The expectation value of the field operator.

Transcription

1 The expecaion value of he field operaor. Dan Solomon Universiy of Illinois Chicago, IL June, 04 Absrac. Much of he mahemaical developmen of quanum field heory has been in suppor of deermining he S-marix in order o calculae scaering cross secions. owever here is also an ineres in deermining how expecaion values of field operaors evolve in ime from an iniial sae. In his paper I will examine some aspecs of his problem.. Inroducion Much of he mahemaical developmen of quanum field heory involves he S-marix which is used o obain scaering cross secions. In his case he mahemaical formulaion aemps o deermine he probabiliy of some iniial sae evolving ino some final sae where boh he iniial and final saes are known. owever here is also ineres in deermining how he expecaion values associaed wih he quanum field evolve in ime [,,,4,5]. In his case we are only ineresed in he iniial sae of he sysem and, in addiion, we need an equaion ha deermines how he expecaion values evolve in ime from some iniial sae. In his paper we will address some aspecs of his problem. The sysem under consideraion will be a real scalar field.. The eisenberg picure We will sar ou in he eisenberg picure. In his case he sae vecors are ime independen and all he ime dependence of he sysem is associaed wih he field operaors. Assume ha he classical acion of he sysem is given by, where V is a polynomial expansion in S d d x m V (.) x where x sands for he quaniy, x in - dimensions. Also naural unis are assumed so ha c. A imes we will consider a model sysem which will be 4 heory in he presence of an exernal source V 4 4! sx. For his case, s. (.)

2 In he eisenberg picure he field operaors are saisfy he equal ime commuaion relaionships, x and x ˆ ˆ. The field operaors x ˆ x i x x x x ˆ x ˆ x ˆ,,, ; ˆ,, ˆ, 0;,,, 0 (.) They obey he following equaions which correspond o he classical equaions of moion, where Dx and V are defined by, Dx ˆ D ˆ ˆ 0; ˆ x V (.4) m and V (.5) V and where. For example for our model sysem, D ˆ ˆ x x x s x (.6)! Le be he sae vecor of he sysem. Assume ha i is normalized, ha is,. Since we are working in he eisenberg picure he sae vecor is ime independen. Le us suppose ha we are ineresed in he expecaion value of he field operaor which is from he lef by and from he righ by. From his we obain, ˆ x. Muliply q. (.6) D ˆ ˆ x x x s x (.7)! Consider he case where 0. In his case D ˆ x x sx soluion is ˆ x x G x, x s x dx where, associaed wih he operaor D x and. This is readily solvable. The x saisfies he equaion D x 0 G x x is he Greens funcion x. owever for he case 0 he siuaion is more complicaed. Tha is because here is one equaion bu wo superficially independen quaniies which are ˆ x and ˆ x x. Of course hey are no really independen because hey are boh dependen on he quaniy which is fully specified by q. (.6). owever is no readily apparen how o solve (.7) in he form presened. ˆ

3 . A differenial equaion for he expecaion value. A possible way o overcome he difficuly discussed above will he presened in his secion. Define he following operaor, where T is he ime ordering operaor and he following convenion, ˆ Z exp ˆ J T i J x xdx (.) Jx is a real valued funcion. In he above we have used, f x dx d x f x d (.) where inegraion over all of -dimensional space is implied by he firs inegral on he righ. I can be shown ha, i J x i J x i J x n T expi dxj ˆ ˆ ˆ ˆ exp ˆ T x x xn i dxj (.) Using his relaionship and he commuaions (.) i is shown in he Appendix ha, ˆ Z ˆ J x T exp i J x ˆ ˆ x dx J x ZJ i J x (.4) Referring again o (.) we obain he following wo relaionships, ˆ x (.5) ˆ Z exp ˆ J T i J x xdx x i J x x ˆ V Z ˆ exp ˆ J T V x i J x x dx i J x (.6) From all his we obain, D ˆ x V Z J ij x ij x T D x V x i J x x dx J x Z J ˆ ˆ ˆ ˆ x exp (.7)

