On commutative and non-commutative quantum stochastic diffusion flows

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1 Jorl of Applid hmic & Bioiformic, vol.5, o.3, 5, 97- ISSN: pri, oli Scipr Ld, 5 O commiv d o-commiv qm ochic diffio flow Pgioi N. Komo, Olivr R. Kik, Evgli S. Ahido 3 d Pioi K. Pvlko 4 Arc I hi work w dvlop qm ochic olio flow of ochic diffio volio qio of h form Lx = F x, > SDE x = x Dprm of hmic, Niol d Kpodiri Uivriy of Ah, Ppiimiopoli GR-5784 Ah, Grc. Dprm of Phyic, Niol d Kpodiri Uivriy of Ah, Ppiimiopoli GR-5783 Ah, Grc. E-mil: pkomo@gmil.com, pkom@phy.o.gr Dprm of hmic, Niol d Kpodiri Uivriy of Ah, Ppiimiopoli GR-5784 Ah, Grc. E-mil: olivrkik@gmil.com 3 Dprm of hmic, Niol d Kpodiri Uivriy of Ah, Ppiimiopoli GR-5784 Ah, Grc. E-mil: h@mh.o.gr 4 Dprm of hmic, Niol d Kpodiri Uivriy of Ah, Ppiimiopoli GR-5784 Ah, Grc. E-mil: ppvlko@mh.o.gr Aricl Ifo: Rcivd : Novmr, 4. Rvid : Frry, 5. Plihd oli : Spmr 5, 5.

2 98 O commiv d o-commiv qm ochic diffio flow o il vo Nm W -, Clifford lgr C of opror wih fii proiliy rglr rc. By h L : = d / d A i i dod lir opror ch A h Hmiloi opror of Qm chicl or Qm Fild Sym i o-giv d lf-djoi lir opror d h ifiiiml gror of h corrpodig lyic migrop cig o L -commiv Bo-Eii of fcio or o L -o-commiv Frmio-Dirc of opror poil odd opror Hilr pc H. By F w m giv H -vld qm ochic proc. Or rl pply o Fock pc grd y Hilr pc K wih cojgio J, i Qm chicl or Qm Fild Sym, icldig ircio ivolvig qizd Bo-Eii d Frmio-Dirc fil pcificlly pi ½ Dirc pricl wih xrl fild vi coff Ykw-yp ircio. hmic Sjc Clificio: 34K3; 47D3; 47D6; 47L3; 8T5; 8T Kywor: Diffio volio qio; o-prmr lyic mi-grop; qm ochic flow; qm mchic; qm fild hory Irodcio Thi ppr i dvod o qm ochic diffio volio qio of h form SDE Lx = F x, > x = x o il Hilr pc H dfid y il vo Nm W -, Clifford lgr C dowd wih proiliy rglr rc. Th jc h roo i h ircio of lmry pricl mly Boo phoo, mo, H 4, moro, pio d Frmio ro,

3 P.N. Komo, O.R. Kik, E.S. Ahido d P.K. Pvlko 99 rio, proo, lcro hv did from vriy of poi of viw cf. [], []. I priclr i hir fmo ppr Crhéodory [3] d Eii [4], ivigd fodio of Thrmodymic which h coqc for r coidrio of modr qm fil modl for h ircio of lmry pricl. Bid, Opphimr d Schwigr [5] xmid ffor o k io cco h rlio of h orc o h moro fild h ihr Blh clicl mho or h priori polio of ior ffordd. orovr, Ykw, Sk d Tki i ri of ppr [6], [7] d [8] followig prvio id of Hirg d Frmi did h miio of ligh pricl, i.. rio d lcro, fr h riio of hvy pricl from ro o phoo. Yr lr, Glimm [9], Glimm d Jff [] coi h ivigio of Ykw-yp ircig coplig pc. O h ohr hd, Accrdi, Aillh d Volrr [], Arold d Sprr [], Cizo, Lopz d Nio [3], Liy [4], Liy d Will [5], Liy d Prhrhy [6], Sprr, Crrillo, Doll d rkowich [7], coidrd cl of qm volio qio, qm dymicl migrop for diffio modl d did o-commiv grlizio of ochic qm diffril qio of Fym-Kc yp drivig ochic qm flow. I h pr work w oi qm ochic diffio flow i commiv c Bo-Eii ircio d i o-commiv c Frmi-Dirc ircio. W dy SDE i h ifii dimiol c, whr L : = d / d A do lir opror ch h A i o-giv lf-djoi lir opror h Hmiloi opror cig o Hilr pc H ch h A i h ifiiiml gror of lyic migrop qm ochic proc kig vl i H. A, R d F i giv

