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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 Â6Ã [ 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 Â6Ã [ ¾ [ WB &Å9Q D Q @ Á6Â Ã [ ¾ [ W 7Æ &½iQ w D Q J iQ w#Ç½ 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 }

8ÙLECTURE 1. QUANTUM FIELD THEORIES: AN INTRODUCTION 6ð ð Ù ð ð g @ \ ð6ð 6ð Ù ð ¿ ¿ ð6ð ð ð g ð Ë iQ - ð g D } D } J g¼We have )Ù ¿ @ û ¿À - ËeiQ - D } iQ D } @ Ë iQ - ¿À - 9Q D } D } J üThis shows that the Hilbert-Schmidt operator is self-adjoint if and only ifË iQ - @ Ë QÜ .outside a subset of measure zero in i.e.,is bounded, and In quantum mechanics one often deals with unbounded operators which are definedonly on a dense subspace of a complete separable Hilbert space . So let us extend the notion of a linear operator by admitting linear mapswhere is a denselinear subspace of (note the analogy with rational maps in algebraic geometry). Forsuch operators we can define the adjoint operator as follows. Letdenote thedomain of definition of . The adjoint operator will be defined on the set( ÙÙ( Ù @ R ( Ù( ÙÙ R \ W k ð Ù ð6iQð QÀ#ð ð ð ( ÙÙ iQ # @ Q( ÙÙ !(JQ Ù iQ #Ù( ÙÛ @ ( Ù K Take. Sinceis dense in the linear functionalextends to a unique bounded linearfunctional on . Thus there exists a unique vector "% " # such that. We take " for the value of at . Note that& & is not necessary dense in . We say that is self-adjoint if and. We shall always assume that cannot be extended to a linear operator ona larger set than. Notice that cannot be bounded onsince otherwise wecan extend it to the whole by continuity. On the other hand, a self-adjoint operatoris always bounded. For this reason self-adjoint linear operators with ('are called unbounded linear operators. ( RÙ Ù p@ ÙÙú @ ( ÙÛÙÙ@ g 1 DQÖÙ ¿ @ G¿ w @ DD ¿ Q JExample 1.3. Let us consider the space ( ÙÛÙand define the operatorObviously it is defined on the space of differentiable functions with square integrablederivative. This space contains the subspace of smooth functions with compact supportwhich is known to be dense in. Let us show that the operator isself-adjoint. Let. Since,¿;R ( Ù Ö g \ a ¿ w 9Q ¿ÀiQ D Q @1 DQ DÙt ( 1k¿wR Ög Q ð ¿À ] ð g F ð ¿À9O ð g F \ a ¿À9Q ¿ w 9Q D Q

9Á6Â Ã [ ¿À ]ð ¿ÀiQ ð g] GF /H ]aÄ¿ R(Ù ¿ @ Á Â6Ã [ \ a G¿ w 9Q 9Q D Q @ Á6Â Ã [ û G¿À ] 9Q ÈÈ [[ F \ a G¿ÀiQ w 9Q D Q @ÈüaÄaÄ @ Á Â6Ã [ \ a ¿ÀiQ iQ D Q @ M¿ Ù pJw@ ( ÙaÄThis shows that ( t( Ù K and Ù is equal to Ù on ( . The proof that (is more subtle and we omit it. D } . Let Ù É be the Hilbert-SchmidtLetbe two copies of the spaceg g defined by a kernel Ë Ö ] Q ç ]Gw Q w which hasa a ] ]Gw as real parameters:operator g Ù '& iQ @ Ë QÅç Q &½iQ D } J É ] ]w w w aaSuppose our kernel has the following properties:*, we see thatexists. Sinceis defined for all . Letting go to )is integrable over, this implies that this limit is equal to zero. Now, for any , we have -,. . É Ëe ] ÅQ ç ] w Q w²w @ a Ë ] Q ç ] w Q w Ëe ] w Q w ç ] w Q w D } D ] w ] ] w ça(N) ð Ë ] Q ç ] w Q w ð g D } @ o ç(T)Ë ] Q ç ] w Q w @ Ë ] g Q ç ] wg Q w if ] wg F ] g @ ] w F ] ç (C) for any &R Ö g D } , the function] ] @ 9Q Ëe ] Q ç ] w Q w &½9Q w D } D }Á6Â6Ã É ] @ 0 & pJis continuous for ]Gw j ] anda Ä propertyai e Å* (M) is 2takenWhen Ë is defined by the path integral,as one of the axioms of QFT. It expresses the property that any pathfrom,w * ,Q 2to Q w isfromequalQ toto * ] ]Gw , from Q to Q w and] a]Gpatha sum of paths]Gw ]Gw to move fromw Qg a particle. Property (N) says that the total probability amplitude ofQtow somewhereis equal to 1. Notice that property (N) implies that the operator Ù É isunitary. In fact,aa Ù KÙ D } @ a a É & a a É (M)/1032/ /0/

