Quantum Field Theory*
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Quantum Field Theory y Frank Wilczek Institute for Advanced Study, School of Natural Science, Olden Lane, Princeton, NJ 08540 I discuss the general principles underlying quantum eld theory, and attempt to identify its most profound consequences. The deep est of these consequences result from the in nite number of degrees of freedom invoked to implement lo cality.Imention a few of its most striking successes, b oth achieved and prosp ective. Possible limitation s of quantum eld theory are viewed in the light of its history. I. SURVEY Quantum eld theory is the framework in which the regnant theories of the electroweak and strong interactions, which together form the Standard Mo del, are formulated. Quantum electro dynamics (QED), b esides providing a com- plete foundation for atomic physics and chemistry, has supp orted calculations of physical quantities with unparalleled precision. The exp erimentally measured value of the magnetic dip ole moment of the muon, 11 (g 2) = 233 184 600 (1680) 10 ; (1) exp: for example, should b e compared with the theoretical prediction 11 (g 2) = 233 183 478 (308) 10 : (2) theor: In quantum chromo dynamics (QCD) we cannot, for the forseeable future, aspire to to comparable accuracy.Yet QCD provides di erent, and at least equally impressive, evidence for the validity of the basic principles of quantum eld theory. Indeed, b ecause in QCD the interactions are stronger, QCD manifests a wider variety of phenomena characteristic of quantum eld theory. These include esp ecially running of the e ective coupling with distance or energy scale and the phenomenon of con nement. QCD has supp orted, and rewarded with exp erimental con rmation, b oth heroic calculations of multi-lo op diagrams and massivenumerical simulations of (a discretized version of ) the complete theory. Quantum eld theory also provides p owerful to ols for condensed matter physics, esp ecially in connection with the quantum many-b o dy problem as it arises in the theory of metals, sup erconductivity,thelow-temp erature b ehavior 3 4 of the quantum liquids He and He , and the quantum Hall e ect, among others. Although for reasons of space and fo cus I will not attempt to do justice to this asp ect here, the continuing interchange of ideas b etween condensed matter and high energy theory, through the medium of quantum eld theory, is a remarkable phenomenon in itself. A partial list of historically imp ortant examples includes global and lo cal sp ontaneous symmetry breaking, the renormalization group, e ective eld theory, solitons, instantons, and fractional charge and statistics. It is clear, from all these examples, that quantum eld theory o ccupies a central p osition in our description of Nature. It provides b oth our b est working description of fundamental physical laws, and a fruitful to ol for investigating the b ehavior of complex systems. But the enumeration of examples, however triumphal, serves more to p ose than to answer more basic questions: What are the essential features of quantum eld theory? What do es quantum eld theory add to our understanding of the world, that was not already presentinquantum mechanics and classical eld theory separately? The rst question has no sharp answer. Theoretical physicists are very exible in adapting their to ols, and no axiomization can keep up with them. However I think it is fair to say that the characteristic, core ideas of quantum eld theory are twofold. First, that the basic dynamical degrees of freedom are op erator functions of space and time { quantum elds, ob eying appropriate commutation relations. Second, that the interactions of these elds are lo cal. Thus the equations of motion and commutation relations governing the evolution of a given quantum eld at a given p oint in space-time should dep end only on the b ehavior of elds and their derivatives at that p oint. One might nd it convenienttouseothervariables, whose equations are not lo cal, but in the spirit of quantum eld theory there must always b e some underlying fundamental, lo cal variables. These ideas, combined with p ostulates of symmetry (e.g., in To app ear in the American Physical So cietyCentenary issue of Reviews of Mo dern Physics, March 1999. y [email protected] IASSNS-HEP 98/20 1 the context of the standard mo del, Lorentz and gauge invariance) turn out to b e amazingly p owerful, as will emerge from our further discussion b elow. The eld concept came to dominate physics starting with the work of Faraday in the mid-nineteenth century.Its conceptual advantage over the earlier Newtonian program of physics, to formulate the fundamental laws in terms of forces among atomic particles, emerges when we takeinto account the circumstance, unknown to Newton (or, for that matter, Faraday) but fundamental in sp ecial relativity, that