DDeformation QuantizationA C Hirshfeld, Universita t Dortmund, Dortmund,Germany 2006 Elsevier Ltd. All rights reserved.IntroductionDeformation quantization is an alternative wayof looking at quantum mechanics. Some of itstechniques were introduced by the pioneers ofquantum mechanics, but it was first proposed asan autonomous theory in a paper in Annals ofPhysics (biBayen et al. 1978). More recent reviewstreat modern developments (HH I 2001,biDito andSternheimer 2002, Zachos 2002).Deformation quantization concentrates on the cen-tral physical concepts of quantum theory: the algebraof observables and their dynamical evolution. Becauseit deals exclusively with functions of phase-spacevariables, its conceptual break with classical mechanicsis less severe than in other approaches. It formulates thecorrespondence principle very precisely which playedsuch an important role in the historical development.Although this article deals mainly with nonrelati-vistic bosonic systems, deformation quantization ismuch more general. For inclusion of fermions andthe Dirac equation see (biHirshfeld et al. 2002b). Thefermionic degrees of freedom may, in special cases, beobtained from the bosonic ones by supersymmetricextension (biHirshfeld et al. 2004). For applications tofield theory, seebiHirshfeld et al. (2002). For therelation to Hopf algebras seebiHirshfeld et al. (2003),and to geometric algebra, seebiHirshfeld et al. (2005).The observables of a physical system, such as theHamilton function, are smooth real-valued functionson phase space. Physical quantities of the system atsome time, such as the energy, are calculated byevaluating the Hamilton function at the pointx0 =(q0, p0) in phase space that characterizes thestate of the system at this time (we assume for themoment, a one-particle system). The mathematicalexpression for this operation isE

Hq. pc2q q0. p p0 dqdp 1where c(2)is the two-dimensional Dirac deltafunction. The observables of the dynamical systemare functions on the phase space, the states of thesystem are positive functionals on the observables(here the Dirac delta functions), and we obtain thevalue of the observable in a definite state by theoperation shown in eqn [1].In general, functions on a manifold are multipliedby each other in a pointwise manner, that is, giventwo functions f and g, their product fg is thefunctionfgx f xgx 2In the context of classical mechanics, the observa-bles build a commutative algebra, called the com-mutative classical algebra of observables.In Hamiltonian mechanics there is another way tocombine two functions on phase space in such a waythat the result is again a function on the phase space,namely by using the Poisson bracketff . ggq. p ni10f0qi0g0pi 0f0pi0g0qi

