~E(A\2) = X2 ( d f(/m)- d2f(i1i2/m2), [3]
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1 Proc. Nat. Acad. Sci. USA Vol. 71, No. 12, pp , December 1974 Photon Propagation Function: A Comparison of Asymptotic Functions (electrodynamics/source they/high energy/spectral fms/field they) JULIAN SCHWINGER University of Califnia, Los Angeles, Calif Contributed by Julian Schwinger, October 3, 1974 ABSTRACT An earlier paper asserts the logarithmic asymptotic identity of various asymptotic functions associated with the photon propagation function, including the spectral weight function the GellMann Low function. This note exhibits explicit expressions f the deviations between pairs of functions. It is emphasized that the differences are quantitatively small, that the various functions have common qualitative characteristics. The discussion refers to a particular method f comparing the functions, which has no special physical status. When that restriction is removed, it becomes possible to prove the general existence of a transfmation of variable that will produce the GellMannLow function from the spectral weight function. This supplies a physical interpretation that has otherwise been missing in the GellMannLow approach. A note published recently in THESE PROCEEDINGS (1) gave a brief discussion of the asymptotic fm of the photon propagation function D(k). F reference convenience we first repeat some significant equations of that publication, hencefth referred to as publication I: k2e2d(k) ex2 1 k2ex2 ( dm2 8 Jx M2 k2 + M2 1 1 (VXdM2 ex2 e2 Jam' M2 8 ( e2 FdM2 k2e2d(k) ex2 exp k2 ji dm2 t X2 dm2k2a 2 e2 exp 2[T 11I M 2 ]pe) 4, (m2 e [I 461, on introducing the new variables To these we add the following immediate consequences of the above equations by M2 d dm2 em2 d M2 em2 dm2 1 em2 oa(em2), M2(eM2), O e Abbreviation: GML, GellAMannLow. [I8] [I7] with their integrals: M2 rw2 dx log ~2( /~m2 XJ e2 Xa(X) log M2 Me~2 ~m2 e dx X Z [122] [148] It was remarked in publication I that these two definitions of ex2, together with the GellMannLow (GAL) definition, k2e2d(k)1k2 X2>0 ex2, [1] are all asymptotically equivalent, leading to an asymptotic identity of the three functions o(x), (l/r) cb(x), t(x), the latter being the GML function* defined analogously to [I21], _ dx2 2 (e>2). [2] The essential basis f this remark was the "logarithmically valid approximation" X2 f1: M2< 2 X2 + M12 0: M2 > X2 [I 50] implying sufficiently large values of log(m2/m2) log(x2/ m2). This context should leave one in no doubt that the cited equivalence of the three functions is an approximate one, [I 13] involving errs that are measured by the smallness of 1/log (X2/m2). However, the absence of any quantitative qualification of the relationships among the three asymptotic functions leaves room f misundersting, which it is the intention of [144] this note to remove. The err associated with the approximation of [I 50] is measured by [145] [121] [I47] 5047 ~E(A\2) X2 ( d f(/m) d2f(i1i2/m2), [3] Jo M2X+MI Jo I2l/2 y log(a2/m2), z log(m12/m2), F(z) f(m11'2/m2), [4] E(y) X dz F(z) _ v dzf(z). 1 ez1 _0 The question thus posed is quite familiar in applications of FermiDirac statistics, but, f completeness, we shall supply * Other, related definitions are commonly used. We prefer here to emphasize the connection between V,(x) cr(x). [5]
