The method of unitary clothing transformations in quantum field theory: the bound-state problem and the S-matrix

Being aimed at reformulating quantum field theory (QFT) within the notion of the so-called clothed particles and interactions between them we consider the problem of finding the simplest eigenstates of the total field Hamiltonian H. Along this guideline H and other operators of great physical meanin...

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Дата:	2007
Автор:	Shebeko, A.V.
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Мова:	English
Опубліковано:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2007
Назва видання:	Вопросы атомной науки и техники
Теми:	Quantum field theory
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/110918
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Цитувати:	The method of unitary clothing transformations in quantum field theory: the bound-state problem and the S-matrix / A.V. Shebeko // Вопросы атомной науки и техники. — 2007. — № 3. — С. 61-65. — Бібліогр.: 6 назв. — англ.

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spelling	irk-123456789-1109182017-01-07T03:05:54Z The method of unitary clothing transformations in quantum field theory: the bound-state problem and the S-matrix Shebeko, A.V. Quantum field theory Being aimed at reformulating quantum field theory (QFT) within the notion of the so-called clothed particles and interactions between them we consider the problem of finding the simplest eigenstates of the total field Hamiltonian H. Along this guideline H and other operators of great physical meaning, e.g., the Lorentz boost generators and the current density operators, which depend initially on the creation and destruction operators for the “bare” particles, are expressed through a new family of their “clothed” counterparts. We are stressing that this transition to the clothed-particle representation (CPR) has been fulfilled via certain unitary (“clothing”) transformations (UT’s) without changing the original Hamiltonian. It is shown how the S-matrix can be evaluated in the CPR. За допомогою переформулювання квантової теорії поля в термінах так званих “одягнених” частинок і взаємодій між ними розглядено проблему пошуку простіших власних станів повного гамільтоніану H. У цьому підході H та інші оператори, що мають глибоке фізичне значення, такі як генератори Лоренц бустів і густина струму, котрі початково залежать від операторів народження і знищення “голих” частинок, подаються через оператори “одягнених” частинок. Перехід до зображення “одягнених” частинок здійснюється за допомогою певного унітарного (“одягаючого”) перетворення без зміни початкового гамільтоніану. Показано, як можна обчислювати S-матрицю в новому зображенні. С помощью переформулирования квантовой теории поля в терминах так называемых “одетых” частиц и взаимодействий между ними рассматрена проблема поиска простейших собственных состояний полного гамильтониана H. В этом подходе H и другие операторы, имеющие глубокий физический смысл, такие как генераторы Лоренц бустов и плотность тока, которые изначально зависят от операторов рождения и уничтожения “голых” частиц, выражаются через операторы “одетых” частиц. Переход к представлению “одетых” частиц осуществляется с помощью определенного унитарного (“одевающего”) преобразования без изменения исходного гамильтониана. Показано, как можно вычислять S-матрицу в новом представлении. 2007 Article The method of unitary clothing transformations in quantum field theory: the bound-state problem and the S-matrix / A.V. Shebeko // Вопросы атомной науки и техники. — 2007. — № 3. — С. 61-65. — Бібліогр.: 6 назв. — англ. 1562-6016 PACS: 21.45.+v, 24.10.Jv, 11.80.-m http://dspace.nbuv.gov.ua/handle/123456789/110918 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Quantum field theory Quantum field theory
spellingShingle	Quantum field theory Quantum field theory Shebeko, A.V. The method of unitary clothing transformations in quantum field theory: the bound-state problem and the S-matrix Вопросы атомной науки и техники
description	Being aimed at reformulating quantum field theory (QFT) within the notion of the so-called clothed particles and interactions between them we consider the problem of finding the simplest eigenstates of the total field Hamiltonian H. Along this guideline H and other operators of great physical meaning, e.g., the Lorentz boost generators and the current density operators, which depend initially on the creation and destruction operators for the “bare” particles, are expressed through a new family of their “clothed” counterparts. We are stressing that this transition to the clothed-particle representation (CPR) has been fulfilled via certain unitary (“clothing”) transformations (UT’s) without changing the original Hamiltonian. It is shown how the S-matrix can be evaluated in the CPR.