q.p f 0

q0

p 0

p0!p g 3in an abbreviated notation.The notation can be further abbreviated by using xto represent points of the phase-space manifold,x=(x1, . . . , x2n), and introducing the Poisson tensorcij, where the indices i, j run from 1 to 2n. Incanonical coordinates cijis represented by the matrixc 0 InIn 0 4where In is the n n identity matrix. Then eqn [3]becomesff . ggx cij0if x 0jgx 5where 0i =00xi.For a general observable,_f ff . Hg 6Because c transforms like a tensor with respectto coordinate transformations, eqn [5] may also bewritten in noncanonical coordinates. In this casethe components of c need not be constants, andmay depend on the point of the manifold at whichthey are evaluated. But in Hamiltonian mechanics,c is still required to be invertible. A manifoldequipped with a Poisson tensor of this kind iscalled a symplectic manifold. In general, the tensorc is no longer required to be invertible, but itnevertheless suffices to define Poisson brackets viaeqn [5], and these brackets are required to havethe properties1. {f , g} = {g, f },2. {f , gh} ={f , g}h g{f , h}, and3. {f , {g, h}} {g, {h, f }} {h, {f , g}} =0.Property (1) implies that the Poisson bracket isantisymmetric, property (2) is referred to as the Leibnitzrule, and property (3) is called the Jacobi identity. ThePoisson bracket used in Hamiltonian mechanics satis-fies all these properties, but we now abstract theseproperties fromthe concrete prescription of eqn [3], anda Poisson manifold (M, c) is defined as a smoothmanifold M equipped with a Poisson tensor c, whosecomponents are no longer necessarily constant, suchthat the bracket defined by eqn [5] has the aboveproperties. It turns out that such manifolds provide abetter context for treating dynamical systems withsymmetries. In fact, they are essential for treating gauge-field theories, which govern the fundamental interac-tions of elementary particles.Quantum Mechanics and Star ProductsThe essential difference between classical andquantum mechanics is Heisenbergs uncertaintyrelation, which implies that in the latter, states canno longer be represented as points in phase space.The uncertainty is a consequence of the noncommu-tativity of the quantum mechanical observables.That is, the commutative classical algebra ofobservables must be replaced by a noncommutativequantum algebra of observables.In the conventional approach to quantummechanics, this noncommutativity is implementedby representing the quantum mechanical observablesby linear operators in Hilbert space. Physicalquantities are then represented by eigenvalues ofthese operators, and physical states are related to theoperator eigenfunctions. Although these entities aresomehow related to their classical counterparts, towhich they are supposed to reduce in an appropriatelimit, the precise relationship has remained obscure,one hundred years after the beginnings of quantummechanics. Textbooks refer to the correspondenceprinciple, which guided the pioneers of the subject.Attempts to give this idea a precise formulation bypostulating a specific relation between the classicalPoisson brackets of observables and the commu-tators of the corresponding quantum mechanicaloperators, as undertaken, for example, by Dirac andvon Neumann, encountered insurmountable diffi-culties, as pointed out bybiGroenewold in 1946 in anunjustly neglected paper (Groenewold 1948). In thesame paper Groenewold also wrote down the firstexplicit representation of a star product (see eqn[11]), without however realizing the potential of thisconcept for overcoming the difficulties that hewanted to resolve.In the deformation quantization approach, thereis no such break when going from the classicalsystem to the corresponding quantum system; wedescribe the quantum system by using the sameentities that are used to describe the classicalsystem. The observables of the system are describedby the same functions on phase space as theirclassical counterparts. Uncertainty is realized bydescribing physical states as distributions on phasespace that are not sharply localized, in contrast tothe Dirac delta functions which occur in theclassical case. When we evaluate an observable insome definite state according to the quantumanalog of eqn [1] (see eqn [24]), values of theobservable in a whole region contribute to thenumber that is obtained, which is thus an averagevalue of the observable in the given state. Non-commutativity is incorporated by introducing anoncommutative product for functions on phasespace, so that we get a new noncommutativequantum algebra of observables. The systematicwork on deformation quantization stems fromGerstenhabers seminal paper, where he introducedthe concept of a star product of smooth functionson a manifold (biGerstenhaber 1964).For applications to quantum mechanics, weconsider smooth complex-valued functions on aPoisson manifold. A star product f g of two suchfunctions is a new smooth function, which, ingeneral, is described by an infinite power series:f g fg i"hC1f . g O"h21n0i"hnCnf . g 7The first term in the series is the pointwise productgiven in eqn [2], and (i"h) is the deformationparameter, which is assumed to be varying con-tinuously. If "h is identified with Plancks constant,then what varies is really the magnitude of the2 Deformation Quantizationaction of the dynamical system considered in unitsof "h: the classical limit holds for systems with largeaction. In this limit, which we express here as "h ! 0,the star product reduces to the usual product. Ingeneral, the coefficients Cn will be such that the newproduct is noncommutative, and we consider thenoncommutative algebra formed from the functionswith this new multiplication law as a deformation ofthe original commutative algebra, which uses point-wise multiplication of the functions.The expressions Cn(f , g) denote functions madeup of the derivatives of the functions f and g. It isobvious that without further restrictions of thesecoefficients, the star product is too arbitrary to be ofany use. Gerstenhabers discovery was that thesimple requirement that the new product be asso-ciative imposes such strong requirements on thecoefficients Cn that they are essentially unique inthe most important cases (up to an equivalencerelation, as discussed below). Formally, Gerstenhaberrequired that the coefficients satisfy the followingproperties:1. jk=nCj(Ck(f , g), h) =jk =n Cj(f , Ck(g, h)),2. C0(f , g) =fg, and3. C1(f , g) C1(g, f ) ={f , g}.Property (1) guarantees that the star product isassociative: (f g) h=f (gh). Property (2) meansthat in the limit "h ! 0, the star product f g agreeswith the pointwise product fg. Property (3) has at leasttwo aspects: (i) mathematically, it anchors the newproduct to the given structure of the Poisson manifoldand (ii) physically, it provides the connection betweenthe classical and quantum behavior of the dynamicalsystem. Define a commutator by using the newproduct:f . g f g g f 8Property (3) may then be written aslim"h!01i"hf . g ff . gg 9Equation [9] is the correct form of the correspon-dence principle. In general, the quantity on the left-hand side of eqn [9] reduces to the Poisson bracketonly in the classical limit. The source of themathematical difficulties that previous attempts toformulate the correspondence principle encoun-tered was related to trying to enforce equalitybetween the Poisson bracket and the correspondingexpression involving the quantum mechanical com-mutator. Equation [9] shows that such a relation ingeneral only holds up to corrections of higher orderin "h.For physical applications we usually require thestar product to be Hermitean: f g = g f, where fdenotes the complex conjugate of f. The starproducts considered in this article have thisproperty.For a given Poisson manifold, it is not clear apriori if a star product for the smooth functions onthe manifold actually exists, that is, whether it is atall possible to find coefficients Cn that satisfy theabove list of properties. Even if we find suchcoefficients, it it still not clear that the series theydefine through eqn [7] yields a smooth function.Mathematicians have worked hard to answer thesequestions in the general case. For flat Euclidianspaces, M=R2n, a specific star product has longbeen known. In this case, the components of thePoisson tensor cijcan be taken to be constants. Thecoefficient C1 can then be chosen antisymmetric,so thatC1f . g 12cij0if 0jg 12ff . gg 10by property (3) above. The higher-order coefficientsmay be obtained by exponentiation of C1. Thisprocedure yields the Moyal star product (biMoyal1949):f M g f exp"i"h2 cij0