2 5048 Physics: Schwinger our own discussion. It is convenient to first differentiate Eq. [5], thereby obtaining d co1' E(y) I dz F(z) dy J, 1+ ezy)2 F (y) J dt (1 + F(y + t) F(y). [6] The "logarithmically valid approximation" is a good one if F(y) changes relatively little f displacements of y of der unity, since e t e t (1 + et)2 (1 + et)2 decreases exponentially f large It, I dt +e 2t' dt() 1 1. J (1+(1+t)2Jodte' + 1 To obtain an err estimate, we exp F(y + t) in powers of t, where only even powers contribute, in view of Eq. [7 ]: d yf)) (y) o t2ni dy E~) n (2n 1)! te + 1 here we have used the partial integration illustrated in Eq. [8] introduced a notation f multiple derivatives of F(y). With the aid of the integral representation of the Bernoulli numbers (B1 1/6, B2 1/30,...) cit we find immediately that (2,)",n Io t2n1 B et 1 4n f cdt ~+Ī (22nI B_,r21 n E(y) (22n_1)r2r nl (2n 1)! 2n [7] [8] [9] [10] [11] F'(y) +..., [12] 6 which incpates the fact that E(y) vanishes as F(y) approaches a constant. To assist in comparing the three functions, we introduce the variable t 1/eX2 [13] define ao(d,) a(exl), 02(02) 1(exl)/ex, 93(Q3),P(ex2), [14] where the subscripts on the t variables are a reminder that three different meanings are ascribed to these variables f given X2. A unified presentation of the various relations (Eqs. [I 21], [I 47], [2]) is now given by d X2 ta aa(ta), a 1,2,3. logx2 [15] If we accept that each t can be expressed as a function of any Proc. Nat. Acad. Sci. USA 71 (1974) other I, the various a functions can be connected by qa(wa) Ob(6) a a({) [1 + [a] 116] The latter fm emphasizes that, to the extent the various differences (a Ēb are small slowly varying, the related functions oa ab agree quantitatively. It also undersces the qualitative similarity of the a functions: where one is positive, all are positive, where one vanishes, all vanish, at the cresponding values of t. Another way of presenting [16], which is based explicitly on the smallness of {a emerges from the expansion Ob(ta) ' b (W) + 66 ( a W) [17] since it permits us to compare different functions at a common value of their arguments. Thus, we have d(ada, Sb)_ do(t) ( a i) [18] f a aa(t) ab(t) ' WO) 2d [ ] [19] where the appearance of t a(r) indicates the complete neglect of a,, e We first illustrate this general discussion by comparing 1 *2 d 2 [20] e2 Jo M2 with p;3 2r2 dll2 e2 Jo M2 X2+ M2 Using just the leading term of Eq. [12], we find that, 2 d [X d log(x2/m2) [m~2] w2 ci.2 d ( 2 6 d ~ 12d from which is derived (Eq. [16]) (Eq. [19]) Ui(~~i) ~.2 d2 0U3(t3) i d ( ())2 (())2 t 0(0) Next we use the phase representation [I 45] to get t3 exp 2 X 2 2 [21] [22] [23] [24] 1 fx dm2 0t][251 The argument of this exponential function is approximated by dc 1 FX2 1 4 I, 2 d e2i 0,( at) 6 d log(x2/m2) Lm2 J 6 a(1[2) ~ [6
3 Proc. Nat. Acad. Sci. USA 71 (1974) Propagation Function 5049 thus t 262t d vr(t) _ K2 d (t) 2 6 A~~ 12 A kt which combines with Eq. [221 to produce T2 ( (t))2 41t 6 The immediate consequences of Eq. [271 are O3(43) a2(42) 0( Fd2 d _(a _2 12(t) <)2 d(t) [29] 12 Ldt2 0a3(t) a2(t) (a(t))2d d ] T 2 d ~Fd2 r~ 6 [(2 2 (O ] [30] while Eq. [28] can also be used directly to give al(s) 02(5) 6(? 