A projection operators technique in calculations of Green function for quantum system in external field

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1 Noliear Dyamics ad Applicatios. Vol. 4 (2007) 52-6 A projectio operators techique i calculatios of Gree fuctio for quatum system i exteral field H.V. Grushevskaya Physics Departmet, Belarusia State Uiversity, 4 Nezavisimosti Av., Misk , BELARUS Properties of projective operatioal fuctio of Gree for quatum system was aalysed. The diagrammatic techique to fid a Gree fuctio of quatum system i a exteral field is offered. PACS umbers: Tb, b Keywords: quatum system, projectio operator, Gree fuctio. Itroductio The method of Gree fuctios is oe of the most powerful methods of computig physics. I [] we have show that a Gree fuctio of quatum system is represeted by matrix elemets of the operator fuctio expressig through projectio operators. We shall call this operator fuctio as a projectio operator Gree fuctio ad further we shall examie its properties. The goal of the article is to develop a diagram techique which allows to fid the projectio operator Gree fuctio describig a quatum system i a exteral field. 2. Formalism of Gree fuctio Gree fuctio G( r, r ; t, t ) satisfies to a equatio with i a δ-fuctio i the right part, for example, as [ ı ] t Ĥ( r, r, r 2,... ; t) G( r, r ; t, t ) = δ( r r )δ(t t ), () where r, r i, r is space variables, t, t is time. Sice G( r, r ; t, t ) should describe propagatio of a sigal from a poit r to a poit r for a time iterval (t t ), it depeds o a differece of times (t t ): G( r, r ; t, t ) G( r, r ; t t ). (2) The time iterval τ = (t t ) should be positive as owig to the wave ature the sigal is propagated accordig to a Huyges priciple, i.e. a excitatio is trasferred sequetially i time from oe poit of space to the earest oe. Therefore, the Gree fuctio has o physical sese itself. Physical sese has a propagatio fuctio or propagator G ± ( r, r ; τ) determied as G ± ( r, r ; τ) = θ(±τ)g( r, r ; τ), (3) Grushevskaja@bsu.by 52

2 A projectio operators techique i calculatios of Gree fuctio... where the fuctio G + ( r, r ; τ) is called a retarded Gree fuctio, the fuctio G ( r, r ; τ) is called a advaced Gree fuctio. A Laplace trasformatio of Gree fuctio G( r, r ; t t ) has the followig form G( r, r ; t t ) = 2π e ıω(t t ) G( r, r ; ω). (4) Let us examie a case of a hamiltoia Ĥ Ĥ0( r, r, r 2,...) which is idepedet of time t. The the Laplace image of fuctio of Gree G( r, r ; z), beig aalytical cotiued i a complex plae Z, ca be determied as a solutio of the followig o-homogeeous differetial equatio: [z Ĥ0( r, r, r 2,...)]G( r, r ; z) = δ( r r ) (5) which has boudary coditios defied o boudary surface S of area V. Here we assume that z is a complex variable, the time idepedet operator Ĥ0( r) is a liear Hermitia differetial operator which possesses a complete set of eige fuctios {φ ( r)}: Ĥ 0 ( r, r, r 2,...)φ ( r) = ω φ ( r). (6) The set {φ ( r)} ca be orthoormalized φ ( r) φ m ( r)d r = δ m, (7) where " " deotes Hermitia cojugatio. Completeess of the set {φ ( r)} meas that φ ( r)φ ( r ) = δ( r r ). (8) The symbol it is ecessary to iterpret as + d if idex umbers the eige fuctios belogig both discrete, ad a cotiuous spectrum. It is coveiet to work i abstract vector space. This represetatio is amed a represetatio of bra (ket)- Dirac vectors [2], [3], [4], [5]. Usig desigatios of the Dirac bra (ket) - vectors ( r, φ, r, φ ), i such space oe ca costrict operators of projectio (projectors) P r r, φ φ. (9) The, for example, φ ( r) is a projectio of the vector φ oto the vector r : φ = d r r r φ = d rφ ( r) r. (0) The equality (0) gives two followig idetities d r r r = ˆ, () φ φ = ˆ, (2) where Î is uity operator. From the property of completeess (8) ad the expressio (2) the followig proof of orthoormality for the basis { r } is esued r r = r φ φ r φ ( r)φ ( r ) = δ( r r ). (3) 53

