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By N. N. Bogolyubov

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62) where the averages are of the products treated earlier. Thus, of all the products ßh · · · ßf. e. with all creation operators α^" to the left and all annihilation operators onf to the right) we shall only have to consider products «Λ · · · αΛαΛ' · · · <*/*' in which each of the indices fv . . , fk is equal to one of the indices / / , .. ,Λ'. B u t s u c h products can obviously be reduced to the form We put ±«Λ · · · αΛαΛ · · · α/*· (Vk = «£ . . OfiO/i . . 63) and note (all the indices^, ..

38 A METHOD FOR STUDYING MODEL HAMILTONIANS We shall move the operator (Ja—CJ successively to the left. We find D. -CJßA . . ßf,{Ji-Cl)ßZ ... ßj-)r, °'32) 7= 1 where the operator Bj is obtained if the operator ßf in the product ßh ■ ■ ■ ßf. 34) To calculate this difference, we shall examine two separate cases: We have α//α-Λα^ = ßf, = °% and ßf, = «ft· ^YtKWfafflfatf-afatf*/} = 2V Σ W){[«//*/ +β/a/Jöiz-a/Iaz/iiz + ei/a/J}. 35) the only terms which remain are those in which f = fj or —/ = /). 34): t Everywhere below, | .

53) for the (/) under consideration, where γ is a certain positive constant. 53) is always satisfied outside the spherical layer: Pi- μ 2m ^Α. 50). 45) that Ε-2γ/3 — oo Therefore, we obtain Hence, taking into account that (BB++B+B)r^l, JB+iB((o)do>. „ + £). 53) is fulfilled, we can also put x = -^{2Mi+2{ifM]m- (L57) This latter value of DC clearly no longer depends in any way on the temperature 0. 39). § 6. PROOF OF THE CLOSENESS OF AVERAGES CONSTRUC­ TED ON THE BASIS OF MODEL AND TRIAL HAMILTONIANS FOR "NORMAL" ORDERING OF THE OPERATORS IN THE AVERAGES We shall first of all examine an operator product of the form u = αΛ · · · α£ αΛ' .

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