i0!j g 11In canonical coordinates, eqn [11] becomesf M gq. p f q. p exp"i"h2 0

q0!p 0

p0!q gq. p 121m.n0i"h2 mn1mm!n! 0mp 0nqf 0np0mq g 13We now come to the question of uniqueness of thestar product on a given Poisson manifold. Two starproducts and 0 are said to be c-equivalent ifthere exists an invertible transition operatorT 1 "hT1 1n0"hnTn 14where the Tn are differential operators that satisfyf 0 g T1Tf Tg 15It is known that for M=R2nall admissible starproducts are c-equivalent to the Moyal product. Theconcept of c-equivalence is a mathematical one(c stands for cohomology (biGerstenhaber 1964)); itdoes not by itself imply any kind of physicalequivalence, as shown below.Deformation Quantization 3Another expression for the Moyal product is akind of Fourier representation:f M gq. p 1"h22

dq1dq2dp1dp2f q1. p1gq2. p2exp 2i"hpq1 q2 qp2 p1q2p1 q1p2

16Equation [16] has an interesting geometrical inter-pretation. Denote points in phase space by vectors,for example, in two dimensions:r qp . r1 q1p1 . r2 q2p2 17Now, consider the triangle in phase space spannedby the vectors r r1 and r r2. Its area (symplecticvolume) isAr. r1. r2 12r r1 ^r r2 12pq2 q1 qp1 p2 q1p2 q2p1 18which is proportional to the exponent in eqn [16].Hence, we may rewrite eqn [16] asf gr

dr1dr2f r1gr2 exp 4i"h Ar. r1. r2 19Deformation QuantizationThe properties of the star product are well adaptedfor describing the noncommutative quantum algebraof observables. We have already discussed theassociativity and the incorporation of the classicaland semiclassical limits. Note that the characteristicnonlocality feature of quantum mechanics is alsoexplicit. In the expression for the Moyal productgiven in eqn [13], the star product of the functions fand g at the point x =(q, p) involves not only thevalues of the functions f and g at this point, but alsoall higher derivatives of these functions at x. But fora smooth function, knowledge of all the derivativesat a given point is equivalent to the knowledge ofthe function on the entire space. In the integralexpression of eqn [16], we also see that knowledgeof the functions f and g on the whole phase space isnecessary to determine the value of the star productat the point x.The c-equivalent star products correspond to differ-ent quantization schemes. Having chosen a quantiza-tion scheme, the quantities of interest for the quantumsystem may be calculated. It turns out that differentquantization schemes lead to different spectra for theobservables. The choice of a specific quantizationscheme can only be motivated by further physicalrequirements. In the simple example we discuss below,the classical system is completely specified by itsHamilton function. In more general cases, one mayhave to decide what constitutes a sufficiently large setof good observables for a complete specification of thesystem (biBayen et al. 1978).A state is characterized by its energy E; the setof all possible values for the energy is called thespectrum of the system. The states are describedby distributions on phase space called projectors.The state corresponding to the energy E isdenoted by E(q, p). These distributions arenormalized:12"h

