В.И. Фущич, А.Г. Никитин. The Galilei-invariant systems of partial differential equations, which describe the non-relativistic Введение Более 300 лет известны принцип относительности и преобразования Галилея.


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/  !      ,1981, ,.5,:.11571219. 106.%./ ,;.".     ,             !         , -      (  ,  =  ).    ,      [814], -         -  -             .: -   ,        =   ! -  ,     -        -   .      & -&  +" (  &+")         -                . =       -            ,   [1517].:!    ,,                 &  " , -      (  ) ,           '   . :  ,          -            ( -          ),      -   !    .!   , -    ,            -             !      .    ,           ,    -      -    .�        (       &+"),         " ,          -  ,         :     ,   ,                .           !                 -  .                -  " .        - (  )   ,                ! .%         " ,      ?                          " .� -   (    ,              - [4,1820].             107 1.        !            '  -         -           .%        ,   *! +              "        ,            ! .             . %    -            p  =( $  ,...,N      ,    ,            & .A  (1)          ,       QA,L]QALLQA=fAL,(2)      ,   .=   ,    (2),          -  (1)    !  . B         (;%)  (1)            .�            D )  (    ),    -  (2)       & .*    ,!     '       & [3].B-     [21],      [22,23].B [22,23]        .  1.   (1)   !-  "!#! ""  "  #$ %,$ & '""&  *"  p p ) x 3 i,A m x 108.%./ ,;.".   =      ( .[22,23]).$   ,      (1)    (4),(5)       .$  (4)    (2) ,    (5)   D,L A,L $  (4),(5)        Pa,Pb]=0,[Pa,Jb]=iabcPc,(6)[Ga,Gb]=0,[Ga,Jb]=iabcGc,(7)[Pa,Gb]=iabM,[P,M]=[Ga,M]=[Ja,M]=0,(8)[P0,Pa]=[P0,Ja]=0,(9)[P0,Ga]=iPa,a,b,c D,P D,G D,P D,J D,M A,G A,M A,J A,P A,P iD, A,D .e.   & ,  &  '   . ? ,  (5)      (1)     (4)C =(2 =(2 ,  ,    ,   ,  -   ,       .D  ,           (1)  ;%(4)(  &   " ). ;  (4)       (  *  ) MJaabcPbGc][MJaabcPbGc],(13)          !  ,          .  (4)(13), , -  '   (1)       , !     D  ,         '  -    ;%, !        -      -  .$ ,   ;%          , ,   (1)      [24]          [ (1)] -     ,     [    ;%(4),(5)],    '   E   !    2 3.             109   !       ,  -           .  1. +" !& !"! ", "*/ #$(6)(10)!""$ *,2"$- /" ! ",$ "! " . # ! "(13)- " "*" #$%(6)(11). :     ,      -    " (6)(10)(   )        &  '   (6)(12)(,    -        ,      -   ,        [26].      .?;%       -  ,     .D ,  (4),       "      ,    =exp[  )=exp( )exp( )exp[         @    -   C +    [2]C =       (14)(18)     - *! +   )exp( )=exp A,B :  (4),(19) )=exp( )exp[ 110.%./ ,;.".   =exp[ ]exp( )exp[       (14)(18). D  ,       "   -    (1)    (20),                 (14)(18).        ,  (20)-      a v , U 1  , v , a U   , R 1 , R 1 a v a , a b a 1  ,             " .   (15) ,       " ,   " . !  (15)  ,          ! .=   ,       " (14)  C  (21) , =exp(   C %  ,  (20)      -   (22)        ,        .             111 F ,;%  (1),   (4),            !     -    .F          .$  (25)(28)        ,     .    ,  '   (1)         ,   .�, ,     -                (25)(28).!      " ,         (14)(18) (25) (28).     ,            -    ,     ,b,  T  C  (16),(20),a   ,       $  (29)     ,b, ?     (30)     14-      .               (30),               (6)(10)  !   -  .         (6)(10)   112.