format	Article
author	Shebeko, A.V.
author_facet	Shebeko, A.V.
author_sort	Shebeko, A.V.
title	The method of unitary clothing transformations in quantum field theory: the bound-state problem and the S-matrix
title_short	The method of unitary clothing transformations in quantum field theory: the bound-state problem and the S-matrix
title_full	The method of unitary clothing transformations in quantum field theory: the bound-state problem and the S-matrix
title_fullStr	The method of unitary clothing transformations in quantum field theory: the bound-state problem and the S-matrix
title_full_unstemmed	The method of unitary clothing transformations in quantum field theory: the bound-state problem and the S-matrix
title_sort	method of unitary clothing transformations in quantum field theory: the bound-state problem and the s-matrix
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate	2007
topic_facet	Quantum field theory
url	http://dspace.nbuv.gov.ua/handle/123456789/110918
citation_txt	The method of unitary clothing transformations in quantum field theory: the bound-state problem and the S-matrix / A.V. Shebeko // Вопросы атомной науки и техники. — 2007. — № 3. — С. 61-65. — Бібліогр.: 6 назв. — англ.
series	Вопросы атомной науки и техники
work_keys_str_mv	AT shebekoav themethodofunitaryclothingtransformationsinquantumfieldtheorytheboundstateproblemandthesmatrix AT shebekoav methodofunitaryclothingtransformationsinquantumfieldtheorytheboundstateproblemandthesmatrix
first_indexed	2025-07-08T01:20:49Z
last_indexed	2025-07-08T01:20:49Z
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fulltext	THE METHOD OF UNITARY CLOTHING TRANSFORMATIONS IN QUANTUM FIELD THEORY: THE BOUND-STATE PROBLEM AND THE S-MATRIX A.V. Shebeko National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine; e-mail: shebeko@kipt.kharkov.ua Being aimed at reformulating quantum field theory (QFT) within the notion of the so-called clothed particles and interactions between them we consider the problem of finding the simplest eigenstates of the total field Hamil- tonian . Along this guideline and other operators of great physical meaning, e.g., the Lorentz boost generators and the current density operators, which depend initially on the creation and destruction operators for the “bare” particles, are expressed through a new family of their “clothed” counterparts. We are stressing that this transition to the clothed-particle representation (CPR) has been fulfilled via certain unitary (“clothing”) transformations (UT’s) without changing the original Hamiltonian. It is shown how the -matrix can be evaluated in the CPR. H H S PACS: 21.45.+v, 24.10.Jv, 11.80.-m 1. PREAMBLE Finding the eigenstates of the total Hamiltonian for interacting fields or its blockdiagonalization is a primary concern in quantum physics. The UT’s in ques- tion do not blockdiagonalize (except some simple models), but convert it into a form, which enables us to facilitate the initial, extremely complicated problem. We express through new operators of particle creation and destruction and show that this can be regarded as a UT of . The respective particles (these quasiparticles of our approach) may be called “clothed”. They are identified with physical particles. H H H H The Hamiltonian in the new form turns out to be de- pendent on the renormalized particle masses and not the initial “bare” ones [1-3]. In addition, it takes on a spe- cific sparse structure in the Hilbert space (say, of the hadronic states in case of the interacting meson and nucleon fields). Forms of the same kind are derived for all the Poincare group generators. After