Eq. pdq dp 1 20and idempotent:E E0 q. p cE.E0 Eq. p 21The fact that the Hamilton function takes the valueE when the system is in the state corresponding tothis energy is expressed by the equationH Eq. p EEq. p 22Equation [22] corresponds to the time-independentSchro dinger equation, and is sometimes called the-genvalue equation. The spectral decompositionof the Hamilton function is given byHq. p EEEq. p 23where the summation sign may indicate an integra-tion if the spectrum is continuous. The quantummechanical version of eqn [1] isE 12"h

H Eq. pdq dp 12"h

Hq. p Eq. pdqdp 24where the last expression may be obtained by usingeqn [16] for the star product.The time-evolution function for a time-indepen-dent Hamilton function is denoted by Exp(Ht), andthe fact that the Hamilton function is the generatorof the time evolution of the system is expressed byi"h ddtExpHt H ExpHt 254 Deformation QuantizationThis equation corresponds to the time-dependentSchro dinger equation. It is solved by the starexponential:ExpHt 1n01n!it"h nHn26where (H )n= H H H....n times. Because each stateof definite energy E has a time evolution exp (iEt"h),the complete time-evolution function may be writtenin the formExpHt EEeiEt"h27This expression is called the FourierDirichletexpansion for the time-evolution function.Questions concerning the existence and unique-ness of the star exponential as a C1 function and thenature of the spectrum and the projectors againrequire careful mathematical analysis. The problemof finding general conditions on the Hamiltonfunction H which ensure a reasonable physicalspectrum is analogous to the problem of showing,in the conventional approach, that the symmetricoperator H is self-adjoint and finding its spectralprojections.The Simple Harmonic OscillatorAs an example of the above procedure, we treat thesimple one-dimensional harmonic oscillator charac-terized by the classical Hamilton functionHq. p p22m m.22 q228In terms of the holomorphic variablesa m.2

q i pm. ."a m.2

q i pm. 29the Hamilton function becomesH .a"a 30Our aim is to calculate the time-evolution function.We first choose a quantization scheme characterizedby the normal star productf N g f e"h

0 a!0 "ag 31we then have"a N a a"a. a N "a a"a "h 32so thata. "a N "h 33Equation [25] for this case isi"h ddtExpNHt H "h.a0aExpNHt 34with the solutionExpNHt ea"a"hexp ei.ta"a"h 35By expanding the last exponential in eqn [35], weobtain the FourierDirichlet expansionExpNHt ea"a"h1n01"hnn!"ananein.t36From here, we can read off the energy eigenvaluesand the projectors describing the states by compar-ing coefficients in eqns [27] and [36]:N0 ea"a"h37Nn 1"hnn!0"anan 1"hnn"anN N0 N an38En n"h. 39Note that the spectrum obtained in eqn [39] doesnot include the zero-point energy. The projectoronto the ground state (N)0 satisfiesa N N0 0 40The spectral decomposition of the Hamilton func-tion (eqn [23]) is in this caseH 1n0n"h. 1"hnn!ea"a"h"anan .a"a 41We now consider the Moyal quantization scheme.If we write eqn [12] in terms of holomorphiccoordinates, we obtainf M g f exp "h20

a0!"a 0

"a0!a g 42Here, we havea M "a a"a "h2. "a M a a"a "h2 43and againa. "a M "h 44The value of the commutator of two phase-spacevariables is fixed by property (3) of the star product,Deformation Quantization 5and cannot change when one goes to a c-equivalentstar product. The Moyal star product is c-equivalent tothe normal star product with the transition operatorT e"h2

