THE NECESSARY AND SUFFICIENT CONDITIONS FOR TRANSFORMATION FROM DIRAC REPRESENTATION TO FOLDY-WOUTHUYSEN REPRESENTATION. V.P.

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1 THE NECESSARY AN SUFFICIENT CONITIONS FOR TRANSFORMATION FROM IRAC REPRESENTATION TO FOLY-WOUTHUYSEN REPRESENTATION V.P.Neznamov RFNC-VNIIEF, Sarov, Nizhniy Novgorod region The paper decribe condition for tranformation from the irac repreentation to the Foldy-Wouthuyen repreentation. The neceary condition i the block-diagonal tranformation of Hamiltonian relative to the upper and lower component of the wave function. The ufficient condition i the wave function tranformation law decribed by relation (6). It ha been demontrated that the unitary tranformation offered by the author of the paper [4], [5], [6], [7] do not atify the ufficiency condition (6) and, hence, they are not the Foldy-Wouthuyen tranformation. In application, the matrix element of any operator in the repreentation can be calculated, according to (6), uing the normalized two-component wave function in the irac repreentation known for the given problem.. Introduction In the preent work, the baic propertie of wave function in the repreentation are determined and the relation between irac and wave function i invetigated. The paper determine condition for tranformation of the irac repreentation to the repreentation. The effect of the atified condition i illutrated with many example. Application of the obtained reult allow calculating the expectation value of operator correponding to the baic claical quantitie, becaue the

2 explicit form of thee operator can be etablihed in the repreentation, but not in the irac one.. The Foldy-Wouthuyen repreentation The Foldy-Wouthuyen repreentation wa introduced in the paper []. The Foldy-Wouthuyen equation of free motion of a quantum-mechanical particle with pin ha the form p = ( H ) = βe. () 0 0 In equation () and below the ytem of unit with = c = i ued; the inner product of 4-vector i = =, = 0,,,3, =,,3; = ; x µ 0 0 k k µ xy x yµ x y x y µ k p i ( x ) i the four-component wave function; µ i 0 σ 0 i I α =, β i = σ 0 0 I are irac matrice; σ i are Pauli matrice; the operator i E = p + m. energy: The equation () olution are plane wave of both poitive and negative ( x, ) = Ue, ( x, ) = Ve, p = ( p + m) () ( π) ( π) ( + ) ipx ipx ϕ 0 U =, V, ϕ 0 = χ and χ in expreion () are the two-component Pauli normalized pin function. V : The following orthonormality and completene relation are valid for U and UU = VV = δ ; UV = VU = 0 U U = + V V = ( β ) ( β ) ( ) γ( ) δ ; ( ) ( ) γδ γ δ γδ (3)

3 3 In expreion () and (3), γ, δ belong to pinor indexe and belong to pin indexe. In the further ummation with repect to pinor indexe the ummation ymbol and indexe themelve are not ued. The unitary tranformation E+ m βα p H0 = U0 H0 U0 ; U0 = + E E+ m relate Hamiltonian ( H 0 ) and free irac Hamiltonian other. H = α p+ βm 0 (4) to each The problem of tranformation from the irac repreentation to the repreentation become coniderably more complex in a general cae of interaction between a fermion and an arbitrary boon field. Eriken [] found the exact tranformation for the time-independent irac Hamiltonian. Foldy and Wouthuyen [] found Hamiltonian H for interaction with electromagnetic field A ( x, t) µ in the form of a erie in term of power of. Blount [] found Hamiltonian m H in the form of a erie in term of power of electromagnetic field and their time and pace derivative. Cae [3] obtained the exact form of the tranformation in the preence of the time-independent external magnetic field B tranformed to obtain the following: ( ) = rota = = +. The irac equation i therefore p0 H β p ea eσb m. (5) The paper [4] (ee alo the review [5]) give the relativitic Hamiltonian in the form of a erie in term of power of charge e for a general cae of interaction with an arbitrary external electromagnetic field. In each of the above-mentioned cae equation for the wave function in the repreentation are non-covariant and their Hamiltonian are non-local (they contain an infinite et of differential operator) and block-diagonal (i.e. they are diagonal relative to the upper and lower component of the wave function). The Foldy-Wouthuyen repreentation, in pite of it non-locality and relative complexity for Hamiltonian H, i of interet for reearcher until now. A number

