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1 UC San Francisco UC San Francisco Previousl Published Works Title Radiative Corrections and Quantum Chaos. Permalink Journal PHYSICAL REVIEW LETTERS, 77(3) ISSN 3-97 Author Dabaghian, Yuri A Publication Date Peer reviewed escholarship.org Powered b the California Digital Librar Universit of California

2 Radiative Corrections and Quantum Chaos Yuri A. Dabaghian Department of Phsics, Universit of Rhode Island, Kingston, Rhode Island 288 (Received 4 June 996) This Letter discusses the question of how the white noise of radiative corrections affects the chaotic properties of an original quantum sstem. It is shown, b an explicit mathematical analsis, that the radiative corrections, in effect, remove the original chaos in the sstem. [S3-97(96)8-5] PACS numbers: b, 3.65.Sq, 5.4.+j,.. z One of the most interesting questions in the theor of quantum chaotic sstems is the question about the interrelations between the chaos and the quantum fluctuations. This question will be studied below in the example of a new model of quantum chaos, suggested recentl in Ref. []. It will be demonstrated that the radiative corrections can suppress the chaos found in the exact QED Green s functions of potential theor. The possible mechanism of chaos suppression is discussed below in detail. The chaotic behavior of the quantum sstem can be understood b analzing the expression for the Green s function of the sstem: ` G x, j A e 2itm dx F X, t µ dxm 3d 2 m P m g m A m X, () which is equivalent to what was obtained in [] using the Fradkin representation. Here t is the proper time of the particle, and the exact form of F X, t is specified and discussed in Ref. [2]. The important feature of () is that this expression contains explicitl the d dx m 2 m P m 2 g m A m X under the integral, so that the functional integration over the trajectories X m t actuall goes over the solutions to the effective dnamical sstem: m dx m P m t 2 gam X. (2) Analzing the behavior of that dnamical sstem, one can make some interesting statements about the quantum sstem described b the Green s function G A. For instance, if the sstem (2) is chaotic then (see [2]) the Green s function unavoidabl inherits ultrasensitive dependence on the initial conditions, which are expressed in terms of functional variable P m t. This situation, as shown in [] and [2], can be naturall interpreted as chaos in a quantum sstem. The dnamical sstem () describes the motion of a particle of momentum P m t in an electromagnetic field in terms of its vector potential. This is analogous to the relation between the generalized momentum of a particle and it s velocit, m dx m ga m X P m, for the particle that moves along the geodesics of the connection A m. The interesting question is: If this is reall a quantum chaos sstem, then how can the quantum fluctuations (the radiative corrections), which naturall appear in a real phsical sstem, change the result? The sstem with quenched radiative corrections is described b the Green s function G, G e 2 i 2 R d da m D mn d da m G j A!, which, as shown in [], can be presented in the form G R dx e 2itm t e 2 g2 R t 2 dx m 2 m P m 2 g m Am X t jkmn 2jt 2 dx m 2 2 m P m 2 g m A m X 3 F X, t det K2 g mn 2, where t jk mn jt 2 g 2 D mn t, t 2, and D mn t, t 2 is the photon propagator. So the d functional is replaced b a smooth Gaussian distribution. Seemingl, the previous analsis is not applicable anmore to the sstem, as far as we are not restricted to the solutions of an dnamical sstem; and, consequentl, one cannot ascribe dnamic Lapunov exponents to the paths, as in (). But, actuall, using a simple transformation, one can restore the d functional in the expression for the Green s function. Let us write the expression for G in the form R G da e 2itm tr t e 2 g2 2 A m 2A m t jkmn 2jt 2 A n 2An 3 det Kmn dµ 2 2 dxm 2 m P m 2 g m A m X F X, t dx. (3) (3) 2666(4)$. 996 The American Phsical Societ

3 Here we have introduced a new functional variable A m, and an additional integration over it, which due to the d dx m 2 m P m 2 g m A m X is trivial, can be performed explicitl, and evidentl does not change the result. But now the last three terms G x,, t j A R d dx m 2 m P m 2 g m A m X F X, t dx in (3) resemble the initial expression for the Green s function, G A, without the radiative corrections, although defined on the solutions to some other dnamical sstem. It should be mentioned that, as far as we have onl one integration over the time t for the whole ensemble, the G s are not exactl the Green s functions of the sstems. In other words, the sstems in the ensemble are not independent, the Gaussian distribution mixes them up. The variable A introduced in (3) would stand for the force function of that dnamical sstem, dx m m P m 2 g m A m X. (4) The Green s function then becomes R t R t G x, e 2 g2 2 A m 2Am t jk mn jt 2 A n 2A n 3 det K mn 2 G x,, t j A da. (5) One can see from this expression that the total G is a result of the averaging over the infinite ensemble of dnamical sstems, where each sstem has its own force function A m, and that these A m s are Gaussian distributed around the initial sstem () described b A m. To understand the dnamical properties of the ensemble, we have to understand how chaotic a tpical