8.323: QFT1 Lecture Notes

Transcription

1 8.33: QFT1 Lecture Notes Joseph A. Mnahan c MIT, Sprng 11

2 Preface Ths volume s a complaton of eght nstallments of notes that I provded for the students who took Relatvstc Quantum Feld Theory at MIT durng the sprng of 11. Ths s a frst semester course n quantum feld theory for begnnng graduate students or advanced undergraduates. There were 6 lectures and the topcs covered were free scalar felds, path ntegrals scalar φ 4 -theory, free Drac felds, scatterng and QED. Each chapter of ths volume corresponds to one of the nstallments and covers roughly 3 or 4 lectures worth of materal. As for any set of lecture notes, the reader has to take them as s. Whle I have located many typos, I am sure that many more stll exst. Probably some thngs could have been better explaned and f I ever teach such a course agan I wll lkely go back and reformulate many thngs. I would lke to thank Prof. Krshna Rajagopal for gvng me the opportunty to teach ths course. I would also lke to thank the 8.33 students who provded many helpful comments about the notes.

3 Contents 1 Free scalar felds Why Quantum Feld Theory Conventons Free scalar felds Quck revew of the smple harmonc oscllator pont correlators for the harmonc oscllator dmensonal scalar feld as an nfnte set of harmonc oscllators pont correlators for the scalar feld The physcal nterpretaton of G F x y and Causalty Symmetres and Noether s theorem Path Integrals.1 Path ntegrals for nonrelatvstc partcles Generaltes The smple harmonc oscllator agan n-pont correlators and Wck s theorem Path ntegrals for the scalar feld Scalar Perturbaton theory Introducton Perturbaton theory for the smple harmonc oscllator Feynman dagrams pont correlators for the perturbed harmonc oscllator pont correlators for the nteractng scalar feld theory Smlartes and dfferences wth the anharmonc oscllator Dmensonal regularzaton The sunset dagram and the nterpretaton of ts cuts pont correlators and the effectve couplng A few comments Free Drac felds The Drac feld The Drac equaton The Lorentz algebra and ts spnor representatons Solutons to the Drac equaton

4 4. Quantzaton of the Drac feld The Drac propagator The Drac feld Lagrangan Symmetres Lorentz symmetres and partcle spns Internal contnuous symmetres Dscrete symmetres Fermonc path ntegrals Grassmann varables The +1 dmensonal fermonc oscllator dmensonal Drac fermons Scatterng The S-matrx The LSZ reducton formula The cross secton Scatterng for fermons LSZ for fermons Feynman rules for fermons Yukawa nteractons Partcle decays Scalar self-energy Four fermon nteracton Photons and QED The gauge feld U1 gauge nvarance Quantzaton under Lorenz gauge The gauge feld propagator The Feynman rules n QED The rules Electron-electron elastc scatterng e e + µ µ QED Contnued The photon helcty The Ward dentty Ampltudes wth external photons Compton scatterng e + e γγ One loop QED The Ward-Takahash dentty Three one-loop calculatons n QED Vacuum polarzaton

5 8.. Fermon self-energy One-loop vertex functon Applcatons Form factors The runnng of the couplng g : The anomalous Landé g-factor The other form factor Soft photons and nfrared dvergences v

6 Chapter 1 Free scalar felds 1.1 Why Quantum Feld Theory Tradtonally, quantum feld theory QFT has been defned as the combnaton of specal relatvty wth quantum mechancs. Ths s not wrong, but t slghtly msses the pont. QFT has become the essental tool of partcle physcs, but t s also hghly relevant n condensed matter physcs whch s nonrelatvstc. By combnng specal relatvty wth quantum mechancs, one ends up wth a quantum mechancal system wth an nfnte number of degrees of freedom. It s the nfnte degrees of freedom whch s responsble for QFT s features. Ths s s the more modern understandng of QFT and t s relevant for condensed matter physcs because here the number of degrees of freedom s very large and can be more or less treated as nfnte. Havng an nfnte number of degrees of freedom turns out to be very trcky and t took several decades before the stuaton was well understood. The problem s that many calculatons that you can do, and whch we wll explctly perform n ths class, are nfnte. One needs a way to consstently get rd of the nfntes so that we are left wth fnte physcal quanttes. In ths course we wll show how to do ths at the one-loop level. A more detaled explanaton wll be gven n the second semester course. Even though QFT s applcable to nonrelatvstc systems, our emphass wll be on scalar feld theory and quantum electrodynamcs QED, whch are explctly relatvstc. QFT as appled to QED s a theoretcal trumph, where theory has been shown to match wth experment up to 1 sgnfcant dgts. But QFT remans ncomplete and s stll a very actve feld of theoretcal research. Here we lst several ssues stll to be resolved n QFT Whle QFT works extremely well, there are parameters n the theory that need to be set by hand, namely couplng strengths and masses. In QFT & 3 you wll learn how the masses are themselves determned by couplngs to the so-called Hggs feld, but what determnes these couplngs s not well understood. Physcsts are always lookng for a more complete theory where the couplngs are somehow determned. 1

7 At hgh enough energes QED breaks down and some other theory must take over. Actually, at a scale well below ths breakdown, QED unfes wth the weak force to form the electroweak force, but even ths theory must break down at hgh enough energes. In prncple, quantum chromodynamcs QCD, the theory of the strong force, could be consstent at an arbtrarly hgh scale because of a property called asymptotc freedom, where the couplng becomes weaker at hgher energy scales. One possblty to prevent a breakdown of the electroweak theory s that t s unfed wth QCD to make a grand unfed theory that s also asymptotcally free. Even f there s a grand unfed theory, ths theory s stll mssng gravty. It turns out that mergng gravty wth quantum mechancs s a very dffcult busness and has not been truly resolved, because of the aforementoned nfntes. It s generally accepted that strng theory can get around ths problem and lead to a fully consstent theory. 1. Conventons It s assumed that you have had an extensve course or subcourse n specal relatvty. In ths secton we gve our conventons whch we use throughout the course. They match those found n Peskn. We wll use unts where the speed of lght s c = 1. Hence, tme has unts of length. Lkewse, from the relaton E = p c + m c 4, 1..1 we see that mass and momentum have the same unts as energy. Space-tme coordnates n a dmensonal nertal frame S are gven by x µ where µ =, 1,, 3 wth x = t. Now and then we wll consder stuatons where there are d + 1 space-tme dmensons wth the µ labeled accordngly. We wll use the + conventon for the metrc η µν, where η µν = dag+1, 1, 1, The nvarant length squared for a space-tme dsplacement x µ s s = τ = η µν x µ x ν = x µ x µ x, 1..3 where τ s the dsplaced proper tme. Infntesmal dsplacements dx µ have the nvarant length squared ds = η µν dx µ dx ν. x µ and dx µ are examples of contravarant 4-vectors. Under a Lorentz transformaton from an nertal frame S to another nertal frame S wth coordnates x µ, the coordnates transform as x µ = Λ µ ν x ν. Lorentz transformatons have sx ndependent generators gven by boosts n the three spatal drectons and three ndependent rotatons. For example, a boost n the x 1 drecton has Λ gven by Λ = γ vγ vγ γ 1 1, 1..4

