Feynman Diagrams in Quantum Mechanics

Transcription

1 Feynman Diagrams in Quanum Mechanics Timohy G. Abbo Absrac We explain he use of Feynman diagrams o do perurbaion heory in quanum mechanics. Feynman diagrams are a valuable ool for organizing and undersanding calculaions. We firs work several examples for he 1-dimensional harmonic oscillaor, and hen proceed o jusify our calculaions. 1 Inroducion In his paper we inroduce he erminology of Feynman diagrams. We firs give several examples of he applicaion of Feynman diagrams o perurbaive quanum mechanics on he harmonic oscillaor. We hen explain he ineracion picure of quanum mechanics, and Wick s Theorem, culminaing in a jusificaion for he Feynman rules used in our examples. Throughou his paper, we will simplify equaions by using he convenions c = h = m = 1. One can always add hese hree facors back ino equaions a he end of a calculaion, since hey have linearly independen unis, and leaving hem ou vasly simplifies noaion. Feynman diagrams are useful ools for Sysemaically calculaing values associaed o scaering processes ha can occur in several differen ways. Algorihmically consrucing he exac inegrals giving he kh order erm of a perurbaion series. Exracing classical inuiion abou wha even(s) each erm of a perurbaion series for a complicaed ineracion represens.

2 2 Timohy G. Abbo 2 Examples wih Feynman Rules We will sar wih he example of an anharmonic oscillaor wih Hamilonian H = p2 2 + ω2 x 2 + λx3 2 6 and compue firs-order correcion in he perurbaion series of Ω x() 3 Ω, where Ω is he ground sae of he perurbed Hamilonian (we wrie x() 3 raher han x 3 because he ime associaed wih hese operaors is imporan). We will do his by consrucing cerain Feynman diagrams for his problem. A diagram is very similar o a graph: i consiss of a se of verices and connecing lines (edges). There are wo ypes of verices in hese diagrams inernal and exernal verices. Inernal verices in his case have hree lines, corresponding o he perurbaion x 3, and are labeled by a parameer ; each inernal verex has a differen parameer. Exernal verices have a single line, and correspond o he x s in he expecaion we re compuing. Exernal verices are labeled by he ime he x s are evaluaed a. A diagram is consruced when he lines are conneced ogeher in pairs o form edges of a graph, wih no sray lines leaving. The general procedure is o firs figure ou which diagrams are relevan o he problem being solved, and hen use some Feynman rules o calculae a value for each diagram. The values are hen added ogeher, wih cerain weighs, o evaluae he desired expecaion. This process may seem a bi arbirary, bu in he second half of he paper we give a naural explanaion as o where hese diagrams come from. As we go hrough examples, he Feynman rules describing how o calculae will become apparen; bu we will also summarize hem in he nex secion. For our example problem, we are compuing he firs-order erm of he perurbaion, so we have one inernal verex, labeled wih ime. We also have 3 exernal verices, each labeled wih ime, since we re compuing Ω x() 3 Ω. The opologically disinc possibiliies are shown below, since he only informaion relevan o he calculaion of he value of a Feynman diagram is he opological srucure of he diagram. and

3 Feynman Diagrams in Quanum Mechanics 3 In our figures, we denoe verices by black dos, and he edges (called propagaors) by lines. We can assign weighs o he diagrams, which can be compued as he number of ways he diagram can be consruced, divided 3! (his facor comes from he symmery of permuing he hree edges ou of an inernal verex). This gives rise o an equaion of he form Ω x 3 Ω = ( ) 6 6 ( + 9 6) +O(λ 2 ) Now, we assign a value o each diagram, based on he verices i possesses and he edges connecing he pairs, using he following Feynman rules. = exernal verex = 1 = inernal verex = iλ d = propagaor = D(, ) = 1 2ω e iω So, in his example, he diagrams ake on he values shown below. = iλ D(, ) 3 d = iλ 8ω 3 e i3ω d = iλ D(, )D(, )D(,)d = ( iλ 8ω 3 )e iω d Unforunaely, he inegrals we use here are a bi complicaed. The limis of inegraion are from (1 iɛ) o (1 iɛ) (hink of his as inegraing along a line slighly iled off he x-axis in he complex plane). We need hose iɛ s because wihou hem, he inegrals would no converge. However, he resul of his inegral is in fac independen of ɛ, so we can omi ɛ from our noaion. Every inegral we wrie down in his paper will have hose limis of inegraion, unless limis are oherwise explicily given. Wih a bi of complex analysis (see [4]), one can derive he following resuls:

4 4 Timohy G. Abbo Theorem 1 For ω >, T lim de iω = 2 T (1 iɛ) T iω To aid in he calculaion of higher order examples, we use he following heorem, which can be proven using Theorem 1 Theorem 2 If a,b,c are posiive real numbers such ha he sum of any wo of hem is posiive, we have ha under he same limis as in Theorem 1 ds de ia s e ib e ic s 2 dsd = (a + b)(b + c) + 2 (a + b)(a + c) + 2 (a + c)(b + c) We can now evaluae he inegrals in our example o obain Ω x() 3 Ω = 6 6 iλ 2 8ω 3 3iω + 9 iλ 2 6 8ω 3 iω +O(λ2 ) = λ 8ω 4(2 3 +3)+O(λ2 ) = 11λ 24ω 4 +O(λ2 ) (1) The reader can check ha one would ge he same answer using sandard imeindependen perurbaion heory. Le s do anoher example. To compue he firsorder correcion o he expecaion of x() in he perurbed ground sae, we do he same calculaion, bu wih only one exernal verex. So, we consider all possible diagrams conaining one degree-3 inernal verex and one degree-1 exernal verex. The only possibiliy is for he inernal verex o be conneced o he exernal verex, wih he remaining wo edges conneced back o iself. This diagram occurs in hree possible ways, and we sill have he weighing facor of 1 from he inernal verex, so 3! his diagram has weigh 3 = 1. We can hen compue 6 2 = iλ D(, )D(,)d = ( iλ 4ω 2 )e iω d = iλ wω 2 2 iω = λ 2ω 3 which implies ha (remembering he weigh of 1 2 on his diagram) Ω x() Ω = 1 λ 2 2ω + 3 O(λ2 ) = λ 4ω + 3 O(λ2 ) (2) as one would obain using sandard ime-independen perurbaion heory. Now, suppose we were insead compuing he expecaion of x 2 o firs-order. In his case we d have one degree-3 inernal verex and wo degree-1 exernal verices. On inspecing he siuaion, he reader should noice ha i is impossible o connec hese verices ogeher, since he oal degree of any graph is even, bu hese have a

5 Feynman Diagrams in Quanum Mechanics 5 oal degree ha is odd. Thus, he firs-order erm in he perurbaion series is. Le us proceed o compue he second-order erm in his series as well. We now have wo degree-3 inernal verices (labeled by imes s and ) and wo degree-1 exernal verices, boh labeled by ime. The possible opologically disinc diagrams are shown below. s = D 1 = λ 2 dsdd(s,) 3 D(, ) s = D 2 = λ 2 dsdd(s,)d(s,s)d(,)d(, ) s s = D 3 = λ 2 dsdd(s, ) 2 D(s,)D(,) = D 4 = λ 2 dsdd(s,) 2 D(,)D(,s) s = D 5 = λ 2 dsdd(s, )D(s,s)D(, )D(,) The values for diagrams composed of wo subdiagrams we ve seen before (up o opological equivalence) can be calculaed immediaely by muliplying he previous resuls. For example, he diagram D 5 above has value ( ) 2 λ D 5 = = λ2 2ω 3 4ω 6 Using Theorem 2 we can calculae D 3 = λ 2 ds dd(s, ) 2 D(s,)D(,) (3) = λ 2 1 ds de i2ω s e iω s (4) (2ω) ( 4 = λ2 2 16ω 4 2ω ω + 2 ) = λ2 (5) 2 3ω 2 8ω 6 Similarly, he fourh diagram evaluaes o ( D 4 = λ 2 dsdd(s,) 2 D(s, )D(, ) = λ2 2 16ω 4 6ω ω + 2 ) = λ2 2 9ω 2 18ω 6 The firs wo diagrams are a bi more difficul o calculae. One will noice ha Theorem 2 does no apply for calculaing hese diagrams. In fac, he inegrals involved are no even bounded. They key propery of hese diagrams is ha hey have conneced componens ha conain no exernal verices. These diagrams are