4 Use (.4) in he above o obain, D ˆ x V Z J J x Z J i J x i J x ˆ (.8) Nex assume we have wo sae vecors and saisfies,. Referring o (.8) he quaniy Z J ˆ D V Zˆ J J x Zˆ J ij x ij x x (.9) Nex define he quaniy From his we obain, is defined by, where xj ; WJ by, exp iw J Zˆ J (.0) exp iw J exp iw J x; J ij x ij x (.) xj ; T ˆ xexpi J x ˆ x dx W J J x Z J (.) Noe ha xj ; is dependen on he space-ime poin x and he funcion as is WJ. dependen on he sae vecors and Use he above resuls in (.9) o obain, J x. I is also Dx x; J V x; J J x i J x (.) where we have used, i J x (.4) 0 Similar resuls have been obained elsewhere using somewha differen approaches [,6,7,8]. 4

5 For our model sysem V! x s x (see q. (.)). In his case, V x; J x; J s x ij x! ij x x; J x; J ; ;! i J x i J x x J x J s x (.5) Use his in (.) o obain, x J x J ; ; Dx x; J x; J x; J s( x) J x! i J x i J x (.6) Now rearrange erms o obain, Dx x J x J C x J s x J x! ; ; ; ( ) (.7) where,! ; x; J C x J x J x J ; ; i J x i J x (.8) Noe ha if he erm Cx; J was removed from (.7) hen his equaion would be equivalen o he classical equaions of moion in he presence of a source s() x J x. Therefore we can hink of Cx; J as a quanum correcion o he classical equaions of moion. Nex we will show ha (.6) is idenical o q. (.6) if ake he limi Jx 0. From (.) and (.) we obain, T ˆ x ˆ x expi J x ˆ xdx x ; J i J x Z J Take he variaion of his wih respec o Jx o obain, x ; J x ; J (.9) 5

6 T ˆ x ˆ x ˆ x expi J x ˆ xdx ij x i J x Z J x; J T ˆ x ˆ x expi J x ˆ xdx Z x; J i J x J ; x J x; J x; J i J x ; x ; J (.0) In he limi ha 0 J x and x x x x we have he following, x x; J 0 J 0 ˆ x (.) xj i J x x x ; J 0 ˆ ˆ x 0 0 x (.) xj ; ij x ij x J 0 x x x ˆ ˆ ˆ ˆ x ˆ x x x x x (.) Use hese in (.6) and se Jx 0o obain, D ˆ x ˆ x! x s x (.4) Muliply boh sides of he equaion by and rearrange erms o obain, ˆ ˆ Dx x x sx 0! (.5) This is equivalen o he evoluion equaion for he field operaor given by (.6). Nex le where. In his case, 6

7 xj ; T ˆ exp ˆ x i J x xdx Z J (.6) If we se Jx 0 his becomes x x J ˆ x value of he operaor independen quaniies one quaniy xj ; ˆ which is jus he expecaion 0 ; J 0 x x and x. The resul of all his is ha insead of having wo superficially ˆ ˆ in he differenial equaion (.7) we have in he differenial equaion (.6). In should be, in principle, possible o solve his differenial equaion for xj ; and hen ake 0 J x o obain he expecaion value of he field operaor which is he desired resul. This will be explored furher in a laer paper. 4. An alernaive expression for Z [J]. In his secion we consider alernaive forms of he expression for Zˆ J. In paricular we wan o wrie i in erms of he ineracion picure field operaors insead of eisenberg picure operaors. In he eisenberg picure all of he ime evoluion of he sysem is given by he field operaors. In he ineracion picure a porion of he ime evoluion is conained in he field operaors and he res is in he sae vecor. The Ineracion picure field operaors commuaions given in (.) and saisfy, x and x ˆI obey he equal ime I ˆ I ˆ 0; ˆ (4.) D x I I The eisenberg field operaors can be expressed in erms of he Ineracion picure field operaors [9] according o he expression, ˆ x Uˆ, ˆ xuˆ, where Uˆ, is a uniary operaor ha saisfies he relaionships, and obeys, (4.),,, I Uˆ Uˆ Uˆ and U ˆ U ˆ UU ˆ ˆ (4.) i Uˆ Vˆ Uˆ,, (4.4) where Uˆ, and, 7