4 O commiv d o-commiv qm ochic diffio flow Fcio pc d flow I wh follow H will do grl complx Hilr pc wih orm. L A o-giv lf-djoi opror cig o h Hilr pc H d l ifiiiml gror A, R : = [, h lyic migrop cig o H wih A. A i i wll-kow w my m h hr xi poiiv rl mr,δ ch h A δ, for ll R. L C R, H h Bch pc of odd coio fcio : R H dowd wih prmm orm. : = { : R } d l C R, H h Fréch pc of coio fcio : R H. By flow dymicl ym, olir migrop o compl mric pc X w m fmily U = U, joyig h followig propri; R of fcio U : X X,. for vry R, U i coio from X io X.3 for ch x X h fcio U x i coio.4 U = i idiy o X.5 U x = U U x, whvr, R d x X W rcll h h fcio U x i clld h rjcory of x X. I prcic flow ri from oomo diffril qio for which hr r horm cocrig xic iq d coiiy of olio.

5 P.N. Komo, O.R. Kik, E.S. Ahido d P.K. Pvlko 3 i rl 3. Th lir c W r wih h lir iiil vl prolm 3. d A x = f, > d x = x whr f i giv H -vld fcio o R, x H. A fcio : R D A i clld clicl olio o rogly diffril for vry R d ifi 3. for vry i O h ohr hd fcio i C R, H giv y A 3. = A f R of 3. if i i R. i clld h mild olio of 3. o = i H. R, wih iiil d Thorm 3.. L f i h Fréch pc C R, H. Th hr xi xcly o mild olio of 3. i C R, H d if f C R, H h lo C R, H. Proof. L i R. By hypohi h fcio coio. Hc h Bochr igrl f :[, ] H i odd d A 3.3 f = i wll-dfid for vry, ic: A f A δ f f f δ 3.4 = f δ whr : p{ f, [, ] } f =. δ

6 O commiv d o-commiv qm ochic diffio flow Th h fcio : = A A f i h iq coio mild olio of 3. lo [8]. Filly if f C R, H h lo C R, H ic A A = f A A f A f 3.4 f δ 3. Th o-lir c 3.5 W coidr h o-lir iiil vl prolm d A x = F x, > d x = x whr F i giv H -vld fcio o H, x H. A fcio : R D A i clld clicl olio o R of 3.5 if i i rogly diffril for vry R d ifi 3.5 for vry i orovr olio i C R, H of h igrl qio A A 3.6 x = F x R. will clld mild olio of 3.5 o = i H. R, wih iiil d

7 P.N. Komo, O.R. Kik, E.S. Ahido d P.K. Pvlko 3 L Φ h corrpodig Nmykii opror of h o-lir opror F : H H pprig i q. 3.5, i.. for vry y : R H, Φ y i dfid y h forml: Φ. Φ y : = F y, R Now w h followig codiio cocrig h Nmykii opror Codiio Φ : Φy C R, H providd h y C R, H d hr xi rl-vld fcio γ R, R ch h: C 3.7 Φy Φy γ y y, for ll y y C R, d R., H Thorm 3.. L codiio Φ hol. Th for y giv H hr xi xcly o mild olio =, i C R, H of 3.5 ifyig : =. orovr mig h vry mild olio i clicl olio of 3.5, hr xi xcly o olio flow U o H wih rjcori U x i C R, H, x H. Proof. L H. Coidrig h Hmri-yp opror 3.8 Π : C R, H C R, H which o y y C R, H oci ccordig o codiio Φ d o Thorm 3. h iq mild olio A A 3.9 Π y : = Φy, R i C R, H of h lir iiil vl prolm: 3. d A x = Φy d x =

8 4 O commiv d o-commiv qm ochic diffio flow Now l y y C R, d, H R. Th pplyig. d codiio Φ w h: A A y Πy = Φy Φy Π A Φy Φy δ Φy Φy δ γ y y δ γ y y 3. γ δ y y Applyig 3. d idcio w ddc 3. y Π y Π γ δ y y!, for ll N. From 3. d for lrg ogh w cocld h Π i corcio opror o C R, H d h iq fixd poi =, ifyig : A A 3.3 : = Φ, R Thrfor h fcio R, H wih = lo [8]. C Th ig 3.4 U : = : R H i h iq mild olio of 3.5 i whvr R d H d mig h i clicl olio of 3.5 w m ifr h U, rjcori U i C R, H. R, i h iq olio flow o H, wih W hv fir o jify h U ifi codiio. d..