6 e 6 Timohy G. Abbo called vacuum bubbles in quanum field heory. This seems disasrous how can we do anyhing useful wih hese quaniies ha fundamenally diverge? I urns ou ha we ve hus far been ignoring an imporan deail when doing hese Feynman diagram calculaions. The ground sae Ω we ve been using in hese calculaions is no acually normalized. However, he firs order correcion o is normalizaion is, so i has no been a problem in our firs order calculaions. We can in fac use he Feynman diagram echnique o expand he expecaion of 1 in he ground sae, as shown below. For simpliciy, we only show he diagrams, no heir weighs, bu we consruc hese from having no exernal verices and m inernal verices for he order λ m erms. The fac ha here are no diagrams wih only one inernal verex confirms our claim ha he firs-order correcion is. Ω Ω = You may noice ha we did no record he imes aached o he various verices. Since inernal verices always have disinc imes aached o hem, and he value of ha ime is hen inegraed over, many auhors leave ou he ime daa for inernal verices. Similarly, because in his paper we will focus only on expecaions of powers of x(), all he exernal verices can be assumed o be labeled by, as well. These echniques are valid for x() for arbirary imes, bu he answers are much messier. I urns ou ha he vacuum bubbles we find when calculaing Ω x 2 Ω exacly cancel he vacuum bubbles in he normalizaion facor in he denominaor, o all orders. In paricular, if we were o include he vacuum bubbles, our series would acually calculae Ω x2 Ω Ω Ω, and we d have o divide by Ω Ω in order o Ω Ω normalize i. The diagrams conaining vacuum bubbles arise from he produc of he higher-order erms of he series for Ω Ω imes lower-order erms of he series we acually wan o compue. This can be seen in our example by noing ha he h-order erm in his expecaion is given by he diagram wih a single propagaor connecing wo exernal verices, and each of he vacuum bubbles in our example muliplies ha h-order erm. Thus, we can in fac ignore all diagrams conaining vacuum bubbles, and ge a correc value for he expecaion in he rue ground sae. We now complee he calculaion. The weighing facors for each of he remaining diagrams once again equals he number of ways i could be produced, divided by he inernal verex symmeries of 2! (3!) 2 = 72 (3! for he edges on each of he wo verices, and an addiional 2! because we can permue he verices hemselves).

7 Feynman Diagrams in Quanum Mechanics 7 The diagrams D 3 and D 4 can each occur in 36 ways, while D 5 can occur in 18 ways. The oher wo diagrams we can ignore because hey have vacuum bubbles. Thus he weighs are 36, , and. We can now compue ha 72 Ω x 2 Ω = 1 2ω + 1 λ 2 2 8ω + 1 λ ω + 1 λ ω + 6 O(λ3 ) = 1 2ω + 11λ2 72ω + 6 O(λ3 ) (6) Our esimae is in fac correc o order λ 3 as well, because for he same pariy reason ha he erm of order λ equals, he erm of order λ 3 equals. We can also use Feynman rules wih oher Hamilonians. Suppose ha we insead had a perurbaion of he form H = p2 2 + ω2 x 2 + λx4 (7) 2 4! In his seing, we use he same Feynman rules as before, excep ha he inernal verices each have degree 4. The diagrams relevan o he compuaion of Ω x() 2 Ω up o firs order are shown below. We omi he vacuum bubbles, since we know ha hey will no conribue. + ( ) 12 Ω x 2 Ω = 24 +O(λ 2 ) = 1 2ω iλ D(, ) 2 D(,) Ω x 2 Ω = 1 2ω iλ 2 d +O(λ 2 ) = 1 8ω 3e i2ω 2ω iλ 2 16ω 3 2iω +O(λ2 ) = 1 2ω λ ) 16ω 4+O(λ2 Anoher ineresing feaure o noice is ha we can deec from hese rules ha for his Hamilonian, Ω x() Ω has no nonzero erms in is perurbaion series, since here will always be an odd number of edges in every diagram. This observaion is confirmed by he fac ha x is an odd funcion in an even poenial, and hus mus have expecaion in he ground sae! Exercise. Use Feynman diagrams o demonsrae Ω x 2 Ω = 1 2ω λ 16ω + 35λ ω + 7 O(λ3 ). (8) Only hree new diagrams are involved. Check he answer using he perurbaion heory we learned in class, and see which echnique you prefer.