%./ ,;.".     ,           I.c2=m=0,c3=m2s(s+1) =0;   1  !          (6)(10)       . =         ,      " ,        &   " - 1-         -     .-       1-      (         )m s.H            m +  p i        +1) +1) ,         &          . F      ,  (31) -      (6)(10).%   (13)    (31)   C m,C +1)  ,  (31)!            +1) -   @   %  ,  (31) !      -    (6)(10).     (31)       (4),    -        '   .            (31)         -             ,     !   %    (19),        ,    (31)C )=exp( )exp( )=exp[             113    ,         )=exp(    (14),(18). %    "         ,   ,  (33)  [ .(146)].=     (33),        )=exp[ /  (14),(33)[ (14),(35)]        "         .              (6)(10), - !     C p i     ,        &  -  " C a,b]=0,[a,Sb]=iabcc,[Sa,Sb]=iabcSc.(37)/  (36)     "        -    C   , -         )=exp[  (14),(18),a   ,         " C )=exp( )exp( "   # . B          "  ,    (30). I ,      C "  (21),       .%(30) ,  -     !  I,P,C,T,PC,PT,CT,CPT ,        CPT , .?    !        C �  CPT CPT CPT CPT CPT CPT CPT CPT CPT CPT 114.%./ ,;.".   #      (30)       - -          (32).B            ,          -  1-  (  )."         1-      (31), !           ,         -  %(29),(30) ,       -           P,PP P,PM MP, P,PG T,TP T,TG T,TM MT, C,CP C,CG C,CM MC. %(30)  ,        ,CP PC,CT TC,PT TP.         ,          (30),   (46)        [27] =exp( =exp( =exp( ,CT ,PT        ,    %(40)(45) ,           (13)(  ,    ),!   -                       (6)(10)."         -       (31)(                      ,   %(40)(45),(48) ,        P,Pp P,Pt tP, T,Tp T,Tt tT, C,Cp C,Ct tC,             115       .$  -  ,   (49),        . %(40)(45),(48) (49) ,         C =exp(      C D  ,                  (51)(54)   B   (51)(54)    .  2. &4("2"*' "")!  ","*/""5(71)(77),4!-  " $2 s, ,# ,..., 8"""*/  "&  " : 000 000 000 000 000 000 116.%./ ,;.".   000 000 000 000 # "!#!"  #!! +1) -2   ";  ",! "2"*&  "5[39] $   . B     (51) (52).%   '      ,      (51)       ,           (51),(52)    ,      . B      (51)(53).D              (60),  ,          (51)(53)     (60) (61)C  ,        (56),(57),          !     =3)          (51)(54) (56)(59). ? ,         -                117 /  (46),(52)(54) (  !   -  )       ,     (40)(45), (47),(48).%(40)(45),(47),(48) , :  , !!"!" !#!! - (0),"""*/ ,*"2  &  *" (3=),(�?),(70),(76)(7?),(6). $ ,       (   -  (14),(15),(26)        , (27) (28)), [19]. 2.$!!              !       -    -                       . %     . A  '   (1)     -  "          - .I  ,   ,                ,  ,   -  (1),     ,       -    .         . " 1.  (�6)# -  ",! " ""(3),# !"! " ,"*/ #$(6)(10). #     (64)     +1) - -    -     C ?       , !   - . (J)      C   (  !     )   , -           (2),     "   (36), 118.%./ ,;.".              +1) -   C +1) -      @     -     . /  (36),(66)  (  !     )     " ,          (38)   (65).   ,       , -  (64)             ,   (50) (56).   (36),(64)(2), ,  (64)           ,       -   HJs,Pa]=[HJs,Ja]=0,(68)[HJs,Ga]=iPa,(69)   (36).  (JJ)       -      , ,        (6)(10).?     (67)@     (    )@  (66). F ,  (70)  (  !  -    )     ,      -   (13) +1) ,    (60)        ,   (50), (56)  ,  (70) !     -         (32)( +1)   (65)).:       (36)(70)   ,     !      (32),!                             119               ,       (71),      -     .H     .A  -             (38)   ,         �!     , !    (         (13))    . % ,    -     (64)  -           (68),(69)    (36),(66)   (6)(10)   (70). & #    B   1   -  .  3. &4("2"*' "")# "  ,"*/"# " (6),(66)"  "5(6?),(6=),&  *"  iak m C H s a ( m + S 1 !&!  ". $   . $           ,  .              -    =exp[( ]=1+(   (76),(77)    (36)    - (31). %(68),(69) (76)      A,A   (67)@        !  . D  ,    (76)  (64)   C  p 2 m +    V  , 120.%./ ,;.".       -       (79)     , (76),(77).  ,  (78)       C      !  . =   , !              =      C /  (83)  (80) .: (84)  -          " (       ) !    . B    (85).  ,  -   d,V =cos ,c =sin , arctg 2  a 3 a     (78)  (81).I  ,       , =1+( 1 2 ( f    (82)     (80). $  (86) (88)                 121   (66),   !  ,  =exp(2 ,  (87).   ,   - ,     (89)   (80)  (81).    (78),(80),(81)   , (76), -     (73),(74).$  (73),(74), ,     (68),(69).* ,!       -     (36),(66),(68),(69)     !  -    /t ,   (88),      (89)   (36),(66).% ,   . D  ,    -          (64),(73),(74).%  !         " (14),(38)    .=  -  ,     ( .(66))  ,  (73),(74)     )exp[ )]=    ,  (73),(74) %(91) , (38)     ,          (  !      (76)),-  *  (13),     (36),    -  C +1) .% ,  (64),(73), (74)               -      ,    !      !  . /  (73),(74)     -            .�    -         ,      ,  (64),(73),(74)     -       ,                          a,k a,k 122.%./ ,;.".   !  a,k a,k  a,k   ,(56),(59).       ,   (92), -  (64),(73),(74)      ,  ,   ! C ,    -  .$ , ,!            (50)   . :   1  (64),(73),(74)        &  '   , !         (36)    m t  P x 3 i + 1  A = t P 1 $   (36) (93)    (6)(11). "  (73),(74)   (36),(66)!         (32).$   (73)!   (71),-       =1+[  S * ,        (92), -  (73),(74)!       ,  (71)   C  a,k a,a  a,k I   (94),  (64),(73),(74)    -    .=   ,               + a  k       1 m        (73),        + m   a +2   S  1 m    (96)             123      (74),  ,  &  �     (95),(96)     (64),(73),(74)  .=         ,     (73),(74)  -          . * ,   ,  (95),(96)          ,      -  " (38),(66), ,  ,                        & #          - ,         (70)   (6)(10),  4. &4("2"*' "")- ! " ,'""(3)"*/- (6)(10),(90),(93),&  *"  m ( S p ) + + 2sin S p a m + b S p ) =sin2 =sin2 sin2 sin2 / 2 1 1 ,s� !&""!  ". $   . :  ,      -     .=   ,  ,  -       -  .D (72) C                    +1) -   (67)C m + c p + d S p )         !  .%  -        [16,17] ,r,r ,...,s, 124.%./ ,;.".                (100)    rpc $  (101), ,    (68). ,    (77).  (101)(72),             , ,           C $          (102)(   -    !     )    2sin c c d ( c b  (98).F ,  (103)     (104)(100),     (97).=               ,    (70)     (10),(57).  -  ,    U,  E  2 E   m + p m             125    ,     (97)   (70)C E,U D  . D  ,              -  (97).A  (64)      -    &   " (70), ,      &  '   , !       (12),(70).�           -   ,   (50) ?    (67).F ,  (64),(97) -        (50),   ,       ,       -      . $ ,    (64),(73) (64),(97)    C )( )( )=(1+   =  . A  (109),(110)         =   -       ,  , ,    -       ,      -       " . =                   ,    F !     " K [28]    (149),(150).    (64),(73)    C   p m  + p  a m p i p p m  = p 126.%./ ,;.".        C   .   p m   = 2   p p ( p  )  ,  i  p      .A  (112)  (64),(73),  !               . 3.'    ?               -       !   ,..       -  C ,n     . %         (114),            & [28],               ,     " .      -                       !  .: -         ,  ,     &+",   -             - . "        . B    -     (114)     ,    -   !            " . $      '   (1), ,   (114)     " ,       (2),                       127   " . ,   "        (114)    -   (36),         (!   ! -     "       (38). D               . " 2.  (11�) -  """#!!  ,! " (11�)"" # " 5#!! !" - (6): ) ",$ &*/!" ($/2  ' ") #$%#!! (9).     (116)   (114)(115),              (114)    -        ,     (37). D  ,    -         (114)        (37),(117). $            ,    -    .%       , -     ,  (114)  -                     .= !      -   ,      !     - -    (114).  2. +" !" ","*/ (9),(119). #  " !& 4  " ," 4"*" - (9),(119). =           (117)          :   2,        (117)  -     ,        "  128.%./ ,;.".         .  ,  (114)      .$-      ,    (114)(   ,        ),               .%        ,   2 (121) ,    !   (     (120) ).  -                 -   (1+ (1+     (116),        (117)C   (123)(117)     !      -      % ,  ,    -   -   (114),       ,!         (37),(124)(129)   :    ,      -     3. +" " "*""5(9), (13�)(13=). # !" " ,# !&  " ,"*/  V,S V,             129 " 4"" (9),(13 ,-3;I.5;ጶ&#x.300;)(13=). $   . A      (124)(129)    .  , (131),     (124)(129)   ,    (130). I     ,   !          (124)(129).#    !     -     !     ,   (130),(131).   !    #(114).         (37),(124)(129),        (114),    ,     +    ( .[29]),         (114) (    )   (  - ) (      ,  -    (114)          (            ).= !    (144)  -   , =exp( /   , ,     LV,    (114). %    *! +   A,B A,B A{A,B A,B     (124)(128), C , V =  1  1     (136)(133),         C % ,     -        (114)         (137) 130.%./ ,;.".   (   ;+"[57]!     [11]).$  -  (114) (137)  !                !     (..  , 4-  -  !     ). A  (137)        -  " ."   "      (137)      ,   (31),    -          ,      %      (137),       -  ,      .; , ,       m  +   = c  p p        @    =0) A  (138),  (114)       - .          #  .         (37),(124)(129). B       (37),     &    " ,     -   I  .* -     (37),..        (37),    !  . "       ,!     -        .$              &     [30].$   - , ,   ,-  ,   -   .* ,   [30]       !    .                 (37). ;  (37)                   -  ,..             . ;  (37)    ,   .  -  !              B           (37),          !       .!      0 0  ,             131             .. ,       Sa,Sb]=iabcsc,sasa=s(s+1) a,b]=iabcc,aa=s(s+1)              ,   (    )  -   .$      ,          -           ,         (139).$            .  5. &4("2"*' "") &4 2!"  #$ (9),*2 */$ ' " !" ! #$ "*/!- ""$ *,2"$  "! " (1=)$-  "#  $" !!" ",4! "  $ 2 n,m, ,# ,nm 8"""*/  " &  "  s  a  S K K a  "(1�1);  " &" +1) ,!"5  " "2'"  anm1]ij=i1,j,n m,i,j, ,nm, anm2]ij=(2s1)1/2i2,j,n�m,i n,j +1) ,nm, anm3]ij=(2s1)1/2i,j,n�m,i m,j +1) ,nm 132.%./ ,;.".   !&!  ". +"  n,m, =1) n,m, =2) ' """- "2 , $   .        .$  -  ,  (143)      , -        a,Sa]=iabcc   (141).-  ,  (143)      ,     [7]C +1) +1)          +(2 +1) +1) +1) A  ,    (139)     -       ,      ,       .  !         (147)     (142),(145). !     n,m    -              s,s ,        [7]C s,s +1) s,s s,s s,s s,s s,s s,s s,s s,s s,s s,s s,s s,s  ,...,s,a  +1) s 1 s 1  s s , C 1 s 2 1  +1)           (148),(149)             133 & #      (37),(124)(129)      ,         -    (37),  5. B          (37),    ,         (139). ,-        (138). B        .  