Quantum Field Theories: An Introduction

Lecture 1Quantum Field Theories: Anintroduction The string theory is a special case of a quantum field theory (QFT). Any QFT dealsof Riemannian manifolds, the dimension ofiswith smooth mapsthe dimension of the theory. We also have an action function defined on the setMapof smooth maps. A QFT studies integrals % '& )( * &-,(1.1) !#"Here ( * &-, stands for some measure on the space of paths, . is a parameter (usually% Map / 0 21 is an insertion function. Thevery small, Planck constant) and 657 9 8/: shouldnumberas the probability amplitude of the contribution; tobetheinterpreted 4 3of the mapintegral. The integral 0?A@ ED &(1.2) 4 B Cis called the partition function of the theory. In a relativistic QFT, the space has aLorentzian metric of signature GF #HIKJKJ J4/H . The first coordinate is reserved for time,the rest are for space. In this case, the integral (1.1) is replaced with @ 657 98/: % M& G(;* &N, J(1.3) 4 7L 3Let us start with a O -dimensional theory. In this case is a point, so & P isa point QSR and T U1 is a scalar function. The Minkowski partition functionof the theory is an integral V@ WB 98/: D Q J(1.4) 7L 3MapMapMapFollowing the Harvard lectures of C. Vafa in 1999, let us consider the followingexample:1

2LECTURE 1. QUANTUM FIELD THEORIES: AN INTRODUCTIONX @ [e [XY'Z \ ]) 4 K ba D ] @dc \f]Gg/ 4 K baih D ] J(1.5)This integral is convergent for Re MZ kj O but can be meromorphically extended to thewhole plane with poles at ZlRSm n \ . We haveXY'Z HdoB @ ZBXYMZ p XY oB @ o q Xr co @ts u J@ s v ]Gw in (1.5), we obtain the Gauss integral:By substituting ] [ D @ XY @zy u J(1.6)[ 4x/aih ] v 8g g vv Although in the substitution above is a positive real number, one can show thatvvformula (1.6) make sense, as a Riemann integral, for any complex with Re 0{ O .vWhen Re Ej O this is easy to see using the Hankel representation of XY'Z as avcontour integral in the complex plane. When is a pure imaginary, it is more delicateand we refer to [Kratzer-Franz], 1.6.1.2.v @ u , we can useTakingD } @ D ] aihto define a probability measure on 1 . It is called the Gaussian measure. Let us computethe integral 9 @ [ WB D } @ [ W WB D Q J[ 7L [ h) L6 @ o7 . We have Here 9 @ [ )F u Q g [ i Qb D Q J \[ p Obviously, [Q g W h D Q @ O J[ Alsoc F oB @ [o oB K @u@DH Wc uQQ XY9 c [ g h h Example 1.1. Recall the integral expression for the -function:

3cu c # @V c c u g wherec @ c c # @ o ¡ c c; ¡ c c F c K ¡ cc7 cis equal to the number of ways to arrange objects in pairs. This gives us M @ orH [ GF o7 g M c # c u J(1.7) Observe that toarrange objects in pairs is the same as to make a labelled 3-valentcgraph X with vertices by connecting 1-valent vertices of the following disconnectedgraph:b 2nb2b1c2c1a2a1c2na 2nFig. 1This graph comes with labeling of each vertex and an ordering of the three edgesemanating from the vertex. Let be such a graph,be the number of its verticesandbe the number of its edges. We have, so thatfor some . Let% % 9X % 9X @ 9X @ c § MXc § §MXMX @ 9X @ )F % ª « # ! c u o ? ! JMXThen M @ orH MX p where the sum is taken over the set of labeled trivalent graphs. Let e9 be the numberof labelled trivalent graphs which define the same unlabelled graph when we forget @ AM 9X , where A9 is the number ofabout the labelling. We can write M labelling of the same unlabelled 3-valent graph . Thus 9 @ o H p 9 #Xwhere the sum is taken with respect to the set of all unlabelled 3-valent graphs. It iseasy to see thatc p - ² A9 @ ³ AutM g