8 where γ 1 = 1 v. s and ds are nvarant under these transformatons. Note that for any Lorentz transformaton, detλ = 1. We can enlarge the set of transformatons to also nclude constant shfts x µ x µ +a µ, where the a µ are constants. The dsplacements and the dfferentals are clearly nvarant under these transformatons. The combnaton of space-tme translatons and Lorentz transformatons are called Poncaré transformatons. The Lorentz transformatons on contravarant vectors can be generalzed to transformatons on tensors. An Lorentz transformaton as n m tensor has the form T µ 1µ...µ n ν 1 ν...ν m and transforms under a T µ 1 µ...µ n ν 1 ν...ν m = Λµ 1 µ1 Λ µ µ... Λ µ n µn Λ ν 1 ν 1 Λ ν ν... Λ νm ν m T µ 1µ...µ n ν 1 ν...ν m, 1..5 Indces can be rased and lowered wth the metrc, wth We can also make an n m + 1 T µ 1µ...µ nλ ν 1 ν...ν m T µ 1µ...µ n ν 1 ν...ν m,σ = η λσ T µ 1µ...µ n ν 1 ν...ν mσ tensor from T µ 1µ...µ n ν 1 ν...ν m by takng a dervatve x T µ 1µ...µ n σ ν 1 ν...ν m σ T µ 1µ...µ n ν 1 ν...ν m We wll often consder the nner product of two contravarant 4-vectors, A µ and B µ, A B A µ B µ, 1..8 whch has no free ndces and thus s a Lorentz nvarant. One partcular tensor s the 4-momentum of a partcle p µ = E, p. The nvarant made from ths s p p µ p ν η µν = p µ p µ = m Another tensor s the gauge feld of an electromagnetc feld A µ x. The feld strength F µν s F µν x = µ A ν x ν A µ x Quantzaton relates the 4-momentum p µ to a 4-wave vector k µ = ω, k by p µ = k µ. We wll choose unts where = 1, although we wll occasonally make the explct n our equatons. Wth these unts a length wll have the dmenson of an nverse momentum whch translates nto an nverse mass when c = 1. Wth our choce of unts, all physcal quanttes have unts whch are mass to some power, D. We wll call D the quantty s dmenson. 3

9 1.3 Free scalar felds Consder a real classcal scalar feld φx where x n the argument refers to all 3+1 spacetme coordnates. A scalar feld s nvarant under Poncaré transformatons, meanng that an observer n an nertal frame S wll see the scalar feld φ x where φ x = φx. We wll want our feld to be dynamcal, meanng that t should satsfy some second order dfferental equaton wth respect to the tme coordnate. Furthermore, we wll want the dfferental equaton to be covarant, so that f φx s a soluton for an observer n S then φ x s a soluton for an observer n S. Therefore, the dervatves n the equaton have to come wth the combnaton = µ µ =. t We wll also assume that φ s a soluton to a lnear equaton, whch we wll later see s relevant for free partcles, that s, partcles that don t nteract wth other partcles. The smplest such equaton we can wrte down s the Klen-Gordon equaton, φx + m φx = Snce a dervatve has dmenson 1, consstency requres that the parameter m s also dmenson 1. For ths reason we wll refer to m as a mass. A general soluton to s φx = d 3 k 1 A k π 3 ω e k x + A k e +k x 1.3. k where A k are constants wth respect to the coordnates x µ and k = ω k = k k + m. The factor of ω k has been nserted for later convenence. Let us now consder the Fourer transform of φx wth respect to the three spatal drectons, φ k, t = d 3 x e k x φx Comparng wth 1.3., we fnd that φ k, t = 1 ω A k e ω kt + A k e+ω kt, k where ω k s defned as above. Ths does not look relatvstc, but nonetheless, let us press on. Substtutng φ k, t nto we mmedately fnd the equaton d dt φ k, t + ω k φ k, t = Hence, for every k we have an equaton for an ordnary harmonc oscllator wth frequency ω k. 4

10 1.3.1 Quck revew of the smple harmonc oscllator Snce our free classcal feld can be thought of as an nfnte collecton of harmonc oscllators, let us revew how one quantzes the smple harmonc oscllator. Gven an oscllator whose poston s xt, ts mass s m and ts frequency s ω, the Hamltonan s gven by H = 1 m ẋ + 1 m ω x Snce the mass s a common factor, t s convenent to defne a new varable φt = m xt so that the Hamltonan becomes and the equaton of moton s H = 1 φ + 1 ω φ, t φt + ω φt = Comparng to 1.3.1, we see that we can nterpret the smple harmonc oscllator as a scalar feld n + 1 dmensons. Notce further that the dmenson of φt s D = 1/. A general soluton to the equaton of moton can be wrtten as φt = 1 ω A e ω t + A e +ω t The acton for ths system s S = dt Lt, where Lt s the Lagrangan The canoncal momentum Πt s then gven by whose solutons are Πt = L = 1 φ 1 ω φ Πt = ω L φt = φt, A e ω t A e +ω t Now let us quantze ths system. In ths case φ becomes an operator that acts on the Hlbert space and whose tme evoluton s gven by φt = e Ht φ e Ht