6. &4("2"*' "")  ,"*/"5(13�)(13=),(1?)(1&#x?14;.60;0)( "7,&  *" (170)(17):  2  ,  i S 1 a  S c 1    S S   ,  1 s S =(2  d 2  ,  i d 1 a 1    S  ,  1 s a =(2 +1)    a  1 s 1 s 1     ,  i ( s a a S c ( s     , 134.%./ ,;.".   S    1 s cK a 1    ;  1     +1) +1)  1  1  1     ,  i dsK a +1) +1)      , S S     1 s a 1 a #  $& 2 " &" +1) +1) +1) ,!"5(1��),(1�6), +1) -2 ", "2   """"*/ &"; !&2 . $   . %   '  ,    ,      (124),(125),(129),(141),     1  x I  )  !        ,       (126).%  (143), (154),          (154) (143)(127),   A  S K a B  K              135  ,  (143),(154),(156)(128),       C  (158)    (144),(146)       !    !     ,      1)+ =2(2 +1)(2 +1)( +1)(2 =2(2 +1)(      (155),(159)    -      !  .B !    ,   (145),    - ,    6. F   !       -      (114),      (150)(153).I -    (150),    ,   (114)                  .D   !     +" [6,7],      & -& [5]. A  (114),(151)      -      .  ,!      -      ,       " . 136.%./ ,;.".   A  (114)          ,  -  (152) (153).�                           .* -   , !   (    &+") -    -     -             !     (     -      ).       (124)(129)  ,        (37)   ,      -    (139).D            . !      (154),    !    C x s ( x 1 x s +1) %    (154),(160)   (143),(145),(124)(128), -             (114),(150)(153)   -  C    @     . *  (163)(165)  (     )  (114)   (151),(150) (152)  .A  (114),(152) ( ,,   (163))         p      a   * =! . '   (114)   #      -   .    (  !     ) -    (124)(129)       (37),               137  5.$ !               (37),    ,      -          ,        A   ,   "          (            F   ), ,  ,   )      ,      -      B          C      @   -    .  ,                C diag .I   (165),  (167)   -    *     ,!   (167)   -   " .=   , (164)  (2 (2   1   x   x  1   x      C (   x  p i   x 1 "       (168)     - ,            - " C  x p i  i    1 ( S a S a . 138.%./ ,;.".   $    (170)    (2).       (168)  -    -     ( (   (168)  -     (114), I,m    . D  ,          (165)           (114),       (169),(171).        (165),          ,     -    [28],   .         (165),    [31].$         -     .D       (165).  (175)(169) (171)  -     I. :  ,         -      -    (114),(173)."    "       (114),(173)    -   (36), /, *       ,     -     ,    ( ),      [28].I   (173)        &   ,  (114),(173)!      & -& [5]            .   -        (114),(152)       . ! (172)    * =! .    (114),(173)              - ,           ,           F , (174)             ,    (114), (173).D  ,                      139   (37)               4.         * +     A                          ,      -      ,        . $           ,   ,   !     .   ,                ,  -        -               ,..      ! ,          -  =  .D          -       . '        , #  /7#  * +    .     '    (64),(73),(74) (64),(97),                   .=       -    !     ,  !      4-  -  !     .             ,  (73),(74) (97) (176)C iak  ( S  ) e S H  , H s  a ( m S  ( 2  ( S  ) e S H  +  m +  m +  m ( S  ) e S H  # a m + b ( S  ) e S H  # + 2sin S  . (180) 140.%./ ,;.".     (178)(180)         -  C A  (177)(180), ,           C )exp[  ,  (177),(178) (177),(179)          " (14),(38)   -    -     [5]       (17).%           ,             (14),(38),(182), -     (38)     ,        ,        (14),(182). A          (185)        ,   [ .(91)]. " 3.  (1?)  """!