BY COMPOSITION OF TRANSLATION S :

eterea,tzo

..

w.

.

-

F. Etc ° \WHERE ± pt GENERATE WITH ± Xw TRÈMITO GROUP

adxw (PI) = ± Pe ⇐a TE E kerladxwees )

BRUNETTI - GUIDO - LONGO CONSTRUCTION✓FREE FIELD

STANDARD SNBSPACE-

p HCId REAL CLOSED SUBSPACE IS STANDARD IF

HI =D,Haiti - {of

SYMPLECTIC Complementi H'= { Ted : In ( gin)

-0,

I 7 EH}

Torna su : Haiti - Haiti cosa

ANTILINEAR| OPERATOR"

ogtiy 1- 7- in invowtion

POLAR DECOMPOSITION : SI fa LI , 2 ANTI - UNNARY MODULAR Conficcati ON

D SELF - ADJOINT MODULAR OPERATOR

Òt MODULAR GROUP

- -

ma telone11 1.1

5h a-0 (3,0 ) with * ) a- tt STANDARD SUBSPACE

Brunetti - Guido - tonico (BGL) Construction-

o Paz PIUPÌ ftp.azpowcaeeapoup ( also not .LI )

= {UN'TARY on P !

ULANT ' -' un'#RYAna , .gr#pyowpI=qpI

POSITIVE ENERGY

HLW ) : ← ( Jo,Ow ) e ( Upn ) ,

è""

) STANDARD SNBSPACE

WHERE Utd )) , eixwt

FIRSTQUANTIZATON ISOTONY LOCALI POINCARÉ COVARIANCE-

a a a U :P! → UNGHIE" ) 70

µ va/ w

,te. iWi- HI ) cld vieni ,⇒H ch'% v. cwi-aHHCHCWJIIHWI.HN/

the Hot. ?Hai o

Brunetti - Guido - tonico (BGL) Construction-

BISOGNANO izùxwt

WKHMANN : UNwhat)) -- e =Dµ,

PROPERTY -

PCT .

Theater: U ( an ) = Jtlw)-

THEN SECOND QUANTI ZATION ii

W 1- Aw ) = ( archi : letta } città

ismchik)-WK)WH ) = e Natale )

,h.k.CH WEYL RELATIONS

li SATISFY FUNDAMENTAL

0 - A = §,

AN) ) QUANTUM $ RELATIVISTI C

PRINCIPLES

=D ON A GENERAL GROUP GWENETD Ti DETERMINE A

FIRST QUANTI ZATION NET ( SECOND IS AFUNCTOR)

Il / . COVARIANT NETS OF STANDARDSUBSPACES

-

LIEWEDCESETTING. G- GRADEDLIE Group

,

la :C- II 1 ),

c' = EÈCI )-

G = C' e 2,= C' x te ,

r) . ci va'.. ciao - a

'

IN THIS TALK we assume 2€ ) = Le /,CONNECTED

9 RettaBra of G'

Action .At? G - Aut )

,

Ad! E,

Ad )-

De lo X E ogEULER ELEMENT II se Cadx ) c f- I. 0,1 )

op=

7 . . ⑦ Io ④ 71

e) ✓ = ka (ad X - v 1) v.Rt , ADJOINT REPIZESENTATIONad :

9-

End'

- × - [× ,.]

↳ WE CAN ASSUME o EULER INVOLUTION FOR X EULER ELEMENT

Ad =eiitadx

THEN BY GUARIRNE 6 ×-

g.X IS EULER INUOLUTION FOR g- XEULER ELEMENTg - -

EUuarweaa.ec-

XEULER ELEMENT• DE . µ=4À* G- ACTION : g.ws/Ad4glx,grg-#

⇒ Wii

* complementi WEDGE N' = fer)

* INK -x ⇒ aww'

' @!Ex

. g= sitter)

,

⇐ f (? ;) Euler Element-

D= RE !) ⑦RX ⑦ RI :) 3- Gradino

| Ad rx = Ad ( f) EULER INVOLUTION

PSL SYMMETRY Group for 51$ ds?SAMEWEDGES

,

DIFFERENT RELATIONS DEFINEDBYINCWSIONS

Euler WEDGES W :( x. sa ) WEDGE-

-

POSITIVE cave :C cod ST. Ad ( = Egcg) C g EG- -WEDGE ENDOMORPHISM SEMIGROUP-

Lw = fatto Io e C- ¥wherectetcnkafas.is )