4 4 of paradoxe typical for the irac repreentation can be reolved in the repreentation. In a free cae, the velocity operator in the repreentation ha a p uual form v = imilar to the claical form (in the irac repreentation v = α ), E electron «trembling» (Zitterbewegung) i abent, the pin operator i kept unchanged with time []. The major operator have a coniderably impler form in the repreentation. For example, operator r in the repreentation i a coordinate operator rather than a complex Newton-Wigner operator in the irac repreentation [6]. Similarly, the polarization operator in the repreentation i the imple operator П = β, where i i σ = 0 0 i σ [7]. The repreentation i very ueful, if one need to obtain emi-claical equation for particle motion and pin [8]. The repreentation allow eaily interpreting the term of the tranformed irac Hamiltonian with external field in the expanion in erie in term of power of the coupling contant and relativitic parameter v. Once the relativitic Hamiltonian c H ha been obtained in [4], it become poible to conider the quantum-field procee in the repreentation [5] within the perturbation theory. 3. The neceary and ufficient condition for tranformation from the irac repreentation to the Foldy-Wouthuyen repreentation Hamiltonian diagonalization relative to the upper and lower component of the wave function ( x ) i the neceary condition for tranformation from the irac repreentation to the Foldy-Wouthuyen repreentation. However, atifaction of thi condition i not ufficient. A it ha been rightly noticed in [9, 0], not each unitary tranformation ued for diagonalization of the irac Hamiltonian lead to the repreentation. It i hown, for example, in [0] that the well-known Eriken-Kolrud tranformation [] i not the repreentation. The econd obligatory condition for tranition from the irac repreentation to ϕ( x, t) the repreentation i the wave function tranformation ( xt, ) = χ( x, t) with

5 5 vanihing the upper and lower component of bi-pinor ( x ) and tranforming the normalization operator of the wave function ( x ) into the unit operator. Thi i a ufficient condition. For a cae, when the irac Hamiltonian i independent of time (the cae of tatic external field) thi condition can be repreented in the following form ( + ) ( + ) ( + ) iet ϕ ( + ) iet ϕ ( xt, ) = e A+ ( xt, ) e + = χ 0 ( ) 0 iet ϕ x iet ( xt, ) = e A ( xt, ) e = χ χ ( x ) Here, E i module of the particle energy operator; A + and A are normalization operator, which are not the ame, in general, for poitive-frequency and negativefrequency olution; function ( + ), ( ϕ x χ ) are normalized to. For a free cae, E+ m E = p + m, A+ = A = ; for the poitive-energy olution we have E ( + ) ipx ( ϕ = e ϕ and for the negative-energy olution - ) ipx χ = e χ ; ϕ and χ are the two-component Pauli pin function (ee ()). Function ϕ ( x ) and χ ( x ) for a particle moving in tatic external field are the appropriate eigen-function of the irac equation. The ufficiency condition implie tranformation of the wave function to the form (6) with the normalization operator equal to. In general, if the irac and Hamiltonian depend on time, the ufficiency condition (6) ha the ame meaning, becaue we ue expanion in the irac equation olution obtained either for free motion of particle, or for motion in the preence of tatic external boon field when olving the particular problem of phyic (at leat, with the ue of the perturbation theory). The ufficiency criterion validity can be proved uing the Eriken tranformation [], which i the exact tranformation for the time-independent Hamiltonian product of two factor H. The tranformation matrix U Er may be repreented, in general, a a (6)

6 6 U Er βλ + λβ U = ( + βλ ) + ; 4 (7) H λ =, λ =± for the poitive-energy and negative-energy olution, ( H ) repectively, where λ H i Hamiltonian in the irac repreentation. It i important that [] =, [ βλ, λβ] = 0 (8) and the operator βλ+ λβ i even, β, ( βλ+ λβ) = 0. (9) Conider the normalized irac wave function, for example, for the poitiveenergy olution. ( + ) ( + ) iet ϕ = e A+ ( + ) (0) χ efinition of the normalized operator A + in (0) implie that pinor ϕ ( + ) normalized to ( + ) ( + ϕ x ϕ ) dv = Apply the Eriken tranformation to equation (0) and obtain + + = Er = + + ( + ). U βλ λβ βλ 4. () The Eriken tranformation in equation (), in contrat to equation (7), ha the changed order of factor, which i admiible in view of relation (8). ( Since λ + ) =, expreion () take the form iet e A ϕ + + = + βλ + λβ + 4. () 0 The wave function normalization requirement may be written a ( + ) ( + ) ( + ) ( + ) + + dv = ϕ A + βλ+ λβ A ϕ dv =. (3) 4 Meeting the requirement (3) mean necearily meeting the requirement A+ + ( βλ + λβ ) A+ =. (4) 4 i