9 P.N. Komo, O.R. Kik, E.S. Ahido d P.K. Pvlko 5 L R. L lo qc i H ch h: 3.5 im = orovr w coidr h corrpodig olio = :,, for vry N, d = :,, ch h: 3.6 Φ = A A, R 3.7 Φ = A A, R Th comiig codiio Φ, 3.6 d 3.7 w hv: U U = Φ Φ = A A Φ Φ A A Φ Φ δ Φ Φ = δ 3.8 γ Th from 3.8 d mkig of Growll iqliy w g: U U = γ 3.9 γ Coqly y 3.5 d 3.9 i follow

10 6 O commiv d o-commiv qm ochic diffio flow 3. imu = U. Nx l H. Coidr lo qc 3. im = d l R wih d 3. =, N. W lo p 3.3 : mx{ } =., Th y 3., 3.5 d 3.3 w ddc U U = R ch h = A A A A Φ Φ A A A Φ Φ A A Φ Φ A Φ Φ A A A A 3.4 γ for vry N. Th y 3., 3.4 d h Lg Domid Covrgc Thorm i follow h: 3.5 imu = U Filly, y drd rgm, w hv U = d U U = U, for ll R,, d h proof of h horm i compl.

11 P.N. Komo, O.R. Kik, E.S. Ahido d P.K. Pvlko 7 4 Applicio 4. Bo-Eii c L E h complxificio Hilr pc of rl Hilr pc E d l E do h Hilr pc of ymmric or ovr E. Th hr xi iomorphim of E vi iry opror oo h Hilr pc L E, B E, d, wih c 4c k 4. d Γ = π dλ x Θ c whr Γ = P Θ, Θ i Borl i h img P E of k -dimiol k k orhogol projcio P o E d R k, B R, λ i h Borl-Lg mr i P E cf. [9]. Thrfor w c k h c k x 4. H = L E, B E, d = E. : c 4. Frmio Frmio-Dirc c I i wll-kow h h Bch lic L p X, S, µ, p wh X, S, µ i mr pc c xdd i o-commiv lgric cox. W r rcllig rifly om wll-kow fc cocrig ocommiv igrio hory i which, id of igrig fcio o mrl pc wih rpc o giv mr, o igr poily odd opror ffilid wih vo Nm lgr V wih rpc o gg or rc o V. W hll rric o proiliy gg ic h gg r rlv for h dy of Frmio.

12 8 O commiv d o-commiv qm ochic diffio flow L E complx Hilr pc h Frmio o-pricl pc d l E do h Hilr pc of iymmric or of rk ovr E, whvr =,, d l E h complx mr C. W hll do y E h Frmio-Dirc Fock pc, h i h Hilr pc dirc m 5. = E d ω will do h complx mr r vcm or o-pricl E. For vry x i E, h crio opror C x i h odd lir opror o E wih orm C x = x ch h: 5. C = P x x whvr E, whr P do h iymmrizio projcio. Th ihilio opror, A x, x E i dfid o h djoi of C x, h i A :. x = C x Now l J cojgio o E. W rcll h fcio J : E E i id o cojgio o E if J i ilir J x y = J x J y, whvr x, y E d for ll complx mr d, J i iiry < J x, J y > = < y, x >, whvr x, y E, whr <, > do h ir prodc o E d J h priod wo J = I. W lo do y C h vo Nm lgr grd y ll opror h Frmio-Dirc fil B x, 5.3 B x = C x AJ x x E o E dfid y h forml: W o h C i h wkly clod Clifford lgr ovr E rliv o h cojgio J.

13 P.N. Komo, O.R. Kik, E.S. Ahido d P.K. Pvlko 9 A rglr proiliy gg pc i ripl K, V, τ, whr K i complx Hilr pc, V i vo Nm lgr of lir opror o K d τ i fihfl, crl, orml rc o V, i.. τ i lir fciol from V io C ch h: τ τ i, i.. τ I =, T V, T impli τ T τ τ i complly ddiiv, mly, if O i y of mlly orhogol projcio i V wih ppr od Y h τ Y = τ P P O τ 3 τ i rglr or fihfl, i.. if T V, T, τ T = impli T = τ 4 τ i crl, i.. τ TS = τ ST, whvr T, S V. E, C, τ i rglr proiliy gg pc, whr τ : C C, d 5.4 τ : = < ω, ω > for vry ω C cf. Sgl [] For y clod lir opror T o E w p 5.5 T : = T T For p <, L p E, C, τ i dfid o h complio of C wih rpc p o h orm p T T = τ T. L E, C, τ p i dfid o h Bch pc C wih rpc o i opror orm. I h how h h Bch pc L p E, C, τ, p r pc of lir poil odd opror o E cf. Sgl []. I priclr h fcio L E, C, τ oo E cf. []. Now w c k h c 5.6 H : = L E, C, τ = E ω x o iry opror from