0a

0"a45We can use this operator to transform the normalproduct version of the -genvalue equation, eqn [22],into the corresponding Moyal product versionaccording to eqn [15]. The result isH M Mn . "a M a "h2 M Mn "h. n 12 Mn 46withM0 TN0 2e2a"a"h47Mn TNn 1"hnn"anM M0 M an48The projector onto the ground state (M)0 satisfiesa M M0 0 49We now have, for the spectrum,En n 12 "h. 50which is the textbook result. We conclude that forthis problem, the Moyal quantization scheme is thecorrect one.The use of the Moyal product in eqn [25] for thestar exponential of the harmonic oscillator leads tothe following differential equation for the timeevolution function:i"h ddtExpMHt H"h.24 0H"h.24 H02H ExpMHt 51The solution isExpMHt 1cos.t2exp 2Hi"h. tan .t2 52This expression can be brought into the form of theFourierDirichlet expansion of eqn [27] by usingthe generating function for the Laguerrepolynomials:11 sexp zs1 s 1n0sn1nLnz 53with s =ei.t. The projectors then becomeMn 21ne2H"h.Ln4H"h. 54which is equivalent to the expression already foundin eqn [48].Conventional QuantizationOne usually finds the observables characterizingsome quantum mechanical system by starting fromthe corresponding classical system, and then, eitherby guessing or by using some more or less systematicmethod, and finding the corresponding representa-tions of the classical quantities in the quantumsystem. The guiding principle is the correspondenceprinciple: the quantum mechanical relations aresupposed to reduce somehow to the classicalrelations in an appropriate limit. Early attempts tosystematize this procedure involved finding anassignment rule that associates to each phase-space function f a linear operator in Hilbert spacef =(f ) in such a way that in the limit "h ! 0, thequantum mechanical equations of motion go over tothe classical equations. Such an assignment cannotbe unique, because even though an operator that is afunction of the basic operators Q and P reduces to aunique phase-space function in the limit "h ! 0,there are many ways to assign an operator to a givenphase-space function, due to the different orderingsof the operators Q and P that all reduce to theoriginal phase-space function. Different orderingprocedures correspond to different quantizationschemes. It turns out that there is no quantizationscheme for systems with observables that depend onthe coordinates or the momenta to a higher powerthan quadratic which leads to a correspondencebetween the quantum mechanical and the classicalequations of motion, and which simultaneouslystrictly maintains the Diracvon Neumann require-ment that (1i"h)[f, g] $ {f , g}. Only within theframework of deformation quantization does thecorrespondence principle acquire a precise meaning.A general scheme for associating phase-spacefunctions and Hilbert space operators, whichincludes all of the usual orderings, is given asfollows: the operator `(f ) corresponding to agiven phase-space function f is`f

~f . iei^Qi^Pe`.id di 55where f is the Fourier transformof f, and ( Q, P) are theSchro dinger operators that correspond to the phase-space variables (q, p); `(, i) is a quadratic form:`. i "h4ci2 u22ii 56Different choices for the constants (c, u, ) yielddifferent operator ordering schemes.6 Deformation QuantizationThe relation between operator algebras and starproducts is given byf g f g 57where is a linear assignment of the kind discussedabove. Different assignments, which correspond todifferent operator orderings, correspond to c-equiva-lent star products. It demonstrates that the quantummechanical algebra of observables is a representa-tion of the star product algebra. Because in thealgebraic approach to quantum theory all theinformation concerning the quantum system maybe extracted from the algebra of observables,specifying the star product completely determinesthe quantum system.The inverse procedure of finding the phase-spacefunction that corresponds to a given operator f is,for the special case of Weyl ordering, given byf q. p

hq 12j^f jq 12ieip"hd 58When using holonomic coordinates, it is convenientto work with the coherent states^ajai ajai. h"aj^ay h"aj"a 59These states are related to the energy eigenstates ofthe harmonic oscillatorjni 1n!p ^aynj0i 60byjai e12a"a"h1n0ann!p jni.h"aj e12a"a"h1n0"ann!p hnj61In normal ordering, we obtain the phase space functionf (a, "a) corresponding to the operator f by just takingthe matrix element between coherent states:f a. "a h"ajf ^a. ^ayjai 62For holomorphic coordinates, it is easy to showNn a. "a 1"hn h"ajnihnjai 1"hnn!"aane"aa"h63in agreement with eqn [38] for the normal starproduct projectors.The star exponential Exp(Ht) and the projectorsn are the phase-space representations of the time-evolution operator exp (i Ht"h) and the projectionoperators ^ jn =jnihnj, respectively. Weyl orderingcorresponds to the use of the Moyal star product forquantization and normal ordering to the use of thenormal star product. In the density matrix formal-ism, we say that the projection operator is that of apure state, which is characterized by the property ofbeing idempotent: ^ j2n = ^ jn (compare eqn [21]). Theintegral of the projector over the momentum givesthe probability distribution in position space:12"h