8 8 Timohy G. Abbo 3 Feynman Rules for he Harmonic Oscillaor In his secion we presen a complee se of Feynman Rules for perurbaions o he Harmonic Oscillaor, of he general form H = p2 2 + ω2 x 2 + λxk 2 k! We wish o calculae he erm of order λ m in he expansion of Ω x n Ω The Feynman rules are as follows. We have m inernal verices, each wih k edges, labeled by imes i, i = 1...m and wih value i λd i. We also have n exernal verices, each wih a single edge of value 1. We connec hese ogeher in all possible pairings of edges, and each propagaor conribues a value D(, ), if he verices conneced by i are labeled by and respecively. Noe ha since we re using he ime-ordered produc, he order of he argumens here is no imporan. The inegral for an inernal verex has he limis of inegraion from Theorem 1. 1 iλ T T d D(, ) All diagrams ha conain vacuum bubbles are given weigh. We weigh each 1 diagram wihou any vacuum bubbles by he quaniy. This value represens k! m m! he symmeries inheren in he inernal verices, wih a facor of m! for he symmery of permuing he verices, and a facor of k! m for permuing he edges of each inernal verex. In our examples, we saved work by considering all he classes of opologically disinc diagrams, and compued a oal weigh for each by muliplying 1 k! m m! by he number of diagrams in he class. These Feynman rules can be exended o handle more complicaed sysems wih wo harmonic oscillaors in coordinaes x and y. For such a sysem we would have wo differen ypes of edges, one corresponding o an x and he oher o a y. We hen could use hese echniques o represen perurbaions of he form x a y b wih a verex wih a x-edges and b y-edges. These echniques furher generalize o he case where here are several perurbaive erms; here is hen more han one ype of inernal

9 Feynman Diagrams in Quanum Mechanics 9 verex in he problem. One can see ha for perurbaions of he harmonic oscillaor, Feynman diagrams can be used in a quie general fashion. The Feynman rules give a nice way o sudy complicaed poenials ha look like he harmonic oscillaor poenial. Harmonic oscillaors are ubiquious because hey are solvable exacly, and many poenials are well approximaed locally by a quadraic. In quanum field heory, we ofen represen a complicaed paricle as a superposiion of harmonic oscillaors. Since Feynman diagrams can be used o calculae perurbaions of harmonic oscillaors, we can use hem o calculae properies of he elemenary paricle ineracions ha quanum field heory describes. For more abou his opic, ake a course in quanum field heory. 4 Correlaion Funcions Now ha we have explained how o calculae wih Feynman Diagrams, we will proceed o explain why hese calculaions should be correc. In his secion we inroduce some of he erminology ha will be used hrough he res of he paper. The objecs in his secion are of paricular imporance when we work in he ineracion picure, which we will discuss in he nex secion. Define he N-poin correlaion funcion as x( N )x( N 1 ) x( 1 ) (9) You can hink of his as measuring he correlaion beween he locaion of he paricle a each of he N momens in ime. The quaniies we ve been compuing hroughou his paper are consruced from hese correlaion funcions, hough i should no ye be apparen. A special case of paricular imporance is he 2-poin funcion, or propagaor x( 2 )x( 1 ) = x( 2 )x( 1 ) (1) We also will find i useful o consider he ime-ordered produc of operaors { x( 1 )x( 2 ), 1 > 2 T(x( 1 )x( 2 )) = (11) x( 2 )x( 1 ), 2 > 1 Combining he las wo consrucions, we obain he ime-ordered propagaor D( 2, 1 ) = x( 2 )x( 1 ) (12)