3 H.V. Grushevskaya It is easy to prove that the operator Ĥ0 i this represetatio takes the matrix form r Ĥ0 r : Ĥ 0 = Î2 Ĥ 0 = d rd r r r Ĥ0 r r, (4) ad the Gree fuctio G( r, r ; z) ca be cosidered as a matrix elemet of the operator Ĝ(z): Ĝ(z) = Î2 Ĝ(z) = d rd r r r Ĝ(z) r r d rd r r G( r, r ; z) r. (5) Oe ca coclude from the expressios (4) ad (5) that Ĝ(z), Ĥ0 are projectors i this abstract space. Accordig to (4), (5) ad (3) the equatio (5) ca be rewritte i the form [z H 0 ( r, r, r 2,...)]G( r, r ; z) = [z (H 0 ( r, r, r 2,...)δ( r r ))]G( r, r ; z) = zg( r, r ; z) r [Ĥ0G(z)]( r, r ) r = r zĝ(z) r d r r Ĥ0 r r Ĝ(z) r = r r Takig ito accout the expressio (), the equatio (6) is rewritte i the operator form (6) [z Ĥ0]Ĝ(z) =. (7) If all eigestate of the operator are o-zero, the equatio (7) ca be solved formally as Ĝ(z) =. (8) z Ĥ0 Multiplyig (8) o the expressio (2), we get Ĝ(z) = φ φ = z Ĥ0 φ φ z ω. (9) Hece Ĥ is the hermitre operator, all its eigevalues ω are real. Therefore of Im z 0, the z ω. The last meas that G(z) is a aalytical fuctio i complex Z-plae everywhere except poits sad itervals o real axis that correspod to eigevalues of Ĥ. Let us defie z = ω ± ıɛ. The the expressio (9) gives two real fuctios G + ( r, r ; ω) = lim ɛ 0 G( r, r ; ω + ıɛ) = G ( r, r ; ω) = lim ɛ 0 G( r, r ; ω ıɛ) = φ ( r)φ ( r ) ω (ω ıɛ), (20) φ ( r)φ ( r ) ω (ω + ıɛ). (2) Let us clarify the physical meaig of these fuctios. Let us fic the itegral 2π e ıωt G + ( r, r ; ω + ıɛ) = 2π e ıωt φ ( r)φ ( r ) ω (ω ıɛ). (22) 54