8 I V ˆ V ˆ x, d x (4.5) The soluion o his, ˆ, exp ˆ U T i VI d (4.6) Consider he quaniy, L T ˆ x ˆ x ˆ x (4.7) n n Assume ha,, n hen, Use (4.) in he above o obain, L ˆ x ˆ x ˆ x (4.8) n n, ˆ, ˆ,, ˆ, L Uˆ x Uˆ x Uˆ Uˆ x Uˆ (4.9) n I I n n I n n Use Uˆ, Uˆ, and Uˆ, Uˆ, Uˆ, in he above o obain,,, ˆ, ˆ,, ˆ, L Uˆ Uˆ x Uˆ x Uˆ Uˆ x Uˆ (4.0) n I I n n I n n This can be wrien as, Use his resul in (.) o obain, The resul is ha, where,,, L Uˆ T Uˆ ˆ x ˆ x ˆ x (4.) n I I I n ˆ ˆ ˆ Z,, exp ˆ J U T U i J x I xdx (4.),, ; Zˆ J Uˆ Uˆ J (4.) ˆ ˆ U, ; J T exp i V exp ˆ I x dx i J x I xdx From [0] his can also be expressed as, (4.4) 8

9 ˆ, ; exp ˆ ˆ U J T i dx V I x J x I x (4.5) Consider he acion of he operaor Zˆ J on some sae vecor. Firs he operaor Uˆ, ; J evolves he sysem in he presence of he exernal source. Nex Uˆ, evolves he sysem from back o wih Jx 0 similar o he in-in formulism [4,5]. From resuls in Ref. [0] we can wrie Uˆ, ; J J x from o as expressed by (4.4) as, ˆ U, ; J exp i dxv T exp i J x ˆ I xdx i J x I is also shown in Ref. [0] ha,. This is (4.6) i T exp i J x ˆ : exp ˆ I x dx i J x I xdx : exp J xf x x J xdxdx (4.7) where he colons : : indicae ha he quaniy beween he colons is o be pu in normal order and is he Feynman propagaor and saisfies 4 D x x x x F x x obain, Uˆ, ; J expi dxv i J x i x. Using his resul we F : exp i J x ˆ I xdx: (4.8) exp J xf x x J xdxdx We can use his resul o evaluae Uˆ, which becomes, 5. Conclusion. To summarize, in q. (.) we defined he operaor, ˆ, ; Uˆ U J (4.9) Zˆ j0 J. I was hen shown ha his operaor saisfied he differenial equaion (.8). Using his resul i was shown in he res of Secion how o 9

10 derive he differenial equaion (see (.) or (.6)) for he quaniy xj ;. This differenial equaion consiss of a par ha saisfies he classical equaions of moion o which is added a quanum correcion. In he limi ha Jx 0 he quaniy xj ; will be equal o he expecaion value of he field operaor ˆ x. Nex in Secion 4 we examined some furher properies of he operaor Zˆ J. I was shown ha he effec of acing on a sae vecor wih he operaor Zˆ J is o evolve he sae vecor forward in ime from o and in he presence of he source Jx and hen o evolve backward in ime from o during which Jx 0. Appendix We wan o show ha (.4) is rue. From (.), where, This can be wrien as, Recall I can be shown ha ˆ n Z J i S (5.) S T dx dx dx J x x J x x J x x n n ˆ ˆ ˆ (5.) n n! n n n n ˆ ˆ ˆ S dx J x x dx J x x dx J x x (5.) n n n n x n n, dxf x d x d f x (5.4) S ˆ n T dx dx dx ˆ ˆ ˆ n J x x J x x J xn xn J x n! ij x T ˆ ˆ ˆ dx dx dxnj x x J x x J xn xn n! (5.5) When his is used in (5.) hen we obain (.4) 0