4LECTURE 1. QUANTUM FIELD THEORIES: AN INTRODUCTION M @ GFc # c g u c ³ # - ² g @ c u )F ? ³ « JAut M Autc 9 Given an unlabelled 3-valent graph withvertices, we assign to each vertex a factorc u , then multiplyGFnumber , toofeachedge a factor o7 all the factors and divide by thesymmetries of the graph. This gives the Feynman rules to compute thecontribution of this graph to the coefficient at . For example, the graphg contributes )F g g g @ F µ and the graph@ F·¶ J The total coefficient at is F . Thiscontributes )F g µg at g in G 9 given by the formula (1.7).g G ) ¹coincides with the coefficientRecall that the Principle of Stationary Phase says that the main contributions to theintegral º WB D 7L6»B¼ &½iQ Qwhen ¾ goes to infinity comes from integrating over the union of small comapct neighborhoods of critical points of ¿ÀiQ . More precisely we have the following lemma:%Lemma 1.1. Assume &½9Q has a compact support and ¿ÀiQ has no critical points% on . Then, for any natural number ,Á Â6à [ ¾ [ WB &Å9Q D Q @ O J»7Ä [ L6»B¼@ O . Integrating by parts,Proof. We use induction on . The assertion is obvious for we get [ WB 7Æ &Å9Q w D Q @ Æ &½iQ WB ÈÈ [ [ H [ WB &Å9Q D Q @¾ [ L6»B¼ ¿À9Q w#Ç ¾ ¿À9Q w/Ç L »7¼ È [ L6»B¼ [ 6W D J [ BL6»7¼ &½iQ Q , we getMultiplying both sides by ¾ Å Á Â6à [ ¾ [ WB &Å9Q D Q @ Á6 à [ ¾ [ W 7Æ &½iQ w D Q J iQ w#ǽ 6LB»¼ L7»¼[[»7Ä»BÄ Applying the induction to the function Æ 5 6 W WB i É w we get the assertion.¼ Çso that

¿À9Q¿ÀiQQ KJ JKJ4 Q Ê &½iQ5¿ 9QË QÍÌiQLL L Îj OÐ [ 6W 98/: D @ Ï Ð WB 98/: D HeÑ pJ&½9Q Q 7L6¼ &½iQ Q M.[ 7L ¼L @ D HVo , where DNow let us consider a QFT in dimension 1. Usually we write (oHV@ o is theOis the space-dimension, and is the time-dimension. A QFT in dimension@ 1*Koquantum mechanics. In this case, we take to be equal to , ÒO , or 1 mÓ Ô Õ parametrized by ] . A mapis path in(infinite, or finite, or a loop). Theaction is defined by @) ] Ö ] # Ø ] ) D ]where Ù f Ú1 is a smooth function defined on the tangent space of (a Lap Ø ] ) D ] is a density on equal to the compositionÖ The expressiongrangian).]D; ÛÙ Ù Öof the differentialand .@ 1 so that@ 1 ÝÜ 1 with coordinates 9Þ Þ Ø . ForÙ ÖFor example, take vany 9Þ Þ Ø and a map ß * à , á1 , 9ß ] p ßYØ ] ) is obtained by replacing Þ withØØÖß ] AandcriticalÞ withpointßY ] of. the functional iß Ö ) satisfies the Euler-Lagrange equationâ p Ø ) @ ] D â # Ø )#Jâ ãÖ iß ] ßY ] D ] â ãÖ Ø 9ß ] ßr ](1.8)LFor example, let us take theL Lagrangian ã Ø F % ã KJ JKJ ã (1.9) Lg L Then we get from (1.8) D g D QÅ] ] @ F0ä % 9QÅ ] #Jghas finitely many critical points, we write our functionThus ifas a sum of functionswith support on a compact neighborhoodofand a functionwhich has no critical points on the support ofand obtain, forany,Thus a critical path satisfies the Newton Law; it gives the major contribution to thepartition function.Fixand. Letbe the space of smooth mapssuch that. The integral P Q Q w R] @]Gw Q R G @ Q åæ ] Q ç ]Gw Q w]]Gw è w(1.10)Ëe ] Q ç ] w Q w @ W é É WBÉ! L 3 !ê a i M8 : (;* ] ,a a that a particle in the position Q at thecan be interpreted as the “probability amplitude”moment of time ] moves to the position Q w at the time ]Gw .