in uences travel no farther than a nite limiting sp eed. For then the force on a given particle at a given time cannot b e deduced from the p ositions of other particles at that time, but must b e deduced in a complicated way from their previous p ositions. Faraday's intuition that the fundamental laws of electromagnetism could b e expressed most simply in terms of elds lling space and time was of course brilliantly vindicated byMaxwell's mathematical theory. The concept of lo cality, in the crude form that one can predict the b ehavior of nearby ob jects without reference to distant ones, is basic to scienti c practice. Practical exp erimenters { if not astrologers { con dently exp ect, on the basis of much successful exp erience, that after reasonable (generally quite mo dest) precautions to isolate their exp eriments they will obtain repro ducible results. Direct quantitative tests of lo cality, or rather of its close cousin causality, are a orded by disp ersion relations. The deep and ancient historic ro ots of the eld and lo cality concepts provide no guarantee that these concepts remain relevantorvalid when extrap olated far b eyond their origins in exp erience, into the subatomic and quantum domain. This extrap olation must b e judged by its fruits. That brings us, naturally, to our second question. Undoubtedly the single most profound fact ab out Nature that quantum eld theory uniquely explains is the existence of di erent, yet indistinguishable, copies of elementary particles.Two electrons anywhere in the Universe, whatever their origin or history, are observed to have exactly the same prop erties. We understand this as a consequence of the fact that b oth are excitations of the same underlying ur-stu , the electron eld. The electron eld is thus the primary reality. The same logic, of course, applies to photons or quarks, or even to comp osite ob jects such as atomic nuclei, atoms, or molecules. The indistinguishability of particles is so familiar, and so fundamental to all of mo dern physical science, that we could easily take it for granted. Yet it is by no means obvious. For example, it directly contradicts one of the pillars of Leibniz' metaphysics, his \principle of the identity of indiscernables," according to whichtwo ob jects cannot di er solely in numb er. And Maxwell thought the similarity of di erent molecules so remarkable that he devoted the last part of his Encyclopedia Brittanica entry on Atoms { well over a thousand words { to discussing it. He concluded that \the formation of a molecule is therefore an event not b elonging to that order of nature in whichwe live ... it must b e referred to the ep o ch, not of the formation of the earth or the solar system ... but of the establishment of the existing order of nature ..." The existence of classes of indistinguishable particles is the necessary logical prerequisite to a second profound insight from quantum eld theory: the assignment of unique quantum statistics to each class. Given the indistinguishabilityof a class of elementary particles, and complete invariance of their interactions under interchange, the general principles of quantum mechanics teach us that solutions forming any representation of the p ermutation symmetry group retain that prop erty in time, but do not constrain which representations are realized. Quantum eld theory not only explains the existence of indistinguishable particles and the invariance of their interactions under interchange, but also constrains the symmetry of the solutions. For b osons only the identity representation is physical (symmetric wave functions), for fermions only the one-dimensional o dd representation is physical (antisymmetric wave functions). One also has the spin-statistics theorem, according to which ob jects with integer spin are b osons, whereas ob jects with half o dd integer spin are fermions. Of course, these general predictions have b een veri ed in many exp eriments. The fermion character of electrons, in particular, underlies the stability of matter and the structure of the p erio dic table. A third profound general insight from quantum eld theory is the existenceofantiparticles. This was rst inferred by Dirac on the basis of a brilliant but obsolete interpretation of his equation for the electron eld, whose elucidation was a crucial step in the formulation of quantum eld theory. In quantum eld theory,were-interpret the Dirac wave function as a p osition (and time) dep endent op erator. It can b e expanded in terms of the solutions of the Dirac equation, with op erator co ecients. The co ecients of p ositive-energy solutions are op erators that destroy electrons, and the co ecients of the negative-energy solutions are op erators that create p ositrons (with p ositive energy). With this interpretation, an improved version of Dirac's hole theory emerges in a straightforward way.