$ &  &#!! (1�),(?),! " "" if(t,x)D(,v)L()D1(,v)exp[ )]=  &4"(1?); )=exp )=exp( )exp(  ", &*/!" ($/2  ' ") #$%#!! (9). I   (185),(184)    (14),(38),(182) (183). %      if(t,x)]aexp[if(t,x)]=   (90),    ,  (177)     (185),(186), (39),(66) ,  ,  (177),(178) (177),(179)  -  .             141 ;   (177)      ,   -  (180)(182)    ,..     *    =  ,  (178)(180) -      .  ,         (178),(179) , (180)   -  C eB + + m 1 S (  E E  Q 1 +1) div eBD S (  H H  ) 2 Q + o  1 ,  H s    m + o  1 , H   m +  m + e H  + + e m 1 S (  E E  ) 1 Q +1) div 2sin m  b s a S (  H H  e  b o  1  ,0]     !     @       C Sa,Sb]+2abs(s+1)     !  ,     -   I (179) ,       -  ,  (188). "  (188) (190)   ,    -        !      ,    ,  -    ,    div    /x   .=   (   -            )    -      -           . 142.%./ ,;.".   ;           ,                   [16,  ,    =        (177),(178) (177),(180)(   (188),(190))  -         ,  -  , ,              -       !   .I     (188) (190)    , -       =  [32]. !, , - (188) (190)      (       ),       =  ,          . $ ,  (177),(178) (177),(179)                (92). =   ,    (95),(96)    /x /x ,   [14]     (95),(96),a   C m  i   i  A  e  C  $ , L   A ( S + i m     A  e    ,  & �     (194) (196)  (177),(178) (177),(179).    ,     (178),(179) (180)  (188),(189) (190)  C =exp =exp =exp             143  2sin  , B s m 1 a [ S  ,  [2(  ) e S H S E # , C s 2 S,eSH]++[( ,eA m s B m a b s  ) e S H  2sin S E $ , C m  2sin % S  ,   e  S H &  4  2sin S  )  ) $ + i m  A s ( S  ,  B s k S E ,  C s m S  m  B "  (178)(180) (188)(190)     VHV      (178),(179) (180),      (197),(198) (199)  .      #      . $   (176)  (114),                     , -   , !     . $       (200),     -     - .   ,!    -          (181).%    (124)(128)     ,            (185)( D1(,v)]). ,  (200)     -    ,      C     (120). =      (200)                     144.%./ ,;.".    =exp      !          .%    (134)  -    (124)(128), C  E + e m , V =    m + 3 e m (  H H  )  + e   H , V  +  + 1   e m (  H H  )  + 2     , m  +  + e   E + 1   H  + + e  A  (203),(204)    (137) ,      ,     !    -  .  ,!     ,-        . A  (203),(204)      !     ,        (37)       , !       , -    ,         (139),  (204)  ,     !     .=   ,!    ,  (    (128)) ,   (205)(204),      m  +  + e I ,  !      ,  (206)       ,  ,  (203) -         !     .�         .             145  4. $ "#,2"$ (300),#  - ","*/ (13�)(13=),(9),! & - "! 2 "'"2!,"!"5-  F (150),(151)    (207).:  , -  (200),(150) (200),(151)(     &+")     (  -   ,    ..)  -     !     . =   ,  (200)   (152),(153)(    -   (207)  )       . B      (203),(204)  ,  -     (150)(153).$   +1) - , -   , -  (203) (204)   (150),       m + m  S H   1 D  ,  (150),(200)(  &+")       (208)  +1) -   = ,    (209),(151)(203),(206),  C  m  H   m g m S H + 1 :  ,  (151),(200)                 ,     ( -      &+")    (         ). B    ,       (152).$-     146.%./ ,;.".     (152),(211)(203),(204).          C m + e S H + e sam   S E + 1 m  S (  H H  )  + + e s (  S H ) #  a   1 m %   m a + e  S H +1)  % ,  (203),(204)   (152)     (212)  +1) -   (     -      (213)).=              (212)         ,         ,     !   -       C =exp =exp sam  L i S  %    *! +  (134)     -   [ S 1 i [ S  ,  , i [ S  ,  S (  H H  ) , i [ S  , S E 1 S (  E E  Q +1) div  C m + e H + e 1 S (  E E  1 Q +1) div Q a S (  H H  )  + o  1 $  (214),      (178)(180),    ,     ,     -             !     .             147 :    ,  (203),(204)  -  (153)       m e S H +1) +1)          =exp +1)       m + e S H +1) +1) S (  E E  1 Q +1) div Q b S (  H H  )  + o  1 $  (215)   (214)    ,     (208),(210), (214),(215),          (200),      (188)     !   .%  ,     (179),(180)        ,  (200),             :  ,   '   (177)    "         ,   (200),  -               ,  ,                   #    #  #8  .           -           !     .#   ,             -  ,   ,           .D                ,     !      .            -         H s   H s  s  ( A a B a 148.%./ ,;.".   L     + e         !      @   (178)@  (150),(151)@   (   ) ,     ,   (216) (217)     " . =    .  9. @! " ""*#    ""(1?7)"# ""# ,#  " !&2 ,  ",&    - (66). $   .      7   [14],- !    .A      (185)    -    C       (39),(66) 0,a]=RabEb(vH)a.(220)  (39),(66)(219) (220),        Bsa,Sb]=iabcBsc,[Asa,Sb]=iabcAsc,[a,Asb]=0,[a,Bsb]=iabcAsc,aAsbc+cAsba=aBsbc+cBsba=0.(221)$     (221)     (218).   (218)(216)          =exp  1 m )exp  , D = ( k m ( k 1 S H k ( k S E + 1 m S (  H H  )  + k 1 m [ S ,             149  [14]C m     (186)    :  (186) (223), ,             (177),(178)    -       (      ! -   ,          ,   !  (192) (223)                ). B     (217). ,   -          (187), D1(,v)],       (220),     - ,         k 2       .   (224)(217)    - (150),(151),          +1) -     +1) -   H  (1+ H H m [ S  , S H 2    (226)           1 U s U s U 1 i  =exp =exp +1)      (188), =(1+ s,D D  ,          ,    (150),(151) (217),      150.%./ ,;.".    !     (188),    -    (177),(178).:  ,  &+"(150),(217)  -  (150),(217),              ,    ,     -       .      #       . F-  ,    -    (177),(178),(200), (216),(217)       (   ! -  )          .= M  !      .  5. +" !&! ", & - /" (196)"*/*#   "- "(1?7). # ! " )=exp )exp " 4""*(1?7) $   .   (230)    if(t,x)]exp(iS),     if(t,x)]exp( .%   ,        *! +  , if(t,x)]exp =exp if(t,x)]exp( )]=exp( )exp if(t,x)]     ,        -   (185), C if(t,x)]exp( )exp( )exp[ )]= ..              (185) .& . %  ,     -    (183)      )=exp     ,     "             (34),(35) (!       " ,            ,                      151 B     (177),(178),(200),(218),(219)     (230),(231)   (232),     C m  + A + e B a e B a E 1 m (  H ) 1 m ( H  ) + e Q Q     ,     -         C A,Sa]=0,[Ba,Sb]=iabcBc,[Qab,Sc]=i(acdQbdbcdQad).(234)% ,         -   -      (232),(233),    -    !   (177),(178),(200),(218) I ,                    (191),  (232),(233)   (212).�     -      -     '   (1)  +1) -   ,       -      !     (        [32]).% ,             (152),(200)   !    -       '   .              . B    (216),(217)              .F  ,               ,..(217)    :  (240)  -     eHx   (178),(240)(216) (218)       ,      m + e S H  + 1 a(kS)2ek0SH],(237)      .     ,          .�        (235),       . 152.%./ ,;.".   %        1  1 i      (66), , C    ,   (238),          [34]C m +1) ,..., ,...,s,   ,    C =(2 m + p 3   (239) ,   ,              !      -      (  &).I  , , =[2 (1+ m + 0 3 o  1 /  (240)       ,          ,   ,         . H       (237),     , -    [17],  . : /                (177),(178),(200),(216)(218) (224)     -   "           ,   ,  -              -       !     .;        -      -        -  [10],     &+"  .-     ,  ,         !             -   . [38] [8,9]         -            .             