10 1 Timohy G. Abbo The ime-ordered propagaor is he propagaor ha we have been compuing wih when doing examples. The ime-ordered propery is imporan for preserving causaliy. By convenion, we choose he firs argumen of he propagaor o occur afer he second. In our example of he harmonic oscillaor, he ime-ordered propagaor can be calculaed explicily: D( 2, 1 ) = x( 2 )x( 1 ) (13) { } (ae iω 1 + a e iω 1 ) 1 = 2ω (ae iω 2 + a e iω 2 ) 1 2ω, 1 > 2 (ae iω 2 + a e iω 2 ) 1 2ω (ae iω 1 + a e iω 1 (14) ) 1 2ω, 2 > 1 } = { 1 2ω e iω 1+iω 2, 1 > 2 1 2ω e iω 2+iω 1, 2 > 1 = 1 2ω e iω 2 1 (15) In he hird sep we use ha he aa erm is he only one ha survives he ground sae expecaion. The value we jus compued is exacly he value we used for he propagaor in Secion 2. Noice ha his resul is symmeric under swapping 2 and 1 ; his arises because we ook he ime-ordered produc. As you have already seen, he ime-ordered propagaor appears in he Feynman rules for he Harmonic oscillaor, as he value aached o an edge in a Feynman diagram. 5 The Ineracion Picure The ineracion picure of quanum mechanics is somewhere beween he familiar Schrödinger and Heisenberg picures. The idea is o rea he main erm of he Hamilonian as in he Heisenberg picure, wih a consan wavefuncion and imedependen operaors, bu o handle he small ineracing or perurbaive erm of he Hamilonian in he syle of he Schrödinger picure, where he wavefuncion iself moves. This allows us o isolae he effec of he ineracing Hamilonian from he (presumable well-undersood) base Hamilonian. We will use he ineracion picure o produce a formula for expecaions in he rue ground sae in erms of cerain expecaions in he unperurbed ground sae. While we will explain hings in erms of an absrac Hamilonian, i may help o keep he harmonic oscillaor in mind. Our presenaion here is based on [2], p Suppose ha we have a Hamilonian of he form H = H + H 1 (16)

11 Feynman Diagrams in Quanum Mechanics 11 where H is a ime-independen Hamilonian, and H 1 is a perurbaive ineracion erm. Le Ω be he ground sae of he ineracing heory, and define he ineracing wavefuncion ψ() I = e ih ψ() S Defining he ineracing picure ineracing Hamilonian we find he differenial equaion H I () = e ih H 1 e ih i ψ() I = H ψ() I + ie ih ψ() S (17) = e ih ( H + H) ψ() S (18) = e ih H 1 e ih ψ() I = H I ψ() I (19) The soluion can be presened as a power series for he ineracion picure imeevoluion operaor U(, ). 1 U(, ) = 1 + ( i) d 1 H I ( 1 ) + ( i) 2 d 1 d 2 H I ( 1 )H I ( 2 ) + (2) This expression can be checked by explici differeniaion; each erm gives he previous erm, muliplied by ih I (). Noe ha we use he iniial condiion U(, ) = 1, and ha each erm is ime-ordered. This may be simplified by using he ime-ordering operaor. For example, d 1 1 d 2 H I ( 1 )H I ( 2 ) = 1 2! d 1 d 2 T(H I ( 1 )H I ( 2 )) (21) where now all he inegrals have he same range. The facors of 1 n! correc for he overcouning. We may conver his o an exponenial by pulling he ime-ordering operaor ou of he inegrals: U(, ) = 1+( i) d 1 T(H I ( 1 ))+ 1 2 ( i)2 d 1 d 2 T(H I ( 1 )H I ( 2 ))+ (22) U(, ) = T ( [ exp i ]) d H I ( ) (23) Why did we go hrough all his work o find he ime evoluion operaor? The goal here is o find he ground sae of he ineracing (perurbed) Hamilonian, so