11 The constants A and A become the annhlaton and creaton operators a and a whch satsfy the commutaton relaton [a, a ] = It then follows that the equal tme commutaton relatons between φt and Πt have the standard form [φt, Πt] = The normalzed egenstates of the system are bult from the ground state, where a =. Thus, we have n = 1 n! a n pont correlators for the harmonc oscllator Of specal nterest are correlators of operators at dfferent tmes 1. two-pont functon s gven by In partcular, the G F t, t = T [φtφt ], where the F subscrpt stands for Feynman and the symbol T stands for tme-ordered, Substtutng nto we obtan T [φtφt ] = φtφt t > t = φt φt t < t G F t, t = 1 ω e ω t t = G F t t To see why ths s nterestng, let us consder the combnaton t G F t t + ω G F t t = ω G F t t + ω G F t t t sgnt t = δt t, 1.3. where sgnt s the sgn of t. Therefore, G F t t s the Green s functon for the dfferental operator + ω t. It wll often be convenent to Fourer transform the operator φt and G F t t nto frequency space, where φω = dt e +ωt φt For the two-pont functon, we Fourer transform both t and t to get dt dt e +ωt+ω t G F t t = dt dτ e +ω+ω T +ω ω τ/ G F τ, Later n ths course we wll see that operator correlators are mportant for partcle scatterng. 6

12 where T = 1t + t and τ = t t. Integratng over T then gves dt dt e +ωt+ω t G F t t = π δω + ω dτ e +ωτ G F τ = π δω + ω G F ω, where G F ω s the Fourer transform of G F τ, G F ω dτ e +ωτ G F τ = 1 ω dτ e +ωτ ω τ The δ-functon n ensures energy conservaton and s a consequence of the tme translaton nvarance n the Green s functon. Strctly speakng ths ntegral n s not well-defned, but we can make t so by shftng ω by a small negatve magnary part, ω ω ɛ. Lookng at G F τ n we see that +ɛ gradually sends G F τ to zero for τ ± whch s a usual procedure for dstrbutons: we assume that G F τ s turned off n the nfnte past and future. We then fnd dτ e +ω ω+ɛτ + dτ e +ω+ω ɛτ = ω ω ω + ɛ ω + ω ɛ = ω ω + ɛ, G F ω = 1 ω where we have absorbed a factor of ω nto ɛ we are assumng that ω >. We can go backward to obtan Gt t from Gω, where G F t t = + dω π e ωt t GF ω To do ths ntegral we can close off the contour and pck off the contrbutons from the poles. Note that Gω has smple poles at ω = ±ω ɛ, hence one pole s just above the real lne and the other s just below t see fgure 1. If t > t then we can close off the contour n the lower half plane, where we end up encrclng the pole clockwse at ω ɛ to fnd G F t t = π 1 π e ωt t = 1 e ωt t ω ω If t < t then we close off the contour n the upper half plane, fndng G F t t = π 1 π e +ωt t = 1 e +ωt t ω ω Ths s not the only way to get a Green s functon. We could also consder the combnaton G R t, t θt t [φt, φt ] = θt t e ω t t e +ω t t = G R t t, ω

13 t<t!="! + # x x!=! " # t>t Fgure 1.1: Integraton contour for G F t t n the case t > t lower and t < t upper. where θt s the Heavsde functon, θt = 1, t >, θt =, t <. Thus, t G Rt t + ω G R t t = ω G R t t + ω G R t t t θt t = δt t In frequency space we have that G R ω = dτe ωτ 1 ω e ω τ e +ω τ In order to do the ntegral we have to shft ω ω ɛ n the frst term and ω ω + ɛ n the second term. Hence we have G R ω = ω ω + ɛω + ω + ɛ = ω ω + ɛ sgnω Clearly, both poles are below the real lne see fgure. The Green s functon allows us to compute φt n the presence of a source. Suppose we have a source Jt for our + 1 dmensonal scalar feld, such that t φt + ω φt = Jt Ths equaton of moton can be derved from the acton by ncludng the term dt Jtφt

14 t<t x!="! " #!=! " # x t>t Fgure 1.: Integraton contour for G R t t n the case t > t lower and t < t upper. The soluton to s φt = φ hom t + dt Gt t Jt, where the homogeneous part s the soluton n and Gt t s one of the Green s functons. By constructon, G R t t only has support when t s n the past of t, hence ths s a retarded Green s functon. However, the Feynman Green s functon has support both n the future and the past, so ths s not retarded nor s t advanced. To understand ts sgnfcance, let us rewrte G F t t as G F t t = θt t ω e ω t t θt t ω e ω t t, whch has a form closer to G R t t. Here, however whle the frst term s the same as the frst term n G R t t, the second term s advanced and the coeffcent s that for a negatve energy. Hence, sources n the past add postve energy modes, n other words they act as emtters. Sources n the future add negatve energy modes, whch s equvalent to removng postve energy modes, so they act as absorbers. Let us look at ths a dfferent way. Suppose that the system s n the ground state untl tme t when t s suddenly put n the frst excted state va a source. As a functon of t the ampltude s proportonal to e ω t t f t > t. Ths s the behavor of φtφt. But by the symmetry of ths correlator there s a source term at t to take the system back down to the ground state. If t < t then the source at t can only add energy to the system and the source at t removes t, n whch case the correlator s φt φt. In other words, we should use G F t t f sources can equally act as emtters or absorbers. 9