Mn q. pdp 12"h

hq 2jnihnjq 2ieip"hd dp hqjnihnjqi jnqj264and the integral over the position gives the prob-ability distribution in momentum space:12"h

Mn q. pdq hpjnihnjpi j~npj265The normalization is12"h

Mn q. pdq dp 1 66which is the same as eqn [20]. Applying theserelations to the ground-state projector of theharmonic oscillator, eqn [47] shows that this is aminimum-uncertainty state. In the classical limit"h!0, it goes to a Dirac c-function. The expecta-tion value of the Hamiltonian operator is12"h

H M Mn q. pdq dp

hqj^H^ jnjqidq tr^H^ jn 67which should be compared to eqn [24].Quantum Field TheoryA real scalar field is given in terms of the coefficientsa(k), a(k) bycx

d3k232 2.kp akeikx "akeikx 68where "h.k ="h2k2 m2

is the energy of a single-quantum of the field. The corresponding quantumfield operator isx

d3k232 2.kp ^akeikx ^aykeikx 69where a(k), ay(k) are the annihilation and creationoperators for a quantum of the field with momen-tum "hk. The Hamiltonian isH

d3k"h.k^ayk^ak 70Deformation Quantization 7N(k) = ay(k) a(k) is interpreted as the number opera-tor, and eqn [70] is then just the generalization ofeqn [39], the expression for the energy of the harmonicoscillator in the normal ordering scheme, for an infinitenumber of degrees of freedom. Had we chosen theWeyl-ordering scheme, it would have resulted in (bythe generalization of eqn [50]) an infinite vacuumenergy. Hence, requiring the vacuum energy to vanishimplies the choice of the normal ordering scheme infree field theory. In the framework of deformationquantization, this requirement leads to the choice ofthe normal star product for treating free scalar fields:only for this choice is the star product well defined.Currently, in realistic physical field theoriesinvolving interacting relativistic fields we are limitedto perturbative calculations. The objects of interestare products of the fields. The analog of the Moyalproduct of eqn [11] for systems with an infinitenumber of degrees of freedom iscx1 cx2 cxn exp 12i>>>>>>>>:[15[The obstructions to passing from a PL to a DIFFstructure on M now lie inHk1(M; k(PLDIFF)) [16[and the number of distinct liftings comprises thecohomology groupHk(M; k(PLDIFF)) [17[Four-Manifold Invariants and Physics 387As an illustration of all this, consider the caseM=S7; then the first nontriviality occurs when n =7and so the obstruction to smoothing S7lies inH8(S7; 7(PLDIFF)) [18[which is of course zero this means that S7can besmoothed, a fact which we know from firstprinciples. However, by the obstruction theoryintroduced above, the resulting smooth structuresare isomorphic toH7(S7; 7(PLDIFF)) =H7(S7; Z28) =Z28 [19[Hence, we have the celebrated result of Milnor andKervaire and Milnor that S7has 28 distinctdifferentiable structures, 27 of which correspond towhat are known as exotic spheres.Lastly, if dimM_ 3, then PL and DIFF coincide this leaves us with the case of greatest interestnamely dimM=4.The Strange Case of Four DimensionsIn four dimensions there are phenomena which haveno counterpart in any other dimension. First of all,there are topological 4-manifolds which have nosmooth structure, though if they have a PL structure,then they possess a unique smooth structure. Second,the impediment to the existence of a smooth structureis of a completely different type to that met in thestandard obstruction theory it is not the pullback ofan element in the cohomology of a classifying space,that is, it is not a characteristic class. Also the four-dimensional story is far from completely known.Nevertheless, there are some very striking resultsdating from the early 1980s onwards.We begin by disposing of the difference betweenPL and DIFF structures: our earlier results togetherwith the vanishing statementn PLDIFF ( ) = 0. n _ 6 [20[mean that every PL 4-manifold possesses a uniqueDIFF structure. Thus, we can take the crucialdifference to be between DIFF and TOP.In Freedman (1982) all, simply connected, topo-logical 4-manifolds were classified by their intersec-tion form q.We recall that q is a quadratic form constructedfrom the cohomology of M as follows: take twoelements c and u of H2(M; Z) and form their cupproduct c ' u H4(M; Z); then we define q(c, u) byq(c. u) = (c ' u)[M[ [21[where (c ' u)[M] denotes the integer obtained byevaluating c ' u on the generating cycle [M] of thetop homology group H4(M; Z) of M. Poincareduality ensures that such a form is always non-degenerate over Z and so has det q =1; q is thencalled unimodular. Also we refer to q, as even ifall its diagonal entries are even, and as oddotherwise.Freedmans work yields the following:Theorem (Freedman). A simply connected4-manifold M with even intersection form q belongsto a unique homeomorphism class, while if q isodd there are precisely two nonhomeomorphicmanifolds M with q as their intersection form.This is a very powerful result the intersectionform q very nearly determines the homeomorphismclass of a simply connected M, and actually onlyfails to do so in the odd case where there are stilljust two possibilities. Further, every unimodularquadratic form occurs as the intersection form ofsome manifold.As an illustration of the impressive nature ofFreedmans work, choose M to be the sphere S4,since H2(S4; Z) is trivial, then q is the zero quadraticform and is of course even; we write this as q=O.Now recollect that the Poincare conjecture in fourdimensions is the statement that any homotopy4-sphere, S4h say, is actually homeomorphic to S4.Well, since H2(S4h; Z) is also trivial then any S4h alsohas intersection form q=O. Applying Freedmanstheorem to S4h immediately asserts that S4h belongs toa unique homeomorphism class which must be thatof S4thereby establishing the Poincare conjecture.Freedmans