smart von Neumann Theorem A Covariant Stone U Nevada Reno Joint with Leonard Huang 117 147 in Math Phys Vol Issue 1 Comm 378 Publication Some History a Hilbert every pair of s.a on Is operators OpenQuestion the the equivalent to which satisfies HCR space Schrodinger pair Heisenberg pair Schrodinger pair A dom H CA S.a P Dom P 724112 Vfe SUR s a Pf i H B dom B EUR7 Q 7K Edom A ndome B xfCx Qf x fescipy V H II Sit PQ f V fESCIR if V Kek ik A B e K that guarantee Determine sufficient conditions on Yainstrategye Integrability of CHAIB such that to CH R S S TR UCHI unitary rep's R IR UCH Weyl relation X EIR ylxIRxSy SyRx y YG IK eiyx and classical Stone von Neumann Theorem then apply
Ismet The Classical Stone von Neumann Theorem Let G be a t.ca group R G UCH and S are unitary theorem Suppose UCH such that representations gp SyRErCx1RxSrV rEEa.xEG.ThenFWiH and Saw RT U 11 Y L4G7s.t for each fecoCG where Hye G f y fix y yxeG V red V f y raptly V yeG G Heisenberg repin CH Rss is a peer is the Defy LTG UN G Schrodinger rep'n that integrates Heisenberg pair If H A B is Corollary a then H Ris Heisenberg to a rep ECG P Q H A B w
IS mert3 Work Our unit gp repins on Goat Extend S VN Theorem to Hilbert c module setting covariant rep'n of G dynamical Good to u systems a unitary gp representations for pairs of an integrability criterion Goal Determine module Hilbert c sua on a operators E dynamical system G I c a Throughout A G a w is a Huang Ismert 20207 is unitarily KCH Kia Heisenberg representation Every sum of of direct to the a copies equivalent IR D KCH Schrodinger repin
Ismet 4 A G D Representations a quadruple AGM is Define An Heisenberg representation X f rid where full Hilbert A module is X a LK nondeg p A rep strongly its unriepta.gg I and recx UCH S G r V xEG SEE r So r x Sore for covariant pair CA Gr is a p r covariant pair for CA E 1 is a pro Define The is the AG a Schrodinger representation LTA G D M u v where quadruple completion of a right A module CDG A KCA G a is the as VAEA it f geccCG A where fix d la f x a fad ffCxTglx diehl flog LILYA G x M A af Mla f V feccCG.tt UNA Gm viG u G ULLCA.G.at L Hx y text ly olylfly and vs f y is Proposition The AG D a Schrodinger representation A G A Heisenberg representation
Ismert Classical to Covariant Step 1 theorem Coca Xe G KCMG MOU where via U G UNCG M 6CGl BCLEG Ux g y _gCx y f Ms of form a covariant rep'n Coca G et Mxu5 Cola tetG Kutch ColG A xe qG theorem K KLAGa Proof Outline then imprinutivity bimodule B A If X is a BE KCXA LTA G a is Greens Imprimitivity Theorem gives a Coca A Aetna G A imprimitivity bimodule IX is implemented by u This isomorphism fg LUKA G n fi g Coca A covariant pair I u Exa Cola A Xero G K L Afar f X I my
smart classical to Covariant Step 7 Proposition s 7nesenmimsHf T.i f ei Y Heini.im 5 oof s its you M Tv Tls R covariant rep CCoCG CG et of TSAR Tir x U w Coca Tet G BCH KILTED Proposition Huang Ismert 2020 Heisenberg representations gain fc.ofanjgnt.arep.ms X f ris a G CGA lt pas oFa where Fa CocciA ftp.s Cocoa A s form Mps r I a MM v covariant rep for G let a Coca A Mmi Ms Exa ftp.sxr Cola A xetoaG f.CH K ELA G H F
Is mert 7 Classical to Covariant Step 3 repin of KCH Theorem Arneson non degenerate Every a direct sum of copies of unitarily equivalent to is the representation identity H LTG y o Tvxu BIH y HsxR KCH Coca xe G BCH Kacey a id new id r new v BIH't BLECG 2 Em Huang Ismert 20201 Every non degenerate rep n where X is Hilbert module KCH is of k x a a direct sum of copies of unitarily equivalent to id kCX klx Coca.AM aG Lex KCHAG.at id I 7 II Katy r n w G IR LILYA G x
Ismert8 Conclusion such that There's a unitary W X ECA.GR o MM v idk u ftp.sxr CECAG x w w OMM.vxu ftp.sxr and Mm u r 17ps new w s nfv and u r p nut M w ME Directions Future Uniqueness statement for pairs of 6 operators that form the appropriate analogue S.a of Heisenberg pair a More general 1 co actions G Non abelian Quantum gps
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