14 O commiv d o-commiv qm ochic diffio flow ic L E, C, τ c rgrdd ordrd Hilr pc of opror o E. Nx l S for-dimiol complx pi pc wih poiiv dfii ir prodc, d l K h Hilr pc of S -vld fcio o wih ψ = ψ x, ψ x dλ <. K x 3 R 3 R Th w c lo k H h Hilr pc Z ovr h Hilr pc Z of fr pi ½ Dirc pricl wih xrl fild vi coff Ykw-yp ircio ch h 5.8 Z = K K whr K i h irrdcil pr of K wh h ifiiiml gror of im rlio i poiiv o K. Rfrc [] Bor,. d Jord, W., Zr Qmchic, Z. Phy., 34, 95, [] Bor,., Hirg, W. d Jord, W., Zr Qmchic II, Z. Phy. 35, 96, [3] Crhéodory, C., Urchg ür di Grdlg dr Thrmodymik, h. A., 67, 99, [4] Eii, A., Űr di vo dr molklrkimich Thori dr Wӓrm gfordr Bwgg vo rhd Flüigii pdir Tilch, A. Dr Phyik, 7, 95, [5] Opphimr, J.R. d Schwigr, J., O h Ircio of oro d Ncli, A. Phy. Rv., 6, 94, 5-5.

15 P.N. Komo, O.R. Kik, E.S. Ahido d P.K. Pvlko [6] Ykw, H., O h Ircio of Elmry Pricl. I, Proc.-h. Soc. Jp, 7, 935, [7] Ykw, H. d Sk, S., O h Ircio of Elmry Pricl. II, Proc.-h. Soc. Jp, 9, 937, 4-3. [8] Ykw, H., Sk, S. d Tki,., O h Ircio of Elmry Pricl. III, Proc.-h. Soc. Jp,, 938, [9] Glimm, J., Ykw coplig of qm fil i wo dimio, Comm. h. Phy., 5, 967, [] Glimm, J. d Jff, A., Slf-djoi for Ykw Hmiloi, A. Phy., 6, 97, [] Accrdi, L., Ailh,. d Volrr, C.V., O h Srcr of Clicl d Qm Flow, J. Fc. Al., 35, 996, [] Arold, A. d Sprr, C., Corviv Qm Dymicl Smigrop for m-fild qm diffio modl, Comm. h. Phy., 5, 4, [3] C ~ izo, J.A., Lópz, J.L. d Nio, J., Glol L hory d rglriy for h 3D olir Wigr-Poio-Fokkr-Plck ym, J. Diff. Eq., 98, 4, [4] Liy, J.., O h gror of Qm Sochic Flow, J. Fc. Al., 58, 998, [5] Liy, J.. d Will, S.J., Exic, poiiviy d corciviy for qm ochic flow wih ifii dimiol oi, Pro. Thory Rl. Fil, 6,, [6] Liy, J.. d Prhrhy, K.R., O h gror of qm ochic flow, J. Fc. Al., 58, 998, [7] Sprr, C., Crrillo, J.A., Doll, J. d rkowich, P., O h Log Tim hvior Qm Fokkr-Plck Eqio, oh. f. h., 4, 4, [8] Sgl, I., No-lir migrop, A. h., 78, 963,

16 O commiv d o-commiv qm ochic diffio flow [9] Sgl, I., Tor lgr ovr Hilr pc I, Tr. Amr. h. Soc., 8, 956, [] Sgl, I., A o-commiv xio of rc igrio, A. h., 57, 953, 4-457; Corrcio, A. h., 58, 953, [] Sgl, I., Tor lgr ovr Hilr pc II, A. h., 63, 956, 6-75.

Approximately Inner Two-parameter C0

Approximately Inner Two-parameter C0 urli Jourl of ic d pplid Scic, 5(9: 0-6, 0 ISSN 99-878 pproximly Ir Two-prmr C0 -group of Tor Produc of C -lgr R. zri,. Nikm, M. Hi Dprm of Mmic, Md rc, Ilmic zd Uivriy, P.O.ox 4-975, Md, Ir. rc: I i ppr,