September 2, 2017

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By Yu B.J., Xu M.

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5) we introduce in place of {p1 , p2 , p3 , p4 } the new variables {p1 , p2 , p3 , m}, where m = E + εp4 . I. G. 17c) ˆ (G− a ) = i[εpa (∂/∂m) − P0 (∂/∂pa )] − ε(Sab pb + S4a κ)/m, where κ ≤ m < ∞, m0 = ε κ 2 /2m . 3). So we reach the following result: Theorem. The Hilbert space of the IR Dε (κ, j, τ ) of the P (1, 4) algebra, corresponding to P 2 = κ 2 > 0, is expanded into the direct integral of the subspaces, which correspond to the IR of the G(3) algebra with the following values of the invariant operators: C1 = κ 2 , C2 = m2 s(s + 1), C3 = εm, |κ| ≤ m < ∞, |j − τ | ≤ s ≤ j + τ .

15. , Teor. Mat. , 1976, 26, 202–220 (transl. Teor. and Math. , 1976, 26, 138–147). 16. , Ukr. Fiz. , 1975, 20, 1730–1732. 17. , Dokl. Akad. Nauk USSR, 1954, 94, 857–861. 18. , Dokl. Akad. Nauk USSR, 1954, 99, 737–739. I. Fushchych, Scientific Works 2000, Vol. 2, 47–54. И. Н. А. РЕДЧЕНКО Введение. В. Остроградский [1] обобщил вариационный принцип Гамильтона на случай, когда лагранжиан L зависит от обобщенных координат qi , обобщен(r) ных скоростей q˙i и высших производных q¨i , . . , qi , и одновременно решил задачу о приведении соответствующей системы уравнений dr ∂L ∂L d ∂L d2 ∂L − + 2 − · · · + (−1)r r (r) = 0, ∂qi dt ∂ q˙i dt ∂ q¨i dt ∂q i (1) N — число степеней свободы, к каноническому виду.

6) a new realisation: Pˆ0 = U1 Pˆ0 U1† = Pˆ0 , Pˆa = U1 Pˆa U1† = Pˆa , Ja = U1 Ja U1† = Ja , M = U1 M U1† = M, + † (G+ a ) = U1 Ga U1 = x4 pa − xa M − − † (G− a ) = U1 Ga U1 = K = U1 kU1† 1 2 εSab pb − S4a (E + κ + εp4 ) , E+κ εS pb − S4a (E + κ − εp4 ) −x4 pa − 2Pˆ0 xa − ab E+κ = −Pˆ0 x4 − εS4a pa /(E + κ), where xk = U1 xk U1† , Skl = U1 Skl U1† . 11) Reduction of the representations of the generalised Poincar´e algebra 39 Using the Hausdorf–Campbell formula ∞ exp(A)B exp(−A) = 1 {A, B}n , n!

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