2 6 d [27] 62 d () [31] 2 [32] 18 d~ ~ We now compare these relations with the only hard evidence that is available. It refers to the purely electrodynamic situation of spin 1/2 charged particles gives the initial terms of a power series in a e2/47r f the asymptotic fm of the inverse propagation function, as stated by k2e2 dda s( k2 \+ a2 ±f k wr rn2 3 4r.2 k2 4~( ) + a' lgk 2 ( log m2 6) 247ra [ m2) + 3) +47 2) 4 log /2 +(8~ y) lgm+.* * + 108t4 [log2) 47 Ic F2 I2 + (12 (3) 4) (log 2) +..]. [33] Here, t(3) is the zeta function of the indicated argument dots appear where a constant the coefficient of a logk2/m2 term is not yet evaluated. The use of the GML definition of Eq. [1] thus gives e2 a X )2 5\ ex2 3wr m2 3J a2 ( 2 42 \log+ 4 (3) )* * [34] with only the leading terms exhibited, then Eq. [2 ] yields 4wV'(ex2) ±+ a + 2 (log m +4(3) _ ) 3T l2ta\ 4T2 ) 2 + F/3 X \2 / 5 ~ log i. 35 ±36w4 m(lo 2~ + The elimination of log(x2/m2), thereby of e2, finally [28] produces the first terms of a power series in x ex2 f the function 4r k(x): 1 1 fx\ 4 r V,(x) 3r+ 4r 4r 3w 4+2\4 ( + 1 ~~(3) l x 2) 3xwI +..., [36] 96 ) which can also be presented as a function of t 1/x. Instead of the spacelike choice of k2 used in the GML definition, we now let k2 be timelike [k2 A2 jo, log (k2/m2) log(a2/m2) ri] take the imaginary part of Eq. [33 ] to obtain 4rs ( e2 47r (ex2) a3 4 \m2 /108w4 [37] since extracting the imaginary part is equivalent to differentiation, except f the third power of log(k2/m2). The additional term of der a3 in [37] is indeed small compared to those already present in 47r 1(eX2), as measured by [log(a2/m2) ] 2 << 1. We can restate Eq. [37] as 1 (x )' V~'(X3) 'LT(XI) 6 w \4w [38] which is to be compared with Eq. [23], where the righth side is evaluated by the appropriate leading terms in the x power series expansion (Eq. [36]) w2 1 d c73(53) u,(5t) 12 12w2d2 6w3 (47r) I w2 1(1'\~ [39] 6 wr5 k4w5/ This agreement justifies the cresponding approximate application of the other general relationships. Thus, Eq. [22] asserts that 5l 650, X Xi while Eq. [24] consistently yields w2d dt 6w3 (4w) I [40] 11/1 /X 4 18 t4 (2 (12 d2 1 1 ~() 6 ~12w2} d~16w 34w4 [41] T(4 ) [42] coinciding with [39] in its x3 dependence. We note that, to the extent that t(x) is known (Eq. [36]), it agrees with o(x). As f the phase functionj (M2/m2, e2), it can be obtained
4 5050 Physics: Schwinger Proc. Nat. Acad. Sci. USA 71 (1974) from the relation i f dm2 M2 k2 + M2 o 11[43] [1 k2e2 f dm2 Ll2 k2 + M2 by taking the imaginary part, with k2 M2 _ O. Crect to der a3, we get 1e2xi [44] 7 3 \3T/ 1 ~~~~~~ 1 a (X2) Xla(Xl) 4 ) [4$] ir~ 4/ we observe that the additional term of der a3 is small, of der [log(x2/m2) ] 2 in comparison with the a' term ija the expansion of e2xi s. To compare with the general relations, we first note frqm Eq. [28] that which enables us to convert [45] into ;6 (+1)2) [46] 2r x \ XI X2 27 4ir ' [47] b(x) xo(z) 147) That is also the inference of Eq. [31]. The example just considered refers to small values of ex2. Although nothing is known about large values of x shall illustrate the possibilities by using the particular fun( ction of [I 36], which we state as «<K 1: v(t) Ct2el/. The dominant terms in Eqs. [22], [23], [24], are ther tained as 1 3 C22e2/t 6 0~3~3),~e/ 3 l(t) 0a3() C 3/r which differ unifmly from the Eqs. [501 by an additional fact of t on the righth side. We have been making the simple point that the various asymptotic functions differ only slightly