4 A projectio operators techique i calculatios of Gree fuctio... If oe assumes t < 0 ad choses a coutour C with couterclockwise directio that is close it i upper semi-plae (Fig. 3a) the there are o poles iside C ad therefore the itegral (23) is equal to zero C 2π eız t If oe assumes t > 0, the itegrad i (23) should be replaced o φ ( r)φ ( r ) z (ω ıɛ). (23) e ı( z) t φ ( r)φ ( r ) z (ω ıɛ). (24) The, accordig to (24) phase of the itegratio variable z = ρ exp(ıφ) is chaged o π at chage t t, therefore for positive time iterval we have the itegral C 2 2π dρd[exp(ı(φ + φ ( r)φ ( r ) π))]eı( z) t z (ω ıɛ) = φ ( r)φ ( r ) C 2 2π dρd[exp(ıφ)]eı( z) t z (ω ıɛ) = φ ( r)φ ( r ) C 2 2π e ızt z (ω ıɛ), (25) where C 2 is arbitrary cotour with the clockwise directio. From that it follows that i compariso with the case t < 0 cotour is closed i lower semi-plae is show i i Fig.3b. Therefore the poles are iside the cotour ad the itegral (25) does ot vaish. Thus we fially get 2π e ıωt G + ( r, r ; ω + ıɛ) = θ(t) C 2π e ızt G( r, r ; z), (26) where C is a arbitrary cotour with couterclockwise directio icluded poles of itegrad i the expressio (). I a similar way, usig Fig.4, oe ca fid the itegral 2π e ıωt G ( r, r ; ω ıɛ) = θ( t) C 2π e ızt G( r, r ; z), (27) where C is a arbitrary cotour with couterclockwise directio icluded poles of itegrad i the expressio (27) as show i i Fig. 4b. Comparig (26, 27) with (3), we coclude that G + ( r, r ; ω + ıɛ) is the Laplace image of the retarded, ad G ( r, r ; ω ıɛ) is the Laplace image of the advaced Gree fuctios. Let us place oe part of the poles of the fuctio G(z) usig shifts o +ıɛ, whereas the secod part by shifts o ıɛ, as it is show i Fig 5a. The the poles of G(z, λ ıɛ, λ 2 ıɛ, λ 3 + ıɛ,..., λ + ıɛ, ) are iside itegratio cotour for all t. Fuctio G(z, λ ıɛ, λ 2 ıɛ, λ 3 + ıɛ,..., λ + ıɛ, ) is called the Laplace image of the causal Gree fuctio. As oe ca see from Fig. 5 the differece of cotours C, C 2 ca be deformed to cotour that iclude the real axis ReZ. Therefore, at itegratio o C C 2 oe ca omit shift of poles o ±ɛ. Therefore the causal Gree fuctio G( r, r ; t) ca be writte as G( r, r ; t) = C 2 C 55 2π e ızt G( r, r ; z). (28)

5 H.V. Grushevskaya (a) (б) Fig. 3. Cotour of itegratio for retarded Gree fuctio. (a) Itegratio cotour for egative time t, t < 0. (b) (a) Itegratio cotour for positive time (a) (б) Fig. 4. Itegratio cotour for advaced Gree fuctio. (a) Itegratio cotour for egative time t, t < 0. (b) (a) Itegratio cotour for positive time If we deform cotours C, C 2 i Fig. 5a i such a way that the resultig cotour C icludes poles λ 3 + ıɛ,..., λ + ıɛ, ad cotour C 2, obtaied from C 2, icludes the poles λ, λ 2, the the causalfuctio G( r, r ; t) ca be expressed through fuctios G ± : θ(t) C 2 G( r, r ; t) = θ( t) C 2π e ızt G( r, r ; z ıɛ) 2π e ızt G( r, r ; z + ıɛ) = G + ( r, r ; t) G ( r, r ; t). (29) 3. Evolutio operator Let us demostrate that the expressios (28) ad (29) are equivalet. With this goal ivmid we use the Sohotsky idetity lim ɛ 0 x ± ıɛ = P ıπδ(x), (30) x 56

6 A projectio operators techique i calculatios of Gree fuctio... to fid the differece of Laplace images of retarded ad advaced Gree fuctios. The the Laplace images G ± ( r, r ; ω) have the form G ± ( r, r ; ω) = P φ ( r)φ ( r ) ω ω ıπ φ ( r)φ ( r )δ(ω ω ) (3) (a) (б) үис. 5. үонтур интегрирования для причинной функции +рина. (а) үонтура интегрирования для отрицательнvх времен t, t < 0 и положительнvх времен t, t > 0. (б) Cotour of itegratio i the case of poles positio at real axis. Substitutig (3) ito (29), we get = 2πı 2π e ıωt G( r, r ; t) = G + ( r, r ; t) G ( r, r ; t) δ(ω ω )φ ( r)φ ( r ) = ı e ıω t φ ( r)φ ( r ). (32) Usig the expressios (2), (9) ad (32), oe ca easily show that the fuctio G( r, r ; t) is a matrix elemet of the operator Ĝ(t), defied by the expressio Ĝ(t) = ı e ıω t φ φ = ıe ıĥ0t. (33) Let us put t = t 2 t. The the matrix form of the expressio (33) gives the expressio for the Gree fuctio G( r, r ; t 2, t ) = ı r e ıωt 2 φ e ıωt φ r = ı φ ( r, t 2 )φ ( r, t ). (34) Here i the right had side of the expressio (28) we chage matrix elemet G( r, r ; z) of projective operator Ĝ(z) o operator itself Ĝ(z), give by the expressio (8). The usig the residue theorem at formal itegratio we obtai the projective operator Ĝ 0 (t) = ı C 2 C 2πı e ızt = ıe ı tĥ0, (35) z Ĥ0 57