,

es . ✓ = ✓

⇐ e

t

Su = exp Ce exp go exp C-

WEDGE INCLUSION-

g.W cui :* g. e Sw

ALL THE incursione AREGENERATED BY

HALFSIDED INCLUSIONS

IHSI : e#wcwx.tec.to

CHOICE Of Positive CONE ⇒ POET of WEDGES

HAAG-KASTLERNETLETWI.BEA GITRANSITNELOGALPOSET OFWEDGES W.R.T.tt CONE C-- -

LET U DE UNITA RY REPRESENTATI ON OF G"on Id

Wta W - HLWJEID

te ISOTONY W, cui ⇒ tifi , ) ch 2)

+ locali ty W, ch' ⇒ HLW . ) ctllwr)

"

+ GIGVARIANE Ucg , Hd ) - tkg.nl ) ge G'

* POSNTNNY OF THE ENERGY ( E cu = { tag / -i 204120 )

* CycleCity TKW ) STANDARD SUBSPACES

* BW.pro#Rty ucèwt ) = ÒÌ (⇒ HLW") -- HCW)

" )Hai

* PCT THEOREN fu extends U to a G COUARIANT reinseritela

un sua

MODULAR CONSTRUCTION-

U #NTIJUNITARY POSITIVE ENERGY RIPRESENTATION of G- = CÌU È

W,

aw i-HCW-kw.ru ) ) ± ( è" au" '

,

Usa)

BGL CONSTRUCTION is possiede AND SATISFY ALL THE ASSUMPTIONS

SPECTRAL CONDITION

IN PARTICOLAR :C' covarianza ISOTONY-

B- wpropeetyÈ -

A

EI C' = PSLZLR )

+ ( = ((I.f)Esteri : bzo.ca , aI.be/(alfoto):teRDSElERALYOS-

- DISCRETE Series REPREENTATIONS HAVE DOSITNITY OFTHE ENERGY (RE )

+ ( = (01, towEDa

=D NET ONTHE CIRCLE

- PRINCIPALICOMPLEMENTARI SERIES REPRESENTATLONS HAVE te POSITIVE ENERGY

⇒ NEI ON DE SITTER (see also Guido -Longo 203 )

LOOKINGFOTLNEWMODELSLETof BE A NON - COMPACT REAL Simple LE ALGEBRA

,

-

-0g= le ⑦ p CARTARI DECOMPOSITION,

=D STUDYOFTTLERESTRICTEDThat Spacea cp CARTANSNBALGEBRA

,

-

10 Look FOR WEDGE ORBITS-

HETLE THEORBITS ARE CONSIDERED W - R-T THE Inn ) : LEI ) ACTION ON of

Thin NUMBER OF ORBITSOFEULER ELEMENTS-

pel

An : h - ORTSITS Bn : 1 -ORBIT Cn : 1 ORBIT

| Dn : 3 ORBITS Eg : 2-ORBNS Ez : 1 ORBIT

o Wing il w! g.W.ge G" @Dei #yaR

Io INP

Thin NUMBER OF ORBITS OF SYMMETRIC EULER ELEMENTS-

ptzArno, i 1-ORBIT Bn : 1-ORBIT Ch : 1- ORBIT

| Du : 1. ORBIT Dzn : 2-ORBITS Fa : 1- ORBIT

ABOUT THE NET STRUCTURE-

-

DEI ORTHOGONALEULER ELEMENTS (X. Y ) : T×LY)=

±

wi-fxek-s.lt/lxil.i--1R .

TI XYEGEUERORThocon.tl ELEMENTS ( xp )

* X. Y ARE SYMMETRIC

* L'e# y )= Il# I

* ( Y ,X) ortogonali Elements

( symmetric propetg)

COMMENDOIN CASE : 24 ) ftef ,

* SEVERAL WEDGE ORBITS

* SEVERAL WEDGE COMPLEMENT PARAMETRIZED BY CENTRAL ELEMENT

* TWISTED LOCALI TY RELATION

EVERYTHING COMPLIES Ok with MODELS→ ⇒

With THE H-K Pictureson s

' COVERINGS

OR FERMIONK NETS

( ITH Modificato» )

Outlook

↳ POSSIBLE NEW MODELS ON ALGEBRA IL GROUP S

↳ Possibile DEFINITION OF SPACELIKE CONES

ANK You FOR YOUR

ATTENTI0N !