MATHEMATICS Prime Formula Weds Number Theory and Quantum Physics When Pythagoras declared that "All is num- ber," he didn't exactly ha ve in mind th e be- havior of, say, an ex cited hydro gen atom in a magnetic fi eld. But two-and -a-half mille nni a later, an unlikely co ll aboration of pure math - ematician s-th e specialists known as ana- lytic numb er theo ri sts- and qu antum ph ys i- cists is bearing out the phil osopher's prescient words. Bo th groups are hop ing th at a prob- lem in number the ory and a problem in phys- ics will turn out to be two sides of the same nume ri cal coin. On the one side are the ordinary prime numb ers of arithmetic-and a mathematical beast known as the Riemann ze ta function, which encodes info nnation about how primes are distributed among the o ther int ege rs. On the o th er sid e is th e effort to understand th e behavior of complex atomic systems, ranging from the response of a hydrogen atom to a magnetic field to the oscillations of large nu- cle i. Class ical ph ys ics suggests th at such sys- tems should behave chaotically- but ch aos , a hair-trigger instability that makes such sys- tems effective ly unpredictable, is anathema to the o rd erly, linear mathema ti cs of quan- tum mecha ni cs. Beca use qua ntum mechan- ics prov id es the "true" description of th e atom-scale worl d, ph ys icists so mehow have to reconcile the cl ass ica l, c haotic description with the qua ntum-me cha nic al on e. Ent er the ze ta function. Th eo ri sts on both sides say there are good reasons for believing th at th e ze ta function not onl y houses information abo ut prime numb ers but may also provide a way to simu- late quantum chaos on a co mputer-a nd thereby test ideas a bout how to bridge the appare ntl y incompatible chaotic and quan- tum-mecha nical d es criptions of the micro- scopic world. Many are hopeful that the con- n ec tion with ph ys ics will, in turn, break a long tim e logjam in pure ma th ema tic s by leading to a proof of a century-old problem known as th e Riema nn hypo the sis. Just wh y numb er theor y a nd qua ntum chaos should be soulmates is a myste ry for the gods to unve il. But rese archers aren't waiting for the why befo re exploiting th e how. A l- ready, says ph ys icist Mich ael Berry of th e Uni versity of Bristol in the Unit ed Kingdom, "we have been able to tell mathematicians properti es of the Riemann zeta function they didn't know" using ideas from quantum me- cha nics. "And on th e o th er ha nd, using mathematics from the Riema nn ze ta func- tion, we've been able to find new ways of 2014 doing calculations in quantum mechanics. " Theorists already had an id ea of how to begin reconciling cl ass ical chaos with quan- tum mechanics. They suspected th at the wild jumb le of trajecto ri es th at characte ri zes cl ass i- cal chaos is refl ected in the statistics of the infinite se t of energy level s, or "spectrum," of the quantum-mechanical co unterpart. More precisely, qua ntum chaol og ists such as Berry and Martin Gut zwiller at IBM's Wat so n Re- se arch Center in Yo rkt own Height s, New (3 --- ---« 2 ( ( Prime Numbers ---- Compound Nucleus Zeta Zeros Telling resemblance? Energy levels of an ex- cited heavy nucleus are compared with the distribution of prime numbers in the interval 7,791 ,097 to 7,791 ,877 and a "spectrum" of zeta function zeros. Yo rk, have proposed th at each classical- quantum pair satisfi es a "trace formula," a kind of formula that pops up a ll over mathematics a nd th at distills a complex relationshi p to its mathematical essence. In th e quantum-chaos case, the trace formula relat es the lengths of peri odic o rbit s as represe nt ed in "ph ase space"- which keeps track of each particle 's changing position and mome ntum-to com- binations of quantum energy levels. Using the trace formul a, phys icists can convert a SCIENCE· VOL. 274 • 20 DEC EMBER 1996 sys tem's chaotic behavior in th e cl ass ica l wo rld into predictions about the statistics of its quantum-mechanical spectrum. Ph ys icists have more than just th eoreti ca l computations convincing them th ey're on th e ri ght track with the trace formula h y- po th es i s. Achim Ric hter and colleagues at the Technical University of Darmstadt in Germany have do ne experime nt s with mi - crowaves in resona tor s shaped to generate a proliferation of differe nt fr equ enci es-a model for the be havior of electrons in quan- tum chaos. O th er researchers ha ve measured energy levels of actual compound nucl e i. Howeve r, while th e res ult s agree we ll with certain predictions of the trace formula, th e best experime nts to date me as ure at most a few hundr ed energy leve ls-too few to rea lly explore the predictions of th e theory. Distilling the essence of chaos. Because it's so hard to t es t the statistical predictions of the trace formula in actual experiment s, qua ntum chaologists have been looking fo r ways to simulate the energy levels of these sys tems on a co mput e r. "The question is, is there a simple model for the quantum me- chani cs of chaos?" says Berry. The answer may be the Riemann z eta function a nd its co nnection with prime numb ers. Ma th ematicians long ago rea li ze d th at prime numb ers appear with a certain statisti- cal regularity in the sequence of int egers. The first to describe thi s pattern analytically was the Ge rman mathematician Be rnhard Riemann. In 1859, Riemann sketch ed an ex- planation of how the distribution of prime numbers depends on properties of the ze ta function, whi ch had been int roduced a ce n- tur y ea rlier by th e Sw iss ma th emat ic ia n Leo nh ard Eule r. Th e ze ta function is defined as an infinite se ri es, formed by summing reciproca l powers of the positive int egers: s( s) = 1 + 1/2' + 1/3' + 1/4' + .. . + lin' + .... For example, S(2) = 1 + 1/4 + 1 /9 + 1/ 16 + ... + 1/n z + .. . . Euler had also proved th at the zeta function can be written not just as an infinite sum but as an infinite product. SpeCific ally, S(s) = 1/( 1 - 2-' )(1- 3-' )(1 - 5-' )(1 - 7-' ) .. . (1 - p-' ) ... where each term in the product involves a prime number p. What Riemann saw- and the French ma th ematician Jacques Hada- mard ultimately proved-i s that there is a se cond way to write the zeta function as an infinite product, thi s time using the "ze ros" of the function: values P for which s(p) = o. The altemative formula is s( s) = f( s)(1 - s/pl) (1 - s/ pz) (1 - s/ p3) ... (1- s/ Pn) ... where PI, pz, etc ., are the ze ros of the z eta function a nd f( s) is a relatively simple fudge fact o r. As Riema nn observed, the analys is of thi s formula leads to preci se res ults about the d is- tribution of prime numbers. A nd to ph ys i- cist s, the equality between the two infinite products looks suspiciously like th e trace for- on July 6, 2020 http://science.sciencemag.org/ Downloaded from