11 As an example show ha (5.5) hold for S 4. ˆ ˆ ˆ ˆ S dx J x x dx J x x dx J x x dx J x x (5.6) From his expression we obain, S4 J x x dx J x x dx J x x dx J x x ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ dx J x x x dx J x x dx J x x ˆ ˆ ˆ ˆ dx J x x dx J x x x dx J x x ˆ ˆ ˆ ˆ dx J x x dx J x x dx J x x x (5.7) Nex ake he firs derivaive wih respec o. The ime appears in he ˆ x ˆ x, and in some of he limis on he inegrals. owever he final resul of he derivaives on he limi of he inegrals is o produce erms of he form ˆ x,, d x J x, ˆ x, due o he fac ha he equal ime commuaor ˆ x ˆ x ˆ x ˆ x o obain,. These erms are zero,,, 0. Use his fac and S4 J x x dx J x ˆ x dx J x ˆ x dx J x ˆ x ˆ ˆ ˆ ˆ ˆ dx J x x x dx J x x dx J x x ˆ ˆ ˆ ˆ dx J x x dx J x x x dx J x x ˆ ˆ ˆ ˆ dx J x x dx J x x dx J x x x (5.8) Take he derivaive of his equaion wih respec o o obain, Sˆ 4 J x Fˆ Fˆ (5.9)

12 where, Fˆ ˆ x dx J x ˆ x dx J x ˆ x dx J x ˆ x ˆ ˆ ˆ ˆ dx J x x x dx J x x dx J x x ˆ ˆ ˆ ˆ dx J x x dx J x x x dx J x x ˆ ˆ ˆ ˆ dx J x x dx J x x dx J x x x (5.0) where ˆ x ˆ x and, Fˆ ˆ x,, d x J x, ˆ x, dx J x ˆ x dx J x ˆ x ˆ ˆ,,, ˆ, ˆ dx J x ˆ ˆ ˆ x dx J x x x,, d x J dx J x x x d x J x x dx J x x x, ˆ, x (5.) Rearrange erms and relabel some of he dummy variables o obain, ˆ F ˆ,,, ˆ, ˆ ˆ x d xj x x T dx dx J x J x x x! Use (.) o obain, Also, from (5.0) we obain, (5.) ˆ F ij x T dx dxj x J x ˆ ˆ x x! x (5.) ˆ ˆ F T dx dx dx J x ˆ ˆ ˆ x J x x J x x! These resuls are consisen wih (5.5). (5.4)

13 References. G. A. Vilkovisky. xpecaion values and vacuum currens of Quanum Fields arxiv:07.79 (007).. V. Branchina,. Faivre, D. Zappala, ffecive acion and he quanum equaions of moion. ur.phys.j. C6 (004) 7-8. arxiv:hep-h/006050v.. N.C. Dias, A. Mikovic, and J.N. Praa. Coheren Saes xpecaion Values as Semiclassical Trajecories J.Mah. Phys. 47:080 (006). arxiv:hep-h/ S. Weinberg. Quanum Conribuions o Cosmolagical Correcions. Phy. Rev. D7:0454 (005). arxiv:hep-h/05066v. 5. R.D. Jordan. ffecive field equaions for expecaion values. Phy. Rev D. Vol. (986), P. Dua, K. A. Meissner,. Nicolai. The DeWi quaion in Quanum Field Theory. arxiv: 0.497v. (0). 7. B. A. Faizullaev and M.M. Musakhanov. Two-Loop ffecive Acion for heories wih Fermions. Annals Phys. 4 (995) arxiv:hep-h/94066v. 8. P. Dua. The DeWi equaion in Quanum Field Theory and is applicaions. PhD hesis. Free Universiy of Berlin (0). hp:// 9. M.. Peskin and D.. Schroeder. An inroducion o quanum field heory Addison-Wesley Publishing, Reading, Massachuses (995). 0. D. Solomon. Deriving Z[J] from he ime evoluion operaor. arxiv: v. (04).