constructing interactions [1-4] between the clothed particles we de- rive the approximate eigenvalue equations for the sim- plest bound and scattering states. Keeping in mind the forthcoming applications of the method of clothing UT’s in describing nuclear reactions (in particular, the meson production in nucleon-nucleon collisions), aspecial attention is paid to expressing the -matrix in terms of the clothed-particle interactions responsible for physical processes in the system under consideration. It is proved that such a reduction be- comes possible if the corresponding UT’s in the Dirac (D) picture satisfy certain asymptotic conditions in the distant past and future [5]. S As a whole, this talk is devoted to a simultaneous exposition of the key points of our approach with an emphasis on its practical aspects and perspectives. 2. UNDERLYING FORMALISM Our departure point is a total Hamiltonian 0( ) ( ) ( )IH H a H a H a≡ = + 0 ( ) ( ) ( )c I cK a K a K a= + ≡ c ) k ,∀k R (1) expressed through “clothed” particle creation (destruc- tion) operators such that († c ca a ( ) ( ) ( ) ( )† † 0 00 ,c c ca Ha k a k kΩ = , Ω = Ω, ∀ = ,k k k (2) where is the physical vacuum (the lowest eigen- state). They obey the same algebra as “bare” operators do. Clothing itself is implemented via Ω ) H (†a a ( ) ( ) ( ) ( )† ,c c ca W a a W a=k k (3) where unitary transformation ( ) ( ) exp ,cW a W a R= = (4) with removes from some undesirable (“bad”) terms that prevent the no-clothed-particle state and the one-clothed-particle states to be eigen- vectors. In the context, but coincides with , viz., †R = − 0 ( )cH a H ≠ Ω H 0 0( ) ( )cK a H a ( ) ( )† 0 0 0( ) ( )c c c cK a H a k a a d= = + ...,∫ k k k (5) where 2 0k µ= +k 2 R   ren    with the physical mass . The operator µ † 0 0( ) ( ) ( ) ( ) ( )I c c c c cK a K a H a WH a W H a≡ − = − (6) consists of interactions between clothed particles, re- sponsible for processes with physical particles. At the beginning of our clothing procedure we could consider such (see {1, 2] for details 1R R= { } 0 1I bad H H= , (7) to eliminate the “bad” terms of the -order, if any, from , where V denotes primary interac- tions between bare particles, are neces- sary mass counterterms. After eliminating these bad terms via W R , 1g IH V M= + 1 1exp= 2 renM O g  = [ ]0 1 1( ) ( ) ( ) 2c c ren cK a K a M a R V= + + , [ ]1 1 1 1[ ] 3renR M R R V     + , , + ..+ , . . (8) PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2007, N3 (1), p. 61-65. 61 Four-operator ( -order) interactions between clothed particles stem from 2g [1 12 R V, ] t 0− c . Two-operator contribu- tions to it can be compensated by species carrying definition of the latter (see, e.g., another con- tribution to the Conference by Korda et al.). ( )2 ( )ren cM a 3. GENERATORS FOR SPACE TRANSLA- TIONS AND OTHER SYMMETRIES We have expressed time translation generator through the set of . How do the total linear and angu- lar momenta and the Lorentz boost generators depend on the clothed particle operators? The answer will allow us to formulate the transformation properties of the clothed no-particle and one-particle states with respect to the Poincaré group. It is convenient, but not neces- sary, to use H ca 00 lim ( ) tR i V t e dε ε ∞ − → + = − ,∫ (9) ≡ = where the interaction op- erator in the D picture. 