We will define a class A of such functions whose asymptotic behavior for high p

We will define a class a of such functions whose

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We will define a class A of such functions, whose asymptotic behavior for high p may be specified in a certain manner, by means of cer- tain "asymptotic coefficients. " (The integrands of covariant perturbation theory, constructed according to the Feynman rules, with the I real variables taken as all Furthermore, it is obvious that for any Li, L2, the vector Lizi+L2 will not be orthogonal to V or V' for sufficiently large g~, so that for qj large enough, n(Ligi+L, ) =— 3. (21) What is not obvious, and indeed is not contained in (7), is that we can find numbers bi(Li, L2, W), b~(L, , L2, W), and 3E(Li, L2, W), such that (12) holds, or alternatively, by comparison with (19) that M(Liyi+L2, W) ~& M(Li)Lg, W)gi iL», b(Liin+L2, W) ~& b2(Li)L2, W), (22) n(Ligi+L, ) = 3. for all qi& bi(Li, L2, W). The statement (7) alone would of course allow us to determine the convergence of Z(p'); however we need (11) for alternatively (22)) to determine its asymptotic behavior. The proof that the asymptotic behavior (4) of Z(p') can be directly obtained from (8) and (11) alone, with- out knowing any other properties of the function f(p', p"), will be reserved until the next section. We shall show there that (4) follows immediately in a simple application of the general asymptotic theorem. LSee (III-15). ]
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HIGH ENERGY BEHAVIOR IN QUANTUM FIELD THEORY components of all internal and external momenta, are shown in Sec. U to belong to A„, providing that all energy integration contours may be rotated up to the imaginary axis. ) The exact definition of the classes A„ and. of the "asymptotic coe%cients" is chosen in just such a way that we will be able to prove the asymptotic theorem, which says that if a function belongs to A „, any sufficiently convergent integral over k of its argu- ments belongs to A „~, and which provides a rule for calculating the convergence properties and asymptotic coeKcients of the integral in terms of the asymptotic coeKcients of the integrand. Definition The function f(P) is said to belong to the class A if to every subspace Sf R„ there corresponds a pair of coefTicients, a "power" u(S) and a "logarithmic power" p(S), and for any choice of 222~&22 independent vectors Ll L„and finite region W in R we have f(L~~" ~+L~ ~+" +~ L. +C) 0{sf a(L&)(lnrf1)S(L1)rf aif Ll, L2f)(inrf2)S(f Ll, Lsl) X ' ' 'g 0'(j LI, Lg ~ ~ L2rb)) (ln lP(( LI L2 ' Lm))4 '''nm ( if sf 1 .. rf tend independently to infinity with C confined to W. [Here n({L1 L, }) and p({L, L„}) are the asymptotic coeKcients associated with the subspace {Ll L„} spanned by the vectors Ll . L, . ] More pre- cisely, there exists a set of numbers b& - b )1 and M) 0 (depending on L, L and W but not of course on the 211 rf„), such that coe%cients of f(P) for P ~ ~ along typical directions in 5. In the special example given in Sec. II, we saw that n(R2)= 3, where R2 was the whole p', p" plane. Furthermore, this was also the value of n(L) for almost all vectors L in R2, the only exceptions being L (1, 0) (with n= 2) and L (1, 1) (with n= 1), as shown in Fig.
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