15 dmensonal scalar feld as an nfnte set of harmonc oscllators We now apply the deas from a sngle harmonc oscllator to the scalar feld n dmensons. As we showed, the Fourer transformed components φ k, t satsfy the equaton of moton for an harmonc oscllator whose classcal soluton s expressed n We should then proceed as we dd for the sngle harmonc oscllator, although there a couple of wrnkles we need to ron out. Frst, unlke φt, φ k, t s not real. Hence, after quantzaton, φ k, t becomes φ k, t = 1 ω a k e ω kt + a k k e+ω kt, where a k s the annhlaton operator for a mode wth momentum k and a s the creaton k operator for a dfferent mode wth momentum k. Here we can draw an analogy to a crcularly symmetrc smple harmonc oscllator n two dmensons. In ths case we have two coordnates φ 1 t and φ t whch after quantzaton have the form φ 1 t = φ t = 1 a 1 e ωt + a 1 e +ω t ω 1 a e ωt + a e +ω t ω If we now let φ ± = 1 φ 1 t ± φ t, then we have that φ + t = φ t = 1 a + e ωt + a e +ω t ω 1 a e ωt + a + e +ω t, ω where a ± = 1 a 1 ± a, a ± = 1 a 1 a. The other wrnkle s that we now have a contnuous set of oscllators, so t wll be necessary to modfy the commutaton relaton n Snce the standard measure n three-dmensonal momentum space s d3 k π 3, t s natural to normalze the relatons to [a k, a k ] = π3 δ 3 k k The ground state for a collecton of oscllators, satsfes a k = for all k. The Hamltonan should be a sum, or n ths case an ntegral, over all the ndvdual oscllators d 3 k 1 d H = π ω k a k a 3 k + a k a k 3 k = π ω ka 3 k a k + C The expresson on the rght hand sde s the normal ordered verson where all creaton operators le on the left of the annhlaton operators. The constant C s an nfnte 1

16 constant whch we can gnore snce we wll only be nterested n relatve energes between the states. From now on we wll call the ground state the vacuum. The states are then constructed by actng wth the creaton operators a k on. The full Hlbert space for ths system s called a Fock space. We defne the states as follows: k 1, k... k n = n =1 ω k a k, where we assume that all k are dfferent, whch s a reasonable assumpton snce k s contnous. The rather strange normalzaton factors are to ensure Lorentz nvarant nner products, as we wll demonstrate below. Actng wth H on ths state we fnd H k 1, k... k n = n ω k k 1, k... k n, =1 whle f we defne the momentum operator P d P 3 k π k a 3 k a k we have that P k 1, k... k n = n k k 1, k... k n =1 Hence the state n the Fock space has the energy and momentum of n partcles wth rest mass m, where the ndvdual momenta are k and ther energes are k = k + m. We can now pont out several propertes of these partcles. They are free partcles, meanng that they are nonnteractng. The state k 1, k... k n s an egenstate of H, hence the ndvdual k are not changng n tme. The only way that the ndvdual momenta cannot change s that there are no forces actng on the partcles. Furthermore, snce k 1, k... k n s an egenstate, no partcles are beng created or destroyed as tme evolves. The partcles have no nternal quantum numbers. In partcular, they must have spn. Ths s not too surprsng snce our partcles orgnated from a scalar feld whch has no Lorentz ndces, meanng that t s nvarant under rotatons. The state does not change sgn under the exchange of k and k j. Accordng to our nterpretaton, ths swtches the sngle partcle states of two partcles. Snce there s no change n the state, the partcles are dentcal bosons. In our analyss we have chosen a partcular nertal frame S and then Fourer transformed the spatal coordnates. An observer n a dfferent frame of course would have a dfferent bass of spatal momenta. However, certan thngs should be nvarant. In 11

17 partcular, the vacuum should be nvarant under a Lorentz transformaton. Let s show that t s. The vacuum s dstngushed by a k = for all k. We were able to dstngush between creaton operators and annhlaton operators because the annhlaton operators n φ k, t came wth a factor of e ω kt whle the creaton operators came wth a factor of e +ω kt, where ω k >. Under a Lorentz transformaton ω k can change, but ts sgn s nvarant. Hence a Lorentz transformaton wll map annhlaton operators to annhlaton operators and creaton operators to creaton operators. Hence the state wll contnue to be annhlated by all a k and so contnue to be defned as the vacuum. Let us now consder the one partcle states k. Ths s not nvarant under a Lorentz transformaton snce the momentum wll change. However, a one-partcle state for an observer n S wll be a one partcle state for an observer n any other frame. Moreover, we would lke the nner product between one partcle states to be Lorentz nvarant. We can quckly check that t s. Consder the Lorentz transformaton of the 4-dmensonal δ-functon π 4 δ 4 k q π 4 δ 4 k q = π 4 detλ 1 δ 4 k q = π 4 δ 4 k q, where Λ µ ν s the Lorentz transformaton matrx that takes S to S. The δ-functon π δk q s also clearly nvarant. When k = q then ths δ-functon becomes πδk q = πk 1 δk q k= q If q µ s on the mass-shell, meanng that q = m, then the δ-functon forces k µ to also be on the mass-shell and so we can replace k = ω k. Hence we have the Lorentz nvarant combnaton One can quckly check that π 4 δ 4 k q π δk q = ω kπ 3 δ 3 k q q k = ω k π 3 δ 3 k q, and so s Lorentz nvarant. Let us now return to the scalar feld n coordnate space, whch by 1.3. s wrtten n terms of the oscllators as φx = d 3 k 1 π 3 ω k a k e k x + a k e +k x To fnd the canoncal momentum for ths feld, we need to know the Lagrangan densty Lx that gves the Klen-Gordon equaton n Usng that the functonal dervatve satsfes δφx δφy = δ4 x y,

18 and that the acton s S = d 4 x Lx we have that the equaton of moton comes from varyng φx to φx + δφx so that δs = We assume that L s made up of φx and ts dervatves µ φx. Hence the varaton of S s L δs = d 4 x µ φ µδφ + L φ δφ = d 4 L x µ µ φ + L φ δφ δφ, where we ntegrated by parts n the second step. Hence for general δφx we fnd that the equatons of moton follow from the Euler-Lagrange equaton Therefore, we obtan the Klen-Gordon equaton f µ L µ φ L φ = L = 1 µφ µ φ 1 m φ Note that the acton S s dmesonless n any number of space-tme dmensons, therefore L has dmenson 4 n 3+1 dmensons and so φ has dmenson 1. The canoncal momentum Πx s then Πx = L φ x = φx Therefore, n terms of the oscllators Πx s gven by d 3 k ω Πx = k a π 3 k e k x a k e +k x The equal tme commutators are then d [φx, x, Πx 3 k 1, y] = e k x y + e + k x y = δ 3 x y π 3 Havng L and the canoncal momentum we can also construct the Hamltonan densty. Recall from your courses n classcal mechancs that for a set of coordnates q I and momentum p I, the Hamltonan H s H = p I q I Lp, q, I 13