LECTURE 1. QUANTUM FIELD THEORIES: AN INTRODUCTION10 û û D}D } D } @ ËÅQçQÅ&9QeË QçQiQ ww]]]]ww ü û ü D } J@ D} D} &½9Q Ëe ] QÅç ] w Q w Ëe ] Q ç ] w Q w ü iQ &Å9Q 9Q Now we use the following Stone-von Neumann’s Theorem:Theorem 1.1. Let ] p ] R 1 \ be a family of unitary operators in a Hilbert space. Assume that@ Í ] is continious for ] j O and ÍÌ(i) for]]Á6Â Ã all\ ÌÍR ] @ , the function ;Ä ] ]Gw R 1 \ ] H ]Gw @ ] ]Gw pJ(ii) for aallThen( @ SR Á Â6Ã \ Í ] FÓÒ exists]is dense in and the operator defineda Ä by @ Á Â6Ã \ ] FÓÒaÄ ]is self-adjoint. It satisfies] @ 7L a ] { O JApplying this to our situation, we obtain thatÙ @ ] { ]\a operatora L aª ba is called the Hamiltonian operator assofor some linear operator . Theciated to Ëe ] Q ç ]Gw Q w .We would like to apply the above to our functionËe ] QÅç ] w Q w @ýû c u # ] F ] h Æ ' Óc iQ ] w F FÎ] Q g Jw ü p w ÇUnfortunately we cannot take the function Ëe ] Q ç ]Gw Q w to be the kernel of a HilbertD DSchmidt operator. Indeed, it does not belong to the space g 1 g Q Q w . In particularÖ (T) is obviously true andproperty (N) is not satisfied. One can show that (M) is OK,(C) is true if one restricts to functions &from a certain dense subspace of g 1 .ÖThe way about this is as follows (see [Rauch]).1First let us recall the notion of the Fourier transform in . It is a linear operator1 of smooth functions with all derivativesdefined on the Schwartz space Û 1 gÖtend to zero faster than any power of ð Q½ð as Q . It is given by the formula[ M&½iQ ) @ &À @ s o c u L W &½9Q D Q J[Here are some of the properties of this operator: 4///765 89: ;89;865 58 55 495!8 5.?88@5AA.@./C,BD-EGFH