153 A   (64) (114), ,    -       ,     " - .D ,                -   -        A  (241)         C +1)    *  (13)     (36). !     (36)         - (32)              " .;            ,                -    & ,      !      !              .!          ,     " .        (114),(173),       -                    ,    [37],  ,                !   -                ,        .�          [8][14]. % 1       * #           (6)(10)    - c,p, ,             p, p, p, p, p, p, %            ,      -     -     (6)(10). =      . 154.%./ ,;.".    . #$ %,! " "5  [   ,a,b,c !&""2;! & #$% #!! ,2  #$%#!! ! #$ %#!! $   .   ,   !     (.1)    ,       %  ,    ,    -     C   K  + K 1   1   K 1   + K 1     , )     ,.. ,         Sa,Sb]=iabcSc,[T1,T0]=iT2,[T2,T0]=iT1,[T1,T2]=0,[S01,S02]=iS12,[S01,S12]=iS02,[S02,S12]=iS01.(.5)      ,        (.5),  (.3),(.4) -   (.2),  , (.2)(.4)   (.5).&  . F ,  (.2)               &     .:     .             155  . +&'"!"  #$% 5#- !! (6)(10)4 & "!/*! " p   p p p + p p p G ( p )  p p  p p   ","*/" "5(+.3). $   .       , - (.6)       (6)(10),..         &   "  .  ,           (.2),-    (.6)      (6)(10),          (13).  (.6)(13),- m,C %     (.3),(.4),  (.7)      %(.8) ,          *       " ,  -    (   )    !         (6)(10).D  . D  ,!            (6)(10)    ,          (.6)C +1) %     [35]         (.3) (.4),         p, p, p, p,  p, p, p,  p, p, p, p, 156.%./ ,;.".           (!   (.8)). ! ...,s,   1  +1)   1  +1) ,a ,a  i ) 1 r  1           . /  (.1),(.6),(.9)(.11)(    ) -       "      -      .    J  (           -      ,   !   )  -   [4].D     JJ     (6)(10),             .D-                    .    JJJ    -     .    !    &    " -      . $       (.6)                     (    [4]                 ).                  ( ! :    (.6)    ,  [4],  -   =exp arctg =exp arctg      " J JJ    -  ,[4],     (.6). $ ,  (.2),(.6)       JV  - ,    .�     !  ,  -  !           .$   ,               157        C       (.2),(.6) ,!   -      !                      . % 2 "      * #   7 # %  B  "      -               F -  .�,   ,               ,     . ;  &       , ),        [31] P,P]=0,[P,J]=i(gPgP),[J,J]=i(gJ+gJgJgJ),(.12)    @  (.12)   ( P P ,  P P J J 1  a 1     (  (.12))C P0,Pa]=[P0,M]=[Pa,M]=[Pa,Pb]=0,[P0,Ja]=[M,J G+a,G+b]=[M,G Pa,Jb]=iabcPc,[Pa,G+b]=iabM,[Ja,Jb]=iabcJc,[P0,G+b]=iPb,(.14)[P0,Ga]=[Ga,Gb]=0,[Ga,M]=iPa,[Ga,Jb]=iabcGc,[Ga,Pb]=iabP0,[Ga,G+b]=i(abcJc+abK),[P0,K]=iP0,[Pa,K]=[Ja,K]=0,[M,K iM, 158.%./ ,;.".   *    (.13)  (6)(10),..     &   "  . %    ,   ,    -   ,        " . D ,  ,    * "  (.13)   ,      "  A  (.16)        '      -    .  [36]             -          ,..      ,           -     ,      , -            - ,     *  (13)  . 1.Jn uI.,WignerI.B., NuovoCimento ,1952, ,705. 2.NargmanV., nn.Math. ,1954, ,1. 3.LieS.,OngelP.,TheoriederTransformationsgruppen,Nd.2,Leipzig,1890. 4.Levi-LeblondQ.-R., J.Math.Phys. ,1963, ,776. 5.Levi-LeblondQ.-R., Commun.Math.Phys. ,1967, ,[email protected], ,[email protected] Xpplications,ed.I.R.Loebl,Vol.2,Y.Z.Lond.,Xcademic\ress,1971. 6.^agen:._.,^urleyW.Q.,             159 22.^agen:._., Phys.Rev.D ,1972, ,377. 23.YiedererU., Helv.Phys.Bcta ,1972, ,802. 24./ .%.,:`.^., E. GGGK ,1977, ,801. 25.  ;."., .., .&.4. ,1980, ,618. 26.PushchychW.J.,YikitinX.G.,

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