under the asymptotic circumstances expressed by large values of log(x2/m2). But one can go further by recognizing that the comparison need not be carried out at a common value of X2, which, after all, has a quite different physical meaning in a A. That freedom could be exploited to ensure, f example, that 0a(ta(Xa2)) 0b(tb(Xb2)). [52] We now make this possibility explicit under the assumption that all the required changes are relatively small. Beginning with aa(ga(xa2)) d (Xa2) d logxb2 d d log Xa2 d 10gX a2 d logxb2 X [(b(xb2) + (Ga(Xa2) {b(xb2))], [53] we get, approximately, [d log(x~2/]{ 0ua(~(Xa2)) L + dlog(xb2/ Jy ab(6b(x)) d d1x2a 4b + logx2 L' a log Xa2J X the imposition of [52] then yields Xa2 Xb2 (d(t)) 1 d (a tb). The latter can also be presented as [48] r [ a(xa2) lb(xb2) d/d; (Ga ) v((do(t)\ 1 X ~~~~~df ( 0a(t) Ob( we/e which shows that [49] Ora(Ga\ a2)) ab (Ga(Xa2) + (d{) (ra(o) ab(o) ) [501 Orb(0) Ua + (d ) (0b() aa()) The outcome, which is obviously valid under the assumption of small changes, describes how to produce one a function from another by an appropriate redefinition of the variable. If we take a, a as the stard function, the others are obtained as 7) 2 d /\a\ (e n18 ds lj We note that a,(~ 3i) is small in comparison with a (~) by a fact proptional to exp (2/s)], which is again of the der [log(x2/m2)]2 << 1. The similar use of Eqs. [28], [31], [32] gives 1 22 C26ae2/k ~~I~ ~ ~ 7 al(t1) o2(42) (~ C3~e 3/t [51] (( 2 6 d \ 6 dot [54] [55] [56] [57] [58] [59] in which we have simplified the writing of derivatives by introducing a' daidt, by regarding a, rather than i, as the independent variable. The preceding discussion, based as it is on the assumption of small differences, will not persuade anyone who wants to question the crectness of [52]. We therefe present a generally valid proof f the most imptant pair of functions, a
5 Proc. Nat. Acad. Sci. USA 71 (1974) Propagation Function 5051 A. Let us return to Eqs. [15] [21] to get d2 2 Mo X2a + M2 0'3(~2 a3(2(2)) 2)) 2 A. 2 f dm2 8 Go X2 +M2 m2 dm2(,(2 + M2) 2 ~ M2 e) In the asymptotic limit characterized by m2/x2 << 1, the spectral weight function s(mi2/m2, e2) can be replaced by the asymptotic function (t(m2)). Then the introduction of the variables [41 will convert the righth side of Eq. [60] into rodz ezv I:.,..o (1 + e2z)2 () dt (1 + e1)2 At this point we have only to apply a simple mean value theem based upon the positiveness of the functions entering [611 on a hypothesis of continuity, asserting that any value lying between the extremes of a can be realized f some value of its argument. The conclusion is that numbers I 2 exist such that Eq. [611 becomes 0r(Y+ t) dt (+et)2 () [62] which, in another notation, is the content of [521 f this pair of functions. We have thus shown that VP(x) 0(x4(), [631 where the function x(x) is only explicitly known f the situation of pure electrodynamics, under the circumstances x << 1: 2.t (X) X x I 27 4r [64] I am grateful to Kimball Milton f telling me about his own effts on this subject, in which some of the same results are derived using GML methods. This wk was suppted in part by the National Science Foundation. 1. Schwinger, J. (1974) "Photon propagation function: Spectral analysis of its asymptotic fm," Proc. Nat. Acad. Sci. USA 71,
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