19 where Lp, q s the Lagrangan as a functon of the q I and p I. The equatons of moton follow from the Posson brackets wth the Hamltonan, where the Posson brackets are defned by q I = {q I, H} ṗ I = {p I, H}, {A, B} = I A q I B p I A p I B q I In the case of the scalar feld we have an nfnte number of coordnates, namely φ x for each value of x, and an nfnte number or momenta Π x. The Hamltonan s then H = d 3 x Π x φ x L = d 3 x Π x φ x L = d 3 x H x, where H x s the Hamltonan densty. The equatons of moton agan follow from the Posson brackets, φ x = {φ x, H}, Π x = {Π x, H}, where the Posson brackets are now defned by δa {A, B} = d 3 δb x δφ x δπ x δa δb, δπ x δφ x and the functonal dervatves satsfy δφ y δφ x = δ3 x y, δπ y δπ x = δ3 x y The dervaton of the Klen-Gordon equaton from , and s left as an exercse pont correlators for the scalar feld Consder the -pont correlator φxφy φxφy, where no assumptons are made about x µ and y µ. Usng the commuaton relatons n t s straghtforward to show that d 3 k φxφy = e k x y π 3 ω k Ths should be a Lorentz nvarant, snce φx and the vacuum are Lorentz nvarant. In fact, one can show that the measure factor s Lorentz nvarant snce d 3 k π 3 ω k d 3 k π 3 ω k π3 ω kδ 3 k q =

20 The rght hand sde s obvously Lorentz nvarant and the ntegrand s also Lorentz nvarant as shown n , hence the measure factor s Lorentz nvarant. The tme ordered correlator s G F x y = T [ φxφy ] d 3 k = θx π 3 ω y e k x y + θy x e +k x y. k If we act on G F x y wth the Klen-Gordon operator + m we fnd d + m 3 k 1 G F x y = x θx y e k x y π 3 x θy x e +k x y d = δx y 3 k π 3 e+ k x y = δ 4 x y Hence G F x y s a Green s functon for the Klen-Gordon operator. In the presence of a source Jx, the Klen-Gordon equaton becomes where we get ths equaton by addng the source term φx + m φ x = Jx Ls = Jxφx to the Lagrangan. A general soluton n the presence of the source s φx = φ hom x + d 4 x G F x x Jx, where φ hom s the homogeneous soluton n Fourer transformng φx and G F x y we have φk = d 4 xe k x φx, and d 4 xd 4 ye k x+q y G F x y = π 4 δ 4 k + q d 4 z e k y G F z = π 4 δ 4 k + q G F k, where here the δ-functon ensures conservaton of 4-momentum. After ntegratng the spatal part we fnd G F k = dz e k z 1 θz ω e ω kz + θ z e ω kz k = k ω k + ɛ = k m + ɛ Smlar to the case of a sngle oscllator, has poles at k = ±ω k ɛ 15

21 1.3.5 The physcal nterpretaton of G F x y and Causalty Suppose we have a sngle free partcle at space-tme poston y µ. The correspondng quantum state we wrte as y, y where we have made the spatal and tme components explct. The probablty ampltude to fnd the partcle at a poston x at tme x > y s x, x y, y. Ths ampltude should be a Lorentz nvarant. Now we have that y, y d 3 k = e Hy y = π 3 ω k k k e Hy y, where k s the sngle partcle state defned n Hence, we get y, y d 3 k = π 3 ω ω k e ω ky k y a k k = φy, where we used that y has a normalzaton to cancel off the normalzaton factors n Therefore, x, x y, y = θx y φxφy for x > y. In other words the -pont correlator s the ampltude for a free partcle at y µ to propagate to x µ. For ths reason the -pont correlator s also called the propagator. G F x y s usually called the Feynman propagator, where we allow for a partcle to propagate from y µ to x µ f x > y and from x µ to y µ f y > x. The poles n G F k tell us the physcal mass of the partcle. The presence of the poles at k = m ɛ s responsble for the e ω kx + k x behavor n G F x, whch we expect for the ampltude of a sngle partcle of mass m to go from y to x. Once we consder nteractons the ampltude wll be modfed, but any pole the correlator has wll stll gve us ths same sort of behavor for the ampltude there could be other contrbutons as well lettng us know that there s a physcal partcle wth a mass at the pole. The expresson n and the defnton of a propagator reles on us beng able to tme order the felds. However f x y s spacelke,.e., x y <, then the tme orderng s a relatve concept. In other words there exsts nertal frames where x y > and other nertal frames where x y <. In order to avod a contradcton, we must have that [φx, φy] = [φx, φy] = f x y <. To show that ths s true, we frst observe that [φx, φy] s a Lorentz nvarant. Hence, f t s zero t wll be zero n all frames. So one strategy s to choose a partcular frame where t s relatvely easy to see f ths commutator s zero. In partcular, there exsts a frame where the tme coordnates are smultaneous, x y =. It then follows from that n ths frame d 3 k [φx, φy] = π 3 ω e k x y e k x y = k because of the symmetry of the measure under k k. If φx dd not commute wth φy when x y < then causalty would be volated. To see why, note that φx s an Hermtan operator actng on a Hlbert space and corresponds to a physcal quantty that s n prncple measurable. Let us suppose that measurements are made of φx and φy where x y <. The space-tme 16

22 z Fgure 1.3: Integraton contour wth cuts n the z plane. The branch ponts are at z = ± m x y. postons x and y are out of causal contact, meanng that no physcal sgnal can travel from one space-tme pont to the other. Hence the measurement of φx cannot effect the measurement of φy f causalty s to hold. We know that n ordnary quantum mechancs two measurements do not affect each other only f the correspondng operators commute. Ths s not to say that φxφy s zero f x y <. To see how the correlator behaves when x µ y µ s space-lke, let us choose a frame where x y =. In ths case the propagator s d 3 k φxφy = = k x y π 3 ω k e 1 1 k dk π 3 = 1 4π 1 π dcos θ dφ k dk snk x y k + m x y = 1 k + m 1 8π x y cos θ x y ek z dz z + m x y sn z. We can now restore the correlator to the Lorentz nvarant form by replacng x y wth x y x y. To do the last ntegral n , one observes that there are branch cuts runnng from z = ± m x y to ± see fgure 3. One can then wrte sn z = 1 ez e z and deform the contour around the upper branch cut for the e kz term and around the lower branch cut for the e kz term. Both parts gve the same contrbuton, so the