11 1 0 1 is an unitary operator;) @ M&½)F Q ) ;(ii)M&½9Q @ &½ ;(iii) '&Å9Q w@ GF ªQ &½9Q ) ç(iv) &À wk @ s c u M& , where(v) '& [@& 9Q&½iQ F - - D J[ Let us show that our function Ëe ] Q ç ]Gw Q w is the propagator for the Schrödinger equationââ â ] H c â Q g ½ ] Q k @ H c WBW @ O ½MO Q @ ¿À9Q R Ö g 1 #Jg @ o . Suposea ¿À9Q R Û 1 . Let us find the solution inWe take for simplicity 1 using the Fourier transform. Using property (iii), we get @ F g (weuse the Fourier transform only in the variable Q ). Integrating this equationwithg initiala@@#J8¿À , we get ½ ]condition À9OL a h g ¿½Taking the inverse Fourier transform, we get[ o@@½ ] Q 4 L a h 8 g ¿½ s c u k L a h L W ¿À D J(1.13)[ h@@ ¿À9Q pJ Of course, we have still to show the existenceM¿Clearly, À9O Q of a solution. We skip the check that formula (1.13) gives a solution in 1 . Thisdefines us a linear operator (the propagator) ] k Û 1 Û 1 # ¿ÀiQ r ½ ] Q #J@We would like to show that it is an integral operator and find its kernel. Let Ë ] Q8 g . Then 4 g [ L a h[ û [ o @D8 g W D ¿À D @Ëe ] QÍF ¿À s cu[[ [ La h L ü [ û [ o-D W 8gD @8 g ¿À ) @ À ] Q #JÀ¿s cu L a h L La h[ [ Lü 8 not belongUnfortunately, this computation is wrong since the function L a h g doesto Û 1 . A way about it is to consider this function as a distributionand extend theD(i)BBD DDE FE;FD;F#DJIKD/D//LI84/ 8M88NBBE88EFOEGFH#D8PD8E8EHFE FGFH;FHE;FFDBBDQB8HHR HHHFDRHHFHEBForier transform to distributions.;F88E

LECTURE 1. QUANTUM FIELD THEORIES: AN INTRODUCTION12ï [ 1 \¿ 1Recall that a distribution is a continuous linear functional on the spaceofsmooth functions with compact support (test functions). Any function which can beintegrated over any finite closed interval (but not necessary over the whole ) can beconsidered as a distribution. Its value at a test function is equal to [@À¿ M&À¿ 9Q &½9Q D Q[ where the bar denotes the complex conjugation.Such a distribution is called a regulardistribution or a tempered distribution. The rest are called singular distributions. Weshall denote the value of a distribution ¿ on a test function & by [@¿ÀM&&½iQ ¿ 9Q D Q J[ 1 , thenIf ¿ is a regular distribution defined by a function ¿ÀiQ fromgÖ¿ÀM& @ ¿ & pJv whose value at a testAn example of a singular distribution is the delta-function iQ Fv1 function & is equal to &½ . It is also denoted by . A linear operator Ù ( g[ 1 ((Ùp\Öwith ïextends to the spacex of distributions by the formulaÙ ¿À'& @ ¿À Ù & #JIf ¿;Rdistribution, we havegÖ 1 , viewed as a regularÙ ¿ÀM& @ Ù & ¿ @ & )Ù ¿so the two definitions agree.@ @ W be defined on the space of functions with@ square integrableFor example let ÙÙderivative. We haveF Ù and for any distribution ¿ , ¿ w M& ¿À)F0& . If ¿ is atempered distribution defined by an integrable differential function ¿ such that ¿ w alsodefines a tempered distribution, then the formula of integration by parts shows that thisdefinition agrees with the usual definition of derivative.Sincetransform is an example of an operator defined on 1 with@ the, weFouriertransform of a distribution ¿ by 4 can define the Fourier'¿ M& @ ¿À 4 M& )pJAll the properties (i)-(v) extend to distributions. In property (v) we define the convolution of a regular distribution ¿ and an element & of Û 1 by the formula¿ M& @ ¿À k& pJv with Re v k{ O v @ O ,Lemma 1.2. For any RW @ s oc v W 8 J x h h x S,NTS& VUUD& B DDDBLI//XW'ID

13v Ij OWto 1 and, x h belongsÐ[[ ooW @ s c uW W D Q @ s c u W 8 D Q @ x

Quantum Field Theories: An introduction The string theory is a special case of a quantum ﬁeld theory (QFT). Any QFT deals with smooth maps of Riemannian manifolds, the dimension of is the dimension of the theory. We also have an action function deﬁned on the set Map of smooth maps. A QFT studies integrals Map ! #" % '&)( * &-, (1.1) Here ( * &-, stands for some measure on the space of .

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