7 H.V. Grushevskaya which coicides with (33). Therefore oe cocludes that the expressios (28) ad (29) are equivalet. But the advatage of the first expressio (28) for the causal gree fuctio o respect to the secod oe (29) for the same fuctio is i symmetric etrace of space ad time coordiate ito the first expressio. Let us show that the projective operator Û(t t 0 ) = e ı(t t 0)Ĥ0 = ıĝ(t t 0) (36) is the projective operator of evolutio. Ideed i the represetatio of bra- ket Dirac vectors the fuctio φ(t) = Û(t t ) φ(t ) = e ı(t t )Ĥ φ(t ) (37) satisfies the equatio ı t r φ(t) d r r Ĥ0 r r φ(t ) = 0, =. (38) Accordig to defiitio (4) the equatio (38) describes the systems behaviour i coordiate represetatio [ ı ] t Ĥ0( r) φ( r, t) = 0, = (39) But the equatio (39) describes the evolutio of a wave fuctio i time. therefore the projective operator of evolutio Û(t t 0) propagates φ from the time momet t 0 to the time momet t. Hece the projector Û is the operator expoet it possesses importat property Û(t t 0 ) = Û(t t )Û(t t 0 ), (40) which allows to itroduce the diagram techique i the followig paragraph. 4. Diagram expasio for the Gree fuctio i represetatio of projective operators. The case of a exteral field. Let a system be subjected by time depedet exteral effect. Ĥ ( r, t), is described by the equatio (39). The the Hamiltoia of the system ca be represeted i the form Ĥ( r, t) = Ĥ0( r) + Ĥ( r, t), (4) where H 0 ( r) is a Hamiltoia of the uperturbed system. I represetatio of iteractio the wave fuctio is defied as φ( r, t) = e ıĥ0t φ V ( r, t), (42) where φ V ( r) is a wave fuctio φ( r) i the represetatio of iteractio. Substitutig the expressios (4), (42) ito equatio (39) ad multiplyig this equatio from the left o exp (ıĥ0t), we get [ e ıĥ0t ı ] t Ĥ0( r) Ĥ( r, t) e ıĥ0t φ V ( r, t) = 0, =. (43) 58