0( ) exp( ) exp( )V t iK t V iK t= Further, since , then = 0 and , i.e., the total linear momentum is the same function of clothed operators as of bare ones. Analogous state- ments are valid for the total angular momentum and the baryon (fermion) number operator B . This means that the clothed states Ω , , , and have the following properties: a) they are eigen- vectors of ; b) they are transformed under space rota- tions in the same manner as relevant bare states do; c) they possess definite values. As supplementary requirements: clothed operators and clothed one-particle states must have the same transformation properties with respect to space inversion, time reversal and charge conjugation as their bare counterparts. [ ] 0V, =P ( ))ca W a= B [ ]R,P †( )c ca W = ∫ ca ( )aP ( )caP ( )x x x †( ( )c cW a a W= P † cd Ω P ( ) ( )aP P † †Ω ≡ ( )ψ ψ † cb Ω d to eliminate the bad terms linear in . Now, by using the property of interaction density to be a scalar, i.e., , one can show that Eq. (13) will hold if . Thus, the UT W removes bad terms (for instance, all three-operator (“three-legs”) terns of the -order in the field models with Yukawa- type couplings) simultaneously from the total Hamilto- nian and the boost generators . One should emphasize that this result is valid for any Lor- entz-scalar function V . The explicit expressions for are not required as well. IN 0( ) ( ) ( )iK t iK tV x V t e V e−= , =x x ( ) ( )F Fi ie V x e V Lxβ β− =N N 1R R= 1g ( )cK a ( )t,x FN 1 exp= (aB )c Henceforth, to be more definite let me refer to the model, where the neutral pion field interacts with the nucleon field φ ψ via the Yukawa coupling, 5V igψ γ ψφ= ca r , implying that the set of clothed operators is composed of the meson (pion) operators with the same notation and the fermion operators (nucleons) and (antinucleons) with the polari- zation index (afterwards, it will be omitted for short). ca (c )b r,p (cd ,p )r 4. LORENTZ BOOSTS In the “instant” form of the relativistic QFT (after Dirac) the generators N of the Lorentz boost (here , and , where is the velocity of a reference frame) contains interaction terms, e.g., for the Yukawa model, 1 2 3(N N N= , , β β= n ) ren , es exp( )iβΛ = N v= v v= /n v tanhβ ( )F V d= − +∫N N x x x N (10) where is the free part of : with FN N F ferm m= +N N N ( )[ ] ( ) ( ) ( ) 2ferm ii m d dψ γ ψ ψ γψ= − − ∇ + + ,∫ ∫N x x x x x x x 22 2 21 ( ) ( )( ) 2mes dπ µ φφ = − +∇ + ∫N x x xx x Ω ... dx 0 1R . (11) We have separated contribution to N that comes from the meson and fermion mass counterterms. renN It is reasonable to anticipate that the physical vac- uum and clothed one-particle states (e.g., a k ) should be, respectively, the no-clothed-particle state and one-clothed-particle states from the point of view of a moving observer. More exactly, they should meet the requirements: and , where is the Lorentz transformation. How can one provide them? Again, in the CPR, Ω L † ( )c Ω † † ( )c ca Lk=ΛΩ = Ω ( )a kΛ Ω †( ) ( )ca W a W≡ =N N N ( ) [ ] [ ]c F I F Ia R R+ + , + , +B N N N N (12) and we are looking for such that R [ ] ( )F IR V, = = − ∫N N x x (13) After this operation, 1( ) ( ) ( ) [ ] 2F ren Rα α α= + + ,B N N NI 1[ ] [ [ ]] 3ren IR R R+ , + , , + ...N N (14) that repeats the structure of the corresponding contribu- tions to K . It is important from the physical point of view if we want to have , . The latter does not mean . ( )α † c aΛ Ω = † 1 c c −Λ Λ ΛΩ = Ω †( ) ( )ca k Lk †( ) ( )a k a Lk= Ω 5. RELATIVISTIC INTERACTIONS IN MESON-NUCLEON SYSTEMS IN CPR Within the method of UT’s a huge amount of virtual processes induced by a meson absorption/emission, a N N -pair annihilation/production and other cloud ef- fects can be accumulated in the creation (destruction) operators for the clothed (physical) mesons and nucle- ons. Such a bootstrap reflects the most significant dis- tinction between the concepts of clothed and