23 correlator becomes φxφy = 1 z dz 4π x y m x y z m x y e z The ntegral can be done and gves a result nvolvng specal functons a modfed Bessel functon of the second knd, but we are manly nterested n the general behavor of the correlator. In the lmt where m x y, the ntegral clearly approaches 1. If m x y >> 1 then 1 z dz 4π x y m x y z m x y e z 1 m x y 1/ e m x y z 1/ e z 4π x y = 1 1/ mπ e m x y π x y 3 Hence, we fnd an exponental falloff f x µ and y µ are space-lke separated. Ths should be expected: f the ponts are so separated, then a classcal partcle cannot propagate from one space-tme pont to the other. Quantum mechancally, there can be tunnelng where we expect an exponentally small probablty for the propagaton to occur. Notce further that Compton wave-length of the partcle s m 1, whch determnes the falloff rate. To understand why we have ths falloff, recall that n nonrelatvstc quantum mechancs the exponental fall-off for tunnelng between y and x can be computed usng the WKB method, x ψx exp p d x, y where p p = me V x s the classcal trajectory for p. In the classcally forbdden regon p p s negatve and so p d x s magnary, leadng to the exponental suppresson. In the case we are consderng, let us choose a frame where x y =. The classcal trajectory s determned by p p = m and m ẋ µ = p µ, where refers to the dervatve wth respect to the proper-tme. From ths we see that p = and so p p = m. Hence p s magnary and so s the proper tme. p s drected along x y, from whch t follows that e p x y = e m x y. If x µ y µ s tme-lke, then to evaluate the correlator we choose a frame where x y =. In ths case, assumng x y >, we have φxφy = 1 4π k dk ωk e ωkx y = 1 4π m dω ω m 1/ e ωx y To make the ths expresson Lorentz nvarant we replace x y wth x y x y. If m x y >> 1 then we can approxmate to φxφy 1 1/ mπ e m x y, π x y 3 18

24 thus here we fnd oscllatory behavor. Notce that can also be obtaned from by replacng x y wth x y, whch s what one does to go from a space-lke to a tme-lke separaton. As a fnal thought, note that correlator s s sngular f x µ y µ s lght-lke,.e. x y =, even f x µ y µ. Ths s bascally a consequence of the Lorentz nvarance and the sngular behavor as x µ y µ. If x y = then we can keep boostng to a new frame, makng x µ y µ arbtrarly small. Snce the fnal result cannot depend on the frame, t must be that the correlator s sngular n all frames. 1.4 Symmetres and Noether s theorem Symmetres are an mportant tool n physcs as they allow us to make practcal statements about many physcal quanttes. For example, the mass of the proton s 938 MeV. The mass s almost entrely dependent on the physcs of QCD, but actually computng ths value s extremely dffcult and requres a huge amount of computng power. On the other hand, because of a symmetry called sospn, one can defntvely say that the neutron should have have a mass that s very close to the proton mass, whch t s, 939 MeV. Isospn symmetry s not exact, that s why the masses are slghtly dfferent, but t s close to an exact symmetry. In ths secton we wll explore some aspects of symmetres n scalar feld theores. If a contnuous symmetry exsts, then under the shft φx + ɛfφx, where ɛ s an nfntesmal parameter and fφx s some functon of the felds, the Lagrangan L s nvarant up to order ɛ. The symmetres we consder here are global symmetres, meanng that ɛ has no space-tme dependence. Later n the course we wll see a dfferent type of symmetry called a gauge symmetry, where the transformatons are local, meanng that they can depend on the coordnate x µ. As an example, consder the massless Lagrangan L = 1 µφ µ φ. Ths s nvarant under φx φx + ɛ. If there were a mass term then ths would no longer be the case. As a second example, consder the Lagrangan for a complex scalar feld, L = µ φ x µ φ m φ xφx Under the nfntesmal shft, φx φx + ɛ φx the Lagrangan shfts to L L + Oɛ and so s nvarant to leadng approxmaton. Ths last transformaton s the nfntesmal form of the transformaton of φx e θ φx whch clearly leaves L nvarant. Noether s theorem states that for every contnuous symmetry there s a conserved current. Actually, we can relax our symmetry requrements a bt by allowng L to be nvarant up to a total dervatve term L L + ɛ µ J µ 1.4. snce such a term wll not affect the equatons of moton. Snce the symmetry transformaton s nfntesmal, we can dentfy ɛ fφx wth δφx. If we assume that φx 19

25 satsfes the equatons of moton then If we defne the current j µ δl = L µ φ µδφ + L φ δφ L = µ µ φ δφ = ɛ µ L µ φ fφ j µ = µ L µ φ L δφ φ = ɛ µ J µ L µ φ fφ J µ, t follows from that µ j µ =. Hence, j µ s conserved. In the case of the constant shft, φx φx + ɛ, the current s j µ = µ φx. In the case of the complex feld the current s j µ = µ φ φ φ µ φ If we wrte φx = ρx e θx, then we have that j µ x = ρ x µ θx. From the tme component of the current we can buld the charge Q = d 3 x j x, x whch s constant n tme assumng that j x falls off suffcently fast as x : t Q = d 3 x t j x, x = d 3 x j x, x = ds j = After quantzng, Q becomes an operator that commutes wth the Hamltonan. Therefore, the egenstates n the quantum feld theory can also be egenstates of Q and Q becomes a well-defned quantum number. Of course, f there several ndependent charges, then the ndvdual charges don t necessarly commute wth each other. As an example, let us return to the complex scalar. The canoncal momentum for φx s Πx = φ x, whle that for φ x s Π x = φx. Hence, after quantzaton the equal tme commutaton relatons are [φ x, x, φ y, x ] = δ 3 x y, [φ x, x, φ y, x ] = δ 3 x y, [φ x, x, φ y, x ] = [φ x, x, φ y, x ] = If we now commute Q wth φ x, x and φ x, x, usng and as well as the fact that Q s constant n tme and so we can substtute Q = Qx, we fnd [Q, φ x, x ] = + φ x, x ], [Q, φ x, x ] = φ x, x ] When quantzng ths Q there s an ambguty about operator orderng, snce φx does not commute wth φ x. A standard choce s to assume that the felds n Q are