8 A projectio operators techique i calculatios of Gree fuctio... From (43) we fid that i the represetatio of iteractio the equatio представлении взаимодействия уравнение (39) gets the form [ ı ] t ĤV ( r, t) φ V ( r, t) = 0, =, (44) where ĤV is defied by the expressio Ĥ V = e ıtĥ0 Ĥ e ıtĥ0, =. (45) Let perturbatio Ĥ( r, t) be small eough ad the Gree fuctiog (0) ( r 2, t 2 ; r, t ) of oiteractig system with a Hamiltoia гамильтонианом Ĥ0( r) is kow. Let us call this Gree fuctio as a oe-particle free Gree fuctio as the system with oe degree of freedom is cosidered. The i the represetatio of iteractio i accord with formulae (36) ad (44) the projective operator ĜV (t i t i ) for the Gree fuctio which describes the sigal propagatio for the ifiitely small time iterval t i ad defied by the Hamiltoia Ĥ(t), ca be represeted i the form of perturbatio series expasio as ıĝv (t i t i ) = e ıĥv ( t i ) t i = ( ıĥv (t i ) t i ), (46) t i = t i t i = ɛ 0, t i {t i, t i }. Usig the property (40) of the evolutio operator we rewrite the equatio i the form (46): ıĝv (t k t i ) = e ı(t k t i )ĤV (tk) e ı(t i t i )ĤV (ti) = ıĝv (t k t i ) ıĝv (t i t i ) [ = ı(ĥv (t i ) t i + ĤV (t k ) t k ) ĤV (t i ) t i Ĥ V (t k ) t k ] ĤV (t k ) t k Ĥ V (t i ) t i, t i = t k = ɛ 0. (47) Performig the above described procedure times ad performig, we get the projective Gree fuctio which describes the sigal propogatio startig from time momet t 0 up to t : = [ ı ıĝv (t t 0 ) = e ı j (t j t j )ĤV (t j) ] Ĥ V (t i )dt i Ĥ V (t i )dt i Ĥ V (t k )dt k.... (48) Now, usig the trasformatio (45) ad (42), we ca tur back from the represetatio of iteractio ito a stadard oe ad fid the projective Gree fuctio Ĝ(t t 0 ) as: ıĝ(t t 0 ) = e ı(t t 0 )Ĥ0 ı e ı(t t i )Ĥ0 Ĥ (t i )e ı(t i t 0 )Ĥ0 dt i dt i dt k e ı(t t i )Ĥ0 Ĥ (t i )e ı(t i t k )Ĥ0 Ĥ (t k )e ı(t k t 0 )Ĥ0.... (49) Remarkig that the expressio { ı exp[ ı(t j t l )Ĥ0]} is the projective Gree fuctio of a free particle Ĝ(0) (t j t l ), we fially fid that Ĝ (t t 0 ) = Ĝ(0) (t t 0 ) + Ĝ (0) (t t i )Ĥ(t i )Ĝ(0) (t i t 0 )dt i + dt i dt k Ĝ (0) (t t i )Ĥ(t i )Ĝ(0) (t i t k )Ĥ(t k )Ĝ(0) (t k t 0 ) (50) 59

9 H.V. Grushevskaya Now we rewrite the expressio (50) i matrix form + + G ( r, t ; r 0, t 0 ) = G (0) ( r, t ; r 0, t 0 ) d r i dt i G (0) ( r, t ; r i, t i )Ĥ( r i, t i )G (0) ( r i, t i ; r 0, t 0 ) d r i dt i d r k dt k G (0) ( r, t ; r i, t i )Ĥ( r i, t i )G (0) ( r i, t i ; r k, t k )Ĥ( r k, t k ) G (0) ( r k, t k ; r 0, t 0 ) (5) Now we poit out that the space ad time variables are icluded ito equatio (5) i a symmetric way. The oe ca rewrite the equatio (5) i a covariat form where Ĝ = Ĝ(0) + Ĝ(0) Ĥ = Î2 Ĥ = Ĥ Ĝ 0 + Ĝ(0) Ĥ Ĝ (0) Ĥ Ĝ (0) (52) dxdx x x Ĥ x x, (53) x = { r, t}, x = { r, t } Ĝ = Î2 Ĝ = dxdx x x Ĝ x x dxdx x G ( r, t; r, t ) x. Let us desigate the free gree fuctio Ĝ(0) by a solid lie, perturbed Gree fuctio Ĝ by a double solid lie ad perturbatio Ĥ by a poited lie. The oe ca correspod a diagram to the equatio (52) as show i Fig. 6. I the form represeted the perturbatio Ĥ( r i, t i ) is a exteral filed. Beside, the equatio (52) is precisely coicided i form with the Dyso-S equatio if oe rewrite it as (54) Ĝ = Ĝ(0) + Ĝ(0) Σ Ĝ (0), (55) Σ = ĤĜ(0) + Ĝ(0) Ĥ Ĝ (0) Ĥ +... ; (56) where Σ is the self-eergy operator. therefore the expasio (55) is the Dyso-Schwiger equatio for particles i a exteral filed. Fig. 6. Diagram for calculatio of the oe-particle Gree fuctio Ĝ up to the secod order o exteral perturbatio Ĥ. 5. Coclusio Withi the techique of projectio operators we have offered the rigorous procedure to costruct diagrammatic expasios for operator Gree fuctios. This allows us a propagator of quatum system i exteral field. 60

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