12 12 Timohy G. Abbo ha we can compue expecaions in ha sae. I urns ou ha if we evolve he original ground sae from ime T o ime, we ge somehing from which we can obain Ω, as T. Le n I be he nh sae of he ineracing heory, wih I = Ω. Then he ime evoluion of is e ih = e ien n I n I (24) n e ih = e ie Ω Ω + e ien n I n I (25) We ve no ye used he fac ha is a real-valued variable, so we ll il i ino he complex plane in order o isolae he ground sae. To do his, we need o assume ha ground sae in he ineracing heory Ω is unique. We replace by (1 iɛ), and consider he limi as (so ha erms of lower order in e ɛ vanish, in paricular hose wih energy greaer han E ). Our expression reduces o e ih(1 iɛ) = e ie E ɛ Ω Ω + e ien ɛen n I n I e ie(1 iɛ) Ω Ω n We hen can obain he ground sae iself as ( e ie ) (1 iɛ) e ih(1 iɛ) Ω = lim Ω n = lim (1 iɛ) ( e ie ) e ih Ω Now, i suffices o find a way o calculae such limis. Noe ha Ω is no normalized; in fac Ω Ω = (26) (27) lim ( ( Ω 2 e ) ie 2 1 U(, )U(, ) (28) (1 iɛ) Wih some sraighforward algebra, we can hen calculae he ime-ordered propagaor for 2 > 1 Ω x( 2 )x( 1 ) Ω = U(T, 2 )x( 2 )U( 2, 1 )x( 1 )U( 1, T) lim (1 iɛ) U(T, T) (29) Noice ha boh sides of his equaion are in ime-order. Thus we can inser a ime-ordered produc on boh sides, and obain he final resul ( T x( 2 )x( 1 ) exp[ i ) T dh T I()] Ω T (x( 2 )x( 1 )) Ω = lim ( (1 iɛ) T exp[ ) T dh T I()] (3)

13 Feynman Diagrams in Quanum Mechanics 13 This expression generalizes o higher correlaion funcions in he obvious way simply add exra x( i ) erms on boh sides. Noice ha in Equaion (3) he RHS conains no erms involving Ω, only. This formula is an exac expression for he ground sae expecaion. To do rh order perurbaion heory, we mus expand he exponenial as a power series, and only reain he erms whose degree in λ is a mos r. In each erm of he power series, we will have a ime-ordered correlaion funcion. Wick s Theorem explains how o calculaed hese correlaion funcions. 6 Wick s Theorem for he Harmonic Oscillaor Theorem 3 (Wick s Theorem) Suppose ha i > i 1, so ha he lef-hand side produc below is ime-ordered. Then in a harmonic oscillaor poenial, we have x( 2N )x( 2N 1 ) x(1) = x( in )x( jn ) x( i1 )x( j1 ) (31) i k >j k,i k+1 >i k where he expecaions are aken in he (unperurbed) ground sae. Concepually, i saes ha every 2n-poin funcion is in fac he sum over all possible pairings of he produc of he n 2-poin funcions on he pairs. The i k > j k condiion ensures ha all he 2-poin funcions are ime-ordered. The i k+1 > i k condiion breaks he symmery of he N! possible orderings for he erms in each produc. Example. For N = 2, Wick s Theorem saes ha x( 4 )x( 3 )x( 2 )x( 1 ) = (32) x( 4 )x( 3 ) x( 2 )x( 1 ) + x( 4 )x( 2 ) x( 3 )x( 1 ) + x( 4 )x( 1 ) x( 3 )x( 2 ) (33) Wick s Theorem is ypically saed as a complicaed operaor ideniy. We believe ha he formulaion presened here is much easier o undersand and apply, as i is ypically he one ha is used in pracice. This version follows from he sandard version by noing he exra erms all have ground sae expecaion. A proof of he sandard version can be found in a sandard ex on Quanum Field Theory, such as [2] p The resul can also be proven direcly using a relaively sraighforward inducion. Wick s Theorem is an exremely useful resul for doing calculaions. An immediae consequence is ha in order o calculae arbirary ime-ordered 2N-poin funcions,