sstem in the ensemble is, and how the different sstems contribute to the whole picture. Although the force functions A m are g close to the original chaotic A m in the sense of the distribution (5), some of the sstems in the ensemble are certainl integrable; even more, it is well known that integrable trajectories are dense around each nonintegrable trajector, so these functions A m, which correspond to the integrable sstems, are not rare [3]. According to the general principles of quantum mechanics, in order to find the probabilit of a particle to go from point x to point, one must add the probabilit amplitudes of the transition, and calculate the square of the modulus of this sum: Ç X Ç 2 w c i x! X i i jc i x! j 2 X c i x! cj x!. ifij The first term on the right hand side gives the probabilit flow along each individual amplitude, and the second one is the interference term. In our case, if one takes the square of modulus of the expression (5), and retains onl the term that describes the probabilit flow along each sstem, then w R R R t t R e 2 g2 2 t jk mn jt2 A n 2An g 2 t2 t 2 2 A m 2A m 2 tjk mn jt2 A n 2A n 2 3 det K mn G x,, t j A G x,, t 2 j A 2 da da 2 j A A 2 R R t t e 2 g2 2 A m 2A m t2 t2 t jk mn jt2 A n 2A n det 2 Kmn G x,, t j A G x,, t 2 j A da. (6) The variables da and da 2, which appear in the second order functional integral for w, are defined over the different intervals of time, so for the diagonal part of the sum, when A A 2, the probabilit w can be represented as a single integration over the variable da, which is the same over the whole time scale. The evaluation of the determinant det K mn in (6) is different from that in (5) (see [4]), so it retains its power 2. The interference is not important for most of the amplitudes, since the lengths of the trajectories for different sstems and hence their phases are ver different, so the main contribution to the probabilit sum should come from (6), and this is the onl contribution considered here. In the Fradkin representation [5] the explicit expression for the Green s function G x, j A has the form ` G x, j A i d R R 3 e 2itm e i t m t 2 m 2g m m t e 2ig t R R µ 3 e g t s mn t F mn 2 t d x 2 m t A m 2 R t t. (7) 2667

4 Now, using the substitution m dx m, and the identit x 2 R t (the argument of the d function), one has G x, j A i ` 3 e ir x R e 2itm e i x mdx m m 2gm m t A mdx m R e g x F mn X s mn dx m dg t. The integration in the exponents goes over a path g t such that #t #t, and dg t represents the functional integration over all paths. Combining the first two exponential terms of the previous expression we will have G x, j A i ` e 2itm m 2g m m 3e ir x dx m ga m X dx m 3 e R g x F mn X s mn dx m dg t. (8) The function G x,, t j A has no integration over t, so we finall have for the j G x,, t j A j 2 the following: G x,, t j A G x,, t 2 j A 3e 2iR x t dg dg 2 e i t 2t 2 m jm 2g m m j 2 R x t2 dx m ga m X dx m i dx m ga m X 2 dx m Å e gr x F mn X s mn dx m Å 2. (9) The integration in the two different brackets in the exponent goes over two different paths g t and g 2 t 2. In order to understand where the main contribution to the total probabilit comes from, we shall look at the saddle point of the expression (6). In Fenmann s formalism, a tpical Green s function is obtained from the functional integral G x, e is x, d g, where the saddle point of the functional S is given b the classical solution R ds dx t. In the same R wa, in the expression e 2i x dx x m ga m X dx m 2i e P mdx m the path that corresponds to the solution to dx m ga m X P m will extremize the functional. This means, in turn, that the exponent in the expression (9) finds it s extremum value on the two paths. If we change the order of integration in the second integral t 2 µ dxm i x ga m X dx m x µ dxm! 2 i t 2 ga m X dx m, then the total argument of the exponents will be t µ dxm x ga m X dx m µ x dxm ga m X dx m. () t 2 Here one can see the tpe of solutions needed to extremize the functional (9). One solution should start at the point x and come to the point b the time t, and the other should go back from to x, starting at the time t 2. The back path g 2 is different from g. We did not change the sign of t, so g 2 is not, generall speaking, g 2t. The sstem needs, indeed, two kinds of solutions to minimize (9). The possibilit exists, for example, if the sstem (4) is integrable that it has closed periodic trajectories. In this case, the sstem can return back to the starting point, and then repeat this process infinitel man times. It s solutions simpl wind around tori in the phase space and ield closed loops. Hence the integrable sstem can appear at the point (or x) at different moments of time t 2, t 3,...,t n,...,and these moments of time are not arbitrar, since the sstem performs periodic motion. The characteristic frequencies of the sstem should satisf v n v 2 n 2 v k n k, which relation guarantees that there are closed-loop paths in the phase space. In the case when t t 2 the sum of two exponents () gives a single integration over a closed loop. And when t and t 2 are not equal, then between these moments of time the trajector will simpl run over the tori several times. Thus, for the integrable trajectories the sum of two exponents in () can be presented as an integration over a closed loop, t x µ dxm ga m dx m x t 2 fit µ dxm ga m I µ dxm dx m n C T ga m dx m, 2668

