Dmitry Georetriequantizationudos

D radsacountablebasis

74 separable complex Hilbert space ex Eto D Q

S H densely defined symmetric endomorphisms of 21

A C SCH c domain A is dense in 4 x Ag Ax yit is more generalthan selfadjoint Ex Fee d derivative

uncloudedoperatesymmetric but notselfadjoin

Mio symp mfd

def ff fr Ccdcm is connkle

if g c Ecm f g3m o ti th g const

def A A CSCH is complete

ie B E SCH at ED f o ti th B c ID.irAxonsidefA quarter is a linear nap Q CRM SCH s t

h Q 1 I2 ihQCLE.gl EQCf QCg3 Whenever fi 13 is complete

QQ y Q K is anplete

The Groeneveld Vanhove

Q quantization doesnot exist

TIKI A D CA H is syn il DCAT tenH and G Ay A x y

V x y E AA D A E H H is self adjoint

A AXED AD f z e te f y C D A

Ay x bitif A is synn then A CA't ComphacGraphatEx L2 Io D IR A d

syn Lt ret self adsoutCx

M o synnmfd 21 separableHilbertspacebuttable hasS clone ofteespan ofhares spH

SCH densely definedsymmetricoperators

def I guntnehe map is a brew nap CKM 5121 etCD QCD id

G it Qaf g3m EQCf QCgDG f fig complete set of functions on M

Qf QL complete

The Groenewel Van Hove A quantization doesnotexist C p 56 incondition Gso e smallLie susalgebra

canbe quantized

If 6 satisfies only G it is called a pre quantization

SegalisprequantizatG M a a d0

std synppotential taut l fo n

14 L4T M Cc

nQCf hXftfxfQ ifIf we change 0

1 0 1 dethen QQ change by Lxed9A way to circumvent this issue

If we change thepotential then we changeguts 4 e 4

if Xf tf Lyfe0 with oldsyruprelatedit Xe t f xp O with new synppelutral

then e'Kh E y I yl equivalence

for a general Ms co cannot find aglobal syruppotentialpromote it to a connection in a pre quake tunebundle

Kostent Saurian prequantizationdef a prequaken he bundleE S a Hermitian linebundle E 14nWowith a compatible connection 17 with curvature f w

def Q c m S Te M E

QCf7 itTh s s Jassi's YT

Def Weil integrality conditiona c IITESEanyclosedsurface

G ED c tf CM IR to C t.pt M Rimage of H2 M 4

than a prequantumlinebundle exists set w is f integralHow to construct this line bundle from a t i legal co

tech colonRecall KYM IRIE H'CU IR

N U open over bycontractible opens

as IfOn each Ui cheese a syruppotential G dQiQi Oj du on Vinofiji Vij Uj letUre o Vinu n U 0

locally constant functions figh satisfy Eccl cocycle conditionfgcha

f ke t flee fe O

o k integral z gh C Itransition fundus c e A

C Ghq e fish y

2dtdktfdrcm.IR ITCU Rg ed over

to f Tou ead U G do

on each Uinuj fe Q Oj dUyf.glUijtUjLtUk on Vi n Uj n the 0w is k integral elf z file C I

Thi I preguentum ene bundleAl exists i o is E mtg al

Pref sketchthe transition functions are given by Gj exp iUhthen Ei GhChi _exp L

On each Ui choose SiSi x Cx I E U x

related via Cigs Si

deiday izC0 0

17 Si In Gilas7 is precisely the capetibility condition her 17 s

Exe 1 M o

for any closed 2 surface Z 1 few Iff Jo o

prequantum space of States E T M preguatua omahasact on itwant to cut it half

Polarizations def A real polarisation is an inebitivedistribution P CTM st PmcTn M isLagrangian

EE.IR co dg adp polarizations 9 fixedP fixed

Ee FM e

Foliation is givenbythefibers of M

EI S xg Ii 52 with a area b n

Foliations 5 103 There is no polarization

pyx s If P is a real lie bundle g2P is trivial Foravarsling section

notallowedto east on SEmotivate ex plantations

Prod Anyreal pal is b all is to stand polarization on IR

G tahepoa.no s

def M a PET M ri a ex pot if 1 RCT.cmuisLagr2 d Pn D NTM cord3 P i integrableA pd is Kaehler if I Pnf ATM Ff La cx valuedherLt theirExi Ma Kehler then snacks Xfm 85

Ta Al 110M To M not sane as truebetwityP DCP

K k K

T M Ga al T anti Gar poiProd if a synn mcd admits a Kehler pot thenCmp is Kohler

Ex S'xS E las p

adn't a Keller potReim counterexamplesnot allsyrupmfds admit a ex plantation co stucted via symplecticblowups

Polarizedsectionsdef mid synp nfd E MF prequantum le.iebundleP ex plantation A sectionof E is called polarized if

17 5 0 ht x tangent topHalbertsma

gyp Halfhour offeespace of all polarized sections

Pxblemswrkth.us fnhnwhen the leaves are G pact might happenthat Hp O

if leaves are non compact manysectors but none offleas a e

quantum observables may notpreserve the condition of being apolarizedseeker

Q g y H 74Mu

Hp Hpin Kohler case f mustbeat Xf is killing

he T M f must be at most bear n f

is pda d't17 5 0 for X tangent to P

polarization

EI S2 Curth radius L

Us Un stereographicprog2 X't 1 2

in Us

wt it n dead I rent d Athe standard area L n on 5h

k integrality data IA idealEtSpo c I

CHEET

in Edt K nt logCitzIIt 2 I

G 95k Kehlerpotential

idk

SyE 4127 e EtTholen function

P span Ozthen plantationSy Sy 3 i f da ada

sniff 4Gt 4 e

f di adzt I4a4 14 E

Z R e

DA RdRado7

S m zh e