26 normal ordered. Ths means that all creaton operators are to the left of annhlaton operators. Wth ths condton we have that Q =, that s, the vacuum has zero charge. If we create the sngle partcle state out of the vacuum, φx, then t follows from the commutaton relatons that ths state has charge +1. On the other hand, we can create a dfferent partcle state, φ x whch has charge 1. Once we nclude nteractons the partcle numbers can change. But f the transformaton φx e θ φx contnues to be a symmetry, then the charge Q cannot change n the nteractons. Noether s theorem can also be used for symmetres that do not drectly change the felds, but nstead are symmetres of the coordnates. For example, space-tme translatons are mplemented by x µ x µ = x µ ɛ µ. In a translatonally nvarant theory, any scalar s nvarant under ths transformaton, n the sense that φ x = φx. To leadng order n ɛ µ ths means that the transformaton for φx s φx φ x = φx + ɛ = φx + ɛ µ µ φx Snce apples for any scalar, the transformaton for Lx s snce t too s a scalar. If we let Lx Lx + ɛ µ µ Lx, T µν = L µ φ ν φ η µν L, then followng the same argument as n 1.4.3, we fnd that µ T µν =. T µν = T νµ s the energy momentum tensor and ts conservaton s a statement about the local conservaton of energy and momentum. T s the energy densty and the charge s the total energy, E = d 3 x T The other charges are the components of the total momentum P = d 3 x T We prevously argued that the translatonal nvarance of G F x y led to a conservaton of 4-momentum for the two-pont functon. Here we see that ths s a consequence of Noether s theorem. 1

27 Chapter Path Integrals In ths chapter of the notes we ntroduce the concept of a path ntegral. We frst consder the path ntegral for a sngle nonrelatvstc partcle. We then specalze ths to the harmonc oscllator, whch s a nonnteractng + 1 dmensonal feld theory. We then generalze to the case of path ntegrals for scalar felds n space-tme dmensons..1 Path ntegrals for nonrelatvstc partcles.1.1 Generaltes Suppose we pose the followng problem n nonrelatvstc quantum mechancs: gven that a partcle s at poston x 1 at tme t 1, what s the probablty that the partcle s at poston x at tme t? The wave-functon evolves as 1 where the Hamltonan H s assumed to have the form Ψ x, t = e Ht t Ψ x, t,.1.1 H = p + V x..1. m Wrtng Ψ x, t as an nner product wth poston states x, we then have Ψ x, t = x Ψ, t = x e Ht/ Ψ x, t Ψ..1.3 We have nserted an explct factor of for later pedagogcal convenence. x, t s the state where the partcle s at x at tme t. Hence, the ampltude that the partcle starts at x 1 at t 1 and ends up at x at t s 1 For ths secton we wll keep the factors of. x, t x 1, t 1 = x e Ht t 1 x

28 To evaluate the expresson n.1.4, we splt up the tme nterval between t 1 and t nto a large number of nfntesmally small tme ntervals t, so that e Ht t 1 = t e H t,.1.5 where the product s over all tme ntervals between t 1 and t. Snce, p does not commute wth x, we see that e H t e m p t e V x t,.1.6 however, f t s very small, then t s approxmately true, n that e H t = e m p t e V x t + O t..1.7 Hence, n the lmt that t, we have that x, t x 1, t 1 = lm t x t e m p t e V x t x In order to evaluate the expresson n.1.8, we need to nsert a complete set of states between each term n the product, the states beng ether poston or momentum egenstates and normalzed so that d 3 x x x = 1 x x 1 = δ 3 x x 1 d 3 p π p p = 1 3 p p 1 = π 3 δ 3 p p 1 x p = e x p/.1.9 Insertng the poston states frst, we have that.1.8 s [ ] x, t x 1, t 1 = d 3 xt xt + t e m p t e V x t xt,.1.1 t 1 <t<t t 1 t<t where the frst product s over each tme between t 1 + t and t t and the second product s over each tme between t 1 and t t. We also have that xt 1 = x and xt = x. Note that for each tme nterval, we have a poston varable that we ntegrate over. You should thnk of the tme varable n these ntegrals as a label for the dfferent x varables. We now need to nsert a complete set of momentum states at each tme t. In partcular, we have that xt + t e m p t e V x t xt d 3 p = xt + t p e m p t p xt e V xt t π 3 d 3 p = e xt p/ e π 3 m p t e V xt t

29 where xt = xt + t xt. We can now do the gaussan ntegral n the last lne n.1.11, whch after completng the square gves [ xt + t e m p t e V x t m 3/ m x xt = exp V xt] t. π t t Strctly speakng, the ntegral n the last lne of.1.11 s not a Gaussan, snce the coeffcent n front of p s magnary. However, we can regulate ths by assumng that the coeffcent has a small negatve real part and then let ths part go to zero after dong the ntegral. Shortly we wll gve a more physcal reason for ths regularzaton. The last term n.1.1 can be wrtten as = m π t 3/ exp [ m x V xt] t = m π t 3/ exp Lt t,.1.13 where Lt s the lagrangan of the partcle evaluated at tme t. Hence the complete expresson can be wrtten as m 3N/ x, t x 1, t 1 = d 3 xt exp π t S t 1 <t<t m 3N/ = D x exp π t S,.1.14 where S s the acton S = t t 1 Ltdt N counts the number of tme ntervals n the path and D x tells us to ntegrate over all x for every pont n tme t. Takng the lmt N leads to an nfnte dmensonal ntegral, known as a functonal ntegral. In ths lmt we see that the constant n front of the expresson dverges. It s standard practce to drop t snce t s essentally a normalzaton constant and can always be restored later. The expresson n.1.14 was frst derved by Feynman and t gves a very ntutve way of lookng at quantum mechancs. What the expresson s tellng us s that to compute the probablty ampltude, we need to sum over all possble paths that the partcle can take n gettng from x 1 to x, weghted by e S. For ths reason the functonal ntegral s also called a path ntegral. It s natural to ask whch path domnates the path ntegral. Snce the argument of the exponental s purely magnary, we see that the path ntegral s a sum over phases. In general, when ntegratng over the xt, the phase vares and the phases comng from the dfferent paths tend to cancel each other out. What s needed s a path where varyng to a nearby path gves no phase change. Then the phases add constructvely and we are left wth a large contrbuton to the path ntegral from the path and ts nearby neghbors