14 14 Timohy G. Abbo i suffices o be able o calculae he ime-ordered propagaor. The (2N + 1)-poin funcion is always in his siuaion, since we have an odd number of a and a operaors on each erm. Thus we can in fac calculae any n-poin funcion given only he values of he ime-ordered propagaors. As we will see in he nex secion Wick s Theorem gives rise o he Feynman diagram formalism. 7 Why Feynman Diagrams Work When we combine Wick s Theorem ogeher wih he resul we derived using he ineracion picure, we can jusify he Feynman rules we provided in secion 3. We have a Hamilonian of he form H = p2 + w2 x 2 + λxk. By Equaion (3), if we wish 2 2 k! o compue he expecaion of x n, we use ( T x() n exp[ i ) T dh Ω x() n T I()] Ω = lim ( T (1 iɛ) T exp[ ) (34) T dh T I()] Le us calculae he numeraor (as we ve said before, he denominaor will cancel wih he vacuum bubbles in he numeraor. For a proof, see [2], p ). These inegrals have he same limis of inegraion as hose we ve used in our Feynman rules previously. Thus, we re going o drop he limi, and wrie our inegrals wihou limis of inegraion, o simplify noaion. To do perurbaion heory, we can expand he exponenial as a power series ( ) T x() n exp[ i dh I ()] = (35) [ T (x() n 1 + iλ dx() k + 1 ( iλ) 2 ]) dsdx() k x(s) k + (36) k! 2! k! 2 In general, he mh order erm of he series is of he form ( 1 ( iλ) m ) T d m! k! m 1...d m x( 1 ) k x( m ) k x() n (37) This can be rearranged o read 1 ( iλ) m d m! k! m 1...d m T ( x( 1 ) k x( m ) k x() n) (38) We now apply Wick s Theorem. Wick s Theorem says ha in order o compue ha ime-ordered produc, we need o pair ogeher he x( i ) s and x() s in all

15 Feynman Diagrams in Quanum Mechanics 15 possible ways, and sum over he resuling pairings. We represen diagramaically an x( i ) k by an inernal verex of degree k, and each x() by a an exernal verex of degree 1. The value aached a Wick pairing of wo verices a imes and is exacly he ime-ordered propagaor D(, ) we represen his diagramaically as an edge. Since Wick s Theorem sums over all possible pairings, we have o sum over all possible diagrams ha can be consruced by pairing he edges from hese verices. In pracice, of course, we sum over he opologically disinc diagrams and hen coun how many imes each occurs. An inernal verex from x( i ) k can be assigned he value iλ d i. Giving exernal verices value 1, he only remaining facor we haven accouned for is he scalar 1 1 m! The ineger in his denominaor is exacly he number of ways o permue he inernal verices (m!) and heir edges (k! for each verex), he weighing facor we discussed in Secion 3. In summary, we ve presened he formalism of Feynman diagrams for perurbaions of he 1-dimensional harmonic oscillaor, and explained enough of heir heory o remove some of he mysery as o why hese diagrams work. k! m. Acknowledgmens The auhor is graeful o Michael Forbes for his numerous suggesions on how o improve his paper, especially his effors a cleaning up he discussion of he ineracion picure. We d also like o hank Ryan Hendrickson and Kayla Jacobs for carefully reading drafs of his paper and making many commens ha have resuled in a much easier o undersand paper. References [1] R.L. Liboff, Inroducory Quanum Mechanics, 3rd Ed. (Addison-Wesley, Reading, MA, 1998) [2] M. Peskin, D. Schroeder, An Inroducion o Quanum Field Theory (Wesview Press, USA, 1995) [3] I learned many of hese mehods from aking MIT s course from Washingon Taylor in Spring 25. [4] L. Ahlfors, Complex Analysis (McGraw-Hill, 1979)