result combined with a much earlierresult of Rohlin (1952) also gives us an example of anonsmoothable 4-manifold: Rohlins theorem assertsthat given a smooth, simply connected, 4-manifoldwith even intersection form q, then the signature the signature of q being defined to be the differencebetween the number of positive and negative eigen-values of q o(q) of q is divisible by 16.Now writeq=2 1 0 0 0 0 0 01 2 1 0 0 0 0 00 1 2 1 0 0 0 00 0 1 2 1 0 0 00 0 0 1 2 1 0 10 0 0 0 1 2 1 00 0 0 0 0 1 2 00 0 0 0 1 0 0 20BBBBBBBBB@1CCCCCCCCCA=E8 [22[(E8 is actually the Cartan matrix for the exceptionalLie algebra e8), then, by inspection, q is even, andby calculation, it has signature 8. By Freedmanstheorem there is a single, simply connected, 4-mani-fold with intersection form q=E8. However, by388 Four-Manifold Invariants and PhysicsRohlins theorem, it cannot be smoothed since itssignature is 8.The next breakthrough was due to Donaldson(1983). Donaldsons theorem is applicable to defi-nite forms q, which by appropriate choice oforientation on M we can take to be positive definite.One has:Theorem (Donaldson). A simply connected, smooth4-manifold, with positive-definite intersection formq is always diagonalizable over the integers toq=diag(1, . . . , 1).One can immediately deduce that no, simplyconnected, 4-manifold for which q is even andpositive definite can be smoothed!For example, the manifold with q =E8E8 hassignature 16 (by Rohlins theorem). But since E8 iseven, then so is E8E8 and so Donaldsonstheorem forbids such a manifold from existingsmoothly.In fact, in contrast to Freedmans theorem, whichallows all unimodular quadratic forms to occur asthe intersection form of some topological manifold,Donaldsons theorem says that in the positive-definite, smooth, case only one quadratic form isallowed, namely I.Donaldsons work makes contact with physicsbecause it uses the YangMills equations as we nowoutline.Let A be a connection on a principal SU(2) bundleover a simply connected 4-manifold M with posi-tive-definite intersection form. If the curvature2-form of A is F, then F has an L2norm which isthe Euclidean YangMills action S. One hasS = |F|2= ZMtr(F . +F) [23[where +F is the usual dual 2-form to F. The minimaof the action S are given by those A, calledinstantons, which satisfy the famous self-dualityequationsF = +F [24[Given one instanton A which minimizes S one canperturb about A in an attempt to find moreinstantons. This process is successful and the spaceof all instantons can be fitted together to form aglobal moduli space of finite dimension. For theinstanton which provides the absolute minimum ofS, the moduli space is a noncompact space ofdimension 5.We can now summarize the logic that is used toprove Donaldsons theorem: there are very strongrelationships between M and the moduli space ;for example, let q be regarded as an n n matrixwith precisely p unit eigenvalues (clearly p _ n andDonaldsons theorem is just the statement thatp=n), then has precisely p singularities whichlook like cones on the space CP2. These combine toproduce the result that the 4-manifold M has thesame topological signature Sign(M) as p copies ofCP2; and so they have signature a b, where a ofthe CP2s are oriented as usual and b have theopposite orientation. Thus,Sign (M) = a b [25[Now by definition, Sign(M) is the signature o(q) ofthe intersection form q of M. But, by assumption, q ispositive definite n n so o(q) =n =Sign(M). Hence,n = a b [26[However, a b=p and p _ n so we can say thatn = a b. p = a b _ n [27[but one always has a b _ a b so we haven _ p _ n =p = n [28[which is Donaldsons theorem.Donaldsons Polynomial InvariantsDonaldson extended his work by introducing poly-nomial invariants also derived from YangMillstheory and to discuss them we must introducesome notation.Let M be a smooth, simply connected, orientableRiemannian 4-manifold without boundary and A bean SU(2) connection which is anti-self-dual so thatF =+ F [29[Then the space of all gauge-inequivalent solutions tothis anti-self-duality equation the moduli spacek has a dimension given by the integerdimk=8k 3(1 b2) [30[Here k is the instanton number which gives thetopological type of the solution A. The instantonnumber is minus the second Chern class c2(F) H2(M; Z) of the bundle on which the A is defined.This means that we havek = c2(F)[M[ = 182ZMtr(F . F) Z [31[The number b2 is defined to be the rank of thepositive part of the intersection form q of M.Four-Manifold Invariants and Physics 389A Donaldson invariant qMd, r is a symmetric integerpolynomial of degree d in the 2-homology H2(M; Z)of MqMd.r : H2(M) H2(M)|{z}d factorsZ [32[Given a certain map mi,mi : Hi(M) H4i(k) [33[if c H2(M) and + represents a point in M, wedefine qMd, r(c) by writingqMd.r(c) = md2(c)mr0(+)[k[ [34[The evaluation of [k] on the RHS of the aboveequation means that2d 4r = dimk [35[so that k is even dimensional; this is achieved byrequiring b2 to be odd.Now the Donaldson invariants qMd, r are differentialtopological invariants rather than topological invari-ants but they are difficult to calculate as they requiredetailed knowledge of the instanton moduli spacek. However they are nontrivial and their valuesare known for a number of 4-manifolds M. Forexample, if M is a complex algebraic surface, apositivity argument shows that they are nonzerowhen d is large enough. Conversely, if M can bewritten as the connected sumM= M1#M2where both M1 and M2 have b2 0, then they allvanish.Topological Quantum Field TheoriesTurning now to physics, it is time to point out thatthe qMd, r can also be obtained, Witten (1988), as thecorrelation functions of twisted N=2 supersym-metric topological quantum field theory.The action S for this theory is given byS =ZMd4x g