30 Hence, we look for the path, gven by a parameterzaton xt, such that xt 1 = x 1 and xt = x, and such that the nearby paths have the same phase, or close to the same phase. Ths means that f xt s shfted to xt + δ xt, then the change to the acton s very small. To fnd ths path, note that under the shft, to lowest order n δ x, the acton changes by δs = t t 1 dt [ L x δ x + L ] t [ x δ x = dt d t 1 dt L x + L ] δ x x Hence there would be no phase change to lowest order n δx f the term nsde the square brackets s zero. But of course, ths s just the classcal equaton of moton. A generc path has a phase change of order δ x, but the classcal path has a phase change of order δ x. Next consder what happens as. Then a small change n the acton can lead to a bg change n the phase. In fact, even a very small change n the acton essentally wpes out any contrbuton to the path ntegral. In ths case, the classcal path s essentally the only contrbuton to the path ntegral. For nonzero, whle the classcal path s the domnant contrbutor to the path ntegral, the nonclasscal paths also contrbute, snce the phase s fnte. The path ntegral need not be used only for a transton from one poston state to another, we can also use t for other transton ampltudes. For example, the transton ampltude ψ, t ψ 1, t 1 can be converted to the form above by nsertng the complete set of poston states at t 1 and t, where afterward one fnds m 3N/ ψ, t ψ 1, t 1 = π t t 1 t t d 3 xt ψ xt ψ 1 xt 1 exp S The smple harmonc oscllator agan Let us now return to the one dmensonal harmonc oscllator. Lagrangan as We agan wrte the L = 1 φ 1 ω φ The path ntegral for the transton from the ground state at t = T to the ground state at t = +T s then Z, +T, T = π t N/ = C T t +T s back to = 1. T t +T dφt ψφt ψ φ T exp dφt e 1 ω φ T e 1 ω φ T exp dt 1 φ t 1 ω φ t dt 1 φ t 1 ω φ t,

31 where we have ncorporated all constants nto one constant C, whch wll later drop out of our computatons. The ntegral n.1.19 has the form of an nfnte number of gaussans and hence s solvable. The gaussans are coupled because of the φ term, so t s very helpful to choose new coordnates so that the argument of the exponent s dagonalzed. But before dong ths, let us make a remark about the wave-functon factor e 1 φ T +φ T. Let us suppose that we have not yet taken the lmt N where N s the number of tme ntervals. In ths case, the path ntegral has the form N Z = C dφ j exp 1φ jm jk φ k..1. j=1 Because all φ j are ndrectly coupled to φ±t through the φ term, all egenvalues of M jk wll have a postve real part. Hence, the ntegral s well defned and s gven by Z = Cπ N/ det M 1/,.1.1 where we have used that the determnant of a matrx s the product of ts egenvalues. In the lmt that N, the egenvalues contnue to have a postve real part whch becomes vanshngly small, but contnues to leave the ntegrals well defned. In the end we can gnore the effects of the ground-state wave-functons, except to nclude a small postve real part n the egenvalues of M. In the lmt that t, M becomes the dfferental operator t ω. The determnant of ths mght seem a lttle obscure, so let us go back to the path ntegral n.1.19 and Fourer transform to frequency space, where φω = dt e ωt φt..1. Carryng out the Fourer transform almost dagonalzes M and the path ntegral becomes dω 1 Z = C D φ exp π φ ωω ω + ɛ φω,.1.3 where the + ɛ term gves all egenvalues a postve real part. In transformng the measure Dφ to D φ we used that D φ = det t e ωt Dφ, where the determnant s for an nfnte dmensonal matrx n the lmt where t and T where the columns are over the t space and the rows are over the ω space. The factor s ndependent of φω and so can be absorbed nto C. Note that φ ω = φ ω and that d z e b z = π b, f Re b >. Therefore, the path ntegral s gven by the nfnte product Z = lm ω C ω 4 π 1/ / ω..1.4 ω ω + ɛ The path ntegral has a strong smlarty to a partton functon n statstcal mechancs, so for ths reason Z s often called a partton functon. 6

32 . n-pont correlators and Wck s theorem Our real nterest s n the correlators, whch wll be of partcular mportance when we consder nteractons. We can generate correlators for any number of felds usng source terms. Let us contnue wth our example of the smple harmonc oscllator. In ths case the n-pont tme-ordered correlator s gven by T [φt 1 φt... φt n ] = Z 1 Dφ φt 1 φt... φt n exp dt 1 φ t 1 ω φ t. where Z 1 normalzes the expresson. Notce that the path ntegral automatcally tme orders the felds, snce the path ntegral tself s constructed by tme orderng the vanshng small segments. Usng that δφt δφt = δt t, we can then wrte..1 as n T [φt 1 φt... φt n ] = Z 1 δ δjt j ZJ J=,.. j=1..1 where ZJ s gven by ZJ = C Dφ e SJ = C Dφ exp dt 1 φ t 1 ω ɛφ t + Jtφt,..3 and where we have ncluded the ɛ term dscussed n the prevous secton. Note that the ɛ arses n the lmt where t but also T. As long as ɛ s not exactly zero, then T s not exactly. Hence the correlators wll be zero f t j < T or t j > +T and so they are strctly speakng turned off n the dstant past or dstant future. ZJ can be evaluated by completng the square. Dong an ntegraton by parts we can rewrte ZJ as [ ZJ = C Dφ exp dt 1 φt JM 1 t ] t + ω ɛ [ φt M 1 Jt ] exp dt 1 JM 1 t t + ω ɛm 1 Jt,..4 where and where M 1 φt = dt M 1 t, t φt,..5 t + ω ɛm 1 t, t = δt t...6 7