bare parti- cles. 62 Performing the normal ordering of the clothed- particle operators involved, we get a simple recipe to select the and interaction operators of the - and -orders between the partially clothed pions, nucleons and antinucleons (in particular, , and ) 2 ←→ 3g NN → 2 32 ←→ NN 2g N Nπ π→ NN NNπ↔ 0( ) ( ) ( ) ( )c cK a K a K NN NN K NN NN= + → + → ( ) ( ) (K NN NN K N N K N Nπ π π π+ → + → + → ) ( ) ( ) (K NN K NN NN K NN NNππ π π+ ↔ + ↔ + ↔ ) ( ) ( ) (K NN NN K NN K N Nπ πππ π ππ+ ↔ + ↔ + ↔ ) ( )K N N …π ππ+ ↔ + , (15) 1 2 K where interactions between clothed nucleons ( ), anti- nucleons ( N N ) and pions ( ) have been separated out. π 5.1. NUCLEON-NUCLEON INTERACTION OPERATOR Along this guideline we derive the interac- tion operator within the Yukawa model NN NN→ 1 2 1 2 1 2 1 2( ) (NNK NN NN d d d d V′ ′ ′ ′→ = , ; ,∫ p p p p p p p p ) † † 1 2 1 2( ) ( ) ( ) ( )c c c cb b b b′ ′× ,p p p p 1 2 1 2 2 2 1 2 1 2 1 2 1 2 3 ( )1( ) 2 (2 )NN g mV E E E E δ π ′ ′ ′ ′ ′ ′ + − − , ; , = − p p p p p p p pp p p p 1 5 1 2 5 22 2 1 1 1( ) ( ) ( ) ( ) ( ) u u u u p p γ γ µ ′ ′ ′× − − p p p p . (16) The corresponding relativistic and properly sym- metrized quasipotential is given by NN 1 2 1 2 2 2 1 2 1 2 1 2 1 2 3 ( )1( ) 2 (2 ) 2NN g m V E E E E δ π ′ ′ ′ ′ ′ ′ + − − , ; , = − p p p p p p p pp p p p 1 5 1 2 2 1 1 1 1( ) ( ) 2 ( ) u u p p γ µ ′ ′  ×  − − p p 2 5 22 2 2 2 1 ( ) ( ) (1 2) ( ) u u p p γ µ ′ ′  + − −  p p (17) − ↔ . If we start with the same zeroth approximation to , our description of the clothed H Nπ and NN states will be very similar to that given for and states. A different situation holds in the case of clothed fermion–antifermion and two–meson states, where one has to handle eigenstates of a mixed kind (see [2]). Nev- ertheless, the corresponding equations for the WF’s can be solved in a nonperturbative way using the methods elaborated in the theory of nuclear reactions. Nπ NN Its distinctive feature is the presence of the covariant (Feynman-like) “propagator”, 2 2 2 2 1 1 2 2 1 1 1 2 ( ) ( )p p p pµ µ′ ′  + − − − −  .  f , , r, (18) On the energy shell of the scattering, that is NN 1 2 1 2 iE E E E E E′ ′≡ + = + ≡p p p p (19) this expression is converted into the genuine Feynman propagator which occurs upon evaluating the -matrix in the -order. S 2g 6. EQUATIONS FOR BOUND AND SCATTERING STATES The clothed one–particle states are eigenstates of . There may be other eigenstates, viz., the states with discrete values of the system mass. For the Yu- kawa model the corresponding states may be fermion– fermion states (deuteron–like), etc. They appear with the following zeroth approximation (ZA) H H 2 (2) 2 4ZAK K g K= + (20) which is created by adding to the two–operator (one– body) contribution † † 2 ( ) ( ) ( ) ( )c c c c r K a a d E b r bω = + ,∫ ∫ ∑k pk k k p p † ( ) ( )c cd r d r d+ , , p p p NK Kπ≡ + , (21) the four–operator (two–body) contributions of the - order that arise, in particular, from the commutator 2g [R V, ( )0 cH a ] Ω E E N NN (1 2) ; . The operator has an important ptoperty: it conserves the total number of clothed particles. More- over, the Fock subspace spanned onto the two-particle eigenstates can be divided into several sectors (the NN-, πN-sector, etc.) such that leaves each of them to be invariant (see Subsection 4.2 and Appendix B of Ref. [2] for details). If we consider Now the eigenstates that belong to the sector, ZA ZAK ZAK NN 1 2 1 2 1 1 2 2 , ( )NN NN r r d d r rΦ = Φ , ; ,∫∑ p p p p † † 1 1 2 2( ) ( )c cb r b r× , ,p p , (22) then will be the state vector of the same sector so that the eigenvalue equation yields