tr 14FjiFji14F+jiFji&12cDjDj` iDji.jiiiDjj i8c[.ji. .ji[ i2`[j. j[ i2c[i. i[ 18[c. `[2' [36[where Fji is the curvature of a connection Aj and(c, `, i, j, .ji) are a collection of fields introducedin order to construct the right supersymmetrictheory; c and ` are both spinless while the multiplet(j, .ji) contains the components of a 0-form, a1-form, and a self-dual 2-form, respectively.The significance of this choice of multiplet is thatthe instanton deformation complex used to calculatedimk contains precisely these fields.Even though S contains a metric, its correlationfunctions are independent of the metric g so that Scan still be regarded as a topological quantum fieldtheory. This is because both S and its associatedenergy momentum tensor T =(cScg) can be writtenas BRST commutators S ={Q, V}, T ={Q, V/} forsuitable V and V/.With this theory, it is possible to show that thecorrelation functions are independent of the gaugecoupling and hence we can evaluate them in a smallcoupling limit. In this limit, the functional integralsare dominated by the classical minima of S, whichfor Aj are just the instantonsFji = F+ji [37[We also need c and ` to vanish for irreducibleconnections. If we expand all the fields around theminima up to quadratic terms and do the resultingGaussian integrals, the correlation functions may beformally evaluated.A general correlation function of this theory isgiven by