7 7 Multiply the left-hand and right-hand ide of equation (4) by the operator A + and obtain + ( βλ + λβ ) = A ( βλ + λβ ) A+ = 4 ; Expreion (5) prove the ufficiency condition (6). According to () we, indeed, obtain (5) ( + ) ( + ) iet ϕ = e (6) 0 Similar conideration are valid for the negative-energy olution and the reult i 0 = (7) iet x e χ x In the next ection, the ufficiency condition (6) i applied to ome blockdiagonal Hamiltonian known to the given paper author and obtained by unitary tranformation of the irac repreentation. It i hown that the ufficiency condition (6) i, ometime, not atified and, if it i the cae, the unitary tranformation performed are not the Foldy-Wouthuyen tranformation. 4. Conideration of the ufficiency condition (6) for ome unitarily tranformed wave function. 4.. Free motion

8 8 E+ m βα p H = αp+ βm; U = + ; H = βe; E = p + m ; E E+ m ( + ) ϕ ( + ) ( + ) E+ m iet ( + ) ( + ) iet ϕ ( xt, ) = e σ p ; (, ) (, ) ; xt = U xt = e + E ϕ 0 x E+ m σ p E+ m iet χ ( xt, ) = e E+ m 0 iet ; ( xt, ) = U ( xt, ) = e. E χ χ (8) 4.. Motion in tatic magnetic field E+ m βαπ H = απ + βm; U = + ; H = β E; π = p ea; E E+ m E = ( p ea) eσ B+ m ; ( + ) ϕ ( + ) ( + ) E+ m iet ( + ) ( + ) iet ϕ ( xt, ) = e σπ ; (, ) (, ) ; xt = U xt = e E ϕ + 0 x E+ m σπ E m 0 iet χ + ( xt, ) = e ; (, ) (, ) iet E m x t U + = x t = e E χ χ 4.3. Motion in tatic electrical field (9) Here we demontrate, how the condition (6) i atified, by uing only term ~ e and term including the quadratic one with repect to v c in the U expanion in erie in term of power of the charge e [4]. Apparently, the procedure can be implemented for any order of expanion in term of e and v, if we ue the c mathematical apparatu from [4]. With denotation ued in [4], we obtain, within the accepted accuracy, that

9 9 0 e βα p p ie H = αp+ βm+ ea0; U = ( + δ + δ ) U0 = + ( α A0) m 8 m 4m iβ 3 (( αp)( α A0) ( α A0)( αp) ); H = βe; 6m p i E = m+ + eβ( A0 + (( αp)( α A0) ( α A0)( αp) )); m 8m ϕ σ σ σ m 4m ( + ) ( + ) ( + ) iet ϕ ( xt, ) = U ( xt, ) = e 0 p ie A σ σ 0 iet p ie χ x ( xt, ) = e + 3( σ pσ A0 σ A0σ p ) m 4m 8 m 6m χ 0 iet ( xt, ) = U ( xt, ) = e χ ( + ) ( + ) iet p ie ( xt, ) = e 3( p( A0) ( A0) ( σ p ) σ p ieσ A0 ( + ) 8 m 6m + ϕ x (0) In expreion (8)-(0), p and function f ( p ) imply the correponding operator; ϕ, χ are the two-component function atifying the equation p ea m ( p ea )( p ea + m) ( p ea ) = 0 0 β σ 0 0 β σ 0 () ϕ In () we have = ; χ equation for ϕ and χ are obtained with β = Ι and β = Ι, repectively, (ee [9, 4]); for tatic external field we have ϕ iε t p0 = ε; ( x, t) = e ; χ for β = +Ι ε = E; for β = Ι ε = E. For convenience, the particle energy operator of different form in expreion (8)-(0) are denoted with the ame ymbol E. In (0) we uppoe that in operator E β = + for the poitive-frequency ( olution + ) ( x, t) and β = for the negative-frequency olution ( x, t) (ee [9, 4]). It follow from (8)-(0) that the ufficiency condition (6) i atified in each of the cae above and, thu, the given unitary tranformation are the Foldy- Wouthuyen tranformation.