6LECTURE 1. QUANTUM FIELD THEORIES: AN INTRODUCTION% @ 1Let us compute it for the action defined by the Lagrangian (1.9) with. Weshall assume that the potential function is equal to zero.The spaceis of course infinite-dimensional and the integration oversuch a space has to be defined. Let us first restrict ourselves to some special finitedimensional subspaces of. Fix a positive integerand subdivide thetime intervalintoequal parts of lengthby inserting intermediate points. Let us choose some pointsinand consider the pathsuch thatits restriction to each intervalis the linear functionåæ ] ÅQ ç ] w Q w* ] ]Gw , @ åæ ] QÅç ]Gw Q w @ @ ]Gw F ] @KJ JKJ ]G ë ]GëQ]]]G]w@ ì Û* ] ]Gw , q1 Q Q g KJKJ J4 Q ë Q ë Q g w * 1 , Å Å ] L ] L ½ ]L @ Q L H Q ] L Å FEF Q] L ] F ] L #JL ½ L ë and so we can integrateIt is clear that the set of such paths is bijective with 1 î ë . Now we cana function Ì åæ ] Q ç ]Gw Q w l í1 over this space to get a numberdefine (1.10) as the limit of integrals î ë when goes to infinity. However, this limitmay not exist. One of the reasons could be that î ë contains a factor ï ë for someconstant ï with ð ïÍð jño . Then we can get the limit by redefining î ë , replacing it ëwith ï ë î ë . This really means that we redefine the standard measure on 1 DD the measure Q on 1 by ï 4 Q . This is exactly what we are1 going 4to replacingdo. Also, when we restrict the functional to the finite-dimensional spaceÉ ÀNØ ë D ] byofpiecewise linear paths, we shall allow ourselves to replace the integral òa Ö sideits Riemann sum. The result of this approximation is by definition the a right-handin (1.10). We should immediately warn the reader that the described method of givinga value to the path integral is not the only possible.We have [ JKJKJ [ * ª ë Á 6ÂÃ@c iQ FÎQ g ,óï ë D Q g JKJKJ D Q ë JË ] Q ç ] w Q w ë [Ä [ [ p L L L Å (1.11) DHere Q KJKJ J4 Q ë are vectors in 1 and Q is the standard measure in 1 . The numberï shouldg be chosen to guarantee convergenceL in (1.11). Using (1.6) we have[ [ 6W W W W D Q @ g Æ W Ç h W W D Q @[ x h h# 4x h h g [ x h kô Gõ h ô köh h g [ @ W W Wg D Q @ y c u v W W J öh h [ x h öh h Next [ * vF c 9Q FÎQ g F v iQ FÎQN g , D Q @[ p

7@ [ * F c v Æ Q F Q H Qb g F v iQ FEQb g , D Q @ y c uv W W ù J Ç ö h[ø K Thus [ * vF iQ FÎQ g g F v 9Q g FEQ g F v iQ FÎQ g , D Q g @[ú K y c u v y c uv * F v iQ FEQb g , @ y u v g * F v iQ FEQb g , J K g p Continuing in this way, we findu [ * v ë vF iQ FÎQ g , D Q g J JKJ D Q ë @ y v ë ë 4 * F 9Q FÎQ ë g ,[ K K Å L L Å 4 L v @ c Jwhere If we choose the Ðconstant ï equal to ï @ û Ð h then we willg L6 übe able to rewrite (1.11) in the formË ] QÅç ] w Q w @ýû c u h þ ô Éiÿ ô " h @úû c u # ] F ] h þ ô É É9ÿÿ ô " h J (1.12)w ü h "ü hWe shall use Ëe ] Q ç ]Gw Q w to define a certain Hermitian operator in the Hilbert1 . D }Recall that for any manifold with some Lebesgue measure D } thespaceg Öspace gof square integrable complex valued functions modulo funcÖ to zeroconsiststions equalon the complement of a measure zero set. The hermitian innerproduct is defined by D } J@¿ ¿D } is a Hilbert-Schmidt operator:Example 1.2. An example of an operator in g ÖÙ ¿ÀiQ @ Ëe9Q - ¿À - D } ÜÜ}} where Ë iQ - Ris the kernel of Ù . In this formula we integrategÖkeeping Q fixed. By Fubini’s theorem, for almost all Q , the function d Ë iQ }is -integrable. This implies that Ù M¿ is well-defined. Using the Cauchy-Schwarzinequality, one can easily checks that ð6ð Ù ¿ ð ð g @ ð Ù ¿ ð g D } ð6ð ¿ ð ð g ð Ëe9Q ð g D } D }