ZA NNK Φ E ZA NN NNK EΦ = Φ [ ]( ) E N NK K NN NN E+ → Φ = Φ (23) in the sector. The corresponding equation for the WF looks as (1 2)E NNΦ ; 1 2 ( ) E NNE E E− − Φ ;p p 1 2 (1 2 1 2 ) (1 2 )E NN NNd d V′ ′ ′ ′ ′ ′= , ; , Φ∫ p p . (24) 7. THE -OPERATOR. AN EQUIVALENCE THEOREM FOR THE MATRIX S S e ı After constructing the interaction operators in the CPR it becomes to be indispensable to express the con- ventional -matrix through the clothed-particle interac- tions and states. In this respect, let me recall the defini- tion S 0 2 0 12 1 2 1 ( )lim lim H t H tH t t t t S e e −− − →+∞ →−∞ = ı ı (25) of the -operator for the decomposition . Furthermore, let us introduce the -operator S ( ) ( )S S a≡ 0 ( )IH H a H a= + S 63 0 2 0 12 1 2 1 ( )lim lim K t K tK t t cloth t t S e e −− − →+∞ →−∞ = ı ı e ı c 1c (26) for the decomposition or 0( ) ( ) ( )c c IH K a K a K a= = + 2 1 lim limcloth t t S →+∞ →−∞ = 0 2 2 1 0 1( )( ) ( ) † 2 1( ) ( )c cK t H a t t K a t D DW t e e e W t− − − ,ı ı ı (27) k k where W t . ( ) ( ) ( )0 0exp expD iK t W iK t= − Since ( )lim 1Dt W t →±∞ = or (28) ( )lim 0Dt R t →±∞ = then 0 2 2 1 0 2 1 ( ) ( )( ) ( )lim lim c cK a t H a t t K a t cloth t t S e e e− − − →+∞ →−∞ = ı ı ı . (29) Matrix elements † † 0 0( )a S a a...Ω ...Ω † 0...Ω 〈 of between the bare states a with and matrix elements ( )S S a= 0Ω =0 0H † †( )c c ca S a a...Ω ...Ω † c ...Ω ca 〈 of between the clothed states a with are equal to each other since the -algebra with the vac- uum is isomorphic to the a -algebra with the vac- uum . Thus, all what we need to ensure this equiva- lence is the asymptotic condition (28), i.e., ( )cS S a= 0Ω = cloth 0K Ω 0Ω [ ] [ ]0 0lim exp ( ) ( )exp ( ) 0c c ct iH a t R a iH a t →±∞ − = . (30) As it has been shown in [6] this condition may be ful- filled, at least, for the Yukawa model. 8. LINKS WITH THE IN(OUT) FORMALISM. REDUCTION TO CLOTHED PARTICLE STATES In order to describe collisions bound systems it is pertinent to proceed with the S-matrix fiS f out i i= 〈 ; ; n in the Heisenberg (H) picture, built of the in(out) states, e.g., for the reaction with d NNπ → † ( )d NN inS NN out a k dπ → = 〈 ;  ( )( ) kA k t f ϕ∗, = , † ( )B k t, = 0 2 0( )F x∂ ,x π , k  . Further, one introduces (to be definite for opposite-charged scalar particles) , with re- spect to the scalar product ( for the H-field operator ( )kfϕ ϕ− , i d F ∗≡ ,∫ x x ( )kf, = )1 2F F, 1 0( )x ( ) ( ) ( )Ht Ht Dx t e eϕ ϕ ϕ −= , ≡x xı ı †( ) ( ) ( ) ( ) ( )D kd A k f B k fϕ ∗= +∫x k x x . (31) Here are the respective plane waves (strictly speaking, the wave-packet-like solutions of the Klein- Gordon equation with positive frequencies). Now, con- sidering the similarity transformations and and employing the LSZ prescription ( )kf x cA k ( )k W †( ) ( ) ( )A k W A k W→ = †W B ( ) ( )cB k B k→ = † †lim ( ) ( )int A k t A k →−∞ Φ , Ψ = Φ Ψ (32) to be valid for any normalizable states and , one can show (see [6]) that the one-meson in-state Φ Ψ † †( ) lim ( ) ( )in ct k in A k A k t A k →−∞ ; ≡ Ω = , Ω = Ω† (33) and the two-meson in-states † † 1 2 1 2( ) ( )in ink k in A k A k, ; ≡ Ω 0 0 1 2( ) † † 1 2lim ( ) ( )H k k t c ct e A k A k− − →−∞ = ı Ω . (34) Its trivial consequence is, † † 1 2 1 2( ) ( )c cin A k A k, ; ≠ Ω . Similar relations with can be derived for the out-states. Moreover, such limits in the distant past and future are equivalent to the “Møller” operators t →+∞ ( ) 0 ( ) limE i E Hε ε ε ± →+ Ω = ± ± −ı . For instance, we find that ( ) 0 0 † † 1 2 1 2 1 2( ) ( ) ( )c ck k in k k A k A k+, ; = Ω + Ω . (35) Such a time-independent representation via the Hamilto- nian resolvent is closely connected with the approaches typical to the nonrelativistic quantum theory (see [5,6]). 