5 where H C T is the integration over the period, A m A m Xm. The number of times the solution winds around a torus, n, is just one of the quantum numbers of the sstem. In an case, if we are looking for the saddle point trajectories for the expression (), one must consider the solutions according to which sstem (4) appears twice at the same point, at the moments of time t and t 2. This is actuall a rather strong requirement that is imposed on the sstem. For instance, if a sstem is not completel integrable, then it has solutions which do not wind around an tori, the simpl cover the whole phase space. If that kind of trajector will appear in the vicinit of the same point more than once, it will do it, firstl, fairl irregularl, and, secondl, after a ver long period of time, which is certainl longer than an characteristic time v for most of the integrable trajectories [4]. For these reasons the trajectories that are chaotic will not give a serious contribution to the expression (9). The main contribution of (9) comes from the integrable trajectories of the sstem (). In other words, the larger the amount of integrable trajectories of a sstem, the more will be their contribution in the sum (9). So, at this point, we can see that for each sstem in the expression (9) the majorit of the trajectories will not give an important contribution. According to (9), the main contribution to the probabilit of a sstem which is not completel integrable (where some of it s trajectories are integrable and some are not) should flow along the integrable paths. The amount of integrable trajectories in the original chaotic sstem (2) is probabl not that large, but the fluctuations of A in phase space language produce something like fluctuating tori, which appear and disappear all the time. These tori create additional integrable trajectories, and the biggest contribution in (9) comes exactl from these integrable, fluctuating trajectories in the ensemble. Thus the probabilit flows along the web of the virtual, resonant tori in phase space, the tori for which the condition v n v 2 n 2 v k n k holds. Even the majorit of the irrational tori drop off. An irrational torus should be almost rational in order to give a significant contribution. That is, if the irrationalit of the torus can be approximated b a rational number, then the trajectories can appear in a given small vicinit of certain points rather regularl. On the other hand, one can sa that a dnamical sstem is as much integrable as it has independent closed loop solutions. In this sense, the largest contribution to the integral (9) comes from the integrable sstems of the ensemble. There is a rather beautiful effect in classical mechanics known as Arnold diffusion [6]. A particle whose motion is not completel restricted b the integrals of motion diffuses in phase space in between the web of the rational tori. Formula (9) predicts, in fact, the same thing. A sstem will use the available tori in phase space to minimize the integral (9), but if the number of integrals is less than the number of degrees of freedom, part of it s trajectories will wander all over the phase space and ield some diffusion. But once the radiative corrections are included, the situation becomes ver different. According to the distribution (), the particle could choose among the different trajectories solutions to the sstem (2) but according to the distribution (6) it can now choose the sstem which provides the most suitable trajectories. Thus the particle tends to go along the virtual tori, which appear as the result of quantum fluctuations in phase space. In the limit g!, when the Gaussian distribution shrinks and ields the d function of the original A m sstem, all field fluctuations disappear and so do the virtual tori; the particle starts to diffuse according to the Arnold scenario, and the sstem again becomes chaotic. That means that the white noise of the radiative corrections, in effect, restructures the phase space, and, as a result, removes the original chaos of the sstem. Since these arguments lead to the conclusion that the magnitude jg c x, j [of the two-point function containing quantum fluctuations of the field A m z appearing in the Green s function G c x, j A ] should not be sensitive to chaotic fluctuations, it follows that chaos can appear onl in the phase of this two-point function. I am grateful to Professor H. M. Fried for introducing me to the problem and for stimulating discussions. I have also benefited greatl from the helpful suggestions and discussions of Professor A. E. Meerovich. [] H. M Fried, Y. Gabelini, and B. H. J. McKellar, Phs. Rev. Lett. 74, 4373 (995); H. M. Fried and Y. Gabelini, Phs. Rev. D 5, 89 (995). [2] H. M. Fried, Y. Gabelini, and B. H. J. McKellar, Phs. Rev. D 5, 783 (995). [3] L. E. Reichl, The Transition to Chaos (Springer-Verlag, New York, 992). [4] M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer-Verlag, Berlin, 99). [5] E. S. Fradkin, Nucl. Phs. 76, 588 (966); H. M. Fried, Functional Methods and Eikonal Models (Edition Frontieres, Gif-sur-Yvette, France, 99). [6] V. I. Arnol d, Russ. Math. Surve 8, 9 (963); 8, 85 (963); B. V. Chirikov, Phs. Rep. 52, 265 (979). 2669