10 Super-algebra in the irac equation with tatic external field Uing the uper-ymmetric quantum mechanic concept, the paper [3] dicue a wide range of interaction between a irac particle and tatic external field, which provide the cloed form of the irac Hamiltonian diagonalization. The paper [3] author ue SU () tranformation of the irac Hamiltonian, a the Foldy- Wouthuyen repreentation. It i intereting to know, how the ufficiency condition (6) i atified for thi tranformation. Uing denotation from [3], we have H Q Q λ, = + + where λ i the Hermitian operator, Q and Q are the two fermion operator meeting the following requirement: Q Q { Q λ} { Q λ} = = 0,, =, = 0. Then, the Hermitian operator of SU () -algebra are introduced: Q+ Q iλ( Q+ Q ) λ J = ; J = ; J 3 = ; J, J iε J =,, ({ QQ} ) λ { QQ} The tranformation operator i ( λ ) ij U e ij θ { QQ} i j ijk k θ θ θ λ( Q+ Q ) = = co + in = (+ co ) + ( co θ ) () ( λ, ) In contrat to [3], exponent ij θ in () i taken with it poitive ign. A one can ee below, thi i a neceary tep for the ufficiency condition (6) to be atified. H = e H e = Q + Q coθ + ij λinθ + λcoθ + ij Q + Q inθ (3) ijθ ij θ If { Q, Q } ({ Q, Q }) ( λ ) tgθ = in θ =, co θ =, / ( λ ) ({ Q, Q } + λ ) { Q, Q } + λ expreion (3) i reduced to it diagonal form H ( λ ) ({ } ) Q, Q λ λ = +. (4) If M λ = β m Q = Q M 0 = 0 0 ;, ;

11 M = σ p+ C ic C = A ie I = ;,,,3,5, 5 I I I the irac Hamiltonian H can be written a H = απ + iβγ π + βm; π = p + A + iβ E, I =,,3,5, p = 0. (5) 5 5 I I I I 5 The following interaction are preented uing thee denotation: A 5 i the peudo-calar potential, E 5 i the time component of the axial-vector potential, E i the electrical component of interaction of the abnormal magnetic moment of a particle, A i the minimum magnetic interaction; if A5 = E5 = 0, A= 0, E = r, Hamiltonian H i reduced to Hamiltonian of the irac ocillator. Each of the interaction above admit cloed tranformation to the diagonalized form (4). Let u check, whether the SU() -tranformation () i the tranformation with the atified ufficiency condition (6). So, H = Q+ Q + βm, where Q and Q are defined in (5). ({ } ), ; {, } M M + m 0 H = βe = β Q Q + m E = Q Q + m = 0 MM + m E+ m β ( Q+ Q ) U = + ; E E+ m ( + ) ϕ ( + ) ( + ) iet E+ m iet ϕ ( xt, ) e + + = ; (, ) (, ) xt = U xt = e + E Mϕ 0 x E+ m iet E + m 0 ( xt, ) = e M χ x ; (, ) (, ) iet E m + xt = U xt = e E χ χ One can ee from expreion (6) that, indeed, the SU () - tranformation in [3] (with the changed ign of exponent ij θ ) provide atifaction of the ufficiency condition (6) and i the tranformation at the ame time. If the author ign of + ( Q+ Q ) the exponent in expreion () remain unchanged ij θ e, ee[ 3 ], the tranformation E+ m β in (6) become equal to U =. Such tranformation doe not E E+ m (6)

12 provide atifaction of the ufficiency condition (6), though the requirement of blockdiagonalization of Hamiltonian can be met (ee [3]) A irac particle motion in external gravitational field (Obukhov Hamiltonian [4, 5]). The paper [4, 5] offer the irac Hamiltonian of the form V H = βmv + ( αp), W (7) for tatic metric ds = V dx 0 W dx. Then, diagonalization of the Hamiltonian (7) uing the cloed Eriken-Kolrud tranformation i performed in [4, 5]. The paper [0] prove nonequivalence of the and Eriken-Kolrud tranformation and offer the explicit form of the operator of tranition between the and Eriken-Kolrud repreentation for the free irac equation. Let u, firt, check, whether the ufficiency condition (6) i atified for the Obukhov Hamiltonian (7), uing the method decribed in [4]. For a free cae, H = α p+ βm; H = βe; the tranformation matrix ued in [4] ha the form E K α p+ β m m iβσ p m βγ 5σ p UE K = ( + βγ i 5β) + iγ5β = + + iγ5 + iγ5 ; E E E E E γ5 = iαα α 3; iσ p ( + ) iet E m ϕ x E k( xt, ) = UE k ( xt, ) = e E+ m ; E 0 0 iet E + m E k( xt, ) = UE k ( xt, ) = e iσ p ; E χ E+ m It can be een from (8) that the condition (6) i not atified and one need to perform additional tranformation E k U (8), which reduce the wave function to the form (6) with no change of the form of Hamiltonian HE k= H= β E. Thi tranformation look like [0], a follow:

13 U E k 3 E+ m iβσ p =. (9) E E+ m Now, let u try to tranform the Hamiltonian (7) with the method decribed in [4, 5] and check, whether the ufficiency condition i atified, or not. Similar to the paper [4, 0], our conideration i performed to the firt-order accuracy of potential ( V, ) ( F ) and their firt derivative in pace coordinate; in the expanion in erie in term of relativitic parameter v c we retrict ourelve with value including v c. Uing denotation from [4, 5] we have 0 e E+ m βα p U = ( + δ + δ ) U0; U0 = + ; E = p + m ; E E+ m 0 0 e e 0 0 E+ m δβe βe δ + N = 0; δr+ Rδ = RLδ δlr; R = ; E βα p L = ; E+ m H = βe+ K ; K = δ βe βeδ + C; e e The Hamiltonian (7) i written a H = α p+ βm+ E Q; V = mv ; Q = {, };. F α p F = W + E β Hence, within the accepted accuracy [4, 5] we have N β = ( α p E E α p) 0 β + Q = {( V, ) αp } + {( F, ) αp} ; δ = N. m e δ = (( F V) p p ( F V) ); 6m 0 e βα p p βα p p β U = ( + δ + δ ) + = + + {( F V), αp} m 8 m m 8 m 4m (( F V) p + αp( F V) αp+ p ( F V) ) 6m

14 4 βp βp β, ( ) { } H = βe+ K = βm+ + C= βm+ + βm V pv + m m 4m β β { pf, } β σ( f p) f σ( Φ p) + Φ = βe m 4m 8m In expreion (30), Φ= V; f = F. The expreion for (30) H i identical to the expreion obtained in [0] (ee expreion (4)). The irac equation with Hamiltonian (7) for tatic potential V, W can be written a εϕ = mϕ + mv ( ) ϕ + σ p { σ p, ( F ) } χ ϕ + iε t ( xt, ) = e ; χ εχ = mχ mv ( ) χ + σ p+ { σ p, ( F ) } ϕ It follow from the econd equation of the equation ytem (3) that χ = σ p+ { σ p, ( F ) } ϕ = ε + m+ m V = σ p+ { σ p, ( F ) } ϕ m( V ), σ p ϕ ε + m+ m V ( ε + m) ( + ) ( + For ε > 0, the replacement εϕ = E ϕ ). can be performed in (3). (3) (3) Hence, within the accepted accuracy we have ( + ) σ p {, } ( + χ ) = + F V σ p ϕ (33) m 4m Similarly, for ε < 0 εχ = E χ and ( ) σ p {, } ( ϕ ) = + F V σ p χ (34) m 4m Uing expreion (30) (34) and the wave function normalization requirement, we obtain: ( + ) ( + ) ie t ϕ ( xt, ) = e A+ ( + ) = χ ie t p = e ( ( F V) p + σ p( F V) σ p+ p ( F V) 8 m 6m ϕ σ p ; ( + ) + {( F V), σ p} ϕ m 4m

15 ie t ϕ ( xt, ) = e A = χ ie t p = e ( ( F V) p + σ p( F V) σ p+ p ( F V) 8 m 6m σ p + {( F V) σ p 4 } χ m m ; χ, ( + ) ϕ xt, = U xt, = e ; 0 0 ie t xt = U xt = e χ ( + ) ( + ) ie t (, ) (, ) 5 Expreion (35) how that the ufficiency condition (6) i atified. And vice vera, application of the Obukhov tranformation diagonalizing the Hamiltonian (7) to the wave function + ( x, t), ( x, t) doen t provide atifaction of the ufficiency condition (6) (imilar to the cae of free motion (8)). Really, uing denotation from [4] the Obukhov tranformation can be written a ( ˆ H UE K = + β J + JΛ ) = ( iγ ) + iγ β + iγ iγ ( H ) Within the accepted accuracy (ee alo [4], [0]), expreion (36) can be written a i i { } βγ 5 γ 5 γ 5 UE K = 5, + iγ σ p F p F + F p f + σ f p mv 4m W W V 4m (37) iγ5 β γ5 σф { σ pf, } σф( iγ5 ) m 4m (35) (36) Hence, according to (35) we have ( + ) ( + ) ( xt, ) = U ( xt, ) = E K E K ie t p i = e (( F V) p + σ p( F V) σ p+ p ( F V) ) + { σ p, F} σф 8m 6m 4 mv 4m ( + ) ϕ 0 (38)