9. CONCLUDING REMARKS i) I have tried to show a possible way in finding bridges between the description of some bound and scattering states in QFT and the approach traditional for the nuclear physics. Of course, in spite of a similarity between nonrelativistic quantum mechanics and our clothing procedure, the latter gives rise to the new non- conserving clothed-particle-number interactions (quasipotentials). ii) The method of unitary clothing transformations enables us to obtain (in a combination with nonpertur- bative recipes of QFT) a number of relations helpful both for the evaluation of reaction amplitudes and state vectors. REFERENCES 1. A. Shebeko, M. Shirokov. Clothing procedure in relativistic quantum field theory and its applications to description of electromagnetic interactions with nuclei (bound systems) //Prog. Part. Nucl. Phys. 2000, v. 44, p. 75-86. 2. A.V. Shebeko, M.I. Shirokov. Unitary transforma- tion in quantum field theory and bound states //Phys. Part. Nucl. 2001, v. 32, p. 31-95; nucl-th\0102037, 2001, 69 p. 3. V.Yu. Korda, A.V. Shebeko. The clothed particle representation in quantum field theory: mass renor- malization //Phys. Rev. D. 2004, v. 70, 085011, p. 1-9. See also another contribution to this Conference by Korda et al. 4. V.Yu. Korda, L. Canton, A.V. Shebeko. Relativistic interactions for the meson-two-nucleon system in the clothed-particle unitary representation Title of the journal article //doi:10.1016/j.aop. 2006.07.010, Ann. Phys. 2006 in press; nucl-th/0603025, 2006, 34 p. 5. A. Shebeko. The S-matrix within the method of clothing transformations //Nucl. Phys. A. 2004, v. 737, p. S252-255. 6. A. Shebeko. The S-matrix within the method of uni- tary clothing transformations //Proc of the XVI ISHEPP. 2004, v. 1, p. 35-42. 64 МЕТОД УНИТАРНЫХ ОДЕВАЮЩИХ ПРЕОБРАЗОВАНИЙ В КВАНТОВОЙ ТЕОРИИ ПОЛЯ: ПРОБЛЕМА СВЯЗАННЫХ СОСТОЯНИЙ И S-МАТРИЦА А.В. Шебеко С помощью переформулирования квантовой теории поля в терминах так называемых “одетых” частиц и взаимодействий между ними рассмотрена проблема поиска простейших собственных состояний полного гамильтониана H. В этом подходе и другие операторы, имеющие глубокий физический смысл, такие как генераторы Лоренц бустов и плотность тока, которые изначально зависят от операторов рождения и унич- тожения “голых” частиц, выражаются через операторы “одетых” частиц. Переход к представлению “оде- тых” частиц осуществляется с помощью определенного унитарного (“одевающего”) преобразования без изменения исходного гамильтониана. Показано, как можно вычислять S-матрицу в новом представлении. H МЕТОД УНІТАРНИХ ОДЯГАЮЧИХ ПЕРЕТВОРЕНЬ В КВАНТОВІЙ ТЕОРІЇ ПОЛЯ: ПРОБЛЕМА ЗВ’ЯЗАНИХ СТАНІВ І S-МАТРИЦЯ О.В. Шебеко За допомогою переформулювання квантової теорії поля в термінах так званих “одягнених” частинок і взаємодій між ними розглядено проблему пошуку простіших власних станів повного гамільтоніану . У цьому підході та інші оператори, що мають глибоке фізичне значення, такі як генератори Лоренц бустів і густина струму, котрі початково залежать від операторів народження і знищення “голих” частинок, пода- ються через оператори “одягнених” частинок. Перехід до зображення “одягнених” частинок здійснюється за допомогою певного унітарного (“одягаючого”) перетворення без зміни початкового гамільтоніану. Показа- но, як можна обчислювати S-матрицю в новому зображенні. H H 65

The method of unitary clothing transformations in quantum field theory: the bound-state problem and the S-matrix

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