16 ( xt, ) = U ( xt, ) = E K E K 6 ie t p i = e (( F V) p + σ p( F V) σ p+ p ( F V) ) { σ p, F} σф 8m 6m 4 mv 4m 0 χ (39) One can ee that the ufficiency condition (6) i not atified in expreion (38), (39). For free motion ( F =, V = ), the wave function in (38), (39) are identical to the wave function in (8), within the accepted accuracy. The cloed tranformation of the Eriken-Kolrud type wa fulfilled in [6] with the Obukhov Hamiltonian uing the uper-ymmetric quantum mechanic method, the reultant Hamiltonian coincide with the tranformed Hamiltonian from [4] in each order of the expanion in term of power of m. The tranformation matrix with denotation from [6] look like Q U = + β ( iγ5), Q = { αp, F} + iγ5βmv (40) ( Q ) If we expand in erie the expreion (40) with the firt-order accuracy of potential ( V, ) ( F ) and their firt pace derivative and retrict ourelve to power in the expanion in term of ma, we can ee that (40) coincide with the Obukhov tranformation matrix (37) within the ame accuracy. Hence, the tranformation fulfilled by the author of paper [6] i not the Foldy-Wouthuyen tranformation, becaue the ufficiency condition (6) i not atified, imilar to the tranformation fulfilled in [4]. In [7] the author alo ue the Eriken-Kolrud tranformation, they call it the exact tranformation and don t check the wave function tranformation law. In view of the foreaid, the Eriken-Kolrud tranformation i not the Foldy- Wouthuyen tranformation, becaue the wave function tranformation condition (6) i not atified. m

17 5. Concluion 7 The paper tate and prove the neceary and ufficient condition for tranition from the irac repreentation to the Foldy-Wouthuyen repreentation. The exact connection between the wave function in the both repreentation ha been found. It ha been demontrated by a number of example that in ome cae ([4], [5], [6], [7]) block-diagonalization of the irac Hamiltonian i inufficient for tranformation to the Foldy-Wouthuyen repreentation; uch tranformation necearily require atifaction of the ufficiency condition (6). The reult obtained allow unambiguou tranformation to the repreentation and calculation of the matrix element of operator correponding to the main claical quantitie, becaue the exact form of thee operator can be eaily found in the Foldy-Wouthuyen repreentation, in contrat to the irac repreentation. In application, during calculation of the matrix element of operator in the Foldy-Wouthuyen repreentation it i ufficient, according to (6), to ue the upper or lower component of the normalized irac wave function. ( + ) ( + ) ( + ) ( + ) ( + ) A = ϕ A ϕ ; (4) A χ A χ A = In expreion (4) the four-component operator A i written in it matrix form ( + ) A ( A ) =. ( A ) A The author gratefully acknowledge dicuion with A.Ya.Silenko, who alo draw attention to the Eriken tranformation [] that allowed the author an eaier proof of the ufficiency condition for tranformation to the Foldy-Wouthhyen repreentation.

18 8 Reference [] Foldy L.L., Wouthuyen S.A.// Phy.Rev 950.V.78. P.9. [] Blount E.I.// Phy.Rev 96.V.8. P.454. [3] Cae K.M.// Phy.Rev 954.V.95. P.33. [4] Neznamov V.P.// Voproy Atomnoi Nauki I Tekhniki. Ser.: Teor. I Prikl. Fizika Iue. P.. [5] Neznamov V.P//Phyic of Particle and Nuclei V.37.. [6] Newton T.., Wigner E.P.// Rev.Mod. Phy [7] Fradkin.M., Good R.H.// Rev.Mod. Phy [8] Silenko A.J.// J.Math. Phy [9] E. e Vrie, J.E.Jonker// Nucl. Phy. B [0] Silenko A.J., Teryaev O.V.// Phy. Rev [] E.Eriken, M.Kolrud// Nucl. Phy. B [] E.Eriken // Phy. Rev [3] R.Romero, M.Mareno, A.Zentella// Phy. Rev [4] Yu.N.Obukhov// Phy. Rev.Lett [5] Yu.N.Obukhov// Fortchr.Phy / [6] S.Heidenreich, T.Chrobok, H.-H.v.Borzezkowki// Phy. Rev. 73, [7] B.Goncalve, Yu.N.Obukhov, I.